# Alternating Current: Phasors and Power Factor > A sinusoidal source drives an inductive load, and the current arrives late. This lecture follows that single fact all the way to a factory's electricity bill. We start from the inductor relation and see that the lag is calculus rather than a quirk, replace the two sinusoids by two rotating arrows, and use those arrows to build impedance as a complex Ohm's law. Multiplying voltage by current then splits the power in two: a term that does work, and a term that sloshes back and forth twice every cycle and delivers nothing. That split gives real, reactive and apparent power, and explains why a utility charges for a poor power factor even though the reactive part carries no energy. We finish by putting one capacitor across a motor, watching the current phasor swing back into line, and delivering the same 4.6 kilowatts with 19.2 amps instead of 24. - Canonical watch page: [Alternating Current: Phasors and Power Factor](https://academa.ai/lectures/alternating-current-phasors-power-factor) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Engineering - Published: 2026-08-28T19:21:59.537Z - Updated: 2026-08-28T19:21:59.537Z - Duration: PT1149S (19 minutes 9 seconds) - Chapters: 5 - Views: 2 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14TY2TTRGKQ79KDQV8VY0ZH/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TY2TTRGKQ79KDQV8VY0ZH/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14TY2TTRGKQ79KDQV8VY0ZH/0/dark/poster.jpg) ## Description Phasors, impedance and power factor for AC circuits: why current lags, what reactive power costs, and how a capacitor fixes it. ## Chapters - [00:00–03:56.611 · Out of Step](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=0) - [03:56.611–06:53.067 · The Rotating Picture](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=236.61085416666663) - [06:53.067–10:4.647 · Impedance](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=413.0674166666666) - [10:4.647–15:13.636 · The Power That Works and the Power That Sloshes](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=604.64725) - [15:13.636–19:09 · Bringing the Current Back Into Line](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=913.6355625) ## Transcript ### [00:00 · Out of Step](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=0) In a direct current circuit, a resistor's current is the voltage divided by the resistance, and that is the end of it. Alternating current has a second question hiding inside it, and the whole of this lecture comes out of that question. Here is a sinusoidal source, and the load it is driving is a coil. A big motor winding, say, or a transformer. Time runs across, and both volts and amps are read up the side. The yellow curve is the voltage the source puts across that coil. It is a sine wave, of amplitude V m, going round at angular frequency omega. And here is the current the coil draws. Same shape, same frequency, and that is the first thing worth noticing: a sinusoidal source in a linear circuit gives you sinusoidal everything, at one single frequency. But look where it sits. The voltage reaches its peak here, and the current does not reach its own peak until later, over here. That gap is a quarter of a cycle. Ninety degrees. The current is running late, and in the language of the trade we say it lags the voltage by an angle phi. For a pure coil that lag is exactly ninety degrees. For a real motor, which has resistance in its windings as well, it will be somewhere between zero and ninety, and finding it will be our job shortly. So where does the lag come from? From one relation, and it is the defining property of an inductor. The voltage across a coil is L times the rate of change of the current through it. Read that carefully, because it is not a statement about how much current is flowing. It is a statement about how fast that current is changing. Now put the two together on the picture. Here is the instant the current is climbing at its very steepest. Look at the red tangent: that is the biggest slope anywhere on the green curve. And at that same instant, the yellow voltage is at its maximum. Steepest current, largest voltage, at the same moment. Now take the instant the current is at its own peak. Its slope there is zero: the red tangent is flat. And the voltage at that instant is passing through zero. So the voltage tracks the slope of the current, and the slope of a sine is a cosine, which is a quarter cycle ahead of it. The lag is not a quirk of the coil. It is calculus. Write it out. Take the current as I m sine of omega t minus ninety degrees. Differentiate it, multiply by L, and what comes back is a sine of omega t with no shift at all, which is the voltage we started with. And hold on to that factor out front, omega L. It has the units of ohms, and it is going to become half of the answer. Before we go on, let me tell you where this ends, so that every step has somewhere to be walking towards. First, we replace those two sinusoids by two arrows on a turning wheel. That picture is called a phasor diagram, and it is the single most useful thing in this subject. Second, the angle between those arrows turns Ohm's law into something that works for coils and capacitors as well. That is impedance. Third, we multiply voltage by current and find that the power splits in two. One part does work. The other part sloshes back and forth twice every cycle and delivers nothing at all. And fourth, we put one capacitor across the load, watch the current arrow swing back into line, and deliver exactly the same useful power with noticeably less current in the wires. That last one is worth real money to a factory, and we will see exactly why. ### [03:56.611 · The Rotating Picture](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=236.61085416666663) Two sinusoids at the same frequency carry only three pieces of information between them: one amplitude, a second amplitude, and the angle by which one trails the other. Everything else is the same frequency ticking along underneath. So here is a picture that keeps exactly those three things. A yellow arrow, whose length is the amplitude of the voltage. And a green arrow, shorter, whose length is the amplitude of the current. The green one is set behind the yellow one by the phase angle phi, and that angle is fixed. It is built into the picture and it will not change. Drop each arrow tip onto the vertical axis. Those two dashed projections are the heights we are going to plot, on the time axes over here. Now spin both arrows, counterclockwise, at the angular frequency omega. They turn together, rigidly, like two spokes of one wheel. Watch what the projections trace out as the wheel turns. The yellow height rises, falls, goes negative, comes back. That is a sine wave, and here it is being drawn instant by instant. Keep going for a second turn. The green projection does the same thing, but it arrives at each landmark a quarter of a cycle later, because its arrow is a quarter of a turn behind. So the two waveforms we started this lecture with are the shadows of two arrows on one turning wheel. That is the whole content of the phasor idea, and everything else follows from it. And now the move that makes it useful. Both arrows turn at the same rate, forever. So the turning tells us nothing we did not already know. What carries all the information is the two lengths and the angle between them, and those never change. So stop the wheel. Photograph it at one instant and throw the rotation away. What is left is this. Line the voltage arrow up along the reference direction and call its angle zero. We are free to do that, because only the difference between the two angles ever mattered. The current arrow then sits below it, at minus phi. For an inductive load it is always below: lagging. Two arrows, frozen. That is a phasor diagram, and each arrow is a phasor: a length and an angle, standing in for a whole sinusoid. One more piece of bookkeeping before we use it. From here on, when I write V or I as a length, I mean the root mean square value, not the peak. Two forty volts from a wall socket is an r m s number, and using r m s throughout keeps the power formulas free of stray factors of one half. So we have replaced two moving curves by two still arrows, and lost nothing. Now let us put those arrows to work and get an Ohm's law that a coil obeys. ### [06:53.067 · Impedance](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=413.0674166666666) A real motor winding is a resistance and an inductance in series, and one current flows through both of them. So let us draw that current first, and hang everything else off it. Across the resistance, voltage and current stay in step. So the resistor's voltage phasor lies flat along the current, and its length is R times I. Across the coil it is different, and we did the calculus for this in the opening. The coil's voltage leads its current by ninety degrees, so its phasor stands square to the current, and its length is omega L times I. That quantity omega L is called the inductive reactance, X sub L, and like resistance it is measured in ohms. The j in front of it is the algebra's way of saying: turn it ninety degrees. The voltage law still holds, provided we add the phasors as vectors rather than as numbers. So the source voltage is the resistor's voltage and the coil's voltage joined head to tail. And there is the whole geometry of an inductive load in one triangle. The source voltage is the hypotenuse, and it sits ahead of the current by an angle phi. That angle is what we are about to name. Factor the current out of both terms. What is left multiplying it is R plus j X sub L, and that combination is the impedance, bold Z. It is a complex number, and its two parts are exactly the two sides of the triangle. Its magnitude is the square root of R squared plus X squared. And its angle is the arctangent of X over R, which is the same phi we have been carrying since the first waveform. The reactance sets the phase; the resistance holds it back. So Ohm's law survives into alternating current unchanged in form. Voltage phasor equals impedance times current phasor. The only thing that has grown is that all three are complex, and the angles carry the timing. Let us put numbers on it, because these numbers will follow us to the end of the lecture. A motor on a two hundred and forty volt supply, with eight ohms of winding resistance and six ohms of reactance. The triangle is now measured in ohms rather than volts, which is the same picture divided through by the current. Eight along, six up. So the magnitude of the impedance is the square root of sixty four plus thirty six, which is the square root of a hundred: ten ohms. Now the current is two hundred and forty volts over ten ohms. Twenty four amps. That is the number a clamp meter on the supply lead would read. And the phase angle is the arctangent of six over eight, which is thirty six point eight seven degrees. The current lags the voltage by that much. Impedance ten ohms at thirty seven degrees, current twenty four amps lagging by thirty seven degrees. Hold on to both. In the next section we multiply the voltage by the current, and that angle is about to cost somebody money. ### [10:4.647 · The Power That Works and the Power That Sloshes](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=604.64725) Power at any instant is voltage times current at that instant. No phasors, no averages: just the product of the two numbers, moment by moment. Here are our two waveforms again, the voltage in yellow and the current lagging it by thirty seven degrees in green. And here in magenta is their product, the instantaneous power flowing into the motor. Two things about it. First, it oscillates at twice the supply frequency, because it is a product of two sinusoids: two humps for every one cycle of the voltage. Second, and this is the interesting one, it goes negative. Here, and here again half a cycle later. Negative power means energy flowing out of the motor and back into the supply. The coil's magnetic field is storing energy on one part of the cycle and handing it straight back on the next. Nothing is consumed by that exchange. It just sloshes. Let us separate the sloshing from the working. Expand the product of the two sines with the usual identity, and it falls into exactly two terms. Look at the first term. It is V I cosine phi, multiplied by one minus cosine of two omega t. That bracket never goes negative: it swings between zero and two, and its average is one. So that term has an average value of V I cosine phi, and it is always into the load. That is energy actually leaving the supply and not coming back. The second term is V I sine phi times sine of two omega t. That is a pure sinusoid at double frequency, sitting symmetrically about zero. Its average over a cycle is exactly nothing. Here are those two terms drawn separately. The blue one is the working term. Notice that it never dips below the axis: it is one-directional, always from supply to load. Its average is this dashed level, and that average is what we call the real power P. V times I times cosine phi, measured in watts. That is the number that turns the shaft and heats the windings. And here is the red one. Equal time above the axis and below it, twice every cycle. Energy out, energy back, out, back. Its average is zero. Over any whole number of cycles it delivers not one joule. But its amplitude is not zero, and that amplitude has a name: reactive power Q, V I sine phi, measured in var. And the plain product of the voltage and current magnitudes, with no angle in it at all, is the apparent power S, in volt amps. That is what a voltmeter and an ammeter multiplied together would tell you. The ratio of the real power to the apparent power is the power factor, and by the algebra it is simply cosine phi. A load whose current is in step has a power factor of one. Our motor, lagging thirty seven degrees, has cosine of thirty seven, which is zero point eight. Now put our numbers in, and draw the three powers as a triangle. It is the impedance triangle from the last section, scaled up by the current squared, so it has the same shape and the same angle. Two forty volts times twenty four amps is five point seven six kilovolt amps of apparent power. That is the whole hypotenuse. Multiply by the power factor, zero point eight, and the real power is four point six one kilowatts. That is the useful part, and it is the horizontal side. The vertical side is the reactive power, three point four six kilovar, and the power factor is zero point eight lagging. So here is the puzzle the customer raises. The reactive power delivers no energy at all. Its average is zero. Why should anybody be charged for it? And the answer is not in the energy. It is in the current. The utility does not send a phase angle down its cables. It sends twenty four amps. Losses in a transmission line go as the current squared times the line resistance. Not as the useful power. As the current, squared. A customer taking the same four point six kilowatts at unity power factor would draw only nineteen point two amps. Compare the squares: twenty four squared over nineteen point two squared is one point five six. Fifty six percent more loss in the cables, and fifty six percent more conductor and transformer capacity needed to serve the same useful load. That is why the tariff has a power factor clause. The wires and the transformers are sized by the hypotenuse, and the customer with the poor power factor is asking for a bigger hypotenuse to get the same horizontal side. Which raises the obvious question. If the trouble is that the current arrow is tilted, can we tilt it back? ### [15:13.636 · Bringing the Current Back Into Line](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=913.6355625) Here is the motor's phasor picture as we left it. Voltage along the reference, current lagging by thirty seven degrees, twenty four amps long. Now think about what a capacitor does. Its current is C times the rate of change of its voltage, which is the mirror image of the coil relation. So where the coil's current lagged by ninety degrees, the capacitor's current leads by ninety. It is reactive current pointing the other way. Put one in parallel with the motor. Both see the same supply voltage, because that is what parallel means, and the supply now has to provide the sum of the two currents. The motor's current is unchanged. It has no idea the capacitor is there: same voltage across it, same impedance, same twenty four amps at the same angle. That is the crucial point and it is worth pausing on. What changes is what the supply cable carries, which is the motor's current plus the capacitor's current, added as phasors, head to tail. So watch what happens to that magenta line current as we make the capacitor bigger. The motor's current has a vertical component: twenty four times the sine of thirty seven, which is fourteen point four amps of lagging reactive current. The capacitor's current points straight up, so it cancels part of that vertical component. The line current shortens, and its angle to the voltage closes up. A bigger one takes out more. The magenta line current is getting shorter and flatter, while the green motor current has not moved at all. And at fourteen point four amps of capacitor current, the two vertical components cancel exactly. The line current lies flat along the voltage. What is left in the supply cable is only the horizontal component: twenty four times cosine thirty seven, which is nineteen point two amps. The phase angle is now zero and the power factor is one. The current is back in step with the voltage. And to size the capacitor, we ask for the reactive power it must supply: two forty volts times fourteen point four amps, which is three point four six kilovar. Exactly the reactive power the motor was drawing from the supply, and now drawn from the capacitor instead. That is the physical picture worth keeping. The coil and the capacitor pass the sloshing energy back and forth to each other, locally, twice a cycle. The supply is no longer part of that exchange. So here is the ledger, before and after one capacitor. Line current down from twenty four amps to nineteen point two. Real power unchanged at four point six one kilowatts, because the motor is doing exactly the same job. Apparent power down from five point seven six to four point six one kilovolt amps, and the power factor up from zero point eight to one. And the line loss goes as current squared, so nineteen point two squared over twenty four squared is zero point six four. Thirty six percent less heat wasted in the cables, for the same work done. That is why the capacitor bank sits in the factory switchroom and not at the power station. Correcting at the load takes the reactive current out of every metre of cable between them. So look back at what four ideas did. Two sinusoids became two arrows. The angle between the arrows became an impedance. That angle split the power into a part that works and a part that only sloshes. And one capacitor sent the sloshing part back to the load that wanted it, leaving the wires to carry only the work. Every one of those steps came out of one small observation at the very start: that the voltage across a coil follows the slope of its current, and not its size. