1. Lecture 18 Review: Forced response of first order circuits
First order circuit natural response
Forced response of first order circuits
Step response of first order circuits
Examples
Related educational materials:
Chapter 7.4, 7.5

2. Natural response of first order circuits – review
Circuit being analyzed has a single equivalent energy storage element
Circuit being analyzed is “source free”
Any sources are isolated from the circuit during the time when circuit response is determined
Circuit response is due to initial energy storage
Circuit response decays to zero as t

3. First order circuit forced response – overview
Now consider the response of circuits with sources
Notes:
We will typically write our equations in terms of currents through inductors and voltages across capacitors
The above circuits are very general; consider them to be the Thévenin equivalent of a more complex circuit

4. RC circuit forced response

5. RL circuit forced response

6. First order circuit forced response – summary
Forced RC circuit response:
Forced RL circuit response:

7. General first order systems
Block diagram:
Governing differential equation:

8. Active first order system – example
Determine the differential equation relating Vin(t) and Vout(t) for the circuit below

9. Active first order system – example
Determine the differential equation relating Vin(t) and Vout(t) for the circuit below

10. Step Response – introduction
Our previous results are valid for any forcing function, u(t)
In this course, we will be mostly concerned with a couple of specific forcing functions:
Step inputs
Sinusoidal inputs
We will defer our discussion of sinusoidal inputs until later

11. Applying step input Block diagram: Example circuit:
Governing equation:
Example circuit:

12. First order system step response
Solution is of the form:
yh(t) is homogeneous solution
Due to the system’s response to initial conditions
yh(t)0 as t
yp(t) is the particular solution
Due to the particular forcing function, u(t), applied to the system
y (t) yp(t) as t

13. First order system – homogeneous solution
Assume form of solution:
Substitute into homogeneous D.E. and solve for s :
Homogeneous solution:

14. First order system – particular solution
Recall that the particular solution must:
Satisfy the original differential equation as t
Have the same form as the forcing function
As t:

15. First order system particular solution -- continued
As t, the original differential equation becomes:
The particular solution is then

16. First order system step response
Superimpose the homogeneous and particular solutions:
Substituting our previous results:
K1 and K2 are determined from initial conditions and steady-state response;  is a property of the circuit

17. Example 1
The switch in the circuit below has been open for a long time. Find vc(t), t>0

18. Example 1 – continued
Circuit for t>0:

19. Example 1 – continued again
Apply initial and final conditions to determine K1 and K2
Governing equation:
Form of solution:

20. Example 1 – checking results