Response of First Order Systems to Sinusoidal Input. 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:.
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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:
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:
Using the solution in the S&C for a sinusoidal function, the solution becomes:
,r = 0
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
for a unit sine input
We can simply add terms on the Bode Magnitude plot
and on the Bode Phase plot to get total response