Keep in mind, the conversion is simply the electrical properties. Normally, current leads voltage in a capacitor and lags it in an inductor, and that’s what a gyrator changes. If you use a gyrator and a capacitor to make a virtual inductor, that inductor won’t magnetically couple to another inductor, real or simulated. There’s no magnetic field to do so. You also don’t get big voltage spikes caused by back EMF, which depending on your application could be a plus or a minus. But if you need an ungainly inductor in a circuit for its phase response, a gyrator may be just the ticket.

Magic

So how does a gyrator do its magic? Consider this circuit simulation. The bottom circuit shows the voltage and current in a 5 H inductor. The top circuit shows a gyrator-equivalent circuit providing a similar inductance.

At DC, the bottom circuit will look like a 1 kΩ resistor along with whatever resistance the inductor’s wire presents (probably not much). At some very high frequency, the inductor’s effective resistance will be so high, the circuit might as well be an open circuit. In between, the resistance will increase with frequency.

Before you look at the top circuit, it might be a good idea to review a few analysis tricks for a “perfect” op amp. We can imagine the inputs of the op amp are open circuits. Whatever voltage is on one terminal, the output will do anything it can to make voltage on the other terminal the same. In practice, that isn’t always possible but for our case here, assume it is.

Now consider the gyrator circuit. At DC, the capacitor will be fully charged and be an open circuit. Since the positive terminal of the op amp theoretically draws no current, that terminal will be at ground potential through the 20 kΩ resistor. That means the op amp output is also at 0 V so the 1 kΩ resistor is between the source and ground, just like in the inductor circuit. At a very high frequency, the capacitor will simply vanish and the input voltage will appear the positive terminal. That means the 1 kΩ resistor will have the same voltage on each side of itself and is effectively out of the circuit.

Now let’s think about the phase shift. When the capacitor sees an AC signal, its current will rise before the voltage across it rises. So at the very instant that starts, the resistor will have the same voltage across it and thus no current. As the voltage across the capacitor increases, the voltage at the positive and output terminals will decrease, causing more current to flow. That’s the opposite of the capacitor’s current-voltage response, and exactly like the inductor’s. So even with no real math, you can get a feel for why this works.

Why?

All components we use are not ideal. We pretend wire has no resistance and no limit on current-handling when we draw schematics, but in reality, it has both. Capacitors leak. Resistors can be non-linear or drift. But inductors are especially troublesome. The wire that forms them has resistance. The loops capacitance with the adjacent loops. In addition, inductors tend to be physically large. Big inductors use so much wire that they are not only large but have lots of resistance and other problems. They take up a lot of space. Perhaps not as large as this one from a 1912 radio antenna, but still.

You can usually get high-quality capacitors and you can use them to make large value inductors that are closer to the ideal than a wire-based counterpart. So assuming a gyrated inductor meets your needs, you can usually get a higher-quality device and often in less space by using a gyrator.

One place this is especially important is inside integrated circuits. Due to the way ICs are made it is reasonably easy to make capacitors of particular values, but inductors are very difficult to make in silicon. It is even easier to make multiple capacitors with very precise ratios of capacitance and that can sometimes lead to interesting designs where the exact values of the capacitors and simulated inductors aren’t as important as the ratio between the values.

Since the inductors made with gyrators aren’t actually magnetic, you can’t use them to drive a relay, or a speaker, or sense magnetic fields, or build transformers. You can’t use them in switching power supplies, either. However, you do see them used often in things like filters. Just remember, the op amp (or other active devices) will have to operate at the frequencies involved and still have good properties.

Design

You might wonder why the 250 nF capacitor in the example circuit is the same as a 5 H inductor. After all, the simulation uses a 20 Hz signal which means the inductor has a reactance of about 628 Ω, while the capacitor has a reactance of almost 32 kΩ.

Consider that the 1 kΩ resistor is R1 and the 20 kΩ resistor is R2. Without getting into the derived math, the effective Z of the circuit is:

For the example, that comes out to 628 Ω of capacitive reactance. (If you’d like a refresher on complex impedance, we’ve got you covered.)

The reactance of an inductor with series resistance R1 is , so the above result reduces to

Just remember, the op amp has to function at the frequency of interest and has to be able to output enough current which will limit how low R1 can be.

Odds and Ends

A gyrator is actually a special use of a NIC or negative impedance converter. You can use the same idea to create negative resistance or even negative reactance. You sometimes see negative resistance used in simple oscillators. Of course, that’s often in the form of a device like a tunnel or Esaki diode that actually has negative resistance, but you could use a NIC circuit, as well.

I’m showing gyrators with op amps because that’s the easiest way to understand their operation. However, you can use other active devices. In addition, the gyrators I’ve shown all require one end to be grounded. There are architectures for gyrators that produce a floating virtual inductance, but those are rare because they take more components.

You can, of course, flip an inductor to a virtual capacitor with the same set up. The problem is there is little incentive to do so. Capacitors are generally smaller, lighter, and–for the same price or lower–better quality than inductors.

Learn More

If you search YouTube for gyrator, you might find some not-entirely-safe-for-work content. Who knew? However, you are safe with [Jeffrey Walling’s] lectures on the topic, which appear below. If you want to dig into the math, you could do worse than these videos.

If you prefer reading, or you are interested in how this was done with devices like vacuum tubes and discrete transistors, have a look at [James Kulesz’s] Master’s thesis. Sure, it is from 1969, but the math hasn’t changed and the only real difference from today is that op amps are a lot better these days.

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