Capacitance multiplier

Introduction

A capacitance multiplier does not convert one type of impedance into another type of impedance, like a gyrator does, but it converts a low impedance into a much higher impedance. 

It can for instance convert a 1uF capacitor into a 1mF capacitor, a value that is 1000 times higher.

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

Grounded scaled capacitors

Capacitance multiplier using 1 single OPAMP turns 1uF into 100mF

The Miller effect is a capacitance multiplication as a result of a small capacitance that virtually gets "stretched out and enlarged" because it is connected in-between a small input signal and an amplified inverted large output signal. With a CE (common-emitter) transistor amplifier, the collector-base junction capacitance is in between the small input signal at the base and the large amplified and inverted signal at the collector. This way, the capacitance feels like amplified at the base of the transistor.
We can also use an OPAMP to multiply the capacitance of an external capacitor that is not part of the internal structure of a component.
With a single OPAMP we can multiply capacitance, even by a factor as large as 100 000 as shown in the image below.
A 1uF capacitor is multiplied so it feels like a 100mF capacitor, albeit with a rather large ESR (equivalent series resistance).
The circuit shows one of the methods to achieve capacitance multiplication. It can also be done by transistors, JFETs …
The resulting large capacitance can not compete with a super capacitor because it can't store/deliver energy. It just behaves as a very large capacitance in terms of amplitude and phase when used in a filter, or a timing circuit or when large time averaging is needed.
This technique is used to achieve huge time constants with relative small capacitors.
It is also used to filter out noise (ripple) in a linear power supply, by using the current gain of a transistor to multiply the capacitance of an RC filter connected to the base of an emitter-follower. 

LTSpice simulation of the capacitance multiplier:
Capacitance multiplier with 1 OPAMP 

Left figure above : R2 and C form an integrator. The OPAMP U1 is configured as a voltage follower, so the capacitor voltage at the non-inverting input also appears on the non-inverting input. This means that the voltage over R1 is equal to the voltage over R2. But because R1 is 1000 times lower than R2, R1 will draw extra current away from the input. Because of this extra current, the input feels like being loaded with a much bigger capacitor (1000x bigger in this case).

 Note: The OPAMP can only deliver a limited current, so there is a lower limit to the value of R1 (10E).

Right figure above :
Here the transistor is an emitter follower that will copy the voltage on it's base, which is the voltage over C, to the emitter, connected to R1.

Note: Because the transistor does not make an exact copy but looses 0,7V (base-emitter voltage) on the way, the circuit is not as accurate as the OPAMP circuit on the left.

Note: The transistor has a limited Hfe (current amplification factor, so there is a lower limit to the value of R1. More emitter current means more base current, means more load on the capacitor C.

In the circuits above, a bootstrapping technique (like pulling yourself up from the floor by your bootstraps) is used to make the capacitor at the input feel like a much bigger capacitor. The bootstrapping lies in the fact that the lower node of the capacitor is not connected to ground. Instead, it is connected to the output of an inverting amplifier that amplifies and inverts the voltage at the upper node of the capacitor. So the lower node of the capacitor is pulled down additionally, so more current is needed to charge the capacitor, making it look like a much bigger capacitor is connected to the input.

The right circuit above is an improvement over the left circuit, because the capacitor voltage is first buffered by an OPAMP U1, that is connected as a voltage follower. This prevents that the input impedance of the inverting amplifier influences the voltage on the capacitor C.