# Basic Calculation Capacitance Multiplier Circuit

A capacitance multiplier circuit is a well-known circuit with the purpose of multiplying capacitance in a circuit.

The op-amp circuit in Figure.(1) is known as capacitance multiplier, for reasons that will become obvious.

Such a circuit is used in integrated-circuit technology to produce a multiple of a small physical capacitance C when a large capacitance is needed.

 Figure 1. Capacitance multiplier

## Capacitance Multiplier

The circuit in Figure.(1) can be used to multiply capacitance values by a factor up to 1000. For example, a 10 pF capacitor can be made to behave like a 100 nF capacitor.

In Figure.(1), the first op-amp operates as a voltage follower, while the second one is an inverting amplifier.

The voltage follower isolates the capacitance formed by the circuit from the loading imposed by the inverting amplifier.

Since no current enters the input terminals of the op-amp, the input current Ii flows through the feedback capacitor. Thus, at node 1,

 (1)

Applying KCL at node 2 gives

or

 (2)

Substituting Equations.(2) to (1) results

or

 (3)

The input impedance is

 (4)

where

 (5)

Hence, by a proper selection of the values of R1 and R2, the op amp circuit in Figure.(1) can be made to produce an effective capacitance between the input terminal and ground, which is a multiple of the physical capacitance C.

The size of the effective capacitance is practically limited by the inverted output voltage limitation.

Hence, the larger the capacitance multiplication, the smaller is the allowable input voltage to prevent the op amps from reaching saturation.

A similar op-amp circuit can be designed to simulate inductance. There is also an op-amp circuit configuration to create a resistance multiplier.

## Capacitance Multiplier Circuit Example

For better understanding, let us review the example below :
1. Calculate Ceq in Figure.(1) when R1 =10 kΩ, R2 = 1 MΩ, and C = 1 nF.
Solution :
From Equation.(5)