# Transient response: Difference between revisions

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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

(It starts with) One thing, I don’t know why It doesn’t even matter how hard you try Keep that in mind, I designed this rhyme To explain in due time All I know time is a valuable thing Watch it fly by as the pendulum swings Watch it count down to the end of the day The clock ticks life away It’s so unreal

### 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]

## References

1. ^ Levine, William S. (1996). The control handbook. CRC Press. p. 158. ISBN 0849385709. The rise time is the time required for the response to rise from x% to y% of its final value. For overdamped second order systems, the 0% to 100% rise time is normally used, and for underdamped systems...the 10% to 90% rise time is commonly used.
2. ^ Ogata, Katsuhiko (1987). Discrete-time control systems. Prentice-Hall. p. 344. ISBN 0132161028.
3. ^ Tay, Teng-Tiow (1997). High performance control. Birkhäuser. p. 93. ISBN 0817640045. Unknown parameter `|coauthors=` ignored (`|author=` suggested) (help)
4. ^ Lipták, Béla G. (2003). Instrument Engineers' Handbook: Process control and optimization (4th ed.). CRC Press. p. 108. ISBN 0849310814.