First-Order Circuit Step Response Analysis for Circuit Designers

1. Lecture 19 Review: First order circuit step response Steady-state response and DC gain Step response examples Related educational modules: Section 2.4.5

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: ;