Main content

## Electrical engineering

### 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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# KVL in the frequency domain

Demonstration that Kirchhoff's voltage law applies in the frequency domain. The voltage phase offsets around a loop sum to zero. Created by Willy McAllister.

## Want to join the conversation?

- So, is that the end of AC analysis? It seems like there should be a lot more...(19 votes)
- Hello Anthony,

Yes there is much more. This section of KA brings you about 1/3 of the way through a traditional EE 101 class. There are many follow on classes including systems & circuits, op-amps, electomechanical systems, DSP, and many others.

Regards,

APD(8 votes)

- What's a frequency domain? I don't remember where it was introduced.(3 votes)
- Alexander - You make a good point. I used the term "frequency domain" with no introduction. For most of the EE material prior to the AC circuit analysis section, we've been working on transient analysis in what is called the "time domain". Analysis was based on using t (time) as the independent variable. We figured out voltages and currents as they changed with time. When we begin AC Analysis we make a giant assumption: Let the signals of interest be sinusoids (as opposed to steps or ramps or jumbled up shapes). With this assumption we can do all sorts of marvelous things, like defining the idea of Impedance. What you may have noticed as you study the last several videos in the AC Analysis section is that frequency (lowercase omega) has become an interesting independent variable to think about. When we discover properties of circuits related to frequency, we say we are working in the
**frequency domain**. Thinking in this manner is one of the most powerful ideas in electrical engineering. It is the basis of how every radio and mobile phone works. In advanced classes on signals and systems you will come across techniques called the Fourier Series and Fourier Transform that perfectly capture the frequency domain idea.(10 votes)

- We use AC analysis only for forced response, How ? What if there is a initial energy present in circuit, will entire idea of AC analysis fail there ?(1 vote)
- Yes, AC analysis is for forced response. That is its main application. You can combine it with natural response if you want, using superposition. In most analog electronic systems we focus on the forced response.

For example, a public address system with a microphone, amplifier, and speaker. We study how the amplifier modifies the signal coming from the microphone. That is entirely a forced response analysis.

The natural response of the public address system happens right when you turn it on. The amplifier might produce pops and grunts for a second or two as it comes up to full voltage. Those pops and grunts are the natural response happening and fading away. After that, the system is just doing its forced response.(6 votes)

- In this video every step involves taking the real part of a complex expression. However, the graph at the end depicts the sum of the phasors (complex #'s) equal to zero. I don't understand how you get from [Re(sum of phasors) = 0] to [sum of phasors = 0]. What am I missing?(2 votes)
- Mark and WIlly,

At10:42e^jwt never equals zero so it follows that V0 + V1 + V2 + V3 = 0. Where Vk are all vectors.

At11:59I don’t think we need specify "real" since V0 + V1 + V2 + V3 = 0 holds for real as well as imaginary components.

We can then plot the vectors as show in the video.

Regards,

APD(3 votes)

- What happended to the solve part?

Are there any problems solved somewhere else?(1 vote)- Juan,

This is the last video I created for Khan Academy as part of my fellowship in 2016. I'm sorry there is nothing beyond this at the moment. However, I'm continuing to work on teaching material on my own web site, spinningnumbers.org. At the moment (mid-2017) the site reproduces a lot of this KA EE section, with some new articles at the beginning on Charge and sign convention. There is also a very nice circuit simulator program for anyone to use. I hope to add a lot more in the coming months.

- Willy McAllister willy@spinningnumbers.org(5 votes)

- AC analysis was nicely taught, thanks. Can you recommend other resources to learn about other advanced topics in circuits analysis before I start learning further topics in Electrical Engineering at KA ?(2 votes)
- A good list of prerequisites for studying EE is here: https://www.khanacademy.org/science/electrical-engineering/introduction-to-ee/intro-to-ee/a/ee-preparing-to-study-electrical-engineering?modal=1

Since completing my fellowship at KA I've worked on spinningnumbers.org. All the articles are updated. Videos are the same, so far. I just posted a new section on digital design.

Analog Devices has always placed a high premium on educating their customers. Click on the Education tab here: https://www.analog.com/en/index.html

EdX.org offers college engineering classes. Look for ones you are interested in. I particularly admire the courses from MIT. https://www.edx.org/course/subject/engineering

Digi-key lists many educational kits: https://www.digikey.com/en/resources/edu/ed-kits(3 votes)

- what are natural and forced responses?(2 votes)
- There's a whole giant section of KA EE on Natural and Forced response. See https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic#ee-natural-and-forced-response. Or, check out the updated articles at https://spinningnumbers.org/t/topic-natural-and-forced-response.html

Natural Response is what a circuit does if you turn off the sources and let it have some stored energy (for example, you put some charge on a capacitor). Then you release the charge and watch what the circuit does "naturally".

