A first-order system is a system whose behavior is described by a first-order differential equation. Such systems contain only one energy storage element, such as a capacitor in an RC circuit or an inductor in an RL circuit.
The general transfer function of a first-order system is
\[
H(s) = \frac{K}{\tau s + 1}
\]
where
\(K\) = system gain
\(\tau\) = time constant
\(s\) = Laplace variable
The time constant determines how fast the system responds to an input.
Table of Contents
Step Response of a First-Order System
For a unit step input, the output response of a first-order system is
\[
y(t) = K\left(1 – e^{-t/\tau}\right)
\]
Important characteristics of this response include:
• The output starts at zero
• It increases exponentially
• It approaches the final value asymptotically
At time \(t = \tau\),
\[
y(\tau) = 0.63K
\]
Thus, the output reaches 63% of its final value after one time constant.
Time Constant
The time constant (\(\tau\)) is a key parameter of first-order systems.
Correct interpretation:
• For a decaying exponential, the value reduces to 37% of its initial value
• For a rising step response, the output reaches 63% of its final value
Thus,
\[
e^{-1} \approx 0.37
\]
Rise Time
Rise time is the time taken by the response to rise from 10% to 90% of its final value.
For a first-order system,
\[
t_r \approx 2.2\tau
\]
Settling Time
Settling time is the time required for the response to remain within
a specified percentage of the final value.
Common definitions used in control and signal systems are:
2% criterion:
\[
t_s \approx 4\tau
\]
5% criterion:
\[
t_s \approx 3\tau
\]
These are the standard definitions used in most textbooks and GATE problems.