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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# AC analysis intro 1

Solving circuits with differential equations is hard. If we limit ourselves to sinusoidal input signals, a whole new method of AC analysis emerges. Created by Willy McAllister.

## Want to join the conversation?

- I think the right-hand side of the equation should be dVin/dt if you define Vin as the voltage. And this equation is not KVL; it is the equation for the rate of change of voltage.(24 votes)
- at Ikchen9699 you are right. The equation comes from the eqaution VL+VR+VC = Vin

if we write the VL,VR and VC as differentials. That looks like that

L dI/dt + R*I + 1/c *∫Idt = Vin. This equation is called intregro-differential equation because of the integral. To make this kind of equations to a differential equation take the derivative on both sides. Now we have the equitation L d^2I/dt^2 + R* dI/dt + 1/c * I = dV/dt.(13 votes)

- are these voltages or rate of change of voltages?(5 votes)
- It really seems like it isn't KVL. When we didnt have any voltage source, the equal of KVL was zero, and if we tried to get the rate of change of the KVL equation, this was equal to d0/dt = 0. So now that the second part of the equation is Vin, we should have d(Vin)/dt, not just Vin. Could we have an explanation over this please?(6 votes)

- Why inductor is used in the AC circuit?(2 votes)
- Hello again Sama!

There are a few ways of looking at this:

First from a radio engineer's perspective: Inductors are your best friend. They along with capacitors allow us to form tuned circuits. These circuits are the building bocks upon which all radio transmitters and receivers are built. Without inductors you would not be able to tune a radio.

From a power supply engineer: If you are using a computer be thankful we have inductors. They are the essential to the construction of power supplies. Inductors and their close cousin the transformer are found everywhere. In fact if you are using a computer you a employing dozens of inductors in the chain of power supplies that take the wall outlet power and convert it to a regulated DC voltage for the main CPU. You will also find inductors in the LED lights that light your room.

Finally, lets see what a power grid engineer has to say: It would be nice if we didn't have inductors but they are everywhere. The large generators at the power plant act as inductors, the power lines have inductance. Yes, a simple wire will act as an inductor! Also, almost every AC motor on the grid acts as an inductor. My job is difficult as I must keep track of and supply the imaginary (aka reactive) power that all of these inductors require. This is where phase comes into the equation. All of these inductors make the delay the phase of the current relative to the voltage. That's OK by me as I can talk about cool things like phasors, real power, reactive power, VARS, VA, symmetrical components, lagging power factor, and my personal favorite the syncrophasor!

Please leave a comment below.

Enjoy!

Aaron(11 votes)

- ac signal is the function of voltage or time or current?(3 votes)
- Hello Samananwar,

Both voltage and current vary as a function of time.

Mathematically you could talk about voltage as V(t) = Asin(2πft + Θ). Here A is the peak voltage, f is the frequency and Θ is the phase shift.

We could make a similar equation for current.

Regards,

APD(4 votes)

- plz see Professor Shankar's introduction into AC LCR circuits

your derivations is really confusing , the results are the same but the road leading to them is very hard ,

https://www.youtube.com/watch?v=cS83YUpcEsY(0 votes)- I'm glad Professor Shankar's presentation was helpful to you. Every teacher takes a slightly different approach to difficult problems. I will see what I can learn from his presentation.(8 votes)

- Why phase difference between two alternating quantities is more important than their actual phases?(3 votes)
- There really isn't a strong definition of "actual phase". Phase is always a relative measurement between a signal and a reference signal. If we have one AC signal we get to choose what time we label time = 0. It can be at a peak or trough, or where the signal crosses 0 volts. It is an arbitrary choice. When we start to think about a second AC signal, one of the important things is its phase relationship with the first signal. This is what we call relative phase.(2 votes)

- I have an existential doubt, not about the video but about the alternating current. I do not know where to put the question, excuse me.

Now my question: Leaving away the stationary state, how does the current to go from the positive pole of the source to the negative pole (assuming that the distance is much greater than the wavelength of the signal as in a transmission line for example or a high frequency circuit), if the polarity varies and therefore, the direction of the current also and due to this, the current should be moved same quantity of meters in one direction and the other. So, how the current can arrive to the "final" of the circuit?

I hope someone can help me with this doubt :( thank you very much in advance.(3 votes) - What is meant by positive phase sequence, negative phase sequence, and zero phase sequence in AIR CIRCUIT BREAKER(2 votes)
- This is probably a silly question, but at5:45, what I see in the parentheses looks like a quadratic expression of the form ax^2+bx+c=0. I am obviously missing something here, as Willy McAllister said it would be extremely difficult to solve. Could someone clarify?(1 vote)
- Not silly at all. When the quadratic expression is set equal to zero, as you wrote, that's the description of the natural response problem, and it is relatively straightforward to solve (see https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-natural-and-forced-response/a/ee-rlc-natural-response-intuition). However, when you set the equation equal to some non-zero arbitrary input represented by v_in, the math blows up in complexity. It gets hard because you have to search for a function whose first and second derivatives, and itself, all add up to equal v_in. When the equation is equal to zero (natural response), we know how to do that, the solution is an exponential.. In other cases, it's a big chore.(2 votes)

- 3:48What are the units of A, s and t. Where this equation come from ? What does it mean ?(1 vote)
- The equation is talking about a current, i = Ae^st.