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M14TY2TTRGKQ79KDQV8VY0ZH/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [Out of Step](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=0) Span: 00:00–03:56.611 (0s–236.61085416666663s). #### Objects - card: a Title that says "Electrical Circuits — Alternating Current: Phasors and Power Factor" - coil\_law: a Math \[text\] that says "$v\_L(t) = L frac(dif i, dif t)$" - coil\_note: a Panel that says "The voltage across an inductor is set by how fast its current is changing, not by how large that current is." - coil\_work: a Derivation \[text\] that says "$i(t) &= I\_m sin(omega t - 90 degree) \\ frac(dif i, dif t) &= omega I\_m cos(omega t - 90 degree) \\ v\_L(t) &= omega L thin I\_m sin(omega t)$" - current\_form: a Math \[text\] that says "$i(t) = I\_m sin(omega t - phi)$" - flat\_dot: a PlotPoint \[red\] drawn in waves (target='i\_curve', x=4.71238898038469) - flat\_tangent: a TangentLine \[red\] drawn in waves (target='i\_curve', x=4.71238898038469, length=1.6) - head\_law: a Heading that says "The Coil's Own Relation" - head\_plan: a Heading that says "Where This Is Going" - head\_wave: a Heading that says "One Source, Two Waveforms" - head\_why: a Heading that says "Why the Current Runs Late" - i\_curve: a FunctionPlot \[green\] labelled "i(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)) - i\_peak\_mark: a Point \[green\] labelled "upright("i peaks")" drawn in waves (location=(4.71238898038469, 0.75)) - lag\_span: a Line \[magenta\] labelled "upright("lag")" drawn in waves (start=(1.5707963267948966, -1.15), end=(4.71238898038469, -1.15)) - lag\_value: a Math \[text\] that says "$phi = 90 degree$" - plan: a Block \[text\] that says "Two rotating arrows replace the two sinusoids. The angle between them becomes an impedance. The power splits into a working part and a sloshing part. A capacitor swings the current arrow back into line." - point: a Point \[yellow\] drawn in waves (location=(1.5707963267948966, 1.0)) - source\_form: a Math \[text\] that says "$v(t) = V\_m sin(omega t)$" - steep\_dot: a PlotPoint \[red\] drawn in waves (target='i\_curve', x=1.5707963267948966) - steep\_tangent: a TangentLine \[red\] drawn in waves (target='i\_curve', x=1.5707963267948966, length=1.6) - v\_curve: a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)) - v\_peak\_mark: a Point \[yellow\] labelled "upright("v peaks")" drawn in waves (location=(1.5707963267948966, 1.0)) - v\_zero\_dot: a PlotPoint \[yellow\] drawn in waves (target='v\_curve', x=4.71238898038469) - waves: an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793) #### Beats ##### [00:00](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=0) Narration: In a direct current circuit, a resistor's current is the voltage divided by the resistance, and that is the end of it. Alternating current has a second question hiding inside it, and the whole of this lecture comes out of that question. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=0): card is shown on the screen, written out. - [00:1.5](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1.5): card: enter:write-left-to-right. - [00:13.989](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=13.9895): card is hidden from the screen — left the board. ##### [00:15.189](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=15.189499999999999) Narration: Here is a sinusoidal source, and the load it is driving is a coil. A big motor winding, say, or a transformer. Time runs across, and both volts and amps are read up the side. Board: Unchanged from the preceding beat in this scene. Actions: - [00:15.189](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=15.189499999999999): head\_wave is shown on the screen, written out. - [00:16.885](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=16.884999999999998): waves is shown on the screen, written out. ##### [00:28.201](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=28.201) Narration: The yellow curve is the voltage the source puts across that coil. It is a sine wave, of amplitude V m, going round at angular frequency omega. Board: waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); head\_wave — a Heading that says "One Source, Two Waveforms" Actions: - [00:28.677](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=28.677000000000003): v\_curve is shown on the screen, written out. - [00:32.961](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=32.961): waves moves to a new place on the board. - [00:32.961](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=32.961): source\_form is shown on the screen, written out. ##### [00:39.191](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=39.1915) Narration: And here is the current the coil draws. Same shape, same frequency, and that is the first thing worth noticing: a sinusoidal source in a linear circuit gives you sinusoidal everything, at one single frequency. Board: source\_form — a Math \[text\] that says "$v(t) = V\_m sin(omega t)$"; waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); head\_wave — a Heading that says "One Source, Two Waveforms"; v\_curve — a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)) Actions: - [00:40.109](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=40.108999999999995): i\_curve is shown on the screen, written out. ##### [00:54.13](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=54.129999999999995) Narration: But look where it sits. The voltage reaches its peak here, and the current does not reach its own peak until later, over here. Board: source\_form — a Math \[text\] that says "$v(t) = V\_m sin(omega t)$"; waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); head\_wave — a Heading that says "One Source, Two Waveforms"; v\_curve — a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] labelled "i(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)) Actions: - [00:56.789](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=56.788999999999994): v\_peak\_mark is shown on the screen, written out. - [01:0.968](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=60.96799999999999): i\_peak\_mark is shown on the screen, written out. ##### [01:3.24](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=63.239999999999995) Narration: That gap is a quarter of a cycle. Ninety degrees. The current is running late, and in the language of the trade we say it lags the voltage by an angle phi. Board: source\_form — a Math \[text\] that says "$v(t) = V\_m sin(omega t)$"; waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); head\_wave — a Heading that says "One Source, Two Waveforms"; v\_curve — a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] labelled "i(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); v\_peak\_mark — a Point \[yellow\] labelled "upright("v peaks")" drawn in waves (location=(1.5707963267948966, 1.0)); i\_peak\_mark — a Point \[green\] labelled "upright("i peaks")" drawn in waves (location=(4.71238898038469, 0.75)) Actions: - [01:3.913](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=63.913): lag\_span is shown on the screen, written out. - [01:10.682](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=70.68199999999999): current\_form is shown on the screen, written out. - [01:12.191](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=72.19099999999999): lag\_value is shown on the screen, written out. ##### [01:13.615](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=73.615) Narration: For a pure coil that lag is exactly ninety degrees. For a real motor, which has resistance in its windings as well, it will be somewhere between zero and ninety, and finding it will be our job shortly. Board: source\_form — a Math \[text\] that says "$v(t) = V\_m sin(omega t)$"; current\_form — a Math \[text\] that says "$i(t) = I\_m sin(omega t - phi)$"; lag\_value — a Math \[text\] that says "$phi = 90 degree$"; waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); head\_wave — a Heading that says "One Source, Two Waveforms"; v\_curve — a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] labelled "i(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); v\_peak\_mark — a Point \[yellow\] labelled "upright("v peaks")" drawn in waves (location=(1.5707963267948966, 1.0)); i\_peak\_mark — a Point \[green\] labelled "upright("i peaks")" drawn in waves (location=(4.71238898038469, 0.75)); lag\_span — a Line \[magenta\] labelled "upright("lag")" drawn in waves (start=(1.5707963267948966, -1.15), end=(4.71238898038469, -1.15)) Actions: - [01:13.615](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=73.615): v\_peak\_mark is hidden from the screen. - [01:13.615](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=73.615): i\_peak\_mark is hidden from the screen. - [01:18.085](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=78.08499999999998): lag\_span is hidden from the screen. - [01:26.282](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=86.282): current\_form is hidden from the screen — left the board. - [01:26.282](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=86.282): head\_wave is hidden from the screen — left the board. - [01:26.282](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=86.282): lag\_value is hidden from the screen — left the board. - [01:26.282](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=86.282): source\_form is hidden from the screen — left the board. ##### [01:26.882](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=86.88199999999999) Narration: So where does the lag come from? From one relation, and it is the defining property of an inductor. The voltage across a coil is L times the rate of change of the current through it. Board: waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); v\_curve — a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] labelled "i(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)) Actions: - [01:26.882](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=86.88199999999999): head\_why is shown on the screen, written out. - [01:34.15](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=94.14999999999999): coil\_law is shown on the screen, written out. - [01:36.39](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=96.38999999999999): coil\_law (the "frac(dif i, dif t)" part) is emphasized. ##### [01:39.045](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=99.04549999999999) Narration: Read that carefully, because it is not a statement about how much current is flowing. It is a statement about how fast that current is changing. Board: waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); v\_curve — a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] labelled "i(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); coil\_law — a Math \[text\] that says "$v\_L(t) = L frac(dif i, dif t)$"; head\_why — a Heading that says "Why the Current Runs Late" Actions: - [01:41.542](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=101.54199999999999): coil\_note is shown on the screen, written out. - [01:46.383](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=106.38299999999998): coil\_law (the "frac(dif i, dif t)" part) is no longer emphasized. ##### [01:47.911](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=107.91149999999999) Narration: Now put the two together on the picture. Here is the instant the current is climbing at its very steepest. Look at the red tangent: that is the biggest slope anywhere on the green curve. Board: waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); v\_curve — a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] labelled "i(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); coil\_law — a Math \[text\] that says "$v\_L(t) = L frac(dif i, dif t)$"; coil\_note — a Panel that says "The voltage across an inductor is set by how fast its current is changing, not by how large that current is."; head\_why — a Heading that says "Why the Current Runs Late" Actions: - [01:51.487](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=111.48699999999998): steep\_dot is shown on the screen, written out. - [01:55.458](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=115.45799999999997): steep\_tangent is shown on the screen, written out. ##### [02:0.336](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=120.33649999999999) Narration: And at that same instant, the yellow voltage is at its maximum. Steepest current, largest voltage, at the same moment. Board: waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); v\_curve — a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] labelled "i(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); coil\_law — a Math \[text\] that says "$v\_L(t) = L frac(dif i, dif t)$"; coil\_note — a Panel that says "The voltage across an inductor is set by how fast its current is changing, not by how large that current is."; head\_why — a Heading that says "Why the Current Runs Late"; steep\_dot — a PlotPoint \[red\] drawn in waves (target='i\_curve', x=1.5707963267948966); steep\_tangent — a TangentLine \[red\] drawn in waves (target='i\_curve', x=1.5707963267948966, length=1.6) Actions: - [02:3.558](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=123.55799999999996): point is shown on the screen, grown. - [02:5.558](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=125.55799999999996): point is hidden from the screen. ##### [02:9.22](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=129.22) Narration: Now take the instant the current is at its own peak. Its slope there is zero: the red tangent is flat. And the voltage at that instant is passing through zero. Board: Unchanged from the preceding beat in this scene. Actions: - [02:9.22](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=129.22): steep\_dot is hidden from the screen. - [02:9.22](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=129.22): steep\_tangent is hidden from the screen. - [02:11.763](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=131.76299999999998): flat\_dot is shown on the screen, written out. - [02:15.965](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=135.96499999999997): flat\_tangent is shown on the screen, written out. - [02:19.205](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=139.20499999999998): v\_zero\_dot is shown on the screen, written out. ##### [02:21.361](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=141.3605) Narration: So the voltage tracks the slope of the current, and the slope of a sine is a cosine, which is a quarter cycle ahead of it. The lag is not a quirk of the coil. It is calculus. Board: waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); v\_curve — a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] labelled "i(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); coil\_law — a Math \[text\] that says "$v\_L(t) = L frac(dif i, dif t)$"; coil\_note — a Panel that says "The voltage across an inductor is set by how fast its current is changing, not by how large that current is."; head\_why — a Heading that says "Why the Current Runs Late"; flat\_dot — a PlotPoint \[red\] drawn in waves (target='i\_curve', x=4.71238898038469); flat\_tangent — a TangentLine \[red\] drawn in waves (target='i\_curve', x=4.71238898038469, length=1.6); v\_zero\_dot — a PlotPoint \[yellow\] drawn in waves (target='v\_curve', x=4.71238898038469) Actions: - [02:21.361](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=141.3605): flat\_dot is hidden from the screen. - [02:21.361](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=141.3605): flat\_tangent is hidden from the screen. - [02:21.361](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=141.3605): v\_zero\_dot is hidden from the screen. - [02:33.226](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=153.226): coil\_law moves to a new place on the board. - [02:33.226](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=153.226): waves moves to a new place on the board. - [02:33.226](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=153.226): coil\_note is hidden from the screen — left the board. - [02:33.226](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=153.226): head\_why is hidden from the screen — left the board. ##### [02:34.426](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=154.426) Narration: Write it out. Take the current as I m sine of omega t minus ninety degrees. Differentiate it, multiply by L, and what comes back is a sine of omega t with no shift at all, which is the voltage we started with. Board: waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); v\_curve — a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] labelled "i(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); coil\_law — a Math \[text\] that says "$v\_L(t) = L frac(dif i, dif t)$" Actions: - [02:34.426](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=154.426): head\_law is shown on the screen, written out. - [02:36.26](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=156.26): coil\_work is shown on the screen, written out. - [02:41.206](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=161.206): coil\_work is shown on the screen, written out. - [02:44.306](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=164.30599999999998): coil\_work is shown on the screen, written out. ##### [02:50.293](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=170.29299999999998) Narration: And hold on to that factor out front, omega L. It has the units of ohms, and it is going to become half of the answer. Board: waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); v\_curve — a FunctionPlot \[yellow\] labelled "v(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] labelled "i(t)" drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); coil\_law — a Math \[text\] that says "$v\_L(t) = L frac(dif i, dif t)$"; head\_law — a Heading that says "The Coil's Own Relation" Actions: - [02:52.696](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=172.69599999999994): coil\_work (the "omega L" part) is emphasized. - [02:56.946](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=176.94599999999994): coil\_work (the "omega L" part) is no longer emphasized. - [02:58.292](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=178.2925): coil\_law is hidden from the screen — left the board. - [02:58.292](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=178.2925): coil\_work is hidden from the screen — left the board. - [02:58.292](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=178.2925): head\_law is hidden from the screen — left the board. - [02:58.292](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=178.2925): waves is hidden from the screen — left the board. - [02:58.292](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=178.2925): v\_curve is hidden from the screen — waves left the board. - [02:58.292](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=178.2925): i\_curve is hidden from the screen — waves left the board. ##### [02:59.492](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=179.49249999999998) Narration: Before we go