User:John R. Brews/WP Import

In electronics, the Miller effect accounts for the increase in the equivalent input capacitance of an inverting voltage amplifier due to amplification of the capacitance between the input and output terminals. The additional input capacitance due to the Miller effect is given by

$C_{M}=C(1-A_{v})\ ,$

where A_v is the voltage gain of the amplifier, which is a negative number because it is inverting, and C is the feedback capacitance.

Although the term Miller effect normally refers to capacitance, the amplifier input impedance is modified by the Miller effect by any impedance connected between the input and another node 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 and MOS transistors.

Derivation

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Miller's theorem

Consider a voltage amplifier of gain −A with an impedance Z_μ 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_μ 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 lower impedance Z_μ / (1+A) is connected 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 and the very noticeable prediction that the input impedance is reduced by the coupling between stages in the amount of the reduced impedance Z_μ / (1+A) shunted across the input is called the Miller effect.

Effects

The Miller effect shows up prominently in amplifier design, where the coupling impedance is a parasitic capacitance. If Z_μ represents a capacitor with impedance Z_μ = 1/jωC_μ, the resulting input impedance has a huge capacitance (1+A)C_μ attached in parallel with the nominal input impedance Z₁₁. This gain-enhanced capacitance is called the Miller capacitance, C_M:

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

That is, the effective or Miller capacitance C_M is the physical C_μ 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 Z_Th = R_Th, simple resistors. Application of Kirchhoff's current law at node 1 leads to the result:

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

which displays a roll-off of the input signal amplitude v₁ exciting the output stage 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 ω_C is called the corner frequency. An interesting point is that this frequency becomes infinite (no roll-off) if the Thévenin resistance R_Th = 0. That is why the parasitic resistance r_X 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. It is important to include as well the capacitances contributed by the output stage, and in feedback amplifiers it is this frequency dependence that controls the stability of the amplifier. This frequency dependence can be affected by the dependent current source that the Miller theorem introduces in the output stage.

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 the required capacitance may be too large to practically include in the circuit. This may be particularly important in the design of integrated circuit, where capacitors can consume significant area, increasing