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### Course: Electrical engineering > Unit 2

Lesson 5: AC circuit analysis- AC analysis intro 1
- AC analysis intro 2
- Trigonometry review
- Sine and cosine come from circles
- Sine of time
- Sine and cosine from rotating vector
- Lead Lag
- Complex numbers
- Multiplying by j is rotation
- Complex rotation
- Euler's formula
- Complex exponential magnitude
- Complex exponentials spin
- Euler's sine wave
- Euler's cosine wave
- Negative frequency
- AC analysis superposition
- Impedance
- Impedance vs frequency
- ELI the ICE man
- Impedance of simple networks
- KVL in the frequency domain

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# Impedance vs frequency

The impedance of capacitors and inductors in a circuit depend on the frequency of the electric signal. The impedance of an inductor is directly proportional to frequency, while the impedance of a capacitor is inversely proportional to frequency. Created by Willy McAllister.

## Want to join the conversation?

- Am I correct with the following reasoning?

For an inductor, if the impedance is 0, that means that i >>> v (definition of short circuit) and if the impedance is infinity, then it means v >>> i ( that happens when di/dt is infinite)(4 votes)- Hello Feneva,

Correct, but let's add one more thing. For DC circuits an inductor appears as a short and for high frequency AC circuits the inductor presents as an open circuit.

Regards,

APD(3 votes)

- What is reactance, can you explain about it please ?(3 votes)
- Impedance (Z) is a complex number. Z = R + jX. The real part is R for resistance. The imaginary part is called "reactance" with the symbol X. Reactance has the same units as resistance, it is measured in Ohms.(4 votes)

- If we have an AC circuit with a resistor and a capacitor and we increase the frequency what impact will this have on VR and VC?(2 votes)
- The impedance of a resistor stays constant with frequency. For a capacitor, its impedance get smaller and smaller as frequency goes up.

If R and C are in series, at 0 frequency the capacitor has infinite impedance (1/jwC = infinity), so it acts like an open circuit (we say a capacitor**blocks**DC current). The Resistor will have zero current and therefore zero voltage. All the voltage will appear across the capacitor.

As frequency rises, more voltage will appear on R, and less and less on C, until at super high frequency the capacitor will seem to disappear and all the voltage is across R.(5 votes)

- How to take the magnitude of jwl ? i.e how to get rid of j by taking the magnitude?(3 votes)
- Hello Shrey,

Keep working your way through this series of videos. Willy presents the answer to your question is in this video:

https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-ac-analysis/v/ee-impedance-of-simple-networks

If you are just starting out this is going to get very complex and fast. These are college level sophomore EE-101 concepts. I have prepared this video to present less technical introduction:

https://www.youtube.com/watch?v=kz4DOi3LNdM

You will see that you cant get rid of j. In a resistive circuit there is not j. A resistor only "burns" power. On the other hand a circuit with inductors of capacitors stores energy. In a way that is what "j" is telling us. The inductor and capacitor store energy and then return it to the power source. The "j" tells us it is imaginary power.

May i also recommend you research "phasor" and "Bode plot."

Regards,

APD(3 votes)

- Could someone explain how when omega is 0, then the magnitude of the impedance of a capacitor is infinite? Would it not be undefined, or is this because if you graph this, the value approaches infinite but technically has an asymptote?(1 vote)
- When omega (the frequency) is zero, we call that "DC". At DC, a capacitor conducts no current (no charge passes through the gap between the capacitor plates). Impedance is the ratio of voltage to current, Z = V/I. If current is zero and V is some normal number, then Z = infinity. That means a capacitor at DC acts the same as an open circuit.

This argument holds for an ideal capacitor. If you talk about a real-world capacitor there are of course all the usual excuses about manufacturing flaws and material defects that allow a tiny amount of current to get through. We call that "leakage". Even if you just touch a capacitor with your fingers, the oil you leave behind on the outside of the capacitor is enough to allow a tiny bit of current. This usually doesn't matter to the operation of the circuit.(2 votes)

- Hi, I learnt that the voltage from a sinusoidal supply through a capacitor was:

V(c) = Vmax [cos(wt) + jsin(wt)], where w = 2pi*f. Can you please explain why there is a real and imaginary part..? I can't find an explanation as to why a voltage is written as a complex number.(1 vote)- The equation you wrote is Euler's Formula, but change the left side to read e^(jwt). This is an exponential term with a complex exponent. (I refer to this exponential as a "spinning number".)

With a little algebra you can turn Euler's formula inside out and solve for cos or sin in terms of complex exponentials.

cos(wt) = Re[e^(jwt)] = 1/2 (e^(+jwt) + e^(-jwt))

sin(wt) = Im[e^(jwt)] = 1/2j (e^(+jwt) - e^(-jwt))

You can see an animation of these two equations here: https://spinningnumbers.org/a/eulers-sine-cosine.html

For a long time I shared your confusion about this formula and how it is applied to AC Analysis. I think it is often taught in a rushed manner to stay within the schedule of a compact academic calendar.(2 votes)

- Hi all and Willy,

Great video guys! Question, shouldn't magnitude of |ZL| equal to -WL? since the way to get magnitude is ZL^2 = (jWL)^2, hence |ZL|=-WL. Please enlighten me if this is wrong.(1 vote) - Why is cosine a typical sinusoid to pick? Is that because electrical engineers like the intial voltage to be non-zero, (v_initial = v_0)?(1 vote)
- Had an idea...is it to do with cosine not containing the imaginary component, j, so the math is simpler?

cos(x) = (e^jx + e^-jx)/**2**

sin(x) = (e^jx + e^-jx)/2j.

