How do I refer the transfer function of a system from its Laplace equivalent?

Question

I have a problem, I know how to analyze circuits in Laplace domain to get the transfer Laplace function. However, I don't know how to translate the answer to "real life"

Suppose that I have a transfer system with Vout=1/sC, where C is a capacitance (say of 10uF). And suppose now that I give the system Vin = 2*sin(pai/2 + 2pai). How do I determain the output signal from the laplace form of my transfer function?

Thanks!

Answer 1

1. If both the transfer function and the input are in Laplace domain, then...
   a) Multiply them to get the output in the laplace domain.
   b) Take the inverse laplace transform of the resutl to convert backt time domain.

2. If the transfer function is in the Laplace domain, but the input is in the time domain...
   a) You can take the Laplace transform of the input and then proceed as in case #1 above. b) Or you can take the inverse Laplace transform of the transfer function to get the impulse response. You can then take the convolution of the impulse response and the input to find the output in the time domain.

Answer 2

The Laplace domain transfer function, if properly derived, is Vout/Vin where Vout and Vin are the Laplace transforms of output and input voltages. You multiply the Laplace transform of the input voltage with the transfer function and find the inverse transform of the result. That's your Vout in time domain.

Pure sinusoidal inputs can be handled in frequency domain, you simply calculate the gain or attenuation and the phase shift. Your transfer function is a multiplying phasor when you calculate it as complex number in the operating frequency.

If you calculate the inverse Laplace transform of the Laplace domain transfer function, you get what the system outputs in case the input is the unit impulse function (=Dirac's delta)

Unfortunately often the inverse transform is too complex. Then you must calculate the output numerically. Circuit analyzing programs do it automatically.