1. Response of First-Order Circuits
RL Circuits
RC Circuits
ECE 201 Circuit Theory I

2. The Natural Response of a Circuit
The currents and voltages that arise when energy stored in an inductor or capacitor is suddenly released into a resistive circuit.
These “signals” are determined by the circuit itself, not by external sources!
ECE 201 Circuit Theory I

3. Step Response
The sudden application of a DC voltage or current source is referred to as a “step”.
The step response consists of the voltages and currents that arise when energy is being absorbed by an inductor or capacitor.
ECE 201 Circuit Theory I

4. Circuits for Natural Response
Energy is “stored” in an inductor (a) as an initial current.
Energy is “stored” in a capacitor (b) as an initial voltage.
ECE 201 Circuit Theory I

5. General Configurations for RL
If the independent sources are equal to zero, the circuits simplify to
ECE 201 Circuit Theory I

6. Natural Response of an RL Circuit
Consider the circuit shown.
Assume that the switch has been closed “for a long time”, and is “opened” at t=0.
ECE 201 Circuit Theory I

7. What does “for a long time” Mean?
All of the currents and voltages have reached a constant (dc) value.
What is the voltage across the inductor just before the switch is opened?
ECE 201 Circuit Theory I

8. Just before t = 0 The voltage across the inductor is equal to zero.
There is no current in either resistor.
The current in the inductor is equal to IS.
ECE 201 Circuit Theory I

9. Just after t = 0
The current source and its parallel resistor R0 are disconnected from the rest of the circuit, and the inductor begins to release energy.
ECE 201 Circuit Theory I

10. The expression for the current
ECE 201 Circuit Theory I

11. A first-order ordinary differential equation with constant coefficients.
How do we solve it?
ECE 201 Circuit Theory I

12. ECE 201 Circuit Theory I

13. The current in an inductor cannot change instantaneously
Let the time just before switching be called t(0-).
The time just after switching will be called t(0+).
For the inductor,
ECE 201 Circuit Theory I

14. The Complete Solution
ECE 201 Circuit Theory I

15. The voltage drop across the resistor
ECE 201 Circuit Theory I

16. The Power Dissipated in the Resistor
ECE 201 Circuit Theory I

17. The Energy Delivered to the Resistor
ECE 201 Circuit Theory I

18. Time Constant
The rate at which the current or voltage approaches zero.
ECE 201 Circuit Theory I

19. Rewriting in terms of Time Constant
ECE 201 Circuit Theory I

20. Table 7.1 page 233 of the text
ECE 201 Circuit Theory I

21. Graphical Interpretation of Time Constant
Determine the time constant from the plot of the circuit’s natural response.
Straight Line Approximation
ECE 201 Circuit Theory I

22. Graphical Interpretation
Tangent at t = 0 intersects the time axis at the time constant
ECE 201 Circuit Theory I

23. Procedure to Determine the Natural Response of an RL Circuit
Find the initial current through the inductor.
Find the time constant,τ, of the circuit (L/R).
Generate i(t) from I0 and τ using
ECE 201 Circuit Theory I