1. Response of First Order RL and RC
Natural response
Step response
Lecture 8

2. Response of First Order RL and RC
Natural response:
Happens when the inductor or capacitor is suddenly disconnected from its dc source.
Step response:
Happens when a dc voltage or current source is applied to an inductor or a capacitor.

3. Natural Response of an RL circuit
If the current is constant, the voltage across the inductor is zero.
Thus the inductor behaves as a short circuit in the presence of a constant, or dc, current.
Assume a constant current source.
Assume that the switch has been
closed for long time.
So, no current passes through Ro or R and all source current Is appears in the inductive branch.
We find the natural response after the switch has been opened at t=0 ( i(0-) = Is).

4. Natural Response of an RL circuit
For t > 0, the circuit reduces to the one shown in the figure.
We need to find the voltage and current at the terminal of the resistor after the switch has been opened.

5. Natural Response of an RL circuit

6. Time Constant

7. Natural Response of an RL circuit
Calculating the natural response of an RL circuit can be summarized as follows:
Find the initial current, Io, through the inductor.
Find the time constant of the circuit, τ = L/R.
Use equation Ioe-t/τ, to generate i(t) from Io and τ

9. Natural Response of an RC circuit
If the voltage is constant, the current across the capacitor is zero.
Thus the capacitor behaves as an open circuit in the presence of a constant, or dc, voltage.
Assume a constant voltage source.
Assume that the switch has been
in position a for long time.
Then the circuit made up of Vg, R1 and C reaches a dc steady-state circuit where the capacitor is an open circuit.
The voltage across the capacitor at (t = 0-)is Vg.

10. Natural Response of an RC circuit
Because there can be no instantaneous change in the capacitor voltage, the initial voltage across the capacitor is V(0+) = Vg

11. Natural Response of an RC circuit
Calculating the natural response of an RC circuit can be summarized as follows:
Find the initial voltage, Vo , through the capacitor.
Find the time constant of the circuit, τ = RC.
Use equation Voe-t/τ, to generate v(t) from Vo and τ

13. Step Response of an RL circuit
Finding the currents and voltages generated when either dc voltage or current sources are suddenly applied.
The initial current through the inductor is zero.
The switch closes at t = 0.
The voltage across the inductor is 0 for t < 0 and Vs for t > 0.

14. Step Response of an RL circuit
After the switch has been closed, KVL requires:

15. Step Response of an RL circuit

17. Step Response of an RC circuit
Finding the currents and voltages generated when either dc voltage or current sources are suddenly applied.
The switch closes at t = 0.
The initial current through the capacitor is zero for t < 0 and Is for t >0.
After the switch has been closed, KCL requires:

19. A general solution for natural and step responses

20. Procedure

21. Example