A transfer function is a mathematical representation of the relationship between the output and input of a linear time-invariant (LTI) control system in the Laplace-transform domain, assuming zero initial conditions. It is defined as G(s) = Y(s) / X(s), where is the Laplace transform of the output and is the Laplace transform of the input. The transfer function describes how a system responds to an input and provides important information about its gain, poles, zeros, stability, transient response, and frequency response. For example, if , the system has a gain of 10 and a pole at . Transfer functions are widely used to analyze and design electrical, mechanical, thermal, and other control systems.
Interactive Transfer Function Simulation
A transfer function represents the relationship between the output
and input of a linear time-invariant (LTI) system in the Laplace domain,
assuming zero initial conditions.
G(s) = Y(s) / X(s)
In this simulation we use:
G(s) = K / (s + a)
The gain K controls the magnitude of the response, while
−a is the system pole. The pole location gives important
information about the system’s transient behavior and stability.
Time-Domain Response
Input X(t)
Output Y(t)
Transfer Function
G(s) = 10 / (s + 2)
Pole
s = −2
DC Gain
5
Time Constant
0.5 s
The pole is in the left half of the s-plane, so this first-order
system is stable. For a unit-step input, the output approaches
K/a as time increases.
Pole Location on the Real Axis
Interpretation:
• Pole on the left side → stable response.
• Pole closer to zero → slower response.
• Pole farther to the left → faster response.
• For this simulation, the pole is at s = −a.
• Pole on the left side → stable response.
• Pole closer to zero → slower response.
• Pole farther to the left → faster response.
• For this simulation, the pole is at s = −a.