yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

LC natural response example


4m read
·Nov 11, 2024

So, in previous videos, we worked out an expression for the current ( i ) in an LC circuit like this, and what we found was that ( i ) is the square root of capacitance over inductance times the starting voltage ( v_0 ) times sine ( \omega_0 t ).

And ( \omega_0 ) is the natural frequency, and we said that ( \omega_0 ) was equal to the square root of ( \frac{1}{L \times C} ), and ( v_0 ) is the starting voltage on the capacitor. Our original assumption was that ( i ) was ( 0 ) to begin with, and that's the expression for the current.

So, what I want to do now is a specific example, and we'll do that down here. First, I'm going to install a switch into our circuit so that we can add some energy, and it won't go anywhere.

Then, for ( C ), ( C ) is going to be equal to a quarter of a farad, and ( L ) is going to be equal to one henry. All right, and on the capacitor, I'm going to put enough charge to bring this up to ( 10 ) volts before we close the switch.

Then, at ( t = 0 ), we'll close the switch, and what we want to find is, what is ( i(t) )? We have ( L ), we have ( C ), we have a starting voltage, and so now we have everything we need to work out the current. Let's do that.

Okay, so first off, we'll do ( \omega_0 ). ( \omega_0 = \frac{1}{\sqrt{L \times C}} ), so that equals ( \sqrt{\frac{1}{1 \ \text{henry} \times \frac{1}{4} \ \text{farad}}} ), which equals ( \sqrt{4} ) or ( 2 ).

And that's in units of radians per second. That's the natural frequency. We know the natural frequency right here, and now we can work out the rest of it.

So we can just fill in ( i = \sqrt{C} ) which is ( \frac{1}{4} \ \text{farad} ) divided by ( 1 \ \text{henry} ) times ( v_0 ). ( v_0 ) was ( 10 ) volts times sine ( \omega_0 t ). Sine ( \omega_0 ) is ( 2t ).

And finally, ( i = \sqrt{\frac{1}{4}} ) or ( \frac{1}{2} ), so that's ( \frac{1}{2} ) times ( 10 ) is ( 5 ).

Thus, ( i = 5 \sin(2t) ). So for this specific circuit, that's the answer for this current here.

Now, I want to show you what that actually looks like. So this is a plot of ( i(t) = 5 \sin(2t) ) and this is what a sine wave looks like with time.

So, the axes are time in seconds, and this axis goes up to ( 5 ) amperes and then down to ( -5 ) amperes and continues on that way, and it basically goes on forever.

What I want to do next is work out the voltage in the circuit. We didn't talk about the voltage yet. So, I'll sketch the circuit again here. Here's ( L ) and ( C ), and this is the voltage right here—voltage across both guys.

We've already worked out ( i ) and now we want to find ( v ). What's ( v )? That's what we're looking for.

So, we know that ( i = 5 \sin(2t) ), and to find ( v ), one of the easy ways to find ( v ) is to use the inductor equation.

We know that ( v = L \frac{di}{dt} ); that's just the basic inductor ( i , v ) equation, right? Let's see what happens here.

( v = L ) is ( 1 ) times ( \frac{di}{dt} ), so that's ( \frac{d}{dt}(5 \sin(2t)) ). Now, let's take that derivative.

Okay, I'll go up here. ( v = 5 ) comes out of the derivative. The derivative of ( \sin(2t) ) is ( 2 \cos(2t) ).

All right, so our voltage solution is ( 10 \cos(2t) ). So, something interesting just happened here. Let me show you; we started with the current being a sine function, and we eventually took the derivative of that sine function, and now we have a cosine function.

So, we went from sine to cosine for voltage. That means the voltage doesn't look quite exactly like the current.

Okay, let me show you a plot of the voltage. Here's the voltage: this is ( v(t) ) and that we decided was equal to ( 10 \cos(2t) ). It starts at a value of ( 10 ) at ( t = 0 ) and then forms a cosine wave. It goes up and down between plus and minus ( 10 ) volts.

Now let me show you what it looks like when we plot both ( i ) and ( v ) on the same graph, and we can see the timing relationship between them.

So, this is a plot of ( i(t) ) in blue and ( v(t) ) in orange. One of them is a sine wave; the current is a sine wave, and the voltage is a cosine wave.

So, this is what we were going for. This is the natural response of an LC circuit, and we had two specific component values in it. We saw that we came out with both current and voltage looking like sinusoidal waves.

This LC circuit—this is where sine waves come from in electronics. It's pretty cool!

More Articles

View All
Stop Trying and You'll Succeed
There’s nothing worse than a sleepless night. We’ve all been there, tossing and turning. You focus all your mental power on trying to fall asleep. With all your will, you force yourself to shut your eyes, turn your brain off, and pray to be whisked away i…
40 AI Founders Discuss Current Artificial Intelligence Technology
What is the most maybe unexpected way that you use AI in your regular life? Yeah, I don’t know if I should say this, but uh, writing speeches for weddings. Yeah, we both set our answering machines to our boy’s spot. “Hey, how’s it going?” “Oh, it’s th…
Dave Ramsey Reacts To My $25 Million Dollar Investment
And there’s my debt: uh, four million twenty thousand dollars. Uh, it’s all five mortgages: 30-year fixed between 2.875 and 3.625. I mean, if you’re willing to let that kind of money just evaporate, I personally don’t do anything like that, so I never tho…
Why You Must Be Ruthless in Business or Fail | Kevin O'Leary
[Music] Yeah, well, you’re preaching to the choir here, and I completely agree. That’s why I jumped ship from my, you know, job at the studio table your door, because you were personally motivated to stop living that way. Yeah, but can you talk about the…
Michael Burry's BIG Short Against Tesla Stock REVEALED!
Well, Michael Burry has released Scion Asset Management’s 13F filing for Q1 of 2021, and it was very interesting this time around. Firstly, lots of options. Secondly, big bets on interest rates going north, which is essentially a prediction that we’ll see…
Arthritis Has Me Down | The Boonies
Courtesy out here, I don’t use a whole lot of fold of money. I don’t make a whole lot of fold of money. It’s just a barter system. It’s a whole lot more fun than that. Common pull the wine out of your pocket in the Clearwater mountains of Idaho. Bear Cla…