Gyrators (simulated inductors)

if you need that huge inductor for temporary energy storage in switched power supply applications or you want to create an LC oscillator with a large inductor, a gyrator will not help you out. This is because a gyrator can not emulate the magnetic field build-up of an inductor. It only emulates the voltage/current relation (thus impedance) of an inductor. So regarding frequency and phase relation, the gyrator acts as a real inductor, but not regarding energy storage in a magnetic field caused by the current flowing through the windings of the inductor.

Gyrators are special circuits that transform an impedance into another (mostly inverse) type of impedance. Gyrators are mostly used to simulate a large inductor using a 'normal' value capacitor. In audio filters such as equalisers, you often find gyrators in the bandpass filter sections, where LC filters are needed that operate in the audio frequency range down to f.e. 10Hz. An LC filter with a resonance frequency of 10Hz would require a large inductor with a value close to 1 Henry. Here a gyrator is an ideal solution, because it does not only transform an capacitive impedance into an inductive impedance but meanwhile also multiplies the capacitive impedance. So with a 'normal' value capacitor, we can create a large inductance.

This is a cost-effective solution in many applications where filtering at relative low frequencies is required. Another benefit of a gyrator is that a bulky inductor would pick up and generate electro-magnetic interference.

General note:
In some of the circuit schematics i left out the power supply connections for the OPAMPs.
All of the gyrator circuits with OPAMPs need a symmetrical (positive and negative) power supply voltage.

Gyrator

Principle

In the picture below, the principle of a gyrator is shown. Between the input (left) and output port (right) there is a mutual relation, expressed by the formula's [3] V1 - -G * I2 and [4] V2 = G * I1. G is the gyration constant of the gyrator. Here we have chosen to have the same gyration constant for the 'forward' path (G) as for the 'backward' path (-G), but this is not necessary. They can be different.
In the mathematical derivation, we see that the impedance Z1 is converted into an impedance Z2 that has a reciprocal relation to Z1.

Example

In the picture below we use the gyrator principle to convert a grounded capacitance into a grounded inductance.
This is a theoretical model of a gyrator, but further down we will see some examples of practical gyrators using this principle.

General Impedance converter (GIC)

The pictur above shows a general impedance converter (GIC) that can be used to generate an inductance or a frequency dependent negative resistance (FDNR) using 2 OPAMPs and 5 impedances Z1 to Z5 that are either be resistive or capacitive.
Because the OPAMPs use negative feedback and operated within the linear region (output is not saturating, clipping or distorted), the voltages V1, V2 and V3 are all equal. This also means that the voltage drop over Z1 and Z2 = 0 and that the voltage drop over Z3 and Z4 = 0 and that the voltage drop over Z5 is the same as the voltage at the input.
The formula under the picture shows that the total impedance.that is seen when looking into the circuit. can be any type of impedance : capacitive, inductive or resistive depending on the type of the individual impedances Z1 to Z5. Not so obvious is the fact that we can even generate a negative impedance in the form of a so-called Frequency Dependent Negative Resistor (FDNR).
To create a simulated inductor or a frequency dependent negative resistance we can use one or more capacitors in the circuit above.
But there are some restrictions :
Z1 and Z2 as well as Z3 and Z4 form a group.
The individual impedances in a group can not be both capacitive and the first and last elements of both groups can not be both capacitive.
In other words Z1 and Z2 can not be both capacitive. Z3 and Z4 can not be both capacitive. Z1 and Z3 can not be both capacitive. Z2 and Z4 can not be both capacitive.
When either Z2 OR Z4 is capacitive and all the others are resistive, the resulting total impedance Z will be inductive, because Z2 and Z4 are in the denominator.
When Z1 AND Z5 are capacitive and all the others are resistive or when Z3 AND Z5 are capacitive and all the others resistive, the resulting total impedance will be resistive, but with a negative sign and with a frequency dependent behaviour (FDNR). The FDNR is a special case of the GIC, that we will talk about more later.
See the table below for the different combinations of Z1 to Z5 and the resulting total impedance Z :

