# Step Response of an RC Circuit (Problem)

I have this example below from "Fundamentals of Electric Circuits", example 7.11.

simulate this circuit – Schematic created using CircuitLab

The question goes like:

The switch has been closed for a long time and is opened at t=0. Find i and v for all time.

What I want to ask is about the second part, which is t>0.

For t>0, the circuit becomes:

simulate this circuit

From this circuit above we find V(∞) to be 20V. We have V(0) as 10V. Inserting in the equation, we get :

$$V(t) = 20-10e^{-0.6t} V$$

Then the solution says, to obtain i, we notice from the figure above that i is the sum of the currents through the 20 ohms and the capacitor that is,

$$i = \frac{V}{20} + C\frac{dV}{dt}$$

Then from this we get i to be:

$$i(t) = 1+e^{-0.6t} A$$

My question is that,

• Is there any other way to find the i?
• If I would consider the capacitor to be an open circuit and then calculate the i by dividing 30V with 30ohms, would I be wrong?
• V1 is marked 30u(t); is this a dependent source? – Hearth May 8 '18 at 15:03
• @Felthry, no, that is an independent source. u(t) is indicating a unit step function, which has the definition, when t < 0 30u(t) = 0 V and when t > 0 30u(t) = 30V. So when t < 0 you can consider that source to be a short circuit and contributing nothing to overall voltage. – André Yuhai May 8 '18 at 15:11
• Ah, I see. The connection I made was that u is often used as a symbol for voltage alongside v, but the unit step function makes more sense. – Hearth May 8 '18 at 15:14

Is there any other way to find the i?

I would personally prefer to get $i(t)$ like so:

$$i(t) = \frac{V_s(t) - v(t)}{R_1} = \frac{30 - 20+10e^{-.6t}}{10}u(t) = (1+e^{-.6t})u(t)$$

Current $i(t)$ is the current through $R_1$, which is given by Ohm's law from the voltage accross it $V_s(t) - v(t)$.

If I would consider the capacitor to be an open circuit and then calculate the i by dividing 30V with 30ohms, would I be wrong?

Yes, you would be wrong. The current varies with time until the capacitor voltage settles. This method would result in a current that is constant.

• Exactly! That was the answer I was looking for. I just couldn't get my mind around it. I was trying to sum the two voltage sources, but then I figured the two voltage sources are not in series so that I thought it was not possible to sum them up. But now, if I am not mistaken, I figured from your solution that you just used the voltage across the 20 ohm resistor which is equal to the capacitor voltage anyway. About the second question, so dividing 30 V by 30 ohms would give me the current when the capacitor is fully charged right? – André Yuhai May 8 '18 at 15:56
• @AndréYuhai Yes, exactly as you put it. – Vicente Cunha May 8 '18 at 16:00