Transient response

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Damping oscillation is a typical transient response, where the output value oscillates until finally reaching a steady-state value.

In Electrical Engineering and Mechanical Engineering, a transient response or natural response is the response of a system to a change from equilibrium. Specifically, transient response in Mechanical Engineering is the portion of the response that approaches zero after a sufficiently long time (i.e., as t approaches infinity). (Contrast with steady-state response)

In Electrical Engineering a simple example would be the output of a 5 volt DC power supply when it is turned on: the transient response is from the time the switch is flipped until the output reaches a steady 5 volts. At this time the power supply reaches its steady-state response of a constant 5 volts.

The transient response is not necessarily tied to "on/off" events but to any event that affects the equilibrium of the system. If in an RC circuit the resistor or capacitor is replaced with a variable resistor or variable capacitor (or both) then the transient response is the response to a change in the resistor or capacitor.

In a mechanical system a simple example is a mass/spring/damper system. The transient response is the position of the mass x(t) as the system returns to equilibrium after an initial force or a non zero initial condition.

The impulse response and step response are transient responses to a specific input (an impulse and a step, respectively).

Both mechanical and electrical systems are analogous.

Damping

The response can be classified as one of three types of damping that describes the output in relation to the steady-state value.

Underdamped

An underdamped response is one that oscillates within a decaying envelope. The more underdamped the system, the more oscillations and longer it takes to reach steady-state. Here Damping Ratio is always < 1

Critically damped

A critically damped response is the response that reaches the steady-state value the fastest without being underdamped. It is related to critical points in the sense that it straddles the boundary of underdamped and overdamped responses. Here, damping ratio is always less than or equal to one (≤1) . There should be no oscillation about the steady state value in the ideal case.

Overdamped

An overdamped response is the response that does not oscillate about the steady-state value but takes longer to reach than the critically damped case. Here Damping Ratio is >1

Properties

Rise time

Rise time is defined in 1996's The Control Handbook as "the time required for the response to rise from x% to y% of its final value", with 0%-100% rise time common for overdamped second order systems and 10%-90% for underdamped.^[1]

Overshoot

Maximum Overshoot is defined in Katsuhiko Ogata's Discrete-time control systems as "the maximum peak value of the response curve measured from the desired response of the system."^[2]

Settling time

Tay, Mareels and Moore (1997) defined settling time as "the time required for the response curve to reach and stay within a range of certain percentage (usually 5% or 2%) of the final value."^[3]

Steady-state error

2003's Instrument Engineers' Handbook defines the steady-state error of a system as "the difference between the desired final output and the actual one" when the system reaches a steady state, when its behavior may be expected to continue if the system is undisturbed.^[4]

See also

• Damping ratio

References