Transient / Steady State Response (1st & 2nd order)

Transient / Steady State Response

We may check the responses for 3 characteristics:

Quality of response

Accuracy of response (% error)

x_h : homogenous solution, due to I.C ⇒ transient response; x_p : particular solution, due to external excitation.

To check stability we have to check the homogen solution.

or it can be unstable due to resonans, r=εω t .

First order system

if R(s)=0 → transient solution

( s_1 : position of pole determines the rate of decay.)

T : time constant of the system

First order systems

Assume that there is no initial condition and R(s)≠ 0 .

case 1) impulse input: R(s)=1

case II) step excitation

derivative of response gives us the rate of change (slope of response)

only x has to be continuous

x and x has to be continuous

Impulse response =ddt (step response)

1st order system

III) Ramp Response

Accuracy is defined by `T'.

2nd order response

RLC in parallel

Spring mass Damper System

ω_n = natural frequency; = zeta = damping ratio

no zeros but 2 poles are present

if <1 ⇒ complex conjugate poles (always oscillatory)

critically damped systems

(sorry) if =1 ⇒ repeated real valued poles

if >1 ⇒ seperate and real valued poles overdamped system

Homework (homogen solution)

Response due to external excitation (particular solution)

Specifying the input

The response frequency for damped systems is ω_d

2nd order systems

If poles are CC → response is oscillatory and decaying

for × ω_d=0 no oscillating frequency; decays at the same time

Step response calculation