Main content

## 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

The ratio of a sinusoidal voltage to a sinusoidal current is called "impedance". This is a generalization of Ohm's Law for resistors. We derive the impedance of a resistor, inductor, and capacitor. The inductor and capacitor impedance includes a term for frequency, so the impedance of these components depends on frequency. Created by Willy McAllister.

## Want to join the conversation?

- How is it that at2:44you set I = e^jwt and this also happens at4:33.

But how then, at6:47can U = e^jwt ?

Doesn't that imply that U = I ? How can current and voltage be the same?

Or is the e^jwt simply arbitrarily substituted for I in the case of the resistor and the inductor is substituted for U in the case of the capacitor?(8 votes)- Sorry for the confusion. The value e^jwt can be assigned to anything, just like any other number, like 3 or -9. If I say, "Let v = 3," there is an understood unit of volts based on the context of the equation. It would be better if I said "Let v = 3 V," but I left that part out in the video. Later on, I might starting talking about current, and I could say, "Let i = 3". Again, I neglected to state the units, which is not the greatest teaching technique, but it is a good bet I'm assuming the units are amperes. In these two examples with "3", where I didn't state the units, you probably wouldn't think that 3 volts = 3 ampere. The same thing is happening in the video when I write I = e^jwt and then U (voltage) = e^jwt. The e^jwt is just a numerical function (that happens to be a function of time). Stated by itself, e^jwt does not have units, just like "3" doesn't have units. e^jwt becomes a value with units from the context of the equation where its used.

Let me know if that helped answer your question.(10 votes)

- ohms are a type of measurement for impedance right?(6 votes)
- Yes. The unit of impedance is the ohm. Impedance is a generalized notion of resistance that applies to resistors, capacitors, and inductors.(10 votes)

- We substitute e^(jwt) for the current instead of the cosine function, to figure out impedance, and I get that but I don't understand how we derive general equations for the impedance using only part of our AC signal. Since the AC signal is made up of two exponential functions, shouldn't a general rule for the impedance also involve the superimposed functions rather that just one?(7 votes)
- Good question. This answer evaded me for many years. It's easy to forget the logic behind some of the powerful cleverness that happens right at the moment you are learning a new technique the first time.

Cosine can be broken into two complex exponentials. The exponentials are conjugates. That's important.

The first time you do a problem you draw it up for superposition. You create two versions of the circuit and drive each one with one of the two complex exponentials. At the end you add the two answers together (superimpose) to get the response to cosine. Right at that moment you notice, "Hey! The two circuit outputs were nearly the same, they were just complex conjugates of each other." Then your next thought is, "Wait a minute. The next time I do a problem like this, I will just do one complex exponential and find the output, since I know the circuit response to other one comes out as the conjugate of the first answer. Saves me half the work!".

From then on, we just talk about one exponential input drive signal. All the extra analysis of the conjugate exponential is not done and not mentioned.(5 votes)

- So in this video, the situation is both v and I are sinusoid? When there is no current source and V is sinusoid, does that mean I must be sinusoid too? I am very confused.(3 votes)
- Let's do a simple example. Suppose you connect a resistor to a voltage source. Let the voltage source be defined to produce a sinusoid, v = A sin wt. What is the current in the resistor? Using Ohm's Law, i = v/R gives us i = (A sin wt) / R. So when the voltage is a sinusoid, the current is a sinusoid, too.

In general, if you define either v or i to be sinusoidal, the other will follow along and be sinusoidal, too.

In a resistor, the sinusoidal i and v are lined up with each other in time. The interesting thing about inductors and capacitors is i and v**don't**line up in time, they get a 90 degree offset time delay from each other.