on, let me tell you where this ends, so that every step has somewhere to be walking towards. Board: Empty. Actions: - [02:59.492](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=179.49249999999998): head\_plan is shown on the screen, written out. ##### [03:5.839](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=185.83899999999997) Narration: First, we replace those two sinusoids by two arrows on a turning wheel. That picture is called a phasor diagram, and it is the single most useful thing in this subject. Board: head\_plan — a Heading that says "Where This Is Going" Actions: - [03:6.315](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=186.31499999999997): plan is shown on the screen, written out. - [03:9.02](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=189.01999999999995): plan (the "Two rotating arrows" part) is emphasized. ##### [03:16.9](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=196.89999999999998) Narration: Second, the angle between those arrows turns Ohm's law into something that works for coils and capacitors as well. That is impedance. Board: plan — a Block \[text\] that says "Two rotating arrows replace the two sinusoids. The angle between them becomes an impedance. The power splits into a working part and a sloshing part. A capacitor swings the current arrow back into line."; head\_plan — a Heading that says "Where This Is Going" Actions: - [03:24.748](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=204.74799999999996): plan (the "Two rotating arrows" part) is no longer emphasized. - [03:24.748](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=204.74799999999996): plan (the "an impedance" part) is emphasized. ##### [03:26.474](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=206.47449999999998) Narration: Third, we multiply voltage by current and find that the power splits in two. One part does work. The other part sloshes back and forth twice every cycle and delivers nothing at all. Board: Unchanged from the preceding beat in this scene. Actions: - [03:34.625](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=214.62499999999997): plan (the "a working part and a sloshing part" part) is emphasized. - [03:34.625](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=214.62499999999997): plan (the "an impedance" part) is no longer emphasized. ##### [03:39.195](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=219.19549999999998) Narration: And fourth, we put one capacitor across the load, watch the current arrow swing back into line, and deliver exactly the same useful power with noticeably less current in the wires. That last one is worth real money to a factory, and we will see exactly why. Board: Unchanged from the preceding beat in this scene. Actions: - [03:41.401](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=221.40099999999998): plan (the "a working part and a sloshing part" part) is no longer emphasized. - [03:41.401](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=221.40099999999998): plan (the "swings the current arrow back into line" part) is emphasized. - [03:52.175](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=232.17499999999998): plan (the "swings the current arrow back into line" part) is no longer emphasized. - [03:55.569](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=235.56918749999997): head\_plan is hidden from the screen — left the board. - [03:55.569](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=235.56918749999997): plan is hidden from the screen — left the board. ### Scene 2: [The Rotating Picture](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=236.61085416666663) Span: 03:56.611–06:53.067 (236.61085416666663s–413.0674166666666s). #### Objects - axis\_h: a Line \[gray\] drawn in wheel (start=(-1.35, 0.0), end=(1.35, 0.0), dashed=True) - axis\_v: a Line \[gray\] drawn in wheel (start=(0.0, -1.35), end=(0.0, 1.35), dashed=True) - between: an Angle \[magenta\] labelled "phi" drawn in wheel (sides=(((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((cloc…, radius=0.32) - clock: a VariableNumber (initial\_value=0.4, format\_spec='.1f') - head\_freeze: a Heading that says "Stop the Wheel" - head\_wheel: a Heading that says "Two Arrows on One Wheel" - hub: a Point \[gray\] drawn in wheel - i\_arrow: a Vector \[green\] labelled "I" drawn in wheel (end=((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((clock…) - i\_rider: a Point \[green\] drawn in plot (location=(\, (0.75 \* sin((cloc…) - i\_shadow: a Line \[green\] drawn in wheel (start=((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((clock…, end=(0.0, (0.75 \* sin((clock - 1.5707963267948966)))), dashed=True) - i\_trace: a ParametricCurve \[green\] drawn in plot (function=\, t\_range=(0.0, \)) - phasor\_i: a Math \[text\] that says "$bold(I) = I angle - phi$" - phasor\_note: a Panel that says "A phasor is the arrow with the wheel stopped: one length and one angle, holding the amplitude and the phase of a sinusoid that everything in the circuit shares." - phasor\_v: a Math \[text\] that says "$bold(V) = V angle 0 degree$" - plot: an Axes (x\_range=(-0.35, 12.916370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793) - projection: a Math \[text\] that says "$v(t) = V\_m sin(omega t)$" - projection\_i: a Math \[text\] that says "$i(t) = I\_m sin(omega t - phi)$" - rim: a Circle \[gray\] drawn in wheel - still: a Figure (x\_range=(-0.35, 1.5), y\_range=(-1.15, 0.55), aspect=(1.85, 1.7)) - still\_angle: an Angle \[magenta\] labelled "phi" drawn in still (sides=((1.25, 0.0), (0.29047597471245057, -0.8939931253174442))) - still\_hub: a Point \[gray\] drawn in still - still\_i: a Vector \[green\] labelled "bold(I)" drawn in still (end=(0.29047597471245057, -0.8939931253174442)) - still\_ref: a Line \[gray\] drawn in still (end=(1.4, 0.0), dashed=True) - still\_v: a Vector \[yellow\] labelled "bold(V)" drawn in still (end=(1.25, 0.0)) - v\_arrow: a Vector \[yellow\] labelled "V" drawn in wheel (end=((1.0 \* cos(clock)), (1.0 \* sin(clock)))) - v\_rider: a Point \[yellow\] drawn in plot (location=(\, (1.0 \* sin(clock)…) - v\_shadow: a Line \[yellow\] drawn in wheel (start=((1.0 \* cos(clock)), (1.0 \* sin(clock))), end=(0.0, (1.0 \* sin(clock))), dashed=True) - v\_trace: a ParametricCurve \[yellow\] drawn in plot (function=\, t\_range=(0.0, \)) - wheel: a Figure (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), aspect=(1.0, 1.0)) #### Beats ##### [03:56.611](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=236.61085416666663) Narration: Two sinusoids at the same frequency carry only three pieces of information between them: one amplitude, a second amplitude, and the angle by which one trails the other. Everything else is the same frequency ticking along underneath. So here is a picture that keeps exactly those three things. Board: Empty. Actions: - [03:56.611](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=236.61085416666663): head\_wheel is shown on the screen, written out. - [04:10.938](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=250.93785416666663): wheel is shown on the screen, written out. - [04:10.938](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=250.93785416666663): rim is shown on the screen, written out. - [04:10.938](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=250.93785416666663): hub is shown on the screen, written out. - [04:10.938](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=250.93785416666663): axis\_h is shown on the screen, written out. - [04:10.938](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=250.93785416666663): axis\_v is shown on the screen, written out. ##### [04:14.255](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=254.25485416666663) Narration: A yellow arrow, whose length is the amplitude of the voltage. And a green arrow, shorter, whose length is the amplitude of the current. Board: wheel — a Figure (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), aspect=(1.0, 1.0)); head\_wheel — a Heading that says "Two Arrows on One Wheel"; rim — a Circle \[gray\] drawn in wheel; hub — a Point \[gray\] drawn in wheel; axis\_h — a Line \[gray\] drawn in wheel (start=(-1.35, 0.0), end=(1.35, 0.0), dashed=True); axis\_v — a Line \[gray\] drawn in wheel (start=(0.0, -1.35), end=(0.0, 1.35), dashed=True) Actions: - [04:14.638](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=254.6378541666666): v\_arrow is shown on the screen, written out. - [04:18.144](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=258.14385416666664): i\_arrow is shown on the screen, written out. ##### [04:22.598](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=262.59835416666664) Narration: The green one is set behind the yellow one by the phase angle phi, and that angle is fixed. It is built into the picture and it will not change. Board: wheel — a Figure (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), aspect=(1.0, 1.0)); head\_wheel — a Heading that says "Two Arrows on One Wheel"; rim — a Circle \[gray\] drawn in wheel; hub — a Point \[gray\] drawn in wheel; axis\_h — a Line \[gray\] drawn in wheel (start=(-1.35, 0.0), end=(1.35, 0.0), dashed=True); axis\_v — a Line \[gray\] drawn in wheel (start=(0.0, -1.35), end=(0.0, 1.35), dashed=True); v\_arrow — a Vector \[yellow\] labelled "V" drawn in wheel (end=((1.0 \* cos(clock)), (1.0 \* sin(clock)))); i\_arrow — a Vector \[green\] labelled "I" drawn in wheel (end=((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((clock…) Actions: - [04:25.536](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=265.53585416666664): between is shown on the screen, written out. ##### [04:31.778](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=271.77835416666665) Narration: Drop each arrow tip onto the vertical axis. Those two dashed projections are the heights we are going to plot, on the time axes over here. Board: wheel — a Figure (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), aspect=(1.0, 1.0)); head\_wheel — a Heading that says "Two Arrows on One Wheel"; rim — a Circle \[gray\] drawn in wheel; hub — a Point \[gray\] drawn in wheel; axis\_h — a Line \[gray\] drawn in wheel (start=(-1.35, 0.0), end=(1.35, 0.0), dashed=True); axis\_v — a Line \[gray\] drawn in wheel (start=(0.0, -1.35), end=(0.0, 1.35), dashed=True); v\_arrow — a Vector \[yellow\] labelled "V" drawn in wheel (end=((1.0 \* cos(clock)), (1.0 \* sin(clock)))); i\_arrow — a Vector \[green\] labelled "I" drawn in wheel (end=((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((clock…); between — an Angle \[magenta\] labelled "phi" drawn in wheel (sides=(((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((cloc…, radius=0.32) Actions: - [04:31.999](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=271.99885416666666): v\_shadow is shown on the screen, written out. - [04:36.284](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=276.28385416666663): i\_shadow is shown on the screen, written out. - [04:38.942](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=278.94185416666664): wheel moves to a new place on the board. - [04:38.942](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=278.94185416666664): plot is shown on the screen, written out. - [04:38.942](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=278.94185416666664): v\_trace is shown on the screen, written out. - [04:38.942](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=278.94185416666664): v\_rider is shown on the screen, written out. - [04:38.942](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=278.94185416666664): i\_trace is shown on the screen, written out. - [04:38.942](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=278.94185416666664): i\_rider is shown on the screen, written out. ##### [04:41.225](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=281.22535416666665) Narration: Now spin both arrows, counterclockwise, at the angular frequency omega. They turn together, rigidly, like two spokes of one wheel. Board: wheel — a Figure (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), aspect=(1.0, 1.0)); plot — an Axes (x\_range=(-0.35, 12.916370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); head\_wheel — a Heading that says "Two Arrows on One Wheel"; rim — a Circle \[gray\] drawn in wheel; hub — a Point \[gray\] drawn in wheel; axis\_h — a Line \[gray\] drawn in wheel (start=(-1.35, 0.0), end=(1.35, 0.0), dashed=True); axis\_v — a Line \[gray\] drawn in wheel (start=(0.0, -1.35), end=(0.0, 1.35), dashed=True); v\_arrow — a Vector \[yellow\] labelled "V" drawn in wheel (end=((1.0 \* cos(clock)), (1.0 \* sin(clock)))); i\_arrow — a Vector \[green\] labelled "I" drawn in wheel (end=((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((clock…); between — an Angle \[magenta\] labelled "phi" drawn in wheel (sides=(((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((cloc…, radius=0.32); v\_shadow — a Line \[yellow\] drawn in wheel (start=((1.0 \* cos(clock)), (1.0 \* sin(clock))), end=(0.0, (1.0 \* sin(clock))), dashed=True); i\_shadow — a Line \[green\] drawn in wheel (start=((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((clock…, end=(0.0, (0.75 \* sin((clock - 1.5707963267948966)))), dashed=True); v\_trace — a ParametricCurve \[yellow\] drawn in plot (function=\, t\_range=(0.0, \)); v\_rider — a Point \[yellow\] drawn in plot (location=(\, (1.0 \* sin(clock)…); i\_trace — a ParametricCurve \[green\] drawn in plot (function=\, t\_range=(0.0, \)); i\_rider — a Point \[green\] drawn in plot (location=(\, (0.75 \* sin((cloc…) Actions: - [04:41.957](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=281.95685416666663): v\_arrow is redrawn as the numbers it depends on change. - [04:41.957](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=281.95685416666663): i\_arrow is redrawn as the numbers it depends on change. - [04:41.957](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=281.95685416666663): between is redrawn as the numbers it depends on change. - [04:41.957](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=281.95685416666663): v\_shadow is redrawn as the numbers it depends on change. - [04:41.957](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=281.95685416666663): i\_shadow is redrawn as the numbers it depends on change. - [04:41.957](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=281.95685416666663): v\_trace is redrawn as the numbers it depends on change. - [04:41.957](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=281.95685416666663): v\_rider is redrawn as the numbers it depends on change. - [04:41.957](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=281.95685416666663): i\_trace is redrawn as the numbers it depends on change. - [04:41.957](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=281.95685416666663): i\_rider is redrawn as the numbers it depends on change. - [04:41.957](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=281.95685416666663): clock ticks to 1.6. ##### [04:51.961](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=291.96085416666665) Narration: Watch what the projections trace out as the wheel turns. The yellow height rises, falls, goes negative, comes back. That is a sine wave, and here it is being drawn instant by instant. Board: Unchanged from the preceding beat in this scene. Actions: - [04:54.666](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=294.66585416666663): v\_arrow is redrawn as the numbers it depends on change. - [04:54.666](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=294.66585416666663): i\_arrow is redrawn as the numbers it depends on change. - [04:54.666](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=294.66585416666663): between is redrawn as the numbers it depends on change. - [04:54.666](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=294.66585416666663): v\_shadow is redrawn as the numbers it depends on change. - [04:54.666](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=294.66585416666663): i\_shadow is redrawn as the numbers it depends on change. - [04:54.666](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=294.66585416666663): v\_trace is redrawn as the numbers it depends on change. - [04:54.666](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=294.66585416666663): v\_rider is redrawn as the numbers it depends on change. - [04:54.666](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=294.66585416666663): i\_trace is redrawn as the numbers it depends on change. - [04:54.666](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=294.66585416666663): i\_rider is redrawn as the numbers it depends on change. - [04:54.666](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=294.66585416666663): clock ticks to 6.683185307179587. ##### [05:5.216](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=305.21585416666665) Narration: Keep going for a second turn. The green projection does the same thing, but it arrives at each landmark a quarter of a cycle later, because its arrow is a quarter of a turn behind. Board: Unchanged from the preceding beat in this scene. Actions: - [05:5.518](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=305.5178541666666): v\_arrow is redrawn as the numbers it depends on change. - [05:5.518](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=305.5178541666666): i\_arrow is redrawn as the numbers it depends on change. - [05:5.518](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=305.5178541666666): between is redrawn as the numbers it depends on change. - [05:5.518](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=305.5178541666666): v\_shadow is redrawn as the numbers it depends on change. - [05:5.518](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=305.5178541666666): i\_shadow is redrawn as the numbers it depends on change. - [05:5.518](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=305.5178541666666): v\_trace is redrawn as the numbers it depends on change. - [05:5.518](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=305.5178541666666): v\_rider is redrawn as the numbers it depends on change. - [05:5.518](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=305.5178541666666): i\_trace is redrawn as the numbers it depends on change. - [05:5.518](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=305.5178541666666): i\_rider is redrawn as the numbers it depends on change. - [05:5.518](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=305.5178541666666): clock ticks to 12.016370614359172. ##### [05:16.195](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=316.1953541666666) Narration: So the two waveforms we started this lecture with are the shadows of two arrows on one turning wheel. That is the whole content of the phasor idea, and everything else follows from it. Board: Unchanged from the preceding beat in this scene. Actions: - [05:17.171](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=317.17085416666663): projection is shown on the screen, written out. - [05:19.597](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=319.5968541666666): projection\_i is shown on the screen, written out. ##### [05:27.906](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=327.90585416666664) Narration: And now the move that makes it useful. Both arrows turn at the same rate, forever. So the turning tells us nothing we did not already know. What carries all the information is the two lengths and the angle between them, and those never change. Board: wheel — a Figure (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), aspect=(1.0, 1.0)); projection — a Math \[text\] that says "$v(t) = V\_m sin(omega t)$"; projection\_i — a Math \[text\] that says "$i(t) = I\_m sin(omega t - phi)$"; plot — an Axes (x\_range=(-0.35, 12.916370614359172), y\_range=(-1.35, 1.35), x\_ticks\_every=3.141592653589793); head\_wheel — a Heading that says "Two Arrows on One Wheel"; rim — a Circle \[gray\] drawn in wheel; hub — a Point \[gray\] drawn in wheel; axis\_h — a Line \[gray\] drawn in wheel (start=(-1.35, 0.0), end=(1.35, 0.0), dashed=True); axis\_v — a Line \[gray\] drawn in wheel (start=(0.0, -1.35), end=(0.0, 1.35), dashed=True); v\_arrow — a Vector \[yellow\] labelled "V" drawn in wheel (end=((1.0 \* cos(clock)), (1.0 \* sin(clock)))); i\_arrow — a Vector \[green\] labelled "I" drawn in wheel (end=((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((clock…); between — an Angle \[magenta\] labelled "phi" drawn in wheel (sides=(((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((cloc…, radius=0.32); v\_shadow — a Line \[yellow\] drawn in wheel (start=((1.0 \* cos(clock)), (1.0 \* sin(clock))), end=(0.0, (1.0 \* sin(clock))), dashed=True); i\_shadow — a Line \[green\] drawn in wheel (start=((0.75 \* cos((clock - 1.5707963267948966))), (0.75 \* sin((clock…, end=(0.0, (0.75 \* sin((clock - 1.5707963267948966)))), dashed=True); v\_trace — a ParametricCurve \[yellow\] drawn in plot (function=\, t\_range=(0.0, \)); v\_rider — a Point \[yellow\] drawn in plot (location=(\, (1.0 \* sin(clock)…); i\_trace — a ParametricCurve \[green\] drawn in plot (function=\, t\_range=(0.0, \)); i\_rider — a Point \[green\] drawn in plot (location=(\, (0.75 \* sin((cloc…) Actions: - [05:39.644](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=339.6438541666666): between is indicated — a transient flash. