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 modules: –Section 2.5.5

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

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

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

Second order system step response – plot

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

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

Rise time Rise time is the time required for the response to get from 10% to 90% of y ss Rise time is closely related to the natural frequency:

Maximum overshoot, M P M P is a measure of the maximum response value M P is often expressed as a percentage of y ss and is related directly to the damping ratio:

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

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

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

Step 1: Determine differential equation

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

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

Example 2 Determine the differential equation governing i L (t) and the initial conditions i L (0 + ) and v c (0 + )

Example 2 – differential equation, t>0

Example 2 – initial conditions

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.

Example 3 – find t r, M P, y ss from plot

Example 3 – find differential equation From plot, we determined: – M P  0.25 – t r  0.05 – y ss  0.002

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