Any time we talk about Impedance, there is an underlying assumption that we've restricted the driving signal to a sinusoid. When we make this restriction, all of a sudden the i-v equations for inductors and capacitors start to look just like Ohm's Law for resistors. This is fantastic because it gives us a way to analyze RLC circuits using the same methods we learned about for just-Resistor circuits.(8 votes)

- Quick question, throughout the video you use jwt, and so far I know that t stands for time, w stands for frequency, but I never caught on as to what j stood for. Could you elaborate on that please? :)(1 vote)
- In electrical engineering, we use the letter j instead of i as the imaginary unit, sqrt(-1). In our field, i is used as the variable for current (from the French intensité du courant électrique, coined by Ampère).(7 votes)

- The current and voltages for passive devices are assumed to be e^jwt but for a complete sinusoid, we also need e^-jwt. Both exponents of combine to form sinusoidal, but just taking one exponents would make it something else, won't it ?(2 votes)
- In this video (https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/modal/v/ee-ac-analysis-superposition) we showed how the circuit response to e^-jwt always comes out as the complex conjugate of whatever e^+jwt causes.

That means if I know what e^+jwt does then I know what e^-jwt does. AND, if I know what both of those do then I know what happens if I apply sine or cosine to the same circuit.

We focus on the response to e^+jwt for 2 reasons.

1. We know how to solve forced differential equations when the forcing in put is an exponential.

2. We know finding the response to e^+jwt instantly leads to the answer for sine and cosine inputs, which is what we care about.

3. (Bonus feature) We know how to construct many other types of signal shape (square, triangle, any periodic function, most non-periodic functions) as a sum of sines and cosines. (This is what Fourier series and Fourier transforms and Laplace transforms are all about.)

When you work with e^+jwt you pretty much learn everything about an analog electronic system.(5 votes)

- The part between2:28and3:45seems a circular reasoning to me.

At2:28`v = i*R`

. Then after some steps we get that, indeed,`R = v/i`

.

(I mean we would get that no matter how`i`

is defined, it does not matter that we used`e^jwt`

in this case, it would work for any other kind of`i`

as well.)(2 votes)- Yes. The discussion of the resistor is circular. It's just two forms of Ohm's Law, and it is valid for any form of i. This is intended to emphasize the perspective of Ohm's Law as the ratio of v to i. It helps set up what comes next for L and C.

When we consider L and C, AND we assume the current or voltage is a complex exponential, the concept of Impedance emerges. It applies to L, C,*and*R. The R case is what you already know (Ohm's Law) so it seems super simple.(2 votes)

- A graph with a Euler circuit, and how did u arrive at that, this is my first seeing this(2 votes)
- Is AC resistance measured in reactance or impedance?(1 vote)
- Hi Chloe. Impedance (Z) is defined as the ratio of Voltage to Current (V/I). In the most general sense impedance has a complex value. Z = real part + j imaginary part. For a resistor, it has a real part and no imaginary part. The real part is called Resistance and it has units of Ohms.

A capacitor or inductor have imaginary impedance (no real part, just an imaginary part. The imaginary part is called "Reactance", and L and C are called "reactive elements". Reactance is the ratio of V/I, so it has the units of Ohms, just like resistance.

The impedance of an inductor is Z = jwL. The reactance of an inductor is the imaginary part, wL. To get the reactance you multiply frequency (w) times Henrys to get Ohms of reactance.

I never found it very useful to talk about a capacitor or inductor in terms of its reactance in Ohms. This value is different at every different frequency. It was much more helpful to model a capacitor as 1/jwC and an inductor as jwL. You end up making plots with w as the independent horizontal axis (frequency response plots).(3 votes)

- For inductor and resistor, current was assmed to be e^jwt. And for Capacitor, voltage was assumed to be e^jwt.

What are intuitions behind these assumptions, how were these assumptions made ? And how can the assumed value used interchangeably between voltage and current ?(1 vote)- Here is a high-level review of why we do AC analysis this way.

