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ECE 201 Circuit Theory I1 General Solution for Natural and Step Responses of RL and RC Circuits Final Value Initial Value Time Constant Determine the initial and final values of the variable of interest and the time constant of the circuit. Substitute into the given expression.

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ECE 201 Circuit Theory I2 Example 7.7 What is the initial value of v C ? What is the final value of v C ? What is the time constant when the switch is in position b? What is the expression for v C (t) when t>=0? + v C (t) -

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ECE 201 Circuit Theory I3 Initial Value of v C The capacitor looks like an open circuit, so the voltage @ C is the same as the voltage @ 60Ω. + v C (0) - + V 60 -

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ECE 201 Circuit Theory I4 Final Value of v C 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. + v C (∞) -

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ECE 201 Circuit Theory I5 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

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ECE 201 Circuit Theory I6 The expression for v C (t) for t>=0

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ECE 201 Circuit Theory I7 The expression for i(t) for t>=0 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 i(t) - 30V +

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ECE 201 Circuit Theory I8 The expression for i(t) (continued)

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ECE 201 Circuit Theory I9 How long after the switch is in position b does the capacitor voltage equal 0?

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ECE 201 Circuit Theory I10 Plot v C (t)

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ECE 201 Circuit Theory I11 Plot i(t)

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