Forced Response is what a circuit does when you connect an input source (like drive the circuit with a voltage step or sine wave). The input source "forces" the circuit to do things.(3 votes)

- What ever happened to the "good old days" where : Xl = 2x pi x F x L , or

Xc = 1 / 2 x pi x F x C ? OBEARCC(2 votes)- We do all this work with complex exponentials because it leads directly into Fourier analysis (frequency domain). In "the old days" not many of us had access to powerful computers that could figure out Fourier transforms, but now we do. Hence the change in emphasis.(3 votes)

- In your earlier video sir you have demonstrated that the complex exponential form of cosX= e^jx-e^-jx/2 but in this video at 6.18 acc to that the complex exponential form of V0cos(wt+¤pi) should be = V0(e^j(wt+¤pi)+e^-j(wt+¤pi)/2 you have written something else how? explain this please i am confused(2 votes)
- How come all voltages have same frequency(w) ?(1 vote)
- When you think about voltage and current in a circuit element you typically drive the element with a voltage or current and measure the other (the current or voltage). To find the impedance you take the ratio Z = V/I.

Think about a resistor. A resistor obeys Ohm's Law, v = i R. If you apply a static (not changing) voltage you get a static current. What if you apply a voltage that looks like a sine wave? What does the current look like? It also looks like a sine wave. Here's an example,

let R = 1 kohm

let v = 3sin wt

then i = v / R = 3sin wt / 1000 = .003 sin wt

If you look at how Ohm's Law works it guarantees the current has a similar shape to the voltage, and that includes having the same frequency. The amplitude of the current is determined by Ohm's law, but the frequency goes through Ohm's law without being modified. The same thing happens for the*i-v*equations for the capacitor and inductor. The frequency is always preserved.(2 votes)