1. Circuits with L and C have a natural response modeled with differential equations.

2. The solutions to these equations are exponential functions. Therefore, we love exponential functions.

3. In the real world a large number of signals resemble sine waves (talking, music) and we want to amplify those signals or put them into filters to modify their frequency content. (Side note: sine waves can be used to construct a lot of other signals that don't look like sines, see for example Sal's video sequence on Fourier Series.)

4. We build amplifiers where we apply external sine-like signals (a forced input) and we want to know what the amplifier (with its R's, L's, and C's modeled as differential equations) does to the signal. The answer will be a natural response plus a forced response.

5. By the miracle of Euler's formulas we know sine and cosine can be represented as exponential functions. Those exponentials happen to have imaginary exponents, which makes them totally weird, but it makes the math totally work because now the forced input has the same general shape as the natural response (both are exponentials) and we can solve the differential equations.

That is why we picked e^jwt for our assumed input. We know how to solve the differential equation, and we know how to reconstruct a sine wave. That's the winning combination at the heart of AC analysis.

As for picking voltage or current interchangeably, it isn't too tricky. For a resistor that obeys Ohm's Law, the shape of the voltage and current are the same, they are just scaled versions of each other, so I could start with either v or i. For L and C, when signals are exponentials, I know I get another exponential when I take a derivative. I like to work with the derivative form of the i-v equation rather than the integral form, so I chose to assume the variable inside the derivative. For L that would be current (v = L di/dt), and for I that would be voltage (i = C dv/dt).(3 votes)

