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 1.Short-circuit inductors or open-circuit capacitors and analyze resulting circuit to determine y(t  ) 2.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 1.Write governing differential equation 2.Determine initial condition 3.Determine final condition (from circuit or diff. eqn) 4.Check differential equation Check time constant (circuit vs. differential equation) Check final condition (circuit vs. differential equation) 5.Solve the differential equation

10. Example 1 The circuit below is initially relaxed. Find v L (t) and i L (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 K 1 and K 2 Governing equation: Form of solution:

15. Example 1 – sketch response

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

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

18. Example 2 – continued

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

20. Example 4 For the circuit shown: determine: 1.The differential equation governing v(t) 2.The initial (t=0 + ) and final (t  ) values of v(t) 3.The circuit’s DC gain 4.C so that  =0.1 seconds 5.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: ;