Response of first order systems to sinusoidal input
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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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First order system 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:

Solving then for a,b, and c:

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First order system sinusoidal input1
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 input2
First Order System - Sinusoidal Input

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

Where:

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First order system sinusoidal input3
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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Adding terms of frequency response
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

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