Contents

- Single Differential Equation to Transfer Function
- Transfer Function to Single Differential Equation
- Example

If a system is represented by a single n^{th }order differential
equation, it is easy to represent it in transfer function form. Starting
with a third order differential equation with x(t) as input and y(t) as output.

To find the transfer function, first take the Laplace Transform of the differential equation (with zero initial conditions). Recall that differentiation in the time domain is equivalent to multiplication by "s" in the Laplace domain.

The transfer function is then the ratio of output to input and is often called H(s).

Note: This notation takes increasing subscripts for the a_{n}
and b_{n} coefficients as the power of s (or order of derivative
decreases) while some references use decreasing subscripts with decreasing
power. This notation was chosen here in part because it is consistent with
MatLab's use.

For the general case of an n^{th} order differential equation with m
derivatives of the input (superscripted numbers in parentheses indicate the
order of the derivative):

This can be written in even more compact notation:

Going from a transfer function to a single nth order differential equation is equally straightforward; the procedure is simply reversed. Starting with a third order transfer function with x(t) as input and y(t) as output.

To find the transfer function, first write an equation for X(s) and Y(s), and then take the inverse Laplace Transform. Recall that multiplication by "s" in the Laplace domain is equivalent to differentiation in the time domain.

For the general case of an n^{th} order transfer function:

This can be written in even more compact notation:

Consider the system shown with f_{a}(t) as input and x(t) as
output.

The system is represented by the differential equation:

Find the transfer function relating x(t) to f_{a}(t).

**Solution:** Take the Laplace Transform of both equations
with zero initial conditions (so derivatives in time are replaced by
multiplications by "s" in the Laplace domain).

Now solve for the ration of X(s) to F_{a}(s) (i.e, the ration of
output to input). This is the transfer function.

Find the differential equation that represents the system with transfer function:

**Solution:** Separate the
equation so that the output terms, X(s), are on the left and the input
terms, Fa(s), are on the right. Make sure there are only positive
powers of s.

Now take the inverse Laplace Transform (so multiplications by "s" in the Laplace domain are replaced by derivatives in time).

© Copyright 2005 to 2019 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College