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): head\_wheel is hidden from the screen — left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): plot is hidden from the screen — left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): v\_trace is hidden from the screen — plot left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): v\_rider is hidden from the screen — plot left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): i\_trace is hidden from the screen — plot left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): i\_rider is hidden from the screen — plot left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): projection is hidden from the screen — left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): projection\_i is hidden from the screen — left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): wheel is hidden from the screen — left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): rim is hidden from the screen — wheel left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): hub is hidden from the screen — wheel left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): axis\_h is hidden from the screen — wheel left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): axis\_v is hidden from the screen — wheel left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): v\_arrow is hidden from the screen — wheel left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): i\_arrow is hidden from the screen — wheel left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): between is hidden from the screen — wheel left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): v\_shadow is hidden from the screen — wheel left the board. - [05:42.36](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=342.36035416666664): i\_shadow is hidden from the screen — wheel left the board. ##### [05:43.56](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=343.5603541666666) Narration: So stop the wheel. Photograph it at one instant and throw the rotation away. What is left is this. Board: Empty. Actions: - [05:43.56](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=343.5603541666666): head\_freeze is shown on the screen, written out. - [05:49.76](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=349.7598541666666): still is shown on the screen, written out. - [05:49.76](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=349.7598541666666): still\_hub is shown on the screen, written out. - [05:49.76](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=349.7598541666666): still\_ref is shown on the screen, written out. ##### [05:51.544](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=351.5443541666666) Narration: Line the voltage arrow up along the reference direction and call its angle zero. We are free to do that, because only the difference between the two angles ever mattered. Board: still — a Figure (x\_range=(-0.35, 1.5), y\_range=(-1.15, 0.55), aspect=(1.85, 1.7)); head\_freeze — a Heading that says "Stop the Wheel"; still\_hub — a Point \[gray\] drawn in still; still\_ref — a Line \[gray\] drawn in still (end=(1.4, 0.0), dashed=True) Actions: - [05:52.322](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=352.3218541666666): still\_v is shown on the screen, written out. - [05:55.701](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=355.7008541666666): still moves to a new place on the board. - [05:55.701](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=355.7008541666666): phasor\_v is shown on the screen, written out. ##### [06:2.106](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=362.10585416666663) Narration: The current arrow then sits below it, at minus phi. For an inductive load it is always below: lagging. Board: phasor\_v — a Math \[text\] that says "$bold(V) = V angle 0 degree$"; still — a Figure (x\_range=(-0.35, 1.5), y\_range=(-1.15, 0.55), aspect=(1.85, 1.7)); head\_freeze — a Heading that says "Stop the Wheel"; still\_hub — a Point \[gray\] drawn in still; still\_ref — a Line \[gray\] drawn in still (end=(1.4, 0.0), dashed=True); still\_v — a Vector \[yellow\] labelled "bold(V)" drawn in still (end=(1.25, 0.0)) Actions: - [06:2.768](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=362.7678541666666): still\_i is shown on the screen, written out. - [06:4.928](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=364.92785416666663): still\_angle is shown on the screen, written out. - [06:9.409](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=369.40885416666663): phasor\_i is shown on the screen, written out. ##### [06:10.833](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=370.8328541666666) Narration: Two arrows, frozen. That is a phasor diagram, and each arrow is a phasor: a length and an angle, standing in for a whole sinusoid. Board: phasor\_v — a Math \[text\] that says "$bold(V) = V angle 0 degree$"; phasor\_i — a Math \[text\] that says "$bold(I) = I angle - phi$"; still — a Figure (x\_range=(-0.35, 1.5), y\_range=(-1.15, 0.55), aspect=(1.85, 1.7)); head\_freeze — a Heading that says "Stop the Wheel"; still\_hub — a Point \[gray\] drawn in still; still\_ref — a Line \[gray\] drawn in still (end=(1.4, 0.0), dashed=True); still\_v — a Vector \[yellow\] labelled "bold(V)" drawn in still (end=(1.25, 0.0)); still\_i — a Vector \[green\] labelled "bold(I)" drawn in still (end=(0.29047597471245057, -0.8939931253174442)); still\_angle — an Angle \[magenta\] labelled "phi" drawn in still (sides=((1.25, 0.0), (0.29047597471245057, -0.8939931253174442))) Actions: - [06:13.922](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=373.92185416666655): phasor\_note is shown on the screen, written out. ##### [06:21.755](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=381.75485416666663) Narration: One more piece of bookkeeping before we use it. From here on, when I write V or I as a length, I mean the root mean square value, not the peak. Two forty volts from a wall socket is an r m s number, and using r m s throughout keeps the power formulas free of stray factors of one half. Board: phasor\_v — a Math \[text\] that says "$bold(V) = V angle 0 degree$"; phasor\_i — a Math \[text\] that says "$bold(I) = I angle - phi$"; phasor\_note — a Panel that says "A phasor is the arrow with the wheel stopped: one length and one angle, holding the amplitude and the phase of a sinusoid that everything in the circuit shares."; still — a Figure (x\_range=(-0.35, 1.5), y\_range=(-1.15, 0.55), aspect=(1.85, 1.7)); head\_freeze — a Heading that says "Stop the Wheel"; still\_hub — a Point \[gray\] drawn in still; still\_ref — a Line \[gray\] drawn in still (end=(1.4, 0.0), dashed=True); still\_v — a Vector \[yellow\] labelled "bold(V)" drawn in still (end=(1.25, 0.0)); still\_i — a Vector \[green\] labelled "bold(I)" drawn in still (end=(0.29047597471245057, -0.8939931253174442)); still\_angle — an Angle \[magenta\] labelled "phi" drawn in still (sides=((1.25, 0.0), (0.29047597471245057, -0.8939931253174442))) Actions: - [06:29.034](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=389.0338541666666): phasor\_v (the "V" part) is emphasized. - [06:33.573](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=393.5728541666666): phasor\_v (the "V" part) is no longer emphasized. ##### [06:42.648](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=402.6483541666666) Narration: So we have replaced two moving curves by two still arrows, and lost nothing. Now let us put those arrows to work and get an Ohm's law that a coil obeys. Board: Unchanged from the preceding beat in this scene. Actions: - [06:52.026](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=412.0257499999999): head\_freeze is hidden from the screen — left the board. - [06:52.026](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=412.0257499999999): phasor\_i is hidden from the screen — left the board. - [06:52.026](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=412.0257499999999): phasor\_note is hidden from the screen — left the board. - [06:52.026](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=412.0257499999999): phasor\_v is hidden from the screen — left the board. - [06:52.026](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=412.0257499999999): still is hidden from the screen — left the board. - [06:52.026](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=412.0257499999999): still\_hub is hidden from the screen — still left the board. - [06:52.026](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=412.0257499999999): still\_ref is hidden from the screen — still left the board. - [06:52.026](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=412.0257499999999): still\_v is hidden from the screen — still left the board. - [06:52.026](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=412.0257499999999): still\_i is hidden from the screen — still left the board. - [06:52.026](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=412.0257499999999): still\_angle is hidden from the screen — still left the board. ### Scene 3: [Impedance](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=413.0674166666666) Span: 06:53.067–10:4.647 (413.0674166666666s–604.64725s). #### Objects - current\_ref: a Vector \[green\] labelled "bold(I)" drawn in tri (start=(0.1, 0.25), end=(1.6500000000000001, 0.25)) - head\_build: a Heading that says "Adding the Two Voltages" - head\_z: a Heading that says "The Impedance" - kvl: a Math \[text\] that says "$bold(V) = bold(V)\_R + bold(V)\_L$" - l\_law: a Math \[text\] that says "$bold(V)\_L = j omega L bold(I) = j X\_L bold(I)$" - ohm: a Math \[text\] that says "$bold(V) = bold(Z) bold(I)$" - ohm\_phi: an Angle \[magenta\] labelled "36.87 degree" drawn in ohms (vertex=(0.1, 0.25), sides=((1.3800000000000001, 0.25), (1.3800000000000001, 1.21)), radius=0.45) - ohm\_r: a Vector \[blue\] labelled "8 thin Omega" drawn in ohms (start=(0.1, 0.25), end=(1.3800000000000001, 0.25)) - ohm\_square: an Angle \[gray\] drawn in ohms (vertex=(1.3800000000000001, 0.25), sides=((0.1, 0.25), (1.3800000000000001, 1.21)), right\_angle=True) - ohm\_x: a Vector \[red\] labelled "6 thin Omega" drawn in ohms (start=(1.3800000000000001, 0.25), end=(1.3800000000000001, 1.21)) - ohm\_z: a Vector \[yellow\] labelled "10 thin Omega" drawn in ohms (start=(0.1, 0.25), end=(1.3800000000000001, 1.21)) - ohms: a Figure (x\_range=(-0.25, 1.85), y\_range=(-0.3, 1.35), aspect=(2.1, 1.65)) - phi\_mark: an Angle \[magenta\] labelled "phi" drawn in tri (vertex=(0.1, 0.25), sides=((1.3800000000000001, 0.25), (1.3800000000000001, 1.21)), radius=0.45) - problem: a Tex \[text\] that says "A motor runs from a $240 thin upright("V")$ supply, with $R = 8 thin Omega$ and $X\_L = 6 thin Omega$." - r\_law: a Math \[text\] that says "$bold(V)\_R = R bold(I)$" - result: a Math \[text\] that says "$bold(Z) = 10 thin Omega angle 36.87 degree$" - result\_i: a Math \[text\] that says "$bold(I) = 24 thin upright("A") angle - 36.87 degree$" - square\_mark: an Angle \[gray\] drawn in tri (vertex=(1.3800000000000001, 0.25), sides=((0.1, 0.25), (1.3800000000000001, 1.21)), right\_angle=True) - tri: a Figure (x\_range=(-0.25, 1.85), y\_range=(-0.3, 1.35), aspect=(2.1, 1.65)) - v\_l: a Vector \[red\] labelled "bold(V)\_L" drawn in tri (start=(1.3800000000000001, 0.25), end=(1.3800000000000001, 1.21)) - v\_r: a Vector \[blue\] labelled "bold(V)\_R" drawn in tri (start=(0.1, 0.25), end=(1.3800000000000001, 0.25)) - v\_total: a Vector \[yellow\] labelled "bold(V)" drawn in tri (start=(0.1, 0.25), end=(1.3800000000000001, 1.21)) - w\_mag: a Derivation \[text\] that says "$abs(bold(Z)) &= sqrt(8^2 + 6^2) thin Omega \\ &= sqrt(100) thin Omega = 10 thin Omega$" - w\_rest: a Derivation \[text\] that says "$I &= frac(240 thin upright("V"), 10 thin Omega) \\ &= 24 thin upright("A") \\ phi &= arctan(frac(6, 8)) = 36.87 degree$" - z\_work: a Derivation \[text\] that says "$bold(V) &= (R + j X\_L) bold(I) \\ bold(Z) &= R + j X\_L \\ abs(bold(Z)) &= sqrt(R^2 + X\_L^2) \\ phi &= arctan(frac(X\_L, R))$" #### Beats ##### [06:53.067](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=413.0674166666666) Narration: A real motor winding is a resistance and an inductance in series, and one current flows through both of them. So let us draw that current first, and hang everything else off it. Board: Empty. Actions: - [06:53.067](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=413.0674166666666): head\_build is shown on the screen, written out. - [06:57.548](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=417.5484166666666): current\_ref is shown on the screen, written out. - [07:0.196](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=420.1964166666666): tri is shown on the screen, written out. ##### [07:4.093](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=424.0929166666666) Narration: Across the resistance, voltage and current stay in step. So the resistor's voltage phasor lies flat along the current, and its length is R times I. Board: tri — a Figure (x\_range=(-0.25, 1.85), y\_range=(-0.3, 1.35), aspect=(2.1, 1.65)); head\_build — a Heading that says "Adding the Two Voltages"; current\_ref — a Vector \[green\] labelled "bold(I)" drawn in tri (start=(0.1, 0.25), end=(1.6500000000000001, 0.25)) Actions: - [07:9.898](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=429.8984166666666): v\_r is shown on the screen, written out. - [07:12.615](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=432.6154166666666): tri moves to a new place on the board. - [07:12.615](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=432.6154166666666): r\_law is shown on the screen, written out. ##### [07:14.643](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=434.6434166666666) Narration: Across the coil it is different, and we did the calculus for this in the opening. The coil's voltage leads its current by ninety degrees, so its phasor stands square to the current, and its length is omega L times I. Board: r\_law — a Math \[text\] that says "$bold(V)\_R = R bold(I)$"; tri — a Figure (x\_range=(-0.25, 1.85), y\_range=(-0.3, 1.35), aspect=(2.1, 1.65)); head\_build — a Heading that says "Adding the Two Voltages"; current\_ref — a Vector \[green\] labelled "bold(I)" drawn in tri (start=(0.1, 0.25), end=(1.6500000000000001, 0.25)); v\_r — a Vector \[blue\] labelled "bold(V)\_R" drawn in tri (start=(0.1, 0.25), end=(1.3800000000000001, 0.25)) Actions: - [07:23.931](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=443.93141666666656): v\_l is shown on the screen, written out. - [07:23.931](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=443.93141666666656): square\_mark is shown on the screen, written out. - [07:26.183](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=446.1834166666666): l\_law is shown on the screen, written out. ##### [07:28.861](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=448.86141666666657) Narration: That quantity omega L is called the inductive reactance, X sub L, and like resistance it is measured in ohms. The j in front of it is the algebra's way of saying: turn it ninety degrees. Board: r\_law — a Math \[text\] that says "$bold(V)\_R = R bold(I)$"; l\_law — a Math \[text\] that says "$bold(V)\_L = j omega L bold(I) = j X\_L bold(I)$"; tri — a Figure (x\_range=(-0.25, 1.85), y\_range=(-0.3, 1.35), aspect=(2.1, 1.65)); head\_build — a Heading that says "Adding the Two Voltages"; current\_ref — a Vector \[green\] labelled "bold(I)" drawn in tri (start=(0.1, 0.25), end=(1.6500000000000001, 0.25)); v\_r — a Vector \[blue\] labelled "bold(V)\_R" drawn in tri (start=(0.1, 0.25), end=(1.3800000000000001, 0.25)); v\_l — a Vector \[red\] labelled "bold(V)\_L" drawn in tri (start=(1.3800000000000001, 0.25), end=(1.3800000000000001, 1.21)); square\_mark — an Angle \[gray\] drawn in tri (vertex=(1.3800000000000001, 0.25), sides=((0.1, 0.25), (1.3800000000000001, 1.21)), right\_angle=True) Actions: - [07:32.135](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=452.1354166666666): l\_law (the "j X\_L" part) is emphasized. - [07:40.68](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=460.6804166666666): l\_law (the "j X\_L" part) is no longer emphasized. ##### [07:42.43](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=462.4299166666666) Narration: The voltage law still holds, provided we add the phasors as vectors rather than as numbers. So the source voltage is the resistor's voltage and the coil's voltage joined head to tail. Board: Unchanged from the preceding beat in this scene. Actions: - [07:42.766](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=462.7664166666666): kvl is shown on the screen, written out. - [07:52.704](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=472.7044166666666): v\_total is shown on the screen, written out. ##### [07:54.431](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=474.4309166666666) Narration: And there is the whole geometry of an inductive load in one triangle. The source voltage is the hypotenuse, and it sits ahead of the current by an angle phi. That angle is what we are about to name. Board: r\_law — a Math \[text\] that says "$bold(V)\_R = R bold(I)$"; l\_law — a Math \[text\] that says "$bold(V)\_L = j omega L bold(I) = j X\_L bold(I)$"; kvl — a Math \[text\] that says "$bold(V) = bold(V)\_R + bold(V)\_L$"; tri — a Figure (x\_range=(-0.25, 1.85), y\_range=(-0.3, 1.35), aspect=(2.1, 1.65)); head\_build — a Heading that says "Adding the Two Voltages"; current\_ref — a Vector \[green\] labelled "bold(I)" drawn in tri (start=(0.1, 0.25), end=(1.6500000000000001, 0.25)); v\_r — a Vector \[blue\] labelled "bold(V)\_R" drawn in tri (start=(0.1, 0.25), end=(1.3800000000000001, 0.25)); v\_l — a Vector \[red\] labelled "bold(V)\_L" drawn in tri (start=(1.3800000000000001, 0.25), end=(1.3800000000000001, 1.21)); square\_mark — an Angle \[gray\] drawn in tri (vertex=(1.3800000000000001, 0.25), sides=((0.1, 0.25), (1.3800000000000001, 1.21)), right\_angle=True); v\_total — a Vector \[yellow\] labelled "bold(V)" drawn in tri (start=(0.1, 0.25), end=(1.3800000000000001, 1.21)) Actions: - [08:3.161](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=483.1614166666666): phi\_mark is shown on the screen, written out. - [08:7.771](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=487.7714166666666): kvl moves to a new place on the board. - [08:7.771](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=487.7714166666666): tri moves to a new place on the board. - [08:7.771](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=487.7714166666666): head\_build is hidden from the screen — left the board. - [08:7.771](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=487.7714166666666): l\_law is hidden from the screen — left the board. - [08:7.771](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=487.7714166666666): r\_law is hidden from the screen — left the board. ##### [08:8.971](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=488.9714166666666) Narration: Factor the current out of both terms. What is left multiplying it is R plus j X sub L, and that combination is the impedance, bold Z. Board: kvl — a Math \[text\] that says "$bold(V) = bold(V)\_R + bold(V)\_L$"; tri — a Figure (x\_range=(-0.25, 1.85), y\_range=(-0.3, 1.35), aspect=(2.1, 1.65)); current\_ref — a Vector \[green\] labelled "bold(I)" drawn in tri (start=(0.1, 0.25), end=(1.6500000000000001, 0.25)); v\_r — a Vector \[blue\] labelled "bold(V)\_R" drawn in tri (start=(0.1, 0.25), end=(1.3800000000000001, 0.25)); v\_l — a Vector \[red\] labelled "bold(V)\_L" drawn in tri (start=(1.3800000000000001, 0.25), end=(1.3800000000000001, 1.21)); square\_mark — an Angle \[gray\] drawn in tri (vertex=(1.3800000000000001, 0.25), sides=((0.1, 0.25), (1.3800000000000001, 1.21)), right\_angle=True); v\_total — a Vector \[yellow\] labelled "bold(V)" drawn in tri (start=(0.1, 0.25), end=(1.3800000000000001, 1.21)); phi\_mark — an Angle \[magenta\] labelled "phi" drawn in tri (vertex=(0.1, 0.25), sides=((1.3800000000000001, 0.25), (1.3800000000000001, 1.21)), radius=0.45) Actions: - [08:8.971](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=488.9714166666666): head\_z is shown on the screen, written out. - [08:9.447](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=489.4474166666666): z\_work is shown on the screen, written out. - [08:17.876](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=497.8764166666666): z\_work is shown on the screen, written out. ##### [08:20.67](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=500.6704166666666) Narration: It is a complex number, and its two parts are exactly the two sides of the triangle. Its magnitude is the square root of R squared plus X squared. Board: kvl — a Math \[text\] that says "$bold(V) = bold(V)\_R + bold(V)\_L$"; tri — a Figure (x\_range=(-0.25, 1.85), y\_range=(-0.3, 1.35), aspect=(2.1, 1.65)); current\_ref — a Vector \[green\] labelled "bold(I)" drawn in tri (start=(0.1, 0.25), end=(1.6500000000000001, 0.25)); v\_r — a Vector \[blue\] labelled "bold(V)\_R" drawn in tri (start=(0.1, 0.25), end=(1.3800000000000001, 0.25)); v\_l — a Vector \[red\] labelled "bold(V)\_L" drawn in tri (start=(1.3800000000000001, 0.25), end=(1.3800000000000001, 1.21)); square\_mark — an Angle \[gray\] drawn in tri (vertex=(1.3800000000000001, 0.25), sides=((0.1, 0.25), (1.3800000000000001, 1.21)), right\_angle=True); v\_total — a Vector \[yellow\] labelled "bold(V)" drawn in tri (start=(0.1, 0.25), end=(1.3800000000000001, 1.21)); phi\_mark — an Angle \[magenta\] labelled "phi" drawn in tri (vertex=(0.1, 0.25), sides=((1.3800000000000001, 0.25), (1.3800000000000001, 1.21)), radius=0.45); head\_z — a Heading that says "The Impedance" Actions: - [08:26.603](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=506.6034166666666): z\_work is shown on the screen, written out. - [08:27.462](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=507.46241666666657): z\_work (the "sqrt(R^2 + X\_L^2)" part) is emphasized. ##### [08:31.069](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=511.0694166666666) Narration: And its angle is the arctangent of X over R, which is the same phi we have been carrying since the first waveform. The reactance sets the phase; the resistance holds it back. Board: Unchanged from the preceding beat in this scene. Actions: - [08:31.069](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=511.0694166666666): z\_work (the "sqrt(R^2 + X\_L^2)" part) is no longer emphasized. - [08:32.486](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=512.4864166666666): z\_work is shown on the screen, written out. - [08:35.481](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=515.4814166666665): z\_work (the "arctan(frac(X\_L, R))" part) is emphasized. - [08:41.042](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=521.0424166666666): z\_work (the "arctan(frac(X\_L, R))" part) is no longer emphasized. ##### [08:43.349](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=523.3494166666666) Narration: So Ohm's law survives into alternating current unchanged in form. Voltage phasor equals impedance times current phasor. The only thing that has grown is that all three are complex, and the angles carry the timing. Board: Unchanged from the preceding beat in this scene. Actions: - [08:48.678](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=528.6784166666666): ohm is shown on the screen, written out. - [08:54.599](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=534.5994166666666): A box is drawn around ohm. - [08:57.624](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=537.6239166666666): head\_z is hidden from the screen — left the board. - [08:57.624](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=537.6239166666666): kvl is hidden from the screen — left the board. - [08:57.624](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=537.6239166666666): ohm is hidden from the screen — left the board. - [08:57.624](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=537.6239166666666): tri is hidden from the screen — left the board. - [08:57.624](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=537.6239166666666): current\_ref is hidden from the screen — tri left the board. - [08:57.624](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=537.6239166666666): v\_r is hidden from the screen — tri left the board. - [08:57.624](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=537.6239166666666): v\_l is hidden from the screen — tri left the board. - [08:57.624](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=537.6239166666666): square\_mark is hidden from the screen — tri left the board. - [08:57.624](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=537.6239166666666): v\_total is hidden from the screen — tri left the board. - [08:57.624](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=537.6239166666666): phi\_mark is hidden from the screen — tri left the board. - [08:57.624](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=537.6239166666666): z\_work is hidden from the screen — left the board. ##### [08:58.824](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=538.8239166666666) Narration: Let us put numbers on it, because these numbers will follow us to the end of the lecture. A motor on a two hundred and forty volt supply, with eight ohms of winding resistance and six ohms of reactance. Board: Empty. Actions: - [08:58.824](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=538.8239166666666): problem is shown on the screen, written out. - [09:6.898](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=546.8984166666665): ohms is shown on the screen, written out. - [09:6.898](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=546.8984166666665): ohm\_r is shown on the screen, written out. - [09:8.907](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=548.9074166666666): ohm\_x is shown on the screen, written out. - [09:8.907](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=548.9074166666666): ohm\_square is shown on the screen, written out. ##### [09:11.504](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=551.5039166666666) Narration: The triangle is now measured in ohms rather than volts, which is the same picture divided through by the current. Eight along, six up. Board: ohms — a Figure (x\_range=(-0.25, 1.85), y\_range=(-0.3, 1.35), aspect=(2.1, 1.65)); problem — a Tex \[text\] that says "A motor runs from a $240 thin upright("V")$ supply, with $R = 8 thin Omega$ and $X\_L = 6 thin Omega$."; ohm\_r — a Vector \[blue\] labelled "8 thin Omega" drawn in ohms (start=(0.1, 0.25), end=(1.3800000000000001, 0.25)); ohm\_x — a Vector \[red\] labelled "6 thin Omega" drawn in ohms (start=(1.3800000000000001, 0.25), end=(1.3800000000000001, 1.21)); ohm\_square — an Angle \[gray\] drawn in ohms (vertex=(1.3800000000000001, 0.25), sides=((0.1, 0.25), (1.3800000000000001, 1.21)), right\_angle=True) Actions: - [09:18.435](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=558.4354166666666): ohm\_r is indicated — a transient flash. - [09:19.271](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=559.2714166666666): ohm\_x is indicated — a transient flash. ##### [09:20.788](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=560.7884166666665) Narration: So the magnitude of the impedance is the square root of sixty four plus thirty six, which is the square root of a hundred: ten ohms. Board: Unchanged from the preceding beat in this scene. Actions: - [09:21.357](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=561.3574166666666): ohms moves to a new place on the board. - [09:21.357](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=561.3574166666666): w\_mag is shown on the screen, written out. - [09:27.707](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=567.7074166666666): ohm\_z is shown on the screen, written out. - [09:27.707](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=567.7074166666666): w\_mag is shown on the screen, written out. ##### [09:29.341](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=569.3409166666665) Narration: Now the current is two hundred and forty volts over ten ohms. Twenty four amps. That is the number a clamp meter on the supply lead would read. Board: ohms — a Figure (x\_range=(-0.25, 1.85), y\_range=(-0.3, 1.35), aspect=(2.1, 1.65)); problem — a Tex \[text\] that says "A motor runs from a $240 thin upright("V")$ supply, with $R = 8 thin Omega$ and $X\_L = 6 thin Omega$."; ohm\_r — a Vector \[blue\] labelled "8 thin Omega" drawn in ohms (start=(0.1, 0.25), end=(1.3800000000000001, 0.25)); ohm\_x — a Vector \[red\] labelled "6 thin Omega" drawn in ohms (start=(1.3800000000000001, 0.25), end=(1.3800000000000001, 1.21)); ohm\_square — an Angle \[gray\] drawn in ohms (vertex=(1.3800000000000001, 0.25), sides=((0.1, 0.25), (1.3800000000000001, 1.21)), right\_angle=True); ohm\_z — a Vector \[yellow\] labelled "10 thin Omega" drawn in ohms (start=(0.1, 0.25), end=(1.3800000000000001, 1.21)) Actions: - [09:31.535](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=571.5354166666666): w\_mag is hidden from the screen — left the board. - [09:31.535](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=571.5354166666666): w\_rest is shown on the screen, written out. - [09:32.975](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=572.9754166666665): w\_rest is shown on the screen, written out. - [09:33.602](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=573.6024166666666): w\_rest (the "24 thin upright("A")" part) is emphasized. - [09:35.738](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=575.7384166666666): w\_rest (the "24 thin upright("A")" part) is no longer emphasized. ##### [09:38.927](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=578.9274166666667) Narration: And the phase angle is the arctangent of six over eight, which is thirty six point eight seven degrees. The current lags the voltage by that much. Board: Unchanged from the preceding beat in this scene. Actions: - [09:40.738](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=580.7384166666666): w\_rest is shown on the screen, written out. - [09:43.176](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=583.1764166666666): ohm\_phi is shown on the screen, written out. ##### [09:49.129](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=589.1289166666666) Narration: Impedance ten ohms at thirty seven degrees, current twenty four amps lagging by thirty seven degrees. Hold on to both. In the next section we multiply the voltage by the current, and that angle is about to cost somebody money. Board: ohms — a Figure (x\_range=(-0.25, 1.85), y\_range=(-0.3, 1.35), aspect=(2.1, 1.65)); problem — a Tex \[text\] that says "A motor runs from a $240 thin upright("V")$ supply, with $R = 8 thin Omega$ and $X\_L = 6 thin Omega$."; ohm\_r — a Vector \[blue\] labelled "8 thin Omega" drawn in ohms (start=(0.1, 0.25), end=(1.3800000000000001, 0.25)); ohm\_x — a Vector \[red\] labelled "6 thin Omega" drawn in ohms (start=(1.3800000000000001, 0.25), end=(1.3800000000000001, 1.21)); ohm\_square — an Angle \[gray\] drawn in ohms (vertex=(1.3800000000000001, 0.25), sides=((0.1, 0.25), (1.3800000000000001, 1.21)), right\_angle=True); ohm\_z — a Vector \[yellow\] labelled "10 thin Omega" drawn in ohms (start=(0.1, 0.25), end=(1.3800000000000001, 1.21)); ohm\_phi — an Angle \[magenta\] labelled "36.87 degree" drawn in ohms (vertex=(0.1, 0.25), sides=((1.3800000000000001, 0.25), (1.3800000000000001, 1.21)), radius=0.45) Actions: - [09:49.349](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=589.3494166666666): result is shown on the screen, written out. - [09:52.379](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=592.3794166666667): result\_i is shown on the screen, written out. - [09:55.955](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=595.9554166666666): A box is drawn around result\_i. - [10:3.606](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=603.6055833333332): ohms is hidden from the screen — left the board. - [10:3.606](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=603.6055833333332): ohm\_r is hidden from the screen — ohms left the board. - [10:3.606](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=603.6055833333332): ohm\_x is hidden from the screen — ohms left the board. - [10:3.606](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=603.6055833333332): ohm\_square is hidden from the screen — ohms left the board. - [10:3.606](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=603.6055833333332): ohm\_z is hidden from the screen — ohms left the board. - [10:3.606](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=603.6055833333332): ohm\_phi is hidden from the screen — ohms left the board. - [10:3.606](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=603.6055833333332): problem is hidden from the screen — left the board. - [10:3.606](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=603.6055833333332): result is hidden from the screen — left the board. - [10:3.606](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=603.6055833333332): result\_i is hidden from the screen — left the board. - [10:3.606](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=603.6055833333332): w\_rest is hidden from the screen — left the board. ### Scene 4: [The Power That Works and the Power That Sloshes](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=604.64725) Span: 10:4.647–15:13.636 (604.64725s–913.6355625s). #### Objects - app\_def: a Math \[text\] that says "$S = V I quad \[upright("VA")\]$" - compare: a Math \[text\] that says "$frac(24^2, 19.2^2) = 1.56$" - head\_bill: a Heading that says "Why the Utility Charges for It" - head\_prod: a Heading that says "Multiply the Two Waveforms" - head\_split: a Heading that says "Two Terms, One Average" - i\_curve: a FunctionPlot \[green\] drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)) - instant: a Math \[text\] that says "$p(t) = v(t) thin i(t)$" - loss\_law: a Math \[text\] that says "$P\_upright("line") = I^2 R\_upright("line")$" - mean\_line: a Line \[blue\] labelled "P" drawn in split (start=(0.0, 0.2879996141334631), end=(12.566370614359172, 0.2879996141334631), dashed=True) - numbers: a Table \[text\] that says "Quantity Value Apparent power $S$ 5.76 kVA Real power $P$ 4.61 kW Reactive power $Q$ 3.46 kvar Power factor 0.80 lagging" (rows=(('Quantity', 'Value'), ('Apparent power $S$', '5.76 kVA'), ('R…, header=True) - p\_curve: a FunctionPlot \[magenta\] drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)) - p\_work: a Derivation \[text\] that says "$p(t) &= V\_m I\_m sin(omega t) sin(omega t - phi) \\ &= V I cos phi thin (1 - cos 2 omega t) \\ & thin - V I sin phi thin sin 2 omega t$" - pf\_def: a Math \[text\] that says "$upright("pf") = cos phi = frac(P, S)$" - point: a Point \[yellow\] drawn in waves (location=(3.491592653589793, -0.07142599256291865)) - point\_2: a Point \[yellow\] drawn in waves (location=(6.633185307179586, -0.07142599256291868)) - pt: a Figure (x\_range=(-0.25, 2.05), y\_range=(-0.3, 1.45), aspect=(2.3, 1.75)) - pt\_p: a Vector \[blue\] labelled "P = 4.61 thin upright("kW")" drawn in pt (start=(0.12, 0.22), end=(1.6400000000000001, 0.22)) - pt\_phi: an Angle \[magenta\] labelled "36.87 degree" drawn in pt (vertex=(0.12, 0.22), sides=((1.6400000000000001, 0.22), (1.6400000000000001, 1.36)), radius=0.46) - pt\_q: a Vector \[red\] labelled "Q = 3.46 thin upright("kvar")" drawn in pt (start=(1.6400000000000001, 0.22), end=(1.6400000000000001, 1.36)) - pt\_s: a Vector \[yellow\] labelled "S = 5.76 thin upright("kVA")" drawn in pt (start=(0.12, 0.22), end=(1.6400000000000001, 1.36)) - pt\_square: an Angle \[gray\] drawn in pt (vertex=(1.6400000000000001, 0.22), sides=((0.12, 0.22), (1.6400000000000001, 1.36)), right\_angle=True) - react\_curve: a FunctionPlot \[red\] labelled "upright("sloshes")" drawn in split (function=\, x\_range=(0.0, 12.566370614359172)) - react\_def: a Math \[text\] that says "$Q = V I sin phi quad \[upright("var")\]$" - real\_curve: a FunctionPlot \[blue\] labelled "upright("works")" drawn in split (function=\, x\_range=(0.0, 12.566370614359172)) - real\_def: a Math \[text\] that says "$P = V I cos phi quad \[upright("W")\]$" - split: an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-0.75, 1.05), x\_ticks\_every=3.141592653589793) - split\_zero: a