## Video transcript

- [Voiceover] Now we're gonna talk about the idea of impedance. This is a really important
idea in electronics and it's something that comes
from the study of AC analysis. And AC analysis is where we
limit ourselves to inputs to our circuits that look like
sinusoids, cosines or sines. And of all the signals
that we could possibly have in the entire universe,
we're gonna limit ourselves just for the moment, to sine waves. And there's some great simplifications that emerge from this. So in this video, we're gonna look at, we're gonna develop,
basically the i-v equations for our three favorite passive components, resistor, inductor, and capacitor. And we're gonna look at those
when the input is a sinusoid. So that means that i or v,
the voltage or the current is in the shape of a sinusoid. And we're gonna see what that
means for the i-v equations for our favorite devices. So as we look at i-v equations
with sinusoid inputs, we're actually gonna break down sinusoids into complex exponentials. And when we studied
sinusoids, we found out that we could disassemble sinusoids into complex exponentials
using Euler's equation. So, for example, if we have a cosine wave, if we have a cosine as a function of time, omega t, we can express that in terms of complex
exponentials like this: 1/2 e to the plus j omega t plus e to the minus j omega t. Like that. And what I'm gonna do
now is I'm gonna say, let's look at what happens when we use this as an input signal. This is not a real input signal, it's an imaginary, rotating vector. But if I have two of these, I can reassemble them into a cosine wave. And we like to use these exponentials, because they go through
the differential equations of a circuit really easily. These are the inputs
that we know how to solve when we do differential equations. So what I'm gonna do is
develop the i-v equations for the resistor, inductor, and capacitor in terms of this kind of an input, when the voltage or the
current looks like this, what do those equations look like? So we're gonna start with,
we'll start with the resistor. Here's a resistor. And we know from that
that just Ohm's law is v equals i times R. And just for the moment I'm
gonna assume that the current, let's assume that i equals
e to the plus j omega t. So if this is i, what
is v for our resistor? Well, we just plug i in here and we get v equals R times e to the plus j omega t. All right. Now I'm gonna do something that looks like it's a little too simple, but it's gonna get interesting soon. I wanna look at the ratio
of voltage to current. In this situation where we're driving with this complex exponential. So voltage turned out to be
R e to the plus j omega t and that's the voltage
I get if I put current though the resistor of
e to the plus j omega t. And what does that equal to? Well, these two are the
same so they cancel, and I get the ratio of voltage to current is R for a resistor. So for our resistor we just proved that v over i equals R. So this isn't news, we
didn't make a discovery here, this is just Ohm's law. And for a resistor, the
voltage over the current is always equal to the resistance. All right. This is gonna get more
interesting now as we go do inductors and capcitors. So let's do an inductor. It has a value of L henries. And for an inductor we know
that voltage equals L di dt. All right, and let's do
the same thing again. Let i equal e to the plus j omega t. So it's a complex exponential current that we're forcing through our inductor. And let's go ahead and work out what v is. So v equals L times d dt of this value here, e
to the plus j omega t. Or v equals, now we take the derivative and the j omega term
comes down to multiply L, so we get j omega L times what? Times the same thing, e
to the plus j omega t, this is the beautiful
thing about exponentials, they give us back themselves. All right, now let's do this. Let's take, once more, what's the ratio of voltage to current? And that equals, here's the voltage, j omega L times e to the plus j omega t, and let's divide that by i
which is i is right here, i is e to the plus j omega t. So, those cancel. And we get v over i equals j omega L. So now we have an equation
for v over i for an inductor. And this is interesting, this time we have the inductance value which we expected, and
there's also this omega, j omega term that comes in. So this tells us this is
frequency, omega is frequency. So this tells us that the
ratio of v to i for an inductor is dependent on frequency. Now we'll do the same
thing for a capacitor. So here's a capacitor. And there's it's
capacitance value in farads. And for the capacitor, we know
that i equals C times dv dt. And this time let's let v of t, let's let v equal e to the plus j omega t. So this time we're gonna force a voltage across our capacitor that is this imaginary,
this complex exponential. And that gives us, let's
plug that into here, i equals C times d dt of e to the plus j omega t. Now let's take that derivative i equals, same thing happens, j omega comes down to
multiply out in front with C, and we get the same thing over here. So we get j omega C times e to the plus j omega t. And now we'll ask the same question again that we did before, which is
what is v over i for capacitor? And we can fill this in, v is sitting right here,
v is e to the j omega t, e to the plus j omega t and the current is, we
worked that out down here, that's j omega C times e to the plus j omega t. And once again, we get this
nice cancellation here, this cancels with this,
and for capacitor we get v over i equals one over j omega C. And we put a box around that one too. So this says that the
ratio of voltage to current in a capacitor, it depends on
the value of the capacitor, of course, and it also
depends on frequency. So just like over with the inductor, we have a frequency term in here. So now we're gonna give this
ratio of voltage to current in all three cases, we're
gonna give a special name and that name is impedance. And the symbol we often
use for impedance is a Z. So this word impedance
is the general notion of the ratio of voltage to current. And we can do that for all
three of our passive components. For a resistor, the impedance
is its resistance, R. So the word impedance is
like the word resistance, except it's a more general concept. It's the general concept of
voltage divided by current. For a resistor, the
impedance is the resistance. For an inductor, the impedance,
v over i is j omega L. And down here for
capacitor, the impedance is one over j omega C, for capacitance. So this is where the idea of
impedance, this word impedance, this is where it comes from. And the idea includes both
the values of the components and the effect that frequency has on the voltage to current ratio. So both of those things
are combined into one idea. So just a quick summary of impedance, if we say the impedance of
a resistor, that equals R. The impedance of an
inductor equals j omega L. And the impedance of a capacitor equals one over j omega C. And as a reminder of
the assumptions we made, we said that we're only gonna
consider sinusoidal inputs. And what we did is we
broke up our sinusoid, our cosine wave, into these, we looked at how these
rotating vectors passed through our components in the
form of voltages and currents. So there's no new physics here. What happened is we took these j omegas that came out of the exponentials when we took the derivatives, we associated those with
the component itself, we did that here and we did that here, and you can see it here. So we sort of done, it's somewhat of a notational trick. We associate this frequency dependence, not with the inputs and
the voltages and currents, but with the components themselves. And this is something we call, this is referred to as transforming. We transformed our components. That's the phrase that's used there. But from this comes the idea of impedance and in the general sense of
the voltage to current ratio.