Thanks for the correction.(1 vote)

- Need to learn some more about rotating Impedance vectors(1 vote)
- See Willy's video on how j rotates a vector:

https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-ac-analysis/v/ee-complex-rotation

So if you want to rotate an impedance vector, consider the impedance of the circuit - RL or RC. For an RL circuit, Z(RL) = R + jZ(L) and for an RC circuit, Z(RC) = R - jZ(C). To rotate the impedance vector, you simply change the inductor or capacitor values which will change the impedance of those devices.

Hope this helps!(0 votes)

## Video transcript

- [Voiceover] In this
video we're gonna continue talking about AC analysis
and the concept of impedance as the ratio of voltage to
current and an AC situation, and just as a reminder of
the assumptions we've made for AC analysis we've assumed
that all of our signals are of the form of some kind
of a sinusoid like that, cosine is the typical one we pick, cosine omega T plus some offset angle. And we're also going
to use Euler's equation which says we can decompose our cosine into two complex exponentials, so one of the complex
exponentials is either the plus J omega T plus fi and J multiplies both of
them, plus e to the minus J omega T plus fi. And when we use Euler's
theorem what we do is we love putting these things
through differential equations so we track each of these
one at a time through our circuits and typically
what I'll focus on is the plus term, but the
minus term all the math is the same except there's
that one little minus sign. And so when we made up
the concept of impedance, we assumed that V or I was of this form, this complex exponential, and
that may, when we put that through our components
we came up with the idea of impedance, and as a
review for a resistor, the impedance of a resistor
turned out to be just R, that's the ratio of V to I in a resistor. And for an inductor, we decided Z of an inductor
was equal to J omega L. And finally for a capacitor of value C, Z C equals one over J omega C. So those are the three forms of, those are the three impedences
of our three favorite passive components. And the units of these,
the units are in ohms. R is measured in ohms,
that's the normal ohms law, and we use the same units
for impedance of inductors and capacitors so this is
in ohms and this is in ohms. That may seem a little funny
that because there's this frequency term in here,
but, this is what it is. So in the rest of this
video I want to look at qualitatively look at the
value of these impedences and specifically look at
what happens when there's a range of values of the omega term, what happens at zero
frequency and low frequency and high frequency and infinite frequency. So let's go ahead and do that. So we're gonna look at the
impedance terms at different frequencies and we'll
measure frequency at radiant frequency in this, so, let's
talk about zero frequency and we'll talk about low frequency, these are qualitative boundaries, we'll talk about high frequency, and let's talk about infinite frequency. So we're gonna build a little chart here. Alright so these are
the value of omega here. And now we'll do our
components, so I'm gonna do, first we'll do our resistor, and we're gonna fill in the table for ZR, and we know that equals R. So, at any frequency, R is
just R, couldn't be simpler. At zero frequency which is
just called DC or battery, R is R, at any low frequency, R is R, R is R, at infinite frequency, so there's
no dependence on frequency in R, so now let's do the inductor, and we decided that Z of
an inductor was J omega L. And let me do something
very specific, I'm gonna do, I'm gonna get rid of this
J, I'm gonna basically say I wanna just look at the
magnitude of the impedance. If we just look at the magnitudes, the magnitude of Z inductor is omega L. So now let's fill our
table in for omega L. So when omega is zero, the magnitude of the impedance
is zero for an inductor. And when the frequency is
low, when omega is low, the impedance is going
to be relatively low. And as the frequency gets high, then omega L becomes a larger
number, so it becomes high. And if the frequency, if we let
the frequency go to infinity then omega becomes infinity
and this becomes infinity. So we see an inductor from low
frequency to high frequency, the impedance of the inductor
goes from zero up to infinity. Alright let's fill in the last one here, here's our capacitor, and the Z of a capacitor
equals one over J omega C, and the magnitude of the
impedance is just 1 over omega C. Alright so let's plot that
out, what is 1 over omega C when omega is zero? Well it's infinity. And what if omega is a small number? If omega is a small number
then this is a large number. This is a high number. And if the frequency gets very
high, the higher omega gets the smaller the magnitude of Z gets, so we get low here, you
see it's anti symmetric with the inductor, and finally,
if we let the frequency go to infinity, 1 over
infinity times C is what? Is about, is zero. With this chart I can show you
some of the words that we use some of the sort of slang
words or jargon words that we use in electrical
engineering to describe the behavior of L and C
at different frequencies. So as we said, R is R at any frequency, so resistance is constant. An inductor it changes over frequency, over this big range of
frequency, and at zero frequency or low frequencies the
impedance is very low. So this leads to the
expression we say an inductor looks like a short. An inductor looks like a
short at low frequency, at zero frequency or low frequency. And now let's go to high
frequency, the impedance gets high and it eventually goes to infinity. That's the impedance of an open circuit. So we say L equals an open at high F, high frequency, and it's
a short at low frequency. Let's do the same sort of
jargon investigation here for our capacitor. A capacitor at zero frequency
has an infinite impedance. So it looks like what? It looks like an open at low frequency. So let's go look over at high frequencies, high frequencies, the
impedance becomes very low. And at infinite frequency
it becomes zero and that's the impedance of a perfect short. So the expression, the
slang expression you'll hear is that a capacitor is a short
circuit at high frequency. So the reason I'm telling you about this, very often experienced
engineers will talk to beginners and use these kinds of terms,
that oh an inductor looks like a short at low frequency. And I wanted to show you
where these things come from, where these terms come
from and what they mean. And this is not a simple idea,
you saw how many assumptions we made and how much work
we did to get to this idea of these sort of common
or familiar expressions for how our components work. I hope that helped for you
to see how this comes about.