In the table above we see that a gyrator with a capacitor (Z3 or Z4) transforms this capacitance into an inductance.
A gyrator with 2 capacitors (Z1 & Z5 or Z3 & Z5) transforms the 2 capacitances into something we call a frequency dependent negative resistor (FDNR), also called a super capacitor, because it has a 2nd order character. It is a negative resistance, meaning that it will not sink (consume) current, but instead it will source (push out) current. In the frequency domain it shows a decreasing impedance with increasing frequency with a slope of -12db/octave (a capacitor has a slope of -6dB/octave). The phase behaviour is reversed compared with a normal 2nd order circuit because it is a negative resistance. So the FDNR behaves like a 2nd order circuit but with a inverted phase behaviour.

Grounded synthetic (active) inductors (Antoniou, Riordan, NIC combinations)

Antoniou grounded synthetic inductor

The Antoniou gyrator is the most versatile type of gyrator and has a very good performance (good Q factor), because it is using 2 amplifiers to compensate for all losses in the circuit. Even with general purpose OPAMPs, good results are achieved.
In the figure below the Antoniou gyrator is configured as an active inductor. This kind of inductor is often found in graphic audio equalisers or low frequency filters.

It is not easy to see how the circuit works, because you get the feeling that it's rather black magic than old school physics. In some cases it helps to try to figure out the behaviour of a circuit in the time domain by checking how it responds to a pulse at the input (step-response), instead of trying to understand it's behaviour in the frequency domain by checking how it reacts to varying frequencies.
We will try to figure out the behaviour of the circuit in the time domain and see how it reacts to a step-input. The expectation is that it behaves as a grounded inductor, so when we apply a step to this inductor via a series resistor (RL-circuit), we will see a differentiating effect. It forms a high pass filter: See picture below:

Often it also helps to redraw the circuit, so the building blocks that form the circuit are revealed. In the right circuit below, the Antoniou inductor is redrawn in a way that it makes it easier to comprehend.

In the right circuit above, C4 forms a differentiator with R5, which is buffered by OPAMP U2. This part of the circuit is responsible for the differentiating effect. OPAMP U1 amplifies the difference between the input (node 1) and (node 2) with gain=1. In fact OPAMP U1 will compensate for the current flowing through R1 by controlling the input of the differentiator via it's output (node 3).

Let's see what happens when we have an input (node1) of 0 volts :
OPAMP U1 will control the output (node 3), so the inverting input (node 2) will become equal to the non-inverting input (node 1). To do this, the OPAMP U1 will put node 3 at 0 volts. Because C4 is discharged, it will have 0 volts over it's terminals, so node 4 is also 0 volts. OPAMP U2 will control the output (node 5), so it's inverting input (node 2), will be equal to it's non-inverting input (node 4), so it will bring node 5 to 0 volts as well.

Let's see what happens when we suddenly increase the input voltage :
OPAMP U1 will increase the output voltage at node 3 high to make the difference between node 1 and node 2 equal to 0 volts. When node 3 raises, capacitor C4 will start charging via R5, so the voltage at node 4 will first increase (following node 3) and then drop back to 0, as C4 charges. That is what a differentiator knows how to do. OPAMP U2 will control the output (node 5), so the voltage at node 2 is equal to the voltage at node 4. Node 3 is high, while node 4 drops to 0 over time. So the output of OPAMP U2 will need to drop below zero volts to keep node 2 equal to node 4. OPAMP U1 will try to keep the voltage at node 1 equal to node 2, while OPAMP U2 tries to keep the voltage at node 2 equal to the voltage at node 4. So at node 1, we find the same voltage as at node 4.

Grounded synthetic inductor using 2 NICs (Negative Impedance Converters) in series

How the circuit below works is explained in detail in the section about NICs : NIC (Negative Impedance Converter)

Riordan grounded synthetic inductor

The circuit below is the Riordan version of a GIC. The mathematical derivation is given right of the picture.
The formula for the input impedance Z of the Riordan GIC is exactly the same as the formula for the Antoniou GIC.

