1. Lecture 19 Review: Steady-state response and DC gain
First order circuit step response
Steady-state response and DC gain
Step response examples
Related educational materials:
Chapter 7.5

2. First order system step response
Block diagram:
Governing differential equation and initial condition:

3. First order system step response
Solution is of the form:
Initial condition:
Final condition:

4. Note on previous slide that we can determine the solution without ever writing the governing differential equation
Only works for first order circuits; in general we need to write the governing differential equations
We’ll write the governing equations for first order circuits, too – give us valuable practice in our overall dynamic system analysis techniques

5. Notes on final condition
Final condition can be determined from the circuit itself
For step response, all circuit parameters become constant
Capacitors open-circuit
Inductors short circuit
Final conditions can be determined from the governing differential equation
Set and solve for y(t)

6. 2. Checking the final response
These two approaches can be used to double-check our differential equation
Short-circuit inductors or open-circuit capacitors and analyze resulting circuit to determine y(t)
Set in the governing differential equation and
solve for y(t)
The two results must match

7. Steady-state step response and DC gain
The response as t is also called the steady-state response
The final response to a step input is often called the steady-state step response
The steady state step response will always be a constant
The ratio of the steady-state response to the input step amplitude is called the DC gain
Recall: DC is Direct Current; it usually denotes a signal that is constant with time

8. DC gain – graphical interpretation
Input and output signals:
Block Diagram:
DC gain =

9. Suggested Overall Approach
Write governing differential equation
Determine initial condition
Determine final condition (from circuit or diff. eqn)
Check differential equation
Check time constant (circuit vs. differential equation)
Check final condition (circuit vs. differential equation)
Solve the differential equation

10. Example 1
The circuit below is initially relaxed. Find vL(t) and iL(t) , t>0

11. Determine initial AND final conditions on previous slide.

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

13. Example 1 – checking results

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

15. Example 1 – sketch response

16. Example 1 – Still continued…
Now find vL(t).

17. Example 2 (alternate approach to example 1)
Find vL(t) , t>0

18. Example 2 – continued

19. Example 3 (still another approach to example 1)
Find iL(t) , t>0

20. Example 4 For the circuit shown: determine:
The differential equation governing v(t)
The initial (t=0+) and final (t) values of v(t)
The circuit’s DC gain
C so that =0.1 seconds
v(t), t>0 for the value of C determined above

21. Example 4 – Part 1
Determine the differential equation governing v(t)

22. Example 4 – Parts 2 and 3
Determine the initial and final values for v(t) and the circuit’s DC gain

23. Example 4 – Checking differential equation
Governing differential equation (Part 1):
Final Condition (Part 2):

24. Example 4 – Part 4
Determine C so that =0.1 seconds

25. Example 4 – Part 5
Determine v(t), t>0 for the value of C determined in part 3
Governing equation, C = 0.01F:
Form of solution:
Initial, final conditions: ;