Line \[gray\] drawn in split (end=(12.566370614359172, 0.0), dashed=True) - v\_curve: a FunctionPlot \[yellow\] drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)) - verdict: a Text \[text\] that says "The wires and the transformer are sized by $S$, not by $P$. The utility builds for the current it must actually carry." - waves: an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-0.75, 1.05), x\_ticks\_every=3.141592653589793) - zero\_line: a Line \[gray\] drawn in waves (end=(12.566370614359172, 0.0), dashed=True) #### Beats ##### [10:4.647](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=604.64725) Narration: Power at any instant is voltage times current at that instant. No phasors, no averages: just the product of the two numbers, moment by moment. Board: Empty. Actions: - [10:4.647](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=604.64725): head\_prod is shown on the screen, written out. - [10:11.381](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=611.38125): instant is shown on the screen, written out. - [10:12.008](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=612.00825): waves is shown on the screen, written out. - [10:12.008](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=612.00825): zero\_line is shown on the screen, written out. ##### [10:14.779](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=614.77875) Narration: Here are our two waveforms again, the voltage in yellow and the current lagging it by thirty seven degrees in green. Board: instant — a Math \[text\] that says "$p(t) = v(t) thin i(t)$"; waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-0.75, 1.05), x\_ticks\_every=3.141592653589793); head\_prod — a Heading that says "Multiply the Two Waveforms"; zero\_line — a Line \[gray\] drawn in waves (end=(12.566370614359172, 0.0), dashed=True) Actions: - [10:18.331](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=618.33125): v\_curve is shown on the screen, written out. - [10:21.315](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=621.31525): i\_curve is shown on the screen, written out. ##### [10:22.705](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=622.70475) Narration: And here in magenta is their product, the instantaneous power flowing into the motor. Board: instant — a Math \[text\] that says "$p(t) = v(t) thin i(t)$"; waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-0.75, 1.05), x\_ticks\_every=3.141592653589793); head\_prod — a Heading that says "Multiply the Two Waveforms"; zero\_line — a Line \[gray\] drawn in waves (end=(12.566370614359172, 0.0), dashed=True); v\_curve — a FunctionPlot \[yellow\] drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)) Actions: - [10:23.622](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=623.62225): p\_curve is shown on the screen, written out. ##### [10:28.912](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=628.91225) Narration: Two things about it. First, it oscillates at twice the supply frequency, because it is a product of two sinusoids: two humps for every one cycle of the voltage. Board: instant — a Math \[text\] that says "$p(t) = v(t) thin i(t)$"; waves — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-0.75, 1.05), x\_ticks\_every=3.141592653589793); head\_prod — a Heading that says "Multiply the Two Waveforms"; zero\_line — a Line \[gray\] drawn in waves (end=(12.566370614359172, 0.0), dashed=True); v\_curve — a FunctionPlot \[yellow\] drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); i\_curve — a FunctionPlot \[green\] drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)); p\_curve — a FunctionPlot \[magenta\] drawn in waves (function=\, x\_range=(0.0, 12.566370614359172)) Actions: - [10:32.418](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=632.41825): p\_curve is indicated — a transient flash. ##### [10:40.53](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=640.53025) Narration: Second, and this is the interesting one, it goes negative. Here, and here again half a cycle later. Negative power means energy flowing out of the motor and back into the supply. Board: Unchanged from the preceding beat in this scene. Actions: - [10:44.315](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=644.31525): point is shown on the screen, grown. - [10:45.395](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=645.39525): point\_2 is shown on the screen, grown. - [10:45.915](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=645.91525): point is hidden from the screen. - [10:46.995](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=646.9952499999999): point\_2 is hidden from the screen. ##### [10:52.38](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=652.3802499999999) Narration: The coil's magnetic field is storing energy on one part of the cycle and handing it straight back on the next. Nothing is consumed by that exchange. It just sloshes. Board: Unchanged from the preceding beat in this scene. Actions: - None. ##### [11:3.278](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=663.27825) Narration: Let us separate the sloshing from the working. Expand the product of the two sines with the usual identity, and it falls into exactly two terms. Board: Unchanged from the preceding beat in this scene. Actions: - [11:6.308](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=666.3082499999999): p\_work is shown on the screen, written out. - [11:10.963](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=670.96325): p\_work is shown on the screen, written out. - [11:10.963](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=670.96325): p\_work is shown on the screen, written out. ##### [11:12.748](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=672.74775) Narration: Look at the first term. It is V I cosine phi, multiplied by one minus cosine of two omega t. That bracket never goes negative: it swings between zero and two, and its average is one. Board: Unchanged from the preceding beat in this scene. Actions: - [11:14.756](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=674.7562499999999): p\_work (the "V I cos phi" part) is emphasized. - [11:20.073](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=680.0732499999999): p\_work (the "(1 - cos 2 omega t)" part) is emphasized. - [11:20.073](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=680.0732499999999): p\_work (the "V I cos phi" part) is no longer emphasized. - [11:24.253](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=684.25325): p\_work (the "(1 - cos 2 omega t)" part) is no longer emphasized. ##### [11:25.922](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=685.92175) Narration: So that term has an average value of V I cosine phi, and it is always into the load. That is energy actually leaving the supply and not coming back. Board: Unchanged from the preceding beat in this scene. Actions: - None. ##### [11:35.751](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=695.75125) Narration: The second term is V I sine phi times sine of two omega t. That is a pure sinusoid at double frequency, sitting symmetrically about zero. Its average over a cycle is exactly nothing. Board: Unchanged from the preceding beat in this scene. Actions: - [11:37.179](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=697.1792499999999): p\_work (the "V I sin phi" part) is emphasized. - [11:41.823](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=701.8232499999999): p\_work (the "V I sin phi" part) is no longer emphasized. - [11:41.823](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=701.8232499999999): p\_work (the "sin 2 omega t" part) is emphasized. - [11:47.152](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=707.15225): p\_work (the "sin 2 omega t" part) is no longer emphasized. - [11:49.857](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=709.85725): head\_prod is hidden from the screen — left the board. - [11:49.857](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=709.85725): instant is hidden from the screen — left the board. - [11:49.857](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=709.85725): p\_work is hidden from the screen — left the board. - [11:49.857](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=709.85725): waves is hidden from the screen — left the board. - [11:49.857](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=709.85725): zero\_line is hidden from the screen — waves left the board. - [11:49.857](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=709.85725): v\_curve is hidden from the screen — waves left the board. - [11:49.857](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=709.85725): i\_curve is hidden from the screen — waves left the board. - [11:49.857](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=709.85725): p\_curve is hidden from the screen — waves left the board. ##### [11:51.057](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=711.05725) Narration: Here are those two terms drawn separately. The blue one is the working term. Notice that it never dips below the axis: it is one-directional, always from supply to load. Board: Empty. Actions: - [11:51.057](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=711.05725): head\_split is shown on the screen, written out. - [11:51.057](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=711.05725): split is shown on the screen, written out. - [11:51.057](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=711.05725): split\_zero is shown on the screen, written out. - [11:54.424](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=714.4242499999999): real\_curve is shown on the screen, written out. ##### [12:3.569](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=723.56925) Narration: Its average is this dashed level, and that average is what we call the real power P. V times I times cosine phi, measured in watts. That is the number that turns the shaft and heats the windings. Board: split — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-0.75, 1.05), x\_ticks\_every=3.141592653589793); head\_split — a Heading that says "Two Terms, One Average"; split\_zero — a Line \[gray\] drawn in split (end=(12.566370614359172, 0.0), dashed=True); real\_curve — a FunctionPlot \[blue\] labelled "upright("works")" drawn in split (function=\, x\_range=(0.0, 12.566370614359172)) Actions: - [12:4.997](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=724.9972499999999): mean\_line is shown on the screen, written out. - [12:7.284](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=727.2842499999999): split moves to a new place on the board. - [12:7.284](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=727.2842499999999): real\_def is shown on the screen, written out. - [12:10.744](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=730.74425): real\_def (the "V I cos phi" part) is emphasized. - [12:15.202](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=735.2022499999999): real\_def (the "V I cos phi" part) is no longer emphasized. ##### [12:17.706](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=737.70625) Narration: And here is the red one. Equal time above the axis and below it, twice every cycle. Energy out, energy back, out, back. Board: real\_def — a Math \[text\] that says "$P = V I cos phi quad \[upright("W")\]$"; split — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-0.75, 1.05), x\_ticks\_every=3.141592653589793); head\_split — a Heading that says "Two Terms, One Average"; split\_zero — a Line \[gray\] drawn in split (end=(12.566370614359172, 0.0), dashed=True); real\_curve — a FunctionPlot \[blue\] labelled "upright("works")" drawn in split (function=\, x\_range=(0.0, 12.566370614359172)); mean\_line — a Line \[blue\] labelled "P" drawn in split (start=(0.0, 0.2879996141334631), end=(12.566370614359172, 0.2879996141334631), dashed=True) Actions: - [12:18.612](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=738.6122499999999): react\_curve is shown on the screen, written out. ##### [12:27.409](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=747.4087499999999) Narration: Its average is zero. Over any whole number of cycles it delivers not one joule. But its amplitude is not zero, and that amplitude has a name: reactive power Q, V I sine phi, measured in var. Board: real\_def — a Math \[text\] that says "$P = V I cos phi quad \[upright("W")\]$"; split — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-0.75, 1.05), x\_ticks\_every=3.141592653589793); head\_split — a Heading that says "Two Terms, One Average"; split\_zero — a Line \[gray\] drawn in split (end=(12.566370614359172, 0.0), dashed=True); real\_curve — a FunctionPlot \[blue\] labelled "upright("works")" drawn in split (function=\, x\_range=(0.0, 12.566370614359172)); mean\_line — a Line \[blue\] labelled "P" drawn in split (start=(0.0, 0.2879996141334631), end=(12.566370614359172, 0.2879996141334631), dashed=True); react\_curve — a FunctionPlot \[red\] labelled "upright("sloshes")" drawn in split (function=\, x\_range=(0.0, 12.566370614359172)) Actions: - [12:37.788](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=757.78825): react\_def is shown on the screen, written out. - [12:40.087](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=760.0872499999999): react\_def (the "V I sin phi" part) is emphasized. - [12:41.515](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=761.5152499999999): react\_def (the "V I sin phi" part) is no longer emphasized. ##### [12:42.858](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=762.8577499999999) Narration: And the plain product of the voltage and current magnitudes, with no angle in it at all, is the apparent power S, in volt amps. That is what a voltmeter and an ammeter multiplied together would tell you. Board: real\_def — a Math \[text\] that says "$P = V I cos phi quad \[upright("W")\]$"; react\_def — a Math \[text\] that says "$Q = V I sin phi quad \[upright("var")\]$"; split — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-0.75, 1.05), x\_ticks\_every=3.141592653589793); head\_split — a Heading that says "Two Terms, One Average"; split\_zero — a Line \[gray\] drawn in split (end=(12.566370614359172, 0.0), dashed=True); real\_curve — a FunctionPlot \[blue\] labelled "upright("works")" drawn in split (function=\, x\_range=(0.0, 12.566370614359172)); mean\_line — a Line \[blue\] labelled "P" drawn in split (start=(0.0, 0.2879996141334631), end=(12.566370614359172, 0.2879996141334631), dashed=True); react\_curve — a FunctionPlot \[red\] labelled "upright("sloshes")" drawn in split (function=\, x\_range=(0.0, 12.566370614359172)) Actions: - [12:48.082](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=768.0822499999999): app\_def is shown on the screen, written out. ##### [12:56.101](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=776.10075) Narration: The ratio of the real power to the apparent power is the power factor, and by the algebra it is simply cosine phi. A load whose current is in step has a power factor of one. Our motor, lagging thirty seven degrees, has cosine of thirty seven, which is zero point eight. Board: real\_def — a Math \[text\] that says "$P = V I cos phi quad \[upright("W")\]$"; react\_def — a Math \[text\] that says "$Q = V I sin phi quad \[upright("var")\]$"; app\_def — a Math \[text\] that says "$S = V I quad \[upright("VA")\]$"; split — an Axes (x\_range=(0.0, 12.566370614359172), y\_range=(-0.75, 1.05), x\_ticks\_every=3.141592653589793); head\_split — a Heading that says "Two Terms, One Average"; split\_zero — a Line \[gray\] drawn in split (end=(12.566370614359172, 0.0), dashed=True); real\_curve — a FunctionPlot \[blue\] labelled "upright("works")" drawn in split (function=\, x\_range=(0.0, 12.566370614359172)); mean\_line — a Line \[blue\] labelled "P" drawn in split (start=(0.0, 0.2879996141334631), end=(12.566370614359172, 0.2879996141334631), dashed=True); react\_curve — a FunctionPlot \[red\] labelled "upright("sloshes")" drawn in split (function=\, x\_range=(0.0, 12.566370614359172)) Actions: - [12:59.386](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=779.3862499999999): pf\_def is shown on the screen, written out. - [13:2.405](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=782.40525): pf\_def (the "cos phi" part) is emphasized. - [13:8.477](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=788.4772499999999): pf\_def (the "cos phi" part) is no longer emphasized. - [13:14.247](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=794.2472499999999): app\_def is hidden from the screen — left the board. - [13:14.247](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=794.2472499999999): head\_split is hidden from the screen — left the board. - [13:14.247](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=794.2472499999999): pf\_def is hidden from the screen — left the board. - [13:14.247](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=794.2472499999999): react\_def is hidden from the screen — left the board. - [13:14.247](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=794.2472499999999): real\_def is hidden from the screen — left the board. - [13:14.247](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=794.2472499999999): split is hidden from the screen — left the board. - [13:14.247](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=794.2472499999999): split\_zero is hidden from the screen — split left the board. - [13:14.247](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=794.2472499999999): real\_curve is hidden from the screen — split left the board. - [13:14.247](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=794.2472499999999): mean\_line is hidden from the screen — split left the board. - [13:14.247](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=794.2472499999999): react\_curve is hidden from the screen — split left the board. ##### [13:15.447](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=795.4472499999999) Narration: Now put our numbers in, and draw the three powers as a triangle. It is the impedance triangle from the last section, scaled up by the current squared, so it has the same shape and the same angle. Board: Empty. Actions: - [13:15.447](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=795.4472499999999): head\_bill is shown on the screen, written out. - [13:18.722](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=798.7222499999999): pt is shown on the screen, written out. - [13:18.722](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=798.7222499999999): pt\_p is shown on the screen, written out. - [13:18.722](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=798.7222499999999): pt\_q is shown on the screen, written out. - [13:18.722](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=798.7222499999999): pt\_square is shown on the screen, written out. - [13:25.177](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=805.17725): pt\_s is shown on the screen, written out. - [13:26.268](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=806.26825): pt\_phi is shown on the screen, written out. ##### [13:27.902](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=807.90175) Narration: Two forty volts times twenty four amps is five