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Question

To put it briefly – how do you know when to use complex numbers in RLC circuit calculations, and when it's enough to just use the magnitude of the impedances?

Some context. While trying to understand how the Wien Bridge Oscillator works, I ran into what seems to be an important conceptual gap. Articles on this topic often start by discussing the RC Phase Shift Network, which essentially consists of a combination of a high-pass and a low-pass filter. Then they state that the output gain of this network is equal to 1/3 of Vin. I decided to check this and derive the general formula for Vout/Vin (Example article).

In the first case, I used the following formula for Zc in my calculations:

Zc = 1/(jωC)

And I had no problem getting the expected result – 1/3.

But if I use Zc = 1/(ωC), the result obviously changes. In this case, I ended up with Vout/Vin = 1/5.

I've often seen that when analyzing different types of circuits containing reactive components, their impedance is sometimes written in "complex" form, and other times as just the magnitude of the complex expression. Intuitively, I feel like the correct approach is to always use the complex form, but I keep wondering – when is it okay to use the "simpler" form?

Answer 1

If all the components in the circuit have the same phase relationship between voltage and current, that is they are all resistors, all capacitors or all inductors, you can use the simplified form and do calculations with magnitude only.

If the circuit consists of a mix of phases, so R and C, or R and L, and especially C, L, and R, then you need to use the complex form, to capture these phase differences.

We don't often run into all C or all L circuits. What are drawn as all R circuits will have stray capacitances associated with them. Ideally then, we should always be using complex notation. However, if at the frequencies we are interested in, there is a very large ratio between wanted and stray impedances, the phase shifts will be small, and we can often approximate with magnitude only calculations. If we have an audio amplifier with kΩ resistors and pF stray capacitances, then an all R calculation is often good enough.₁

An interesting case is the x10 'scope probe'. Typically an oscilloscope input is a (standard) 1 MΩ shunted by a (non-standard) 10 to 30 pF or so. At DC, the 10:1 attenuation is provided by a 9 MΩ resistor in the probe. At high frequency, the 10:1 attenuation is provided by a small capacitor in the probe, working into the capacitive load of the scope input. 'Tuning' the probe comprises adjusting this small capacitor so that the squarewave calibration waveform (containing both low and high frequencies) used for the purpose looks correct. At this point the angle of the impedance of both the probe and the input is the same, and we could calculate the attenuation from the magnitudes of the impedances.

₁ If we have a really fast amplifier, even with kΩ level resistors, the approximation is no longer good, and the few pF stray of the inverting input to ground can catch us out. That's why you will often see a capacitor of a few pFs across an opamp feedback resistor. An OP275 caught me out like that!

Answer 2

But if I use Zc = 1/(ωC), the result obviously changes. In this case, I ended up with Vout/Vin = 1/5.

The result doesn't change; your math probably made erroneous simplifications such as not realizing that a resistor and a capacitor in series don't produce an impedance of \$A + B\$ but, in fact produces an impedance of \$\sqrt{A^2+B^2}\$.

It's no hardship to incorporate the complex operator especially if it gives you the correct answer first time. I always use the complex operator just to avoid this problem. You can always derive magnitudes and phase responses later (after doing the algebra).