Miller effect

In electronics, the Miller effect is the increase in the equivalent input capacitance of an inverting voltage amplifier due to a capacitance connected between two gain-related nodes, one on the input side of an amplifier and the other the output side. The amplified input capacitance due to the Miller effect, called the Miller capacitance CM, is given by
 * $$C_{M}=C (1-A)\ ,$$

where A is the voltage gain between the two nodes at either end of the coupling capacitance, which is a negative number because the amplifier is inverting, and C is the coupling capacitance.

Although the term Miller effect normally refers to capacitance, the Miller effect applies to any impedance connected between two nodes exhibiting gain. These properties of the Miller effect are generalized in Miller's theorem.

History
The Miller effect was named after John Milton Miller. When Miller published his work in 1920, he was working on vacuum tube triodes, however the same theory applies to more modern devices such as bipolar transistors and MOSFETs.

Derivation
Consider a voltage amplifier of gain −A with an impedance Z&mu; connected between its input and output stages. The input signal is provided by a Thévenin voltage source representing the driving stage. The voltage at the input end (node 1) of the coupling impedance is v1, and at the output end −Av1. The current through Z&mu; according to Ohm's law is given by:


 * $$i_Z =  \frac{v_1 - (- A)v_1}{Z_\mu} = \frac{v_1}{ Z_\mu / (1+A)}$$.

The input current is:


 * $$i_1 = i_Z+\frac{v_1}{Z_{11}} \ . $$

The impedance of the circuit at node 1 is:


 * $$\frac {1}{Z_{1}} = \frac {i_1} {v_1} = \frac {1+A}{Z_\mu} +\frac{1}{Z_{11}} .$$

This same input impedance is found if the input stage simply is decoupled from the output stage, and the reduced impedance Z&mu; / (1+A) is substituted in parallel with Z11. Of course, if the input stage is decoupled, no current reaches the output stage. To fix that problem, a dependent current source is attached to the second stage to provide the correct current to the output circuit, as shown in the lower figure. This decoupling scenario is the basis for Miller's theorem, which replaces the current source on the output side by addition of a shunt impedance in the output circuit that draws the same current. The striking prediction that a coupling impedance Z&mu; reduces input impedance by an amount equivalent to shunting the input with the reduced impedance Z&mu; / (1+A) is called the Miller effect.

Gain roll-off
The Miller effect shows up prominently in amplifier design, where it can cause drastic reduction of amplifier gain as frequency increases, called gain roll-off. In these amplifiers, the coupling impedance is a parasitic capacitance. If Z&mu; represents a capacitor with impedance Z&mu; = 1/j&omega;C&mu;, the resulting input impedance has a huge capacitance (1+A)C&mu; attached in parallel with the nominal input impedance Z11. This gain-enhanced capacitance is called the Miller capacitance, CM:


 * $$ C_{M}=C_\mu (1+A).$$

That is, the effective or Miller capacitance CM is the physical C&mu; multiplied by the factor (1+A). This huge capacitance seriously degrades the amplifier frequency performance, because this capacitance becomes a short-circuit at high frequencies, effectively preventing any signal from entering the amplifier. The bigger this Miller capacitance, the lower the frequency at which the amplifier fails to work.

To illustrate this point, suppose Z11 = R11 and ZTh = RTh, simple resistors. Application of Kirchhoff's current law at node 1 leads to the result:


 * $$\frac{v_1}{v_{Th}} = \frac{(R_{11}//R_{Th})}{R_{Th})}\ \frac{1}{1+j\omega (1+A)C_\mu (R_{11}//R_{Th})} \ . $$

The leading resistance ratio applies when &omega; = 0, and expresses the simple voltage division caused by the two impedances in series. The second factor, however, displays a roll-off with increasing &omega; of the input signal amplitude v1 exciting the output stage. This roll-off becomes acute for frequencies larger than:


 * $$ \omega= \omega_C = \frac{1}{(1+A)C_\mu (R_{11}//R_{Th})} =  \frac{1}{C_M (R_{11}//R_{Th})} \, $$

where &omega;C is called the corner frequency. An interesting point is that this frequency becomes infinite (no roll-off) if the Thévenin resistance RTh = 0. That is why the parasitic resistance rX in the base lead of the hybrid-pi model for the bipolar transistor can be influential in determining the amplifier roll-off when these transistors are driven with a very low resistance Thévenin voltage source.

It is also important to note that the Miller capacitance is not the only source of amplifier frequency dependence. At higher frequencies, the dependent current source that the Miller theorem introduces in the output stage also becomes frequency dependent. For example, if the amplifier output resistance is included in the analysis, the impact of the frequency-dependent current source on the output side must be taken into account. It is important to include the capacitances contributed by the output stage, and in feedback amplifiers it is these high-frequency effects that control the stability of the amplifier.

Miller approximation
This example assumes A is frequency independent, but more generally A is frequency dependent. Because the Miller capacitance depends upon A, frequency dependence of A makes the Miller capacitance frequency dependent, so it is not possible to interpret CM as an everyday capacitance independent of frequency. However, in many cases, frequency dependence of A arises only at frequencies higher than the corner frequency caused by the Miller effect, so for frequencies up to the corner, roll-off of the gain does not occur, and A is approximated adequately by its low-frequency value. This approximation, the determination of CM using A evaluated at low frequencies, is the so-called Miller approximation. So long as the Miller approximation is accurate, CM is frequency independent, and its interpretation as a capacitance is secure.

Mitigation
The Miller effect can be undesirable, and several approaches to lower its impact are used in the design of amplifiers.

One approach to increase corner frequency is to decrease the resistance multiplying the Miller capacitance. It is important to notice that the corner-frequency limitation due to CM is greatly reduced for low impedance drivers, that is, CM RTh//R11 is small if RTh is small. Consequently, one way to minimize the Miller effect upon bandwidth is to use a low-impedance driver, for example, by interposing a voltage follower stage between the driver and the amplifier This stage acts as an impedance transformer, reducing the apparent driver impedance seen by the amplifier, reducing the Thévenin resistance of the driver.

Another approach to increase corner frequency is to reduce the Miller effect itself. It may be possible to lower the gain A across the coupling impedance Z&mu;, without greatly reducing overall gain. Inasmuch as the Miller effect is directly a result of this gain, its reduction will decrease the Miller effect and avoid limitations upon the amplifier bandwidth. For example, in the two-stage cascode amplifier, a common emitter amplifier may be used despite its tendency toward a Miller effect, as an an input stage to a common base amplifier. The low input impedance of the common base stage kills the gain of the common emitter stage and there is no Miller effect. The common emitter stage does nothing to add to the gain of the overall amplifier, but it has a purpose: it enables use of high impedance drivers.

The Miller effect is not always a nuisance: it may be exploited to synthesize larger capacitors from smaller ones. One such example is in the stabilization of feedback amplifiers, where stability is achieved by adding capacitance. The required capacitance may be too large to include directly in the circuit, so a smaller capacitance is added that is made larger by the Miller effect. This approach is particularly important in integrated circuit, where large capacitors consume significant area, increasing costs.