0:03

The topic of this problem is The Complete Response of RLC Circuits.

The problem is to find the 2nd order differential equation expression for

the voltage, Vc(t) in the circuit shown below.

So, we have a circuit that has a series combination of R, Ls and Cs.

It also has a voltage source, VS sub t.

The voltage Vc(t) is the voltage that's across

the compositor in the top of our circuit.

So we're going to use Kirchhoff's voltage law, and

sum up the voltages around our single loop circuit.

To find an expression which will ultimately give us our route

to finding our second order differential equation for Vc(t).

So if we use Kirchhoff's voltage law, it's a single loop circuit,

so only one loop to choose from, and we're going to travel

clockwise around our circuit starting in the lower left corner.

The first thing we encounter is a negative polarity of the voltage source Vs(t).

As we continue around this, we run into the inductor and

the voltage across the inductor.

We know that the voltage across

the inductor is L di, dt.

And it's an I of t it's a function of time,

and this is our single loop current I of t.

We also have our voltage Vc(t), it's part of this expression

as well because we capture the capacitance next and then we have the resistor.

And the voltage throughout across the resistor is R times i(t),

and so the sum of those is equal to 0.

So if we use our well known expression for the capacitor that is,

that the current through the capacitor is equal to the capacitance times,

2:19

The time derivative of the voltage across a capacitor

then we can rewrite our expression that we have above.

And so we have our Vc (t) here.

We have a Vc(t) in our expression, our Kirchhoff's voltage law expression,

and ultimately we're going to rewrite this, so that we end up with L times C.

And then the second time derivative of the voltage across the capacitor.

2:54

And what we've done is we've plugged our expression for

the current into our second term,

take in the derivative of it and we end up with this.

And then we have still our Vc (t) for

the voltage or up across the capacitor a third term.

We have our fourth term that we can plug in for using our expression that we R for

the current and voltage associated with the capacitor.

And so, that would be Rc(dVc(t) / dt) and

all those are going to be equal to the source

voltage Vs(t) pointing to the other side.

So if we continue to put this in a form that is standard for

second-order differential equations,

just moving terms around, we'll get this expression for

our series combination of R, Ls and Cs.

4:22

Again, it's equal to our forcing function on the right hand side of this circuit.

And the forcing function for us is Vs(t) / LC.

So, here's our standard format for our second order

differential equation expression for the voltage Vc(t).