- Transfer Function Definition: A transfer function is defined as the ratio of the Laplace transform of a system’s output to the input, assuming initial conditions are zero.
- Utilization of Block Diagrams: Block diagrams simplify complex control systems into manageable components, making it easier to analyze and derive transfer functions.
- Understanding Poles and Zeros: Poles and zeros critically influence a system’s behavior by indicating points where the transfer function respectively becomes infinite or zero.
- Laplace Transform in Control Systems: Laplace transform is essential for representing all types of signals in a uniform format, aiding in the mathematical analysis of control systems.
- Impulse Response Insight: The output from an impulse input reveals the transfer function, illustrating the direct relationship between a system’s input and output.
A transfer function is the Laplace ratio of a linear control system output to its input, taken with all initial conditions at zero. A block diagram draws that ratio as a block. Arrows carry the input and output signals.
The reference input (the excitation or cause) passes through that transfer function and becomes the controlled output (the response).
Cause and effect are tied together by that transfer function.
In a Laplace Transform, write the input as R(s) and the output as C(s). Then
C(s) equals the transfer function times R(s), at zero initial conditions.
What is a Transfer Function
Transfer Function Explained: the ratio of the Laplace transform of the output to the Laplace transform of the input, with zero initial conditions.
Steps to find the transfer function of a control system:
- Write the system equations.
- Take the Laplace transform of those equations, with initial conditions set to zero.
- Name the output and the input.
- Divide the Laplace transform of the output by the Laplace transform of the input. That ratio is the transfer function.
The physical input and output need not be the same kind of quantity. electric motors take an electrical command and produce rotation. Generators take mechanical drive and produce electrical output.
For algebra, put every signal into Laplace form. The transfer function is then output C(s) divided by input R(s). A basic block looks like this:
r(t) and c(t) are the time-domain input and output.
Methods of Obtaining a Transfer Function
Two common ways to get a transfer function:
- Block Diagram Method: A full algebra dump of a large plant is clumsy. Draw each element as its own transfer-function block, then reduce the diagram to one block.
- Signal Flow Graphs: A compact cousin of the block diagram is a signal flow graph. Blocks show the layout. A signal-flow graph packs the same paths into nodes and gains.
Poles and Zeros of Transfer Function
A rational function can be written as a ratio of polynomials. For example,
A control-system transfer function takes the same shape:
K is the gain factor of the transfer function.
If s = z1, or s = z2, or s = z3,….s = zn, the transfer function is zero. Those z1, z2, z3,….zn, are the numerator roots. Numerator roots are the zeros of the transfer function.
If s = p1, or s = p2, or s = p3,….s = pm, the transfer function goes to infinity. Denominator roots are the poles.
Write the transfer function as a polynomial ratio:
As s goes to infinity the finite roots drop out next to s itself. Therefore
If s → ∞ and n > m, G(s) → ∞. Those are poles at infinity, order n – m.
If s → ∞ and n < m, G(s) → 0. Those are zeros at infinity, order m – n.
Concept of Transfer Function
In Laplace form the transfer function is still output over input. Take a series resistance (R) and inductance (L) across a voltage source (V).
Current i is the response to applied voltage (V). Treat that voltage as input and the current as output.
From the circuit, we get,
Laplace transform of the loop equation:
G(s) = I(s)/V(s), output over input.
1) Poles and zeros, worked example.
Solution
Zeros: -1, -2. Poles: -3, -4, -5, -2 + 4j, -2 – 4j.
Here n = 2 and m = 5, so n < m and m – n = 3. The function has 3 zeros at s → ∞. The plot is below 
2) Second transfer-function example
Solution
Set the numerator to zero:
Those are the finite zeros.
Set the denominator to zero:
Those are the finite poles.
Counting zeros at infinity so that zeros match poles, the other three zeros sit at s →∞.
Example of Transfer Function of a Network
3)
Solution
From the network,
Assume
Laplace transform at zero initial conditions:
The Effect of Impulse Signal
The unit impulse is defined as
Its Laplace transform is 1.
If the input is that unit impulse, then
The output Laplace transform equals the transfer function.
Transfer Function Example
1) The impulse response of a system is
What is the transfer function?
Solution
For a unit impulse, C(s) equals the transfer function.
The transfer function is therefore
2) Find the poles and zeros of
Solution
3) Poles and zeros on the s-plane below
Find the transfer function.
Solution
Poles at s = – 3, – 1, 0.
Denominator:
Zeros at -4, – 2.
Numerator:
Transfer function:
K is the gain factor of the control system.
4) Find K if G(2) = 2 for
Solution
From that condition,
5) Find the transfer function of the following network.
Solution
From the circuit,
Apply Laplace transformation to both sides:
Applied voltage is the cause and charging current is the effect, so voltage is input and charging current is output.
6) Find the transfer function of the network below.
Solution
Input is v(t). Output is the voltage on capacitor C, written vc(t).
Laplace transformation of those circuit equations is
7) A shaft of inertia J turns through angle θ under torque T against bearing friction f. Find the transfer function.
Solution
The differential equation is
Laplace transform both sides:
Torque is input. Angular displacement is output.
8) Impulse response c(t) = 1 – te-2t + sin3t. Find the transfer function.
Solution
9) Find the transfer function for this impulse response:
Solution
