1. Lecture 23 Second order system step response
Governing equation
Mathematical expression for step response
Estimating step response directly from differential equation coefficients
Examples
Related educational materials:
Chapter 8.5

2. Second order system step response
Governing equation in “standard” form:
Initial conditions:
We will assume that the system is initially “relaxed”

3. Second order system step response – continued
We will concentrate on the underdamped response:
Looks like the natural response superimposed with a step function

4. Step response parameters
We would like to get an approximate, but quantitative estimate of the step response, without explicitly determining y(t)
Several step response parameters are directly related to the coefficients of the governing differential equation
These relationships can also be used to estimate the differential equation from a measured step response
Model parameter estimation

5. Second order system step response – plot

6. Steady-state response
Input-output equation:
As t, circuit parameters become constant so:
Circuit DC gain:

7. On previous slide, note that DC gain can be determined directly from circuit.

8. Rise time
Rise time is the time required for the response to get from 10% to 90% of yss
Rise time is closely related to the natural frequency:

9. Maximum overshoot, MP MP is a measure of the maximum response value
MP is often expressed as a percentage of yss and is related directly to the damping ratio:

10. Maximum overshoot – continued
For small values of damping ratio, it is often convenient to approximate the previous relationship as:

11. Example 1
Determine the maximum value of the current, i(t), in the circuit below

12. In previous slide, outline overall approach:
Need MP, and steady-state value
Need damping ratio to get MP
Need natural frequency to get damping ratio
Need to determine differential equation

13. Step 1: Determine differential equation

14. Step 2: Identify n, , and steady-state current
Governing equation:

15. Step 3: Determine maximum current
Damping ratio,  = 0.54
Steady-state current,

16. Example 2
Determine the differential equation governing iL(t) and the initial conditions iL(0+) and vc(0+)

17. Example 2 – differential equation, t>0

18. Example 2 – initial conditions

19. Example 3 – model parameter estimation
The differential equation governing a system is known to be of the form:
When a 10V step input is applied to the system, the response is as shown. Estimate the differential equation governing the system.

20. Example 3 – find tr, MP, yss from plot

21. Example 3 – find differential equation
From plot, we determined:
MP  0.25
tr  0.05
yss  0.002

22. Example 4 – Series RLC circuit
MP  100%, n = 100,000 rad/sec (16KHz)