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A transfer function depicts the relationship between an input signal and an output signal of a control system for all feasible input values. A block diagram is a representation of the control system that uses blocks to represent the transfer function and arrows to represent the various input and output signals. For any control system, there is a reference input known as excitation or cause that operates through a transfer operation (i.e., the transfer functions) to produce an effect leading to a controlled output or response.
As a result, a transfer functions connects the input and output in a cause-and-effect relationship.
In a Laplace Transform, if the input is represented by R(s) and the output is represented by C(s), then the transfer functions will be:
In other words, the output function of the system is obtained by multiplying the input function by the transfer functions of the system.
What is a Transfer Function
The ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, with all initial conditions set to zero, is known as the transfer function of a control system.
Procedure for determining the transfer function of a control system are as follows:
- We form the equations for the system.
- Now we take Laplace transform of the system equations, assuming initial conditions as zero.
- Specify system output and input.
- Lastly we take the ratio of the Laplace transform of the output and the Laplace transform of the input which is the required transfer functions.
The output and input of a control system do not always belong to the same category. For instance, mechanical signals are produced from electrical signals in electric motors because the motors need electrical energy to rotate. Similar to this, an electric generator requires mechanical energy to produce electricity, so its input is a mechanical signal, and its output is an electrical signal.
For mathematical analysis, all signals from a system should be represented similarly. All signal types are converted to their Laplace forms in order to achieve this. By dividing the output Laplace transfer function by the input Laplace transfer functions, the transfer functions of a system is also represented in the Laplace form. Consequently, the fundamental block diagram of a control system can be described as
Where r(t) and c(t) are time domain function of input and output signal respectively.
Methods of Obtaining a Transfer Function
There are major two ways of obtaining a transfer function for the control system. The ways are:
- Block Diagram Approach: For a complex control system, it is not practical to derive a complete transfer functions. Consequently, a block diagram is used to represent each control system component’s transfer function. Techniques for reducing block diagrams are used to get the desired transfer functions.
- Signal flow graphs: A signal flow graph is a modified block diagram. Using a block diagram, you can visualize a control system. A control system is further condensed in a signal flow graph.
Poles and Zeros of Transfer Function
Generally, a function can be represented to its polynomial form. For example,
Now similarly transfer function of a control system can also be represented as
Where K is known as the gain factor of the transfer function.
Now in the above function if s = z1, or s = z2, or s = z3,….s = zn, the value of transfer functions becomes zero. These z1, z2, z3,….zn, are roots of the numerator polynomial. As for these roots the numerator polynomial, the transfer functions becomes zero, these roots are called zeros of the transfer functions.
Now, if s = p1, or s = p2, or s = p3,….s = pm, the value of transfer function becomes infinite. Thus the roots of denominator are called the poles of the function.
Now let us rewrite the transfer function in its polynomial form.
Now, let us consider s approaches to infinity as the roots are all finite number,. They can be ignored compared to the infinite s. Therefore
Hence, when s → ∞ and n > m, the function will have also value of infinity,. That means the transfer function has poles at infinite s, and the multiplicity or order of such pole is n – m.
Again, when s → ∞ and n < m, the transfer function will have value of zero that means the transfer function has zeros at infinite s, and the multiplicity or order of such zeros is m – n.
Concept of Transfer Function
The relationship between a system’s input and output is known as the transfer function, and it is typically expressed using the Laplace Transform. Consider a system where a voltage source (V) is connected in series with a resistance (R) and an inductance (L).
In this circuit, the applied voltage (V) is the cause, and the current (i) is the result. Therefore, the circuit is voltage and current can be viewed as the system’s input and output, respectively.
From the circuit, we get,
Now applying Laplace Transform, we get,
The transfer functions of the system, G(s) = I(s)/V(s), the ratio of output to input.
1) Let us explain the concept of poles and zeros of transfer functions through an example.
The zeros of the function are, -1, -2 and the poles of the functions are -3, -4, -5, -2 + 4j, -2 – 4j.
Here n = 2 and m = 5, as n < m and m – n = 3, the function will have 3 zeros at s → ∞. The poles and zeros are plotted in the figure below
2) Let us take another example of transfer functions of control system
In the above transfer functions, if the value of numerator is zero, then
These are the location of zeros of the function.
Similarly, in the above transfer functions, if the value of denominator is zero, then
These are the location of poles of the function.
As the number of zeros should be equal to number of poles,. The remaining three zeros are located at s →∞.