The Miller effect is a basic electronic phenomenon associated with feedback circuits. It can occur undesirably in amplifiers, caused by parasitic capacitance, but it can also be applied in capacitance and resistance multiplier circuits. Miller's theorem is derived here and several applications of it are presented.
Miller's Theorem Applied to Op-Amps
The inverting op-amp configuration has a resistor connected from output to inverting input in Figure (a).
In Figure (b), the inverting voltage amplifier has a gain of –K with input quantities of νi and ii. The output voltage is νo. The equivalent input resistance can be found as follows. First, by definition of op-amp open-loop voltage gain, –K,
At the input node, apply KCL (Kirchhoff's current law),
Substituting for νo;
Rearrange the above equation for input resistance, νi /ii , and we have Miller’s Theorem:
For an amplifier with output resistance, the equivalent shunt contribution from Rf is
From the output, Rf appears to be slightly less than its actual value for large K. From the input, Rf appears to be 1/(1 + K) times its actual value, causing input resistance to be much reduced and providing a low-resistance path for ii . For infinite K, the input node is a virtual ground, as it is for the ideal inverting op-amp. The equivalent circuit resulting from Miller's theorem is shown above in (b). The circuit has been converted by the Miller transform to an equivalent circuit that does not have an input-output bridging resistance.
Miller's theorem readily generalizes to amplifiers in which Rf is replaced by a reactance, such as a capacitor. Because of the Miller effect, a common-emitter (CE) transistor stage with base-collector capacitance, Cbc will cause this capacitance to appear as 1 + K greater at the base, where K is the base-to-collector voltage gain of the CE stage. This can be visualized as follows. When a change of one unit of voltage occurs at the base side of Cbc, –K units appear on the other terminal, resulting in a net K + 1 units across the capacitor from the one unit at the input terminal. Consequently, K + 1 times as much current will flow through the capacitance causing it to appear K + 1 times larger. This larger equivalent capacitance forms a time constant with the input resistance that limits input bandwidth. By reducing the gain, the bandwidth increases proportionally. (Hence the stage has a fixed gain-bandwidth product.)
This undesirable appearance of the Miller effect is a nuisance to wideband amplifier designers, but can be put to good use in capacitance multiplier circuits using op-amps. Timing circuits with long time-outs often require large capacitors. For accurate timing, these capacitors are plastic. Large-value plastic capacitors are volumetrically large and expensive. The capacitance multiplier is a circuit that uses the Miller effect to make a small capacitor appear electrically large. One realization is shown below.
The x1 buffer drives an inverting op-amp with a gain of -Rf/Ri. Applying Miller's theorem,
Timer with Capacitance Multiplier
Bias current of the threshold (TH) terminal of a 555 timer limits its useful timing range as a MMV. A current-divider capacitance multiplier is used to extend the time-out tH by connecting it as shown below.
The TH input is now driven by the op-amp output, eliminating bias-current error from TH but introducing op-amp offset current and voltage error. An equivalent circuit shown below is derived as follows.
The op-amp circuit with Rf and -Ri is Thevenized and floated on νC. The op-amp offset voltage VOS is divided by the resistors so that its Thevenin voltage is
in series with Rf||Ri. With a x1 op-amp gain, op-amp VOS also is in series with the TH input. VOS at the op-amp output contributes VOS/Rf to the timing current. From this model, timing error can be calculated. For significant multiplication of C, Ri >> Rf. The series Thevenin voltage source is then about VOS, and the IOS term in the error current dominates.
Capacitance multiplication is achieved in the preceding circuits by applying bootstrapping to a current divider. The idea can be extended to resistance multiplication. A basic instance is the bootstrapped CC (emitter-follower) or FET CD (source-follower) stage.
A more deliberate and precise resistance multiplier, shown above, uses an op-amp buffer instead of a CC or CS. Solving the circuit for rin results in
Passive networks across inverting amplifiers produce the Miller effect. It not only appears as undesirable in transistor amplifiers but can also be applied usefully in capacitance and resistance multipliers.
Miller's theorem was derived here assuming an ideal (frequency-independent) inverting amplifier which increases the equivalent value of feedback or bridging capacitance. It is also valid for noninverting amplifiers, when K is negative. In this case, the equivalent C is reduced (or R increased) in value, and this effect is more commonly referred to as bootstrapping. The resistance multiplier was an instance of its application. Nothing prevents the application of Miller's theorem to inductors either, or to more general amplifiers.