Both sides of the equation have units of amperes.

A is the scaling factor that tells you how large the current is.

t is time in seconds.

s is frequency in 1/seconds, also known as hertz.

As you work through this sequence of videos you will find out the meaning of e^st. When you combine e^st with e^-st you get a sine wave (Euler's formula).(2 votes)

## Video transcript

- [Voiceover] We now
begin a whole new area of circuit analysis called
sinusoidal steady state analysis. You can also call it AC analysis. AC stands for alternating current. It means it is a voltage or a current that where the signal actually changes sign. It is positive sometimes. It is negative sometimes. And the conventional name for that is AC or alternating current. It could've been called
alternating voltage but that's not the name. The name is AC. What I want to do in this video is we will do a quick review of what it is like to solve an equation like the one showed here. This is an RLC circuit. I'm going to show what
it is like to solve this in differential equation form, which is gonna be a lot of work. I want to introduce the idea of a new point of view
or a new analysis method that we refer to as the
sinusoidal steady state. And it is a transformation that we are gonna do on this circuit. And it is gonna be a
big reward at the end. I want to go through what's the reward and what's it gonna look like at the end. Then we are gonna know some of the math that we have to review in
order to fully understand this change that we are gonna go through. This change of point of view. Let's first take a look
at this circuit here. This is a circuit that's
now a driven RLC circuit. So here's the drive function. It is a voltage. It is some waveform. It is driving a sequence of an inductor, a resistor, and a capacitor. Now in an earlier video we derived the natural response of this circuit and to do that, we shorted out we removed the source and shorted it out and added a little bit
of energy to this circuit and saw what it did on its own. What was its natural response. Now we've upgraded this. We've added a source and now we have to solve this again including the source. If we use the differential
equation technique this is how we are gonna go about it. So the first step in a
circuit analysis like this is to write a KVL equation. We are gonna solve for
this current right here. That's the one current that's in this. So I is our independent variable. If I write KVL as you recall when we did this for natural response, we ended up with a differential equation that looked like this. We had L times the second derivative of I plus R times the first derivative of I plus one over C times I. So these are the voltages. Each of these individual terms are the voltages across
these components here. So that's the resistor voltage. This is the capacitor voltage and this here is the inductor voltage so it is inductor voltage, resistor voltage, capacitor voltage. And all those if we add those up, those have to equal VN. So this is now a forced equation, which means this is the forcing function and we are gonna have to solve this and the math for doing this is pretty difficult. It was hard enough to
do the natural response and we add in this and it gets to be even more work. So as we did before, what we do now is we propose a solution and the solution we are in
the habit of doing this now is gonna be sum constant times E to the sum natural frequency times T. So AE to the ST is our proposed solution
for I as a function of time. You remember we called S. S is a frequency term because S times T has to have no units so S has units of one
over time or frequency so that's called the natural frequency. And when we plug in I the way to tell if I is a solution is to plug this into this equation here. And we got an equation
that looks like this. We ended up after factoring our I we ended up with LS squared plus RS plus one over C and all that is equal to for the natural response we put in, so we solve this equation by setting this term here equal to zero and solving for s to find out what the natural frequency is and then we go back and we find out A by looking at the initial
conditions over here. Whatever initial energy
was in this circuit determines the value of A here. The next step in this forced response where VN is driving the circuit is we have to set this back to VN and solve for the forced. Now if we let VN be sort of
any forcing function we want, any kind of waveform, this is gonna be a really
hard piece of mathematics. This is gonna be a really
difficult calculation. It is gonna take a long time and basically I don't want to do it. I'm gonna wish there was
some other way to do these kinds of equations and there is. The way we simplify this
process substantially is we make a little
limitation on ourselves on what VN can be. If we make a rule, if we basically volunteer
to limit ourselves to VN equals sinusoids that means that VN is of the form cosine of omega T plus phi where phi is some angle or it could be sine of omega T plus some angle. Any waveform of this form here is called the sinusoid. So sinusoid is the general
name for cosine and sine and signals that look like that. Because we don't want this
math to blow up on us, with the general input over here, we are gonna develop a really elegant way to solve circuits with this we limit ourselves to sinusoidal inputs. We will take a pause right here and continue on in the next video to introduce the idea
of sinusoidal analysis.