Lecture 23 Second order system step response Governing

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 yss • Rise time is closely related to the natural frequency:

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:

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 vc(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 10 V step input is applied to the system, the response is as shown. Estimate the differential equation governing the system.

Example 3 – find tr, MP, yss from plot

Example 3 – find differential equation • From plot, we determined: – MP 0. 25 – tr 0. 05 – yss 0. 002

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