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Why is it that the real part of \$G(j\omega)\$ (where \$G(s)\$ is transfer function) is related to active power of the circuit?

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In the real world, every complex pole has a conjugate. Real part of these complex conjugate pairs contribute attenuation or amplification, while their imaginary parts cause oscillation. Oscillation means equal rise and fall in magnitude in a period; which corresponds to energy fluctuation on capacitors and inductors in linear circuits.

Therefore, in the real world, imaginary part of frequency response always gives the reactive power, while the real part of it giving the real power.

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