Consider a linear, time-invariant, dynamic system, we can define the transfer function as the ratio of the Laplace transform of the output to the Laplace transform of the input under the assumption that all the initial conditions are zero.
transfer function = LaplaceTransformof output / Laplace Transform of input
When all the initial conditions are zero.
We can write time-invariant differential equations as
We see that system parameters which appear as coefficients and the output, c(t) and e input r(t) also appear throughout the equation.
It is better if we have a mathematical representation, where the input, output and ,stem are distinct and separate parts, as shown in equation. Transfer function method satisfies this relationship. You cannot obtain this convenience using differential equation method. This function also allows you to algebraically combine mathematical representation of subsystems to yield a total system presentation.
It is better if we have a mathematical representation, where the input, output and ,stem are distinct and separate parts, as shown in equation. Transfer function method satisfies this relationship. You cannot obtain this convenience using differential equation method. This function also allows you to algebraically combine mathematical representation of subsystems to yield a total system presentation.
Steps to obtain a transfer function
- Understand the physical system and its components.
- Make appropriate simplifying assumptions.
- Use basic principles or laws to formulate the mathematical model.
- Write differential and algebraic equations which describe the model.
- Apply Laplace transformation to the model.
- Obtain the Input-output relationships (i.e. transfer functions) for systems and components.
Up to now, we have discussed about single input single output systems. Now, let’s consider the case of multivariable systems. A multivariable system is a system which has multiple inputs and multiple outputs. In a multivariable system, we get a differential equation of the form of equation to describe the relationship between input and output, when all other inputs are set to zero. Since the principle of superposition can be applied to linear systems, we can obtain the total effect on any output due to all the inputs acting simultaneously. (by adding up the outputs due to each input acting alone.)
Properties of the Transfer Function
Finally, the properties of the transfer function can be summarised as follows.
4. The transfer function is independent of the input of the system.
Properties of the Transfer Function
Finally, the properties of the transfer function can be summarised as follows.
- Transfer function is defined only for a linear time-invariant system. It is not defmed for non-linear systems.
- The transfer function can be defined in 2 ways.
- Transfer function = LaplaceTransform(output variable) / LaplaceTransform(input variable)
- Transfer function = Laplace Transform of impulse response
4. The transfer function is independent of the input of the system.
5. The transfer function of a continuous data system is expressed only as a function of
the complex variables. It is not a function of the real variable, time or any other variable
that is used as the independent variable. In discrete data system (digital) models
represented by difference equations, the transfer function is a function of z, when the
z-transform is used.
the complex variables. It is not a function of the real variable, time or any other variable
that is used as the independent variable. In discrete data system (digital) models
represented by difference equations, the transfer function is a function of z, when the
z-transform is used.