First-order system time response
v Transient
v Steady-state
Second-order system time response
v Transient
v Steady-state

**System Response**

First-order system time response

v Transient

v Steady-state

Second-order system time response

v Transient

v Steady-state

*Y s / R(s) = K / (1+ K+sT) = K / (1+sT)*

Evolution of the transient response is determined by the pole of the transfer function at s=-1/t where t is the time constant

Also, the step response can be found:

LTI second-order system

Overdamped response: Poles: Two real at

- σ 1 - - σ 2

Natural response: Two exponentials with time constants equal to the reciprocal of the pole location

*C( t)= k1 e*-ζ1+* k2 e*-ζ2

Poles: Two complex at

-σ1±jWd

**Natural response: **Damped sinusoid with an exponential envelope whose time constant is equal** **to the reciprocal of the pole‗s radian frequency of the sinusoid, the damped frequency of oscillation, is equal to the imaginary part of the poles

Poles: Two imaginary at

±jW1

**Natural response: **Undamped sinusoid with radian frequency equal to the imaginary part of the** **poles

C(t) = Acos(w1t-φ)

**Poles: Two real at**

**Natural response: **One term is an exponential whose time constant is equal to the reciprocal of** **the pole location. Another term product of time and an exponential with time constant equal to the reciprocal of the pole location.

**Second order system responses damping cases**

**Second- order step response**

Complex poles

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