Can any AC circuit be treated as DC by using complex numbers?

Question

I am a physics undergrad, looking for ways to solve complex circuits efficiently.

I have recently started to learn about complex number applications in AC circuits and its advantage over phasor methods.

What I want to know is that suppose I am given a complex AC circuit and I am required to find its impedance and phase factors. Can I do this by imagining the AC circuit to be a DC circuit at every instant, treating capacitors and inductors as if they were resistors and following the parallel/series combination methods of resistance addition?

Note: I am mainly interested in sinusoidally varying voltage sources and resistors, capacitors and inductors.

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Please forgive the question to be too obvious as I am from a physics and not electrical engineering background.

Answer 1

Can I do this by imagining the AC circuit to be a DC circuit at every instant, treating capacitors and inductors as if they were resistors and following the parallel/series combination methods of resistance addition?

Yes, but it would be better to use more accurate language, it will be less to unlearn later.

Treat inductors and capacitors as impedances. The combination of impedances in series and parallel follows the same formulae for that of resistances in series and parallel, which are just pure real impedances.

This is how a simulator like SPICE computes the AC gain of a circuit at a particular frequency. It computes the impedance of each component at that frequency, then generates a large impedance matrix to give all the voltages as impedance * currents, and solves the resulting system of equations, all in complex numbers. This is why a zero impedance loop, or a floating node, will give you a 'singular matrix' error.