1. General Solution for Natural and Step Responses of RL and RC Circuits
Final Value
Time Constant
Initial Value
Determine the initial and final values of the variable of interest and the time constant of the circuit.
Substitute into the given expression.
ECE 201 Circuit Theory I

2. Example 7.7 What is the initial value of vC?+
vC(t) -
What is the initial value of vC?
What is the final value of vC?
What is the time constant when the switch is in position b?
What is the expression for vC(t) when t>=0?
ECE 201 Circuit Theory I

3. Initial Value of vC +
V60 - +
vC(0) -
The capacitor looks like an open circuit, so the C is the same as the 60Ω.
ECE 201 Circuit Theory I

4. Final Value of vC +
vC(∞) -
After the switch is in position b for a long time, the capacitor will look like an open circuit again, and the C is +90 Volts.
ECE 201 Circuit Theory I

5. The time constant of the circuit when the switch is in position b
The time constant τ = RC = (400kΩ)(0.5μF)
τ = 0.2 s
ECE 201 Circuit Theory I

6. The expression for vC(t) for t>=0
ECE 201 Circuit Theory I

7. The expression for i(t) for t>=0-
30V +
i(t)
Initial value of i is ( )V/400kΩ = 300μA
Final value of i is 0 – the capacitor charges to +90 V and acts as an open circuit
The time constant is still τ = 0.2 s
ECE 201 Circuit Theory I

8. The expression for i(t) (continued)
ECE 201 Circuit Theory I

9. How long after the switch is in position b does the capacitor voltage equal 0?
ECE 201 Circuit Theory I

10. Plot vC(t)
ECE 201 Circuit Theory I

11. Plot i(t)
ECE 201 Circuit Theory I