Lecture 18

1. Lecture 18 Review: First order circuit natural response Forced response of first order circuits Step response of first order circuits Examples Related educational modules: Section 2.4.4, 2.4.5

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: • 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