Transfer Functions

What are Transfer Functions?

Transfer functions are the Laplace domain representation of signal modification in a system where the input = u(t) or U(s), output = y(t) or Y(s), and the transfer function = h(t) or H(s) where "s" indicates a function in the Laplace domain. The transfer function H(s) is defined as the ratio between the output and the input:

Essentially this means that the output is equal to the input multiplied by the transfer function:

For example, if you have a simple block diagram like this:

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Example Problem: Find the Laplace equation of the output from this system. #undefined

Solution +

The output is \( \frac{4}{(s+3)(s+4)} \)

Pretty easy since it's just a multiplication of two fractions.

Another consideration of transfer functions is that they can be combined and simplified, sort of like the components of an electrical circuit. Transfer functions in series are multipled together while those that are in "parallel" are added or subtracted.

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Example Problem: Find the single block representation of this system. #undefined

Solution +

The simplified transfer function is \( H(s) = G_{1}G_{2}(G_{4}G_{5}-G_{3}) \)

Now try an example problem that combines these aspects and use the table of common Laplace transforms (here) to convert your answer to the time domain.