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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. Second order system step response.
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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
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
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 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 iL(t) and the initial conditions iL(0+) and vc(0+)
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 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 (16KHz)