General Solution for Natural and Step Responses of RL and RC Circuits

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Presentation transcript:

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

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

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

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 voltage @ C is +90 Volts. ECE 201 Circuit Theory I

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

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

The expression for i(t) for t>=0 - 30V + i(t) Initial value of i is (90 - - 30)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

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

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

Plot vC(t) ECE 201 Circuit Theory I

Plot i(t) ECE 201 Circuit Theory I