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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.

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• So, is that the end of AC analysis? It seems like there should be a lot more... • What's a frequency domain? I don't remember where it was introduced. • 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.
• 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.
• 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? • Mark and WIlly,

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

At I 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
• 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
• 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 ? • what are natural and forced responses? • 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 • 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 • 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.