# If capacitors are frequency-dependent resistors, then what are inductors?

I was reading The Art of Electronics 3rd Edition, there it was saying "You can think of capacitors as frequency-dependent resistors."

So in terms of this description, what could inductors be?

As @jonathanjo mentioned:

The fuller quotation, with emphasis added: "To a first approximation, capacitors are devices that might be considered simply frequency-dependent resistors. They allow you to make frequency-dependent voltage dividers, for instance. For some applications (bypass, coupling) this is almost all you need to know, but for other applications (filtering, energy storage, resonant circuits) a deeper understanding is needed." (Horowitz and Hill, Art of Electronics, 3rd ed., s 1.4.1.)

• Think of an x-y chart. Resistors on the x-axis, which can be both positive and negative except that resistors are never actually negative so only one half of the x-axis is usable. Here, capacitors and inductors are on the y-axis. One of them is the positive side and the other of them is the negative side. So the y-axis is fully used. Commented Aug 13, 2023 at 21:43
• seeing as it's a reciprocal result: frequency dependant conductors. Commented Aug 13, 2023 at 23:41
• The fuller quotation, with emphasis added: "To a first approximation, capacitors are devices that might be considered simply frequency-dependent resistors. They allow you to make frequency-dependent voltage dividers, for instance. For some applications (bypass, coupling) this is almost all you need to know, but for other applications (filtering, energy storage, resonant circuits) a deeper understanding is needed." (Horowitz and Hill, Art of Electronics, 3rd ed., s 1.4.1.) Commented Aug 14, 2023 at 0:46
• There are aspects of the capacitor behaviour for which this might be a useful model sometimes. A more fundamental difference is capacitors and inductors store energy, resistors dissipate it. Commented Aug 14, 2023 at 6:23
• I like this question, because (at the time of writing) there are three answer saying: Yes, no, and yes - and I agree with and have upvoted all of them. :) Commented Aug 14, 2023 at 8:59

If you consider capacitors as frequency dependent resistors, then by same logic, inductors are frequency dependent resistors too.

The difference is that they work opposite based on frequency. At DC, capacitor is open or high impedance and inductor is short or low impedance.

• The analogies may be more for analog filters or frequency dependent voltage dividers or trimmers. The inductor version is often termed a "peaking coil" motocraft.com/en/Article_detail/79/What-is-a-peaking-coil ~ ceccoils.com/CECWEB/Catalog/pdf/PEAKING.pdf Commented Aug 13, 2023 at 22:00
• Since the opposite of "resistance" is "conductivity", I guess you could say it's a "frequency dependant conductor" Commented Aug 14, 2023 at 15:33
• @Alexander You could say that, but this is about what a book said. Commented Aug 14, 2023 at 16:02

To the extent that capacitors are frequency-dependent resistors, so are inductors -- just with the opposite frequency dependence.

But be careful with that metaphor -- it's sort of like saying that you can think of a motorcycle as being a car with just two wheels, or a pair of pliers is an infinitely adjustable wrench, or a soldier is a policeman with more weaponry*. They're all true, but each one can get you into a whole lot of trouble if you depend on it too much.

* From the perspective of someone who lives in a well-run democratic country. Your mileage may vary, depending on where you live.

# What capacitors and inductors are

Figuratively speaking, both capacitors and inductors are "opposing rechargeable electric sources". They use the fact that the current in a circuit can be reduced from maximum to zero not only by resistance but also by reverse-polarity voltage.

simulate this circuit – Schematic created using CircuitLab

# How to understand them

Although their operation is considered in both the time and frequency domains, it can best be intuitively understood in the time domain.

# Operation

## Input voltage step

These "batteries" are charged by the input voltage source and oppose it by subtracting their voltage in series manner (KVL) from it in two different ways through time:

• Capacitors initially do not oppose (Vc = 0), but over time their opposition (voltage) gradually increases and becomes maximum (Vc = 0) after a sufficiently long time. So at the beginning they behave as a "virtual short" (an element with zero voltage drop, a "piece of wire") and at the end as a "virtual open circuit" (a source with an equivalent but opposite voltage, zero current).

simulate this circuit

• In contrast, inductors initially resist maximally (VL = Vin) but over time their opposition (voltage) gradually decreases to zero (VL = 0). So, at the beginning they behave as a "virtual open circuit" (a source with an equivalent but opposite voltage, zero current) and at the end as a "true short" (a piece of wire).

simulate this circuit

## Continuously changing input voltage

If we start to continuously vary (for example, increase) the input voltage, these devices will continuously increase their opposition, each in their own way. An analogy can be if we are swimming in the ocean and a swimmer is chasing us.

simulate this circuit

simulate this circuit

## AC input voltage

Theoretically, we can increase the input voltage to infinity (only 1 MV in the simulations above:) but in reality we are limited by the finite value of the supply voltage (in the swimming analogy above, this means swimming in a pool). Then how can we vary the voltage infinitely (swim endlessly)? Obviously by reversing the direction of the "voltage motion" when approaching the supply rails (swimming by circulating between the opposite sides of the pool)... that is, using an AC voltage. And then these devices (the swimmer) will continue to follow the input voltage (us) in an effort to oppose.

Now we can transfer the ideas we have acquired in the time domain to explain the operation of these elements in the frequency domain. When the input voltage begins to oscillate, each of these elements begins to oppose it in its own way - capacitors through a "delayed opposition" and the inductors through an "instantaneous opposition":

• The faster the input voltage "moves" (the frequency increases), the worse the "lazy" capacitors manage to counteract it, and they behave as constant voltage sources that do not disturb the current.

simulate this circuit

• In contrast, nimble inductors do it better in the second case (high frequency) and worse in the first case (low frequency). So, at high frequency they behave as constant current sources.

simulate this circuit

# Summary

Both capacitors and inductors are frequency-dependent "resistors" in the broadest sense of this word (something that impedes). They are not resistors in the strict sense of the word (something that dissipates energy in the form of heat).

These elements reduce the current not by resistance but by a counter voltage that they "steal" from the input source and then use against it. And since the magnitude of the "stolen" voltage depends on the frequency, they are "frequency dependent".

So, they are not resistors; they are "resistors".

• If a capacitor is a "rechargeable voltage source" (which is a reasonable description), then an inductor is not. It is a "rechargeable current source". Commented Aug 14, 2023 at 15:43
• @Ben Voigt, It is... but it maintains a constant current in the circuit by varying the voltage. Commented Aug 14, 2023 at 15:56
• Which is the opposite of what a capacitor does. That is, an inductor has a current determined by its history and a voltage determined by its instantaneous load. A capacitor has a voltage determined by its history and a current determined by its instantaneous load. Commented Aug 14, 2023 at 15:58
• @Ben Voigt, "An inductor has a current determined by its history" and "capacitor has a voltage determined by its history" sound very philosophical (oh, how I love that way of expression). They are actually memory devices (e.g., DRAM). Commented Aug 14, 2023 at 16:57
• @Ben Voigt, I have added more clarity about the behavior of capacitors and inductors. Indeed, capacitors act as voltage sources but provided that the input current change (or voltage through a resistance) is abrupt (or the frequency is high) and their capacity is large; otherwise they behave like "resistors" with high "resistance". Similarly, inductors also act as current sources provided the change in input current is abrupt (or the frequency is high) and their inductance is large. Then they have a high "resistance"; otherwise, they behave like "resistors" with little "resistance". Commented Aug 15, 2023 at 15:48