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Response to Everlasting Exponentials
Suppose we have a linear time-invariant system which has forcing function
is a frequency
and a transfer function is
Normally, we would have to calculate the Laplace transform
of the forcing function, in this case,
then calculate the inverse Laplace transform of
; however, note the following:
Assume we have the impulse response
. In this case, the response to an exponential
. But this integral is nothing more than the Laplace transform of the function and
by definition, the Laplace transform of the impulse function is the transfer function.
Therefore, we have the very important result
The response of a sum of exponential functions
is the sum of the responses to the exponential functions and
the response of each exponential function is the exponential multiplied by
the transfer function evaluated at the frequency: .
What is the response of the circuit with transfer function
when the input/forcing function is
In the first case, the response is .
In the second case, the transfer function is undefined: . This is the
resonant frequency of the system and such a system would blow up.
In the third case, we may consider
and therefore the response is
In the fourth case, we note that this function is the real part of the complex exponential
. The response of the complex exponential is
. The real part of the
response is .