The circuit below is a variant of the Riordan GIC that is configured to generate an inductance.
Another variant, that would generate the same inductance can be created by changing R2 to a capacitor of 10nF and C1 to a resistor of 10k.
The circuit can also be used to create a FDNR (Frequency Dependent Negative Resistor)

Derived from Riordan grounded synthetic inductor

Current driven synthetic inductor

Other grounded synthetic inductor circuits with relative good Q factor

Above circuits can be explained when looking at the behavior in the frequency domain.

For DC, the capacitor C has infinite impedance and forms an open circuit. This means that the non-inverting input of the OPAMP is at 0 volts. The OPAMP will control it's output in such a way that the voltage at the inverting input is equal to the voltage at the non-inverting input, being 0 volts. With the inverting input at 0 volts, this means that R1 is grounded and connected parallel over the input port. So for DC, the input-port presents a low resistance, being R1. A real grounded inductor is a short circuit for DC, so the circuit behaves like a real inductor with a series-resistor with value R1.

When we increase the input frequency, C will start to conduct and the signal at the non-inverting input of the OPAMP will start to increase. This means that the output of the OPAMP will also start to increase. Seen from the input port, this means that less and less current will flow through R1 and more and more current will flow through R2.

For high frequencies, C has a very low impedance and forms a short circuit, so the OPAMP will pass the whole of the input signal to the output. This means that the current through R1 will be 0, because the signal at both sides of R1 will be the same. Seen from the input port, this means that R2 is the only load impedance. So at infinite frequency, the port represents a high resistance, just like a real grounded inductor would do.

So we can state that the circuit behaves like a real grounded inductor.

The circuit above can also be best explained when looking at the behavior in the frequency domain.

For DC, the capacitor C has a very high impedance and forms an open circuit. So the input signals only 'sees' R1 and R2, which are parallel over the input signal. So for DC, the input port presents a load equal to R1 + R2. This can be seen as the series resistance of the inductance that is being generated.

When we increase the frequency of the input signal, the impedance of C will decrease. The OPAMP is connected as a buffer (gain = 1), so the output of the OPAMP follows the signal on the non-inverting input, which is the input signal. So with increasing frequency, more and more of the input signal is passed to the node between R1 and R2 via C. So the OPAMP generates positive feedback via C back to the input. This positive feedback decreases the current that flows through R1, thereby increasing the input impedance that is seen by the input port.
At very high frequencies, C presents a very low impedance and forms a short circuit. Because the OPAMP reflects the input signal at its output, R1 has the same input signal at both legs. This means that the current through R1 will be 0, so it seems like R1 has infinite resistance and does not exist anymore.
In summary this means that for high frequencies, the circuit presents a very high impedance to the input port, just like a real inductor would do.

We can state that the circuit behaves like a real grounded inductor.

Other grounded synthetic inductor circuits with low Q factor

The transistor circuits above (borrowed from an EDN article) simulate an inductor which value depends on the collector current and the Beta (current-amplification factor) of the transistor. Their behaviour can easily be understood when checking in the frequency domain with a signal source connected to it via a resistor, forming a RL circuit :

With a low frequency input, the capacitor will have a high impedance, so most of the current flows into the base of the transistor causing it to conduct. When the transistor conducts, the collector is grounded, so we see a short circuit when we look into the input port. This is the same behaviour as a real inductor.

When increasing the frequency of the input signal, the capacitor impedance will decrease and will steal current from the transistor, causing it to conduct less, so the collector signal increases. So when looking into the input port, while increasing the frequency, we see that the impedance increases. This is also the same behaviour as a real inductor.

Another angle of view to understand the behaviour of the circuit is the phase behaviour :

The integrating RC combination causes a phase shift of maximum -90 degrees. The transistor is an inverter and generates a phase shift of 180 degrees. Both together will cause a phase shift of maximum +90 degrees. A real grounded inductor would generate the same phase shift when connecting a signal source via a resistor (RL circuit).
So concerning phase shift, the circuit behaves as a real inductor.