point seven six kilovolt amps of apparent power. That is the whole hypotenuse. Board: pt — a Figure (x\_range=(-0.25, 2.05), y\_range=(-0.3, 1.45), aspect=(2.3, 1.75)); head\_bill — a Heading that says "Why the Utility Charges for It"; pt\_p — a Vector \[blue\] labelled "P = 4.61 thin upright("kW")" drawn in pt (start=(0.12, 0.22), end=(1.6400000000000001, 0.22)); pt\_q — a Vector \[red\] labelled "Q = 3.46 thin upright("kvar")" drawn in pt (start=(1.6400000000000001, 0.22), end=(1.6400000000000001, 1.36)); pt\_square — an Angle \[gray\] drawn in pt (vertex=(1.6400000000000001, 0.22), sides=((0.12, 0.22), (1.6400000000000001, 1.36)), right\_angle=True); pt\_s — a Vector \[yellow\] labelled "S = 5.76 thin upright("kVA")" drawn in pt (start=(0.12, 0.22), end=(1.6400000000000001, 1.36)); pt\_phi — an Angle \[magenta\] labelled "36.87 degree" drawn in pt (vertex=(0.12, 0.22), sides=((1.6400000000000001, 0.22), (1.6400000000000001, 1.36)), radius=0.46) Actions: - [13:28.25](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=808.2502499999999): pt moves to a new place on the board. - [13:28.25](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=808.2502499999999): numbers is shown on the screen, written out. - [13:32.894](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=812.8942499999999): numbers is shown on the screen, written out. - [13:36.76](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=816.7602499999999): pt\_s is indicated — a transient flash. ##### [13:38.614](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=818.61375) Narration: Multiply by the power factor, zero point eight, and the real power is four point six one kilowatts. That is the useful part, and it is the horizontal side. Board: Unchanged from the preceding beat in this scene. Actions: - [13:43.072](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=823.0722499999999): numbers is shown on the screen, written out. - [13:47.507](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=827.5072499999999): pt\_p is indicated — a transient flash. ##### [13:49.535](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=829.5347499999999) Narration: The vertical side is the reactive power, three point four six kilovar, and the power factor is zero point eight lagging. Board: Unchanged from the preceding beat in this scene. Actions: - [13:50.057](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=830.05725): pt\_q is indicated — a transient flash. - [13:52.287](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=832.2872499999999): numbers is shown on the screen, written out. - [13:56.281](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=836.2812499999999): numbers is shown on the screen, written out. ##### [13:57.659](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=837.6587499999999) Narration: So here is the puzzle the customer raises. The reactive power delivers no energy at all. Its average is zero. Why should anybody be charged for it? Board: Unchanged from the preceding beat in this scene. Actions: - [14:2.314](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=842.3142499999999): pt\_q is emphasized. ##### [14:8.273](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=848.2727499999999) Narration: And the answer is not in the energy. It is in the current. The utility does not send a phase angle down its cables. It sends twenty four amps. Board: Unchanged from the preceding beat in this scene. Actions: - [14:8.273](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=848.2727499999999): pt\_q is no longer emphasized. - [14:15.766](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=855.7662499999999): numbers (the "row=2" part) is emphasized. ##### [14:17.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=857.55675) Narration: Losses in a transmission line go as the current squared times the line resistance. Not as the useful power. As the current, squared. Board: Unchanged from the preceding beat in this scene. Actions: - [14:17.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=857.55675): numbers (the "row=2" part) is no longer emphasized. - [14:17.748](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=857.7482499999999): loss\_law is shown on the screen, written out. - [14:26.003](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=866.00325): loss\_law (the "I^2" part) is emphasized. ##### [14:27.532](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=867.5317499999999) Narration: A customer taking the same four point six kilowatts at unity power factor would draw only nineteen point two amps. Compare the squares: twenty four squared over nineteen point two squared is one point five six. Board: loss\_law — a Math \[text\] that says "$P\_upright("line") = I^2 R\_upright("line")$"; pt — a Figure (x\_range=(-0.25, 2.05), y\_range=(-0.3, 1.45), aspect=(2.3, 1.75)); head\_bill — a Heading that says "Why the Utility Charges for It"; pt\_p — a Vector \[blue\] labelled "P = 4.61 thin upright("kW")" drawn in pt (start=(0.12, 0.22), end=(1.6400000000000001, 0.22)); pt\_q — a Vector \[red\] labelled "Q = 3.46 thin upright("kvar")" drawn in pt (start=(1.6400000000000001, 0.22), end=(1.6400000000000001, 1.36)); pt\_square — an Angle \[gray\] drawn in pt (vertex=(1.6400000000000001, 0.22), sides=((0.12, 0.22), (1.6400000000000001, 1.36)), right\_angle=True); pt\_s — a Vector \[yellow\] labelled "S = 5.76 thin upright("kVA")" drawn in pt (start=(0.12, 0.22), end=(1.6400000000000001, 1.36)); pt\_phi — an Angle \[magenta\] labelled "36.87 degree" drawn in pt (vertex=(0.12, 0.22), sides=((1.6400000000000001, 0.22), (1.6400000000000001, 1.36)), radius=0.46) Actions: - [14:27.532](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=867.5317499999999): loss\_law (the "I^2" part) is no longer emphasized. - [14:35.101](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=875.1012499999999): compare is shown on the screen, written out. ##### [14:42.226](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=882.2262499999999) Narration: Fifty six percent more loss in the cables, and fifty six percent more conductor and transformer capacity needed to serve the same useful load. Board: loss\_law — a Math \[text\] that says "$P\_upright("line") = I^2 R\_upright("line")$"; compare — a Math \[text\] that says "$frac(24^2, 19.2^2) = 1.56$"; pt — a Figure (x\_range=(-0.25, 2.05), y\_range=(-0.3, 1.45), aspect=(2.3, 1.75)); head\_bill — a Heading that says "Why the Utility Charges for It"; pt\_p — a Vector \[blue\] labelled "P = 4.61 thin upright("kW")" drawn in pt (start=(0.12, 0.22), end=(1.6400000000000001, 0.22)); pt\_q — a Vector \[red\] labelled "Q = 3.46 thin upright("kvar")" drawn in pt (start=(1.6400000000000001, 0.22), end=(1.6400000000000001, 1.36)); pt\_square — an Angle \[gray\] drawn in pt (vertex=(1.6400000000000001, 0.22), sides=((0.12, 0.22), (1.6400000000000001, 1.36)), right\_angle=True); pt\_s — a Vector \[yellow\] labelled "S = 5.76 thin upright("kVA")" drawn in pt (start=(0.12, 0.22), end=(1.6400000000000001, 1.36)); pt\_phi — an Angle \[magenta\] labelled "36.87 degree" drawn in pt (vertex=(0.12, 0.22), sides=((1.6400000000000001, 0.22), (1.6400000000000001, 1.36)), radius=0.46) Actions: - [14:42.632](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=882.63225): compare (the "1.56" part) is emphasized. ##### [14:50.872](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=890.8722499999999) Narration: That is why the tariff has a power factor clause. The wires and the transformers are sized by the hypotenuse, and the customer with the poor power factor is asking for a bigger hypotenuse to get the same horizontal side. Board: Unchanged from the preceding beat in this scene. Actions: - [14:50.872](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=890.8722499999999): compare (the "1.56" part) is no longer emphasized. - [14:54.727](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=894.72725): verdict is shown on the screen, written out. - [14:57.002](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=897.00225): pt\_s is emphasized. - [15:1.286](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=901.2862500000001): pt\_s is no longer emphasized. - [15:2.958](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=902.95825): pt\_p is emphasized. ##### [15:5.045](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=905.04475) Narration: Which raises the obvious question. If the trouble is that the current arrow is tilted, can we tilt it back? Board: loss\_law — a Math \[text\] that says "$P\_upright("line") = I^2 R\_upright("line")$"; compare — a Math \[text\] that says "$frac(24^2, 19.2^2) = 1.56$"; verdict — a Text \[text\] that says "The wires and the transformer are sized by $S$, not by $P$. The utility builds for the current it must actually carry."; pt — a Figure (x\_range=(-0.25, 2.05), y\_range=(-0.3, 1.45), aspect=(2.3, 1.75)); head\_bill — a Heading that says "Why the Utility Charges for It"; pt\_p — a Vector \[blue\] labelled "P = 4.61 thin upright("kW")" drawn in pt (start=(0.12, 0.22), end=(1.6400000000000001, 0.22)); pt\_q — a Vector \[red\] labelled "Q = 3.46 thin upright("kvar")" drawn in pt (start=(1.6400000000000001, 0.22), end=(1.6400000000000001, 1.36)); pt\_square — an Angle \[gray\] drawn in pt (vertex=(1.6400000000000001, 0.22), sides=((0.12, 0.22), (1.6400000000000001, 1.36)), right\_angle=True); pt\_s — a Vector \[yellow\] labelled "S = 5.76 thin upright("kVA")" drawn in pt (start=(0.12, 0.22), end=(1.6400000000000001, 1.36)); pt\_phi — an Angle \[magenta\] labelled "36.87 degree" drawn in pt (vertex=(0.12, 0.22), sides=((1.6400000000000001, 0.22), (1.6400000000000001, 1.36)), radius=0.46) Actions: - [15:5.045](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=905.04475): pt\_p is no longer emphasized. - [15:12.594](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=912.5938958333334): compare is hidden from the screen — left the board. - [15:12.594](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=912.5938958333334): head\_bill is hidden from the screen — left the board. - [15:12.594](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=912.5938958333334): loss\_law is hidden from the screen — left the board. - [15:12.594](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=912.5938958333334): numbers is hidden from the screen — left the board. - [15:12.594](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=912.5938958333334): pt is hidden from the screen — left the board. - [15:12.594](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=912.5938958333334): pt\_p is hidden from the screen — pt left the board. - [15:12.594](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=912.5938958333334): pt\_q is hidden from the screen — pt left the board. - [15:12.594](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=912.5938958333334): pt\_square is hidden from the screen — pt left the board. - [15:12.594](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=912.5938958333334): pt\_s is hidden from the screen — pt left the board. - [15:12.594](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=912.5938958333334): pt\_phi is hidden from the screen — pt left the board. - [15:12.594](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=912.5938958333334): verdict is hidden from the screen — left the board. ### Scene 5: [Bringing the Current Back Into Line](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=913.6355625) Span: 15:13.636–19:8.599 (913.6355625s–1148.5990625s). #### Objects - angle\_now: an Angle \[magenta\] labelled "phi" drawn in stage (vertex=(0.12, 0.0), sides=((1.3199999999999998, 0.0), (1.1183986623293385, ((-14.40003429…, radius=0.42) - before\_after: a Table \[text\] that says "Before After Line current 24.0 A 19.2 A Real power 4.61 kW 4.61 kW Apparent power 5.76 kVA 4.61 kVA Power factor 0.80 1.00" (rows=(('', 'Before', 'After'), ('Line current', '24.0 A', '19.2 A'),…, header=True) - c\_work: a Derivation \[text\] that says "$I\_upright("M,react") &= 24 sin 36.87 degree = 14.4 thin upright("A") \\ I\_C &= 14.4 thin upright("A") \\ I\_upright("line") &= 24 cos 36.87 degree = 19.2 thin upright("A") \\ phi' &= 0 degree, quad upright("pf") = 1.00$" - cap\_law: a Math \[text\] that says "$bold(I)\_C = j omega C bold(V)$" - cap\_note: a Panel that says "A capacitor's current leads its voltage by ninety degrees, which is the opposite sense to a coil's. Placed in parallel, it supplies reactive current the coil would otherwise draw from the supply." - closing: a Math \[text\] that says "$frac(19.2^2, 24^2) = 0.64 quad arrow.r quad 36% thin upright("less loss")$" - head\_cap: a Heading that says "A Capacitor Draws the Other Way" - head\_end: a Heading that says "Same Work, Less Current" - head\_fix: a Heading that says "Cancelling the Reactive Current" - i\_c\_arrow: a Vector \[cyan\] labelled "bold(I)\_upright("C")" drawn in stage (start=(1.1183986623293385, -0.7488017835575628), end=(1.1183986623293385, (-0.7488017835575628 + (i\_cap \* 0.052)))) - i\_cap: a VariableNumber (initial\_value=2.0, format\_spec='.1f') - i\_line: a Vector \[magenta\] labelled "bold(I)\_upright("line")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, ((-14.4000342991839 + i\_cap) \* 0.052))) - i\_motor: a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)) - sizing: a Math \[text\] that says "$Q\_C = V I\_C = 240 dot.op 14.4 = 3.46 thin upright("kvar")$" - stage: a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)) - summary: a Block \[text\] that says "Two sinusoids became two arrows. The angle between them became an impedance. It split the power into work and sloshing. One capacitor returned the sloshing to the load." - v\_ref: a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)) #### Beats ##### [15:13.636](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=913.6355625) Narration: Here is the motor's phasor picture as we left it. Voltage along the reference, current lagging by thirty seven degrees, twenty four amps long. Board: Empty. Actions: - [15:13.636](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=913.6355625): head\_cap is shown on the screen, written out. - [15:14.925](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=914.9245625): stage is shown on the screen, written out. - [15:16.747](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=916.7465625): v\_ref is shown on the screen, written out. - [15:18.083](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=918.0825625): i\_motor is shown on the screen, written out. ##### [15:22.502](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=922.5015625) Narration: Now think about what a capacitor does. Its current is C times the rate of change of its voltage, which is the mirror image of the coil relation. Board: stage — a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)); head\_cap — a Heading that says "A Capacitor Draws the Other Way"; v\_ref — a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)); i\_motor — a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)) Actions: - [15:23.698](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=923.6975625): stage moves to a new place on the board. - [15:23.698](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=923.6975625): cap\_law is shown on the screen, written out. ##### [15:32.088](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=932.0880625) Narration: So where the coil's current lagged by ninety degrees, the capacitor's current leads by ninety. It is reactive current pointing the other way. Board: cap\_law — a Math \[text\] that says "$bold(I)\_C = j omega C bold(V)$"; stage — a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)); head\_cap — a Heading that says "A Capacitor Draws the Other Way"; v\_ref — a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)); i\_motor — a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)) Actions: - [15:36.059](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=936.0585625): cap\_note is shown on the screen, written out. - [15:39.275](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=939.2745625): cap\_law (the "j omega C" part) is emphasized. ##### [15:40.896](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=940.8960625) Narration: Put one in parallel with the motor. Both see the same supply voltage, because that is what parallel means, and the supply now has to provide the sum of the two currents. Board: cap\_note — a Panel that says "A capacitor's current leads its voltage by ninety degrees, which is the opposite sense to a coil's. Placed in parallel, it supplies reactive current the coil would otherwise draw from the supply."; cap\_law — a Math \[text\] that says "$bold(I)\_C = j omega C bold(V)$"; stage — a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)); head\_cap — a Heading that says "A Capacitor Draws the Other Way"; v\_ref — a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)); i\_motor — a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)) Actions: - [15:40.896](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=940.8960625): cap\_law (the "j omega C" part) is no longer emphasized. ##### [15:52.027](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=952.0265625) Narration: The motor's current is unchanged. It has no idea the capacitor is there: same voltage across it, same impedance, same twenty four amps at the same angle. That is the crucial point and it is worth pausing on. Board: Unchanged from the preceding beat in this scene. Actions: - [15:53.397](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=953.3965625): i\_motor is indicated — a transient flash. ##### [16:5.548](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=965.5480625) Narration: What changes is what the supply cable carries, which is the motor's current plus the capacitor's current, added as phasors, head to tail. Board: Unchanged from the preceding beat in this scene. Actions: - [16:9.926](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=969.9255625): i\_c\_arrow is shown on the screen, written out. - [16:11.365](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=971.3645625): i\_line is shown on the screen, written out. - [16:11.365](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=971.3645625): angle\_now is shown on the screen, written out. ##### [16:14.368](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=974.3680625) Narration: So watch what happens to that magenta line current as we make the capacitor bigger. Board: cap\_note — a Panel that says "A capacitor's current leads its voltage by ninety degrees, which is the opposite sense to a coil's. Placed in parallel, it supplies reactive current the coil would otherwise draw from the supply."; cap\_law — a Math \[text\] that says "$bold(I)\_C = j omega C bold(V)$"; stage — a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)); head\_cap — a Heading that says "A Capacitor Draws the Other Way"; v\_ref — a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)); i\_motor — a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)); i\_c\_arrow — a Vector \[cyan\] labelled "bold(I)\_upright("C")" drawn in stage (start=(1.1183986623293385, -0.7488017835575628), end=(1.1183986623293385, (-0.7488017835575628 + (i\_cap \* 