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I am currently doing a project which involves taking a circuit of resistors, inductors, and capacitors (combined in any way imaginable) and finding the total impedance of the circuit.

For the latter 2 components the impedance depends on the input voltage's frequency. I was just wondering if the input voltage frequency can vary throughout the circuit.

An example to illustrate what I mean, say I have an AC source with a frequency of 2 Hz, then will the frequency in the impedance expression ALWAYS be 2 Hz or can branches change its value?

I assume it is a constant but just need confirmation.

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2 Answers 2

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In a Linear, Time-Invariant (LTI) system, the frequency will not change. An excitation at 2 Hz will lead to effects at 2 Hz throughout the system. Circuits consisting only of linear, time-invariant elements (DC sources, sine sources, linear controlled sources, resistors, capacitors, and inductors) are themselves LTI systems. Your circuit consists only of these components, so it will indeed only have 2 Hz voltages and currents in its branches.1

These systems also have properties of linearity, and hence superposition. In a more complicated example, you could have an LTI circuit excitations at DC, 2 Hz, and 3 Hz, and they would lead to effects at DC, 2 Hz, and 3 Hz throughout the circuit. Each of these will occur additively, meaning that you can calculate the effect of each excitation independently and add the effects together to estimate the final response.

In contrast, there are non-linear systems where other frequencies are created. For instance, a transistor amplifier will amplify 2 Hz to 2 Hz, but it will add weak signals at 4 Hz, 6 Hz, etc., to the output as a result of the amplifier's nonlinearity. An example of this being done intentionally is mixer circuits, that intentionally convert between high frequencies (radio signals) and low frequencies (baseband) as part of communication systems.

1 In practice, some components (esp. ceramic capacitors and some inductors) can be slightly non-linear, but this effect will be very slight and might not be strong enough to create detectable signals at other frequencies. However, other inductors are intentionally made nonlinear (e.g. by saturating easily), and will produce fairly detectable harmonics. A historical example is the magnetic amplifier.

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    \$\begingroup\$ Inductors can also get quite nonlinear, to the point that magnetic amplifiers are a thing that exists (or at least existed. They're extremely obsolete nowadays.) \$\endgroup\$
    – Hearth
    May 13 at 21:19
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    \$\begingroup\$ @Hearth Completely forgot about saturation effects with inductors - will update to add a brief remark. Thank you for pointing me to magnetic amplifiers; they're a cool bit of history I didn't know about before! \$\endgroup\$
    – nanofarad
    May 13 at 22:06
  • \$\begingroup\$ Heat dissipation affecting temperature affecting component values can also cause nonlinear behavior, although this effect tends to be fairly small at 'sane' frequencies. \$\endgroup\$
    – TLW
    May 15 at 5:18
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Ideal resistors, inductors, and capacitors are linear components. So via the superposition property of linearity, and therefore superposition of frequencies (like Fourier), no new frequencies will be generated if they are the only things that exist in the circuit.

So if a 2Hz AC source is the only frequency you put in an RLC circuit, that's the only frequency that can exist. If you have a 50Hz and 60Hz source in the RLC circuit then those two frequencies are the only frequencies that can exist elsewhere in the circuit. If you put a square wave into the circuit then the only frequencies in the circuit are the frequencies that originally came with the square wave; No new frequencies are produced.

You can sort of intuitively tell (if you know what you are looking for) by looking at the equations that describe the V-I behaviour of capacitors (\$i_c=C\frac{dv}{dt}\$) and inductors (\$v_L = L\int idt\$). Because you change the slope by some amount by adding or subtracting from it, the result just changes in proportion by that amount.

An example to illustrate what I mean, say I have an AC source with a frequency of 2 Hz, then will the frequency in the impedance expression ALWAYS be 2 Hz or can branches change its value?

That's the basis of using complex numbers to analyze an RLC circuit. For each frequency present in the circuit, you have a version of the circuit where the impedance expression of all components in the circuit have that frequency plugged in. Then you superimpose all the circuits to get the final result via Fourier.

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