ac - Under what circumstances can impedances be simplified to simple resistances?

Consider the following circuit:

This is a simple first-order low-pass RC filter. The cutoff frequency is:

\$ f_c = \dfrac{1}{2 \pi (1k\Omega) (1\mu F)} \approx 159 Hz \$

The reactance of C1 at \$f_c\$ is:

\$ X_{C1} = \dfrac{1}{2 \pi (159 Hz) (1uF)} = 1k\Omega \$

So what if we pretend that C1 was just a resistor, with a resistance that's a function of frequency? If you are a 159 Hz sinusoid, then the circuit might as well be this (I think...):

At least, if we only care about the frequency response, and not the phase response, of the circuit. The voltage between T1 and T2 is half of V1, and this is consistent with the calculation of the cutoff frequency above.

I think with this circuit, we can pick any frequency, calculate the reactance, replace C1 with a resistor of that value, and get an equivalent circuit for the purposes of calculating the frequency response.

But what about this circuit?

Maybe not so much. Under what circumstances can we simplify impedances this way?

Answer

Unfortunately, you cannot simply replace reactances by resistances because you are then ignoring the phase shift. In your first example, the cutoff frequency of a low-pass filter is defined as the frequency for which the output voltage is 3 dB down (0.707) from its maximum, not half. So replacing the capacitive reactance with an equal value resistance does not yield the correct voltage output. This is because the voltage and current in the real circuit with the capacitor are not in phase, but you are forcing them to be in phase by replacing capacitive reactance with a pure resistance. I really don't think frequency dependent resistors (which are not available as passive components) are more intuitive than capacitors and inductors which are easily constructed as passive components. In your second circuit, you completely lose the concept of resonant frequency because, unlike capacitive reactance and inductive reactance, resistances are always positive. Thus R3 and R4 can never cancel each other out. I think you need to stick with reactances and learn to apply them to real circuits. It will be worth the effort.