0.052)))); i\_line — a Vector \[magenta\] labelled "bold(I)\_upright("line")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, ((-14.4000342991839 + i\_cap) \* 0.052))); angle\_now — an Angle \[magenta\] labelled "phi" drawn in stage (vertex=(0.12, 0.0), sides=((1.3199999999999998, 0.0), (1.1183986623293385, ((-14.40003429…, radius=0.42) Actions: - [16:19.233](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=979.2325625): stage moves to a new place on the board. - [16:19.233](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=979.2325625): cap\_law is hidden from the screen — left the board. - [16:19.233](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=979.2325625): cap\_note is hidden from the screen — left the board. - [16:19.233](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=979.2325625): head\_cap is hidden from the screen — left the board. ##### [16:20.433](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=980.4325625) Narration: The motor's current has a vertical component: twenty four times the sine of thirty seven, which is fourteen point four amps of lagging reactive current. Board: stage — a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)); v\_ref — a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)); i\_motor — a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)); i\_c\_arrow — a Vector \[cyan\] labelled "bold(I)\_upright("C")" drawn in stage (start=(1.1183986623293385, -0.7488017835575628), end=(1.1183986623293385, (-0.7488017835575628 + (i\_cap \* 0.052)))); i\_line — a Vector \[magenta\] labelled "bold(I)\_upright("line")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, ((-14.4000342991839 + i\_cap) \* 0.052))); angle\_now — an Angle \[magenta\] labelled "phi" drawn in stage (vertex=(0.12, 0.0), sides=((1.3199999999999998, 0.0), (1.1183986623293385, ((-14.40003429…, radius=0.42) Actions: - [16:20.433](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=980.4325625): head\_fix is shown on the screen, written out. - [16:21.873](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=981.8725625): c\_work is shown on the screen, written out. ##### [16:29.827](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=989.8270625) Narration: The capacitor's current points straight up, so it cancels part of that vertical component. The line current shortens, and its angle to the voltage closes up. Board: stage — a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)); v\_ref — a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)); i\_motor — a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)); i\_c\_arrow — a Vector \[cyan\] labelled "bold(I)\_upright("C")" drawn in stage (start=(1.1183986623293385, -0.7488017835575628), end=(1.1183986623293385, (-0.7488017835575628 + (i\_cap \* 0.052)))); i\_line — a Vector \[magenta\] labelled "bold(I)\_upright("line")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, ((-14.4000342991839 + i\_cap) \* 0.052))); angle\_now — an Angle \[magenta\] labelled "phi" drawn in stage (vertex=(0.12, 0.0), sides=((1.3199999999999998, 0.0), (1.1183986623293385, ((-14.40003429…, radius=0.42); head\_fix — a Heading that says "Cancelling the Reactive Current" Actions: - [16:32.654](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=992.6535625): i\_c\_arrow is redrawn as the numbers it depends on change. - [16:32.654](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=992.6535625): i\_line is redrawn as the numbers it depends on change. - [16:32.654](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=992.6535625): angle\_now is redrawn as the numbers it depends on change. - [16:32.654](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=992.6535625): i\_cap ticks to 6.0. ##### [16:39.86](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=999.8600625) Narration: A bigger one takes out more. The magenta line current is getting shorter and flatter, while the green motor current has not moved at all. Board: Unchanged from the preceding beat in this scene. Actions: - [16:40.244](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1000.2435624999999): i\_c\_arrow is redrawn as the numbers it depends on change. - [16:40.244](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1000.2435624999999): i\_line is redrawn as the numbers it depends on change. - [16:40.244](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1000.2435624999999): angle\_now is redrawn as the numbers it depends on change. - [16:40.244](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1000.2435624999999): i\_cap ticks to 11.0. ##### [16:48.727](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1008.7265625) Narration: And at fourteen point four amps of capacitor current, the two vertical components cancel exactly. The line current lies flat along the voltage. Board: Unchanged from the preceding beat in this scene. Actions: - [16:49.424](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1009.4235625): i\_c\_arrow is redrawn as the numbers it depends on change. - [16:49.424](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1009.4235625): i\_line is redrawn as the numbers it depends on change. - [16:49.424](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1009.4235625): angle\_now is redrawn as the numbers it depends on change. - [16:49.424](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1009.4235625): i\_cap ticks to 14.4000342991839. - [16:53.499](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1013.4985624999999): c\_work is shown on the screen, written out. ##### [16:58.952](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1018.9520625) Narration: What is left in the supply cable is only the horizontal component: twenty four times cosine thirty seven, which is nineteen point two amps. Board: Unchanged from the preceding beat in this scene. Actions: - [17:1.495](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1021.4945625): c\_work is shown on the screen, written out. - [17:6.278](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1026.2775625): i\_line is indicated — a transient flash. ##### [17:8.654](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1028.6540625) Narration: The phase angle is now zero and the power factor is one. The current is back in step with the voltage. Board: Unchanged from the preceding beat in this scene. Actions: - [17:10.315](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1030.3145625): c\_work is shown on the screen, written out. ##### [17:15.953](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1035.9530625) Narration: And to size the capacitor, we ask for the reactive power it must supply: two forty volts times fourteen point four amps, which is three point four six kilovar. Exactly the reactive power the motor was drawing from the supply, and now drawn from the capacitor instead. Board: Unchanged from the preceding beat in this scene. Actions: - [17:16.453](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1036.4525624999999): sizing is shown on the screen, written out. - [17:24.475](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1044.4745625): sizing (the "3.46 thin upright("kvar")" part) is emphasized. - [17:26.739](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1046.7385625): sizing (the "3.46 thin upright("kvar")" part) is no longer emphasized. ##### [17:33.713](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1053.7125624999999) Narration: That is the physical picture worth keeping. The coil and the capacitor pass the sloshing energy back and forth to each other, locally, twice a cycle. The supply is no longer part of that exchange. Board: stage — a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)); v\_ref — a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)); i\_motor — a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)); i\_c\_arrow — a Vector \[cyan\] labelled "bold(I)\_upright("C")" drawn in stage (start=(1.1183986623293385, -0.7488017835575628), end=(1.1183986623293385, (-0.7488017835575628 + (i\_cap \* 0.052)))); i\_line — a Vector \[magenta\] labelled "bold(I)\_upright("line")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, ((-14.4000342991839 + i\_cap) \* 0.052))); angle\_now — an Angle \[magenta\] labelled "phi" drawn in stage (vertex=(0.12, 0.0), sides=((1.3199999999999998, 0.0), (1.1183986623293385, ((-14.40003429…, radius=0.42); sizing — a Math \[text\] that says "$Q\_C = V I\_C = 240 dot.op 14.4 = 3.46 thin upright("kvar")$"; head\_fix — a Heading that says "Cancelling the Reactive Current" Actions: - [17:46.112](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1066.1120624999999): stage moves to a new place on the board. - [17:46.112](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1066.1120624999999): c\_work is hidden from the screen — left the board. - [17:46.112](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1066.1120624999999): head\_fix is hidden from the screen — left the board. - [17:46.112](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1066.1120624999999): sizing is hidden from the screen — left the board. ##### [17:47.312](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1067.3120625) Narration: So here is the ledger, before and after one capacitor. Board: stage — a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)); v\_ref — a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)); i\_motor — a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)); i\_c\_arrow — a Vector \[cyan\] labelled "bold(I)\_upright("C")" drawn in stage (start=(1.1183986623293385, -0.7488017835575628), end=(1.1183986623293385, (-0.7488017835575628 + (i\_cap \* 0.052)))); i\_line — a Vector \[magenta\] labelled "bold(I)\_upright("line")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, ((-14.4000342991839 + i\_cap) \* 0.052))); angle\_now — an Angle \[magenta\] labelled "phi" drawn in stage (vertex=(0.12, 0.0), sides=((1.3199999999999998, 0.0), (1.1183986623293385, ((-14.40003429…, radius=0.42) Actions: - [17:47.312](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1067.3120625): head\_end is shown on the screen, written out. - [17:48.311](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1068.3105625): before\_after is shown on the screen, written out. ##### [17:52.417](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1072.4165625) Narration: Line current down from twenty four amps to nineteen point two. Real power unchanged at four point six one kilowatts, because the motor is doing exactly the same job. Board: stage — a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)); v\_ref — a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)); i\_motor — a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)); i\_c\_arrow — a Vector \[cyan\] labelled "bold(I)\_upright("C")" drawn in stage (start=(1.1183986623293385, -0.7488017835575628), end=(1.1183986623293385, (-0.7488017835575628 + (i\_cap \* 0.052)))); i\_line — a Vector \[magenta\] labelled "bold(I)\_upright("line")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, ((-14.4000342991839 + i\_cap) \* 0.052))); angle\_now — an Angle \[magenta\] labelled "phi" drawn in stage (vertex=(0.12, 0.0), sides=((1.3199999999999998, 0.0), (1.1183986623293385, ((-14.40003429…, radius=0.42); head\_end — a Heading that says "Same Work, Less Current" Actions: - [17:52.823](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1072.8225625): before\_after is shown on the screen, written out. - [17:56.643](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1076.6425625): before\_after is shown on the screen, written out. - [17:57.305](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1077.3045625): before\_after (the "row=3" part) is emphasized. ##### [18:3.652](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1083.6515625) Narration: Apparent power down from five point seven six to four point six one kilovolt amps, and the power factor up from zero point eight to one. Board: Unchanged from the preceding beat in this scene. Actions: - [18:3.652](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1083.6515625): before\_after (the "row=3" part) is no longer emphasized. - [18:04](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1083.9995625000001): before\_after is shown on the screen, written out. - [18:9.225](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1089.2245625): before\_after is shown on the screen, written out. ##### [18:12.774](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1092.7735625) Narration: And the line loss goes as current squared, so nineteen point two squared over twenty four squared is zero point six four. Thirty six percent less heat wasted in the cables, for the same work done. Board: Unchanged from the preceding beat in this scene. Actions: - [18:14.584](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1094.5835625): closing is shown on the screen, written out. - [18:20.61](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1100.6095625): closing (the "36% thin upright("less loss")" part) is emphasized. ##### [18:25.517](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1105.5170625) Narration: That is why the capacitor bank sits in the factory switchroom and not at the power station. Correcting at the load takes the reactive current out of every metre of cable between them. Board: stage — a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)); v\_ref — a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)); i\_motor — a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)); i\_c\_arrow — a Vector \[cyan\] labelled "bold(I)\_upright("C")" drawn in stage (start=(1.1183986623293385, -0.7488017835575628), end=(1.1183986623293385, (-0.7488017835575628 + (i\_cap \* 0.052)))); i\_line — a Vector \[magenta\] labelled "bold(I)\_upright("line")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, ((-14.4000342991839 + i\_cap) \* 0.052))); angle\_now — an Angle \[magenta\] labelled "phi" drawn in stage (vertex=(0.12, 0.0), sides=((1.3199999999999998, 0.0), (1.1183986623293385, ((-14.40003429…, radius=0.42); closing — a Math \[text\] that says "$frac(19.2^2, 24^2) = 0.64 quad arrow.r quad 36% thin upright("less loss")$"; head\_end — a Heading that says "Same Work, Less Current" Actions: - [18:25.517](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1105.5170625): closing (the "36% thin upright("less loss")" part) is no longer emphasized. ##### [18:36.531](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1116.5310625) Narration: So look back at what four ideas did. Two sinusoids became two arrows. The angle between the arrows became an impedance. Board: Unchanged from the preceding beat in this scene. Actions: - [18:39.307](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1119.3065625000002): summary is shown on the screen, written out. - [18:40.572](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1120.5715625): summary (the "Two sinusoids became two arrows" part) is emphasized. - [18:43.857](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1123.8565625): summary (the "Two sinusoids became two arrows" part) is no longer emphasized. - [18:43.857](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1123.8565625): summary (the "became an impedance" part) is emphasized. ##### [18:45.502](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1125.5020625) Narration: That angle split the power into a part that works and a part that only sloshes. And one capacitor sent the sloshing part back to the load that wanted it, leaving the wires to carry only the work. Board: stage — a Figure (x\_range=(-0.2, 1.75), y\_range=(-1.15, 0.95), aspect=(1.95, 2.1)); v\_ref — a Vector \[yellow\] labelled "bold(V)" drawn in stage (start=(0.12, 0.0), end=(1.5699999999999998, 0.0)); i\_motor — a Vector \[green\] labelled "bold(I)\_upright("M")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, -0.7488017835575628)); i\_c\_arrow — a Vector \[cyan\] labelled "bold(I)\_upright("C")" drawn in stage (start=(1.1183986623293385, -0.7488017835575628), end=(1.1183986623293385, (-0.7488017835575628 + (i\_cap \* 0.052)))); i\_line — a Vector \[magenta\] labelled "bold(I)\_upright("line")" drawn in stage (start=(0.12, 0.0), end=(1.1183986623293385, ((-14.4000342991839 + i\_cap) \* 0.052))); angle\_now — an Angle \[magenta\] labelled "phi" drawn in stage (vertex=(0.12, 0.0), sides=((1.3199999999999998, 0.0), (1.1183986623293385, ((-14.40003429…, radius=0.42); closing — a Math \[text\] that says "$frac(19.2^2, 24^2) = 0.64 quad arrow.r quad 36% thin upright("less loss")$"; summary — a Block \[text\] that says "Two sinusoids became two arrows. The angle between them became an impedance. It split the power into work and sloshing. One capacitor returned the sloshing to the load."; head\_end — a Heading that says "Same Work, Less Current" Actions: - [18:48.881](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1128.8805625): summary (the "became an impedance" part) is no longer emphasized. - [18:48.881](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1128.8805625): summary (the "work and sloshing" part) is emphasized. - [18:53.42](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1133.4195625000002): summary (the "returned the sloshing to the load" part) is emphasized. - [18:53.42](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1133.4195625000002): summary (the "work and sloshing" part) is no longer emphasized. - [18:54.767](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1134.7665625): summary (the "returned the sloshing to the load" part) is no longer emphasized. ##### [18:57.608](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1137.6075625) Narration: Every one of those steps came out of one small observation at the very start: that the voltage across a coil follows the slope of its current, and not its size. Board: Unchanged from the preceding beat in this scene. Actions: - [19:7.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1147.5573958333334): before\_after is hidden from the screen — left the board. - [19:7.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1147.5573958333334): closing is hidden from the screen — left the board. - [19:7.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1147.5573958333334): head\_end is hidden from the screen — left the board. - [19:7.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1147.5573958333334): stage is hidden from the screen — left the board. - [19:7.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1147.5573958333334): v\_ref is hidden from the screen — stage left the board. - [19:7.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1147.5573958333334): i\_motor is hidden from the screen — stage left the board. - [19:7.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1147.5573958333334): i\_c\_arrow is hidden from the screen — stage left the board. - [19:7.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1147.5573958333334): i\_line is hidden from the screen — stage left the board. - [19:7.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1147.5573958333334): angle\_now is hidden from the screen — stage left the board. - [19:7.557](https://academa.ai/lectures/alternating-current-phasors-power-factor?t=1147.5573958333334): summary is hidden from the screen — left the board.