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please help me understand the significance of substitute s=jw in a transfer function

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liudehua

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could someone help me prove the following statement in my textbook?

"replacing s by jw provides the transfer function for physical frequency w, that is, the transmission magnitude and phase for a sinusoidal input signal of freqency w"

i could some how understand it ambiguously, but cannot prove it by myself.
 

liudehua said:
"replacing s by jw provides the transfer function for physical frequency w, that is, the transmission magnitude and phase for a sinusoidal input signal of freqency w"
i could some how understand it ambiguously, but cannot prove it by myself.
Click to expand...

The complex frequency variable s=sigma+jw is used primarily to compute and to describe some specific properties of the network (pole and zero location in the complex frequency plane).
More than that, the denominator of the transfer function - if expressed using s - is identical to the characteristic polynomial of the corresponding differential equation in the time domain.
This clearly demonstrates the relation between time and frequency domain - and, thus, justifies the usage of s.
However, when you try to measure or to calculate the real frequency response (magnitude and/or phase) you must use the "physical" frequency w=2*Pi*f because the complex variable s is only for mathematical purposes and cannot be generated as a test signal.
 
LvW said:
The complex frequency variable s=sigma+jw is used primarily to compute and to describe some specific properties of the network (pole and zero location in the complex frequency plane).
More than that, the denominator of the transfer function - if expressed using s - is identical to the characteristic polynomial of the corresponding differential equation in the time domain.
This clearly demonstrates the relation between time and frequency domain - and, thus, justifies the usage of s.
However, when you try to measure or to calculate the real frequency response (magnitude and/or phase) you must use the "physical" frequency w=2*Pi*f because the complex variable s is only for mathematical purposes and cannot be generated as a test signal.
Click to expand...
thanks, but i am still blurred about it. my textbook(sedra/smith microelectronic circuits) doesnt mention s=sigma+jw, but jw only?
 

liudehua said:
thanks, but i am still blurred about it. my textbook(sedra/smith microelectronic circuits) doesnt mention s=sigma+jw, but jw only?
Click to expand...

No surprise - when a book primarily deals with circuit design there is no urgent need to use the complex frequency variable (but I am not sure if your claim about Sedra/Smith is true).
However, all books covering system aspects (signals and systems, frequency transformations, Laplace, stability,...) make use of s=sigma+jw because this is the variable used in the Laplace transformation.
 
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