Response of First Order Systems to Sinusoidal Input

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Response of First Order Systems to Sinusoidal
Input
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First Order System - Sinusoidal Input
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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First Order System - Sinusoidal Input
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
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First Order System - Sinusoidal Input
c will be the complex conjugate of b and is
Using the solution in the SC for a sinusoidal
function, the solution becomes
,r 0
Where
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First Order System - Sinusoidal Input
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).
Consider simply substuting iw for s in the
original transfer function and solving for
Note, this gives us the non-transient solution
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Plotting Frequency response
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Plotting Frequency response
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Plotting Frequency response
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Adding terms of Frequency Response
We can simply add terms on the Bode Magnitude
plot and on the Bode Phase plot to get total
response