Response of First Order Systems to Sinusoidal Input

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Response of First Order Systems to Sinusoidal
Input
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Consider the following first order system
Determine the response of the system if input is
a sinusoidal
Which may be transformed to
the system response, O(s), is then
Solving then for a,b, and c
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To solve for a, multiply by s 1/t , and let s
-1/t
To solve for b, multiply by s iw, and let s
-iw
c will be the complex conjugate of b and is
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Using the solution in the SC for a sinusoidal
function, the solution becomes
,r 0
,
,
Where
,
The complete solution is then
It is important to note that the solution is
made-up of a transient (the first term) and a
non-transient part (the second term). Also,
frequency out frequency in, magnitude out is a
fraction of magnitude out, and the output is
delayed (phase lag).
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Graphical Output