## Video transcript

- [Voiceover] As we do AC analysis, and we do operations in
the frequency domain, we need to bring along Kirchhoff's Laws so that we can make sense of circuits. So, in this video, I'm
gonna basically show that Kirchhoff's Voltage Law works in the frequency domain. And what I have here is a circuit that has some voltage
source, an AC voltage source, let's put AC on it like that, and it has three impedances connected. Inside each of these boxes
is an R, an L, or a C, and we're not gonna show which because we're gonna carry these along just as general impedances. So, in AC analysis, the
voltages are all cosine waves. So, v-in equals some voltage amplitude times cos of omega-t plus we'll call it phi-zero, some starting phase shift. So, this is our input signal. Now let me label the
voltages on everything else. We'll call this v-one, and I'm gonna label it this way, here, and this will be v-two, and I'll put it upside-down
like that, the plus, and this will be v-three, minus, plus, v-three. And, oh, now that I have v-one here, let's change the name of this to v-zero just so I don't get i and one mixed up. So, we'll change that to zero. So, the input source is v-zero, and that's that voltage. Now, when we apply KVL to
this, Kirchhoff's Voltage Law, what it says is if we start in a corner, if we start somewhere in the circuit, let's start right here, and go around the loop,
it should add up to zero volts. That's KVL for normal DC circuits, and we're gonna see how that
applies to AC circuits, here. So in time domain, we say that v-zero plus v-one plus v-two plus v-three equals zero. So, let's talk about how
this is gonna turn out. Well, what do we know right now? Well, we know that v-zero is a cosine wave at some phase angle. Now, what do we know about
the other voltages in this? In AC analysis, what we're
doing is we're looking for a forced response. So, we've let the natural
responses to die out. There's no switch in this circuit, and we just assume this circuit has been in this state forever. So, the natural response, the natural response has died out, and that means we're looking
for the forced response. So, what we know is,
we have three voltages. We have three voltages. We know that all these voltages are gonna resemble the input voltage. So, they're all gonna be sinusoids. All the voltages here
are gonna be AC sinusoids because the forcing
function is a sinusoid. The other thing we know, they're gonna all have the same omega. The frequency of this
voltage, and this voltage, and this voltage is gonna
be identical to omega, here. I'm gonna put a big bang there. That's really important. In an AC circuit, when you're driving it from a frequency, every
other frequency in the system is the same frequency. This is a linear system
and linear components. All the analysis we've done, linear components don't
create new frequencies. They're all omega. Now, some other things we know. There's gonna be phase
shifts involved here. Remember when we do impedance, we are multiplying by j and
rotating things by 90 degrees. So, we're gonna have different, different phi, for each one. And, the other thing we're gonna have, is we're gonna have different... The amplitude of our sinusoids
are gonna be different. The amplitude of v-one could be different than the amplitude of v-two. So, this is what an AC
solution is going to look like. Let's move on a little farther here. What I'm gonna do now is we're gonna take this input voltage plus these
things that we know, here, and we're gonna see how
Kirchhoff's Voltage Law works in the frequency domain, when we worked with these
transformed z's, these impedances. Okay, let's go ahead and do that. Okay, let's do a little
more in the time domain. And we'll write out
our KVL equation again. So, the KVL equation was v-naught, cosine omega-t plus phi-zero plus v-one, that's the amplitude of v-one, cosine omega-t plus some different phase angle. We don't know what that is yet. Plus, v-two, amplitude of v-two, cosine omega-t plus phi-three plus v-three, the amplitude of v-three, cosine omega-t plus phi-three all equals zero. And omega, all these omegas,
are the same exact number, the same radian frequency. All the phi's are different, and all the v-twos and
v-threes are different. Okay, now I'm gonna switch to complex exponential notation. We just changing notation here. We can represent this number here as this is the real part of v-naught, v-zero, e to the j times omega-t plus phi-zero. That's exactly the same as this. This cosine can be represented as the real part of a complex exponential with this frequency. And, I can write out the rest of these. V-one, e to the j omega-t plus phi-one plus the real part of v-two, e to the j-omega-t, plus phi-two, parentheses, plus v-three, oops, real part of v-three, e to the j-omega-t plus phi-three equals zero. All right, now, one thing I can do next, we can start to factor this. We can start to take
this apart a little bit. So, I know that if I have the expression, if I have the expression, e to the j-omega-t plus phi, just in general, I can change that just
by exponent properties to e to the j-phi, let's get
the parentheses in there, like that, e to the j-phi times e to the j-omega-t. So, I'm gonna do this transformation on all four of these terms here. Let's keep going. So, we're still working on this. Let's go real part... Now I'm gonna take apart omega-t and phi-zero, here, and I get v-naught e to the j-phi-zero, e to the j-omega-t, plus real part, v-one, e to the j-phi-one, e to the j-omega-t, plus real part, v-two, e to the j-phi-two, times e to the j-omega-t, plus real part v-three, e to the j-phi-three, e to the j-omega-t all equals zero. And here's a nice simplification, we take out this common term. We're gonna factor out this common term across the entire equation. And what do we come up with? We come up with... The result is the real part of v-zero, e to the j-phi-zero, plus v-one e to the j-phi-one plus v-two, e to the j-phi-two. See the pattern. All that times e to the j-omega-t. And we close that and that equals zero. Now, we're getting close. We're getting close. All right, how do we
make this equation zero? Does e to the j-omega-t ever become zero? Well, e to the j-omega-t e to the j-omega-t is a rotating vector. It's never zero. So, that's not gonna do it. So, how do we get it? Well, that means that
this other term, here, has to be equal to zero. So, how am I gonna do that? I'm gonna make one more notational change. This kind of a number, here, is called a phasor. It's some amplitude times e to a complex one angle, and there's no time up here. There's no time. The time is only over here. This is the only place that
time appears in the equation, and this is the only
place that omega appears in the equation, and these
are just phase angles, these are starting phase angles. So, my notation for a
phasor is gonna be... This gonna be called... I'm gonna call it v-zero and
I'm gonna put a line over it to indicate that it's a complex vector, and that equals v-naught, e to the j-phi-naught. So, when you see the vector
symbol and the aught, that's that right there. And we can write now, finally, the real part of V-naught plus V-one phasor plus V-two phasor plus V-three phasor equals zero. So, this is KVL in the frequency domain. And fortunately, it looks like... It looks exactly like KVL that we remember from our DC analysis. The sum of the voltages,
going around the loop, is equal to zero, and, in this case, it's the sum of the phasors
going around the loop is equal to zero. Let's try to give a graphical
interpretation to this. Here's our real and imaginary
plane, the complex plane, and what it says is that these phasors... So, let's say that v-naught, let's say that v-naught look like that. That was our voltage source, okay. This represents a vector spinning around at the frequency, omega, and it's offset phase is
this angle right here. So, this is phi-zero. Each of these other
components is gonna have an AC voltage, a sinusoidal voltage on it, with some phase and some magnitude, and what KVL tells us, it puts a constraint on
what those voltages can be. So, we have three impedances here. I don't know what they are because we didn't fill in the circuit, but there's gonna be some vector associated with each one of those. Let's say that's vector-one, and let's say that this is vector-two, and what it says is that vector-three, by the time we get done, vector-three has to sum back to zero. So, this kind of a constraint where voltage is going around in a circle, have to come back to zero, that's KVL and the frequency domain. That's what that means. So, we've shown that KVL
works in the frequency domain. I could do a similar analysis and show that KCL,
Kirchhoff's Current Law, also works in the frequency domain, and that means, fantastically, that all the tools that we developed for DC analysis of just resistor circuits, all those tools work
for AC analysis as well. Thanks for listening.