Transfer Functions and Block Diagrams of Control Systems

Transfer Functions

A transfer function is a mathematical formulation that relates the output variable of a device to the input variable. For linear devices, the transfer function is independent of the input quantity and solely dependent on the parameters of the device together with any operations of time, such as differentiation and integration that it may possess.

The transfer function can be obtained through the following steps:

• Determine the governing equation for the device expressed in terms of the output and input variables.
• Determine the Laplace transforming the governing equation, assuming all the initial conditions to be zero.
• Rearrange the equation to yield the ratio of the output to input variable.

The properties of transfer functions can be summarized as follows:

• The transfer function is defined only for a linear time-invariant system. It is meaningless for nonlinear systems.
• The transfer function between an input variable and an output variable of a system is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input or as the Laplace transform of the impulse response. The impulse response of the linear system is defined as the output response of the system when the input is a unit impulse function.
• All the initial conditions of the system are assumed to be zero.
• The transfer function is independent of the input.

Control Systems Block Diagrams

The block diagram is pictorial representation of the equations of the system. Each block represents a mathematical operation, and the blocks are interconnected to satisfy the governing equations of the system. Therefore, a block diagram provides a chart of the procedure to be followed in combining the simultaneous equations, from which useful information can be obtained without finding a complete analytical solution.

Let’s consider the diagram below:

The above figure shows a basic building block of a complex block diagram. The arrows on the diagram imply the block has a unilateral property i.e. a signal can only pass in the direction of the arrows. A box is the symbol for multiplication; the input quantity is multiplied by the function in the box to obtain the output.

The circles indicate summing points (in an algebraic sense). Any linear mathematical expression can be expressed by block diagram notation.

R(s) – reference variable (input signal)

C(s) – output signal (controlled variable)

B(s) – feedback signal = H(s)C(s)

E(s) – actuating signal (error variable) = R(s) – B(s)

G(s) – forward path transfer function or open loop transfer function = C(s)/H(s)

M(s) – closed loop transfer function = C(s)/R(s) = G(s)/1+G(s)H(s)

H(s) – feedback path transfer function

G(s)H(s) – loop gain

The expression for the output quantity with negative feedback is given by:

The transfer function M of the closed loop system is given by: