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There are amplifier circuits that have capacitance between the input and output. If the gain is positive, this can lead to oscillations. If the gain is negative, another outcome is the result. We can deduce this from the following theoretical circuit.

An amplifier with a negligibly low output impedance, an infinitely high input impedance and gain A has feedback in the form of capacitor C (refer Figure 1). The gain A is negative. In addition, input current I, input voltage U and output voltage Uo are also drawn in. The input current I is equal to Ic and the input voltage U is equal to Uc + Uo. Uo in turn is equal to the product A U. From this follows that

Uc = U–Uo = U (1–A).

Substituting into the formula the current that flows through a capacitor, Ic = C(dUc/dt) results in

I = C.dU(1-A)/dt

We rearrange this as  I=(1-A).C.dU/dt

Now we can see that the gain determines the relationship between I and C. C appears to be larger by a factor of (1–A) (note: if A is negative, you can actually speak of a factor 1+A larger).

This is called the Miller effect. The apparent (larger) capacitance is called the Miller capacitance. When designing signal amplifiers you need to take this capacitance into account. We can actually use this Miller capacitance in other ways. If we make A variable, with an adjustable resistor for example, we create a variable capacitor. For this purpose we conceived the following schematic (see Figure 2).

Cm is the apparent capacitor between the input of the circuit and ground. If we connect a signal generator via a series resistor to the input and measure the input voltage with an oscilloscope, we can easily determine the corner frequency.

JFET-opamp A1 is necessary to prevent R1 from appearing in parallel with Cm and affecting the corner frequency. A2 is the actual (inverting) amplifier. The gain of A2 is equal to P1/R1. C is the capacitor which is enlarged artificially. The remaining components only serve to set the operating point of the circuit.

Cb blocks any DC voltages and needs to be relatively large, for example 25 times the maximum Cm. From the test results it appears that Cm is indeed equal to (1+P/R1) C. Cm can be varied with a potentiometer from about 560 pF to 12 nF.

As is usually the case, there are a few limitations in practice. The input signal may not be too large. Otherwise, the AC voltage across Cm will cause clipping at the output of the second opamp. At maximum Cm, the gain of A2 is about 20 times. The peak-to-peak value of the input voltage may therefore not be more than about 1/20 of the power supply voltage. The circuit will always work well for smaller signals, provided the frequency is not too high.

For A1 and A2 we used an LF356 and TL081 respectively. These are mainly used for frequencies not exceeding 100 kHz. Very fast JFET opamps could extend the useful frequency range to applications in the RF-range. For LF applications we could also use a dual
opamp for A1 and A2, such as the TL082.

The value of capacitor C can be changed to suit the application. With opamps of the type AD8099 with a C of 22 pF we can make a (tuning) capacitor with a value from 22 to 440 pF, for use up to 30 MHz. The alternative, a varicap diode that can be varied in capacitance over a range of 20 times (or more) is not used in practice very much any more. Other applications for this circuit are, for example, adjustable LC-filters for audio applications.

Author: Gert Baars

(Elektor Electronics Magazine – 2006)

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