HOW TO FIND THE TRANSFER FUNCTION

	Time Domain	Frequency Domain
Linearity	f(t) + g(t)
Function	x(t)
1st Derivative	x'(t)
2nd Derivative	x"(t)
nth Derivative	xn(t)

Note: While linearity allows Laplace Transforms to be added, the same does not hold true for multiplication. f(t)g(t) does not equal F(s)G(s). The solution to multiplication requires convolution, please refer to a differential equations book.

In order to convert the time dependent governing equation to the frequency domain, perform the Laplace Transform to the input and output functions and their derivatives. These transformed functions must then be substituted back into the governing equation assuming zero initial conditions. Because the transfer function is defined as the output Laplace function over the input Laplace function, rearrange the equation to fit this form.

Find the transfer function of the second order tutorial example problem:

From the free body diagram we were able to extract the following governing equation:

f(t) - kx - bx' - mx" = 0

The notation of the Laplace Transform operation is L{ }.

When finding the transfer function, ‘zero’ initial conditions must be assumed, so x(0) = x'(0) = 0.

Taking the Laplace Transform of the governing equation results in:

F(s) - k[X(s)] - b[sX(s)] - m[s²X(s)] = 0

Collecting all the terms involving X(s) and factoring leads to:

[ms² + bs + k] X(s) = F(s)

The transfer function is defined as the output Laplace Transform over the input Laplace Transform, and so the transfer function of this second order system is:

X(s)/F(s) = 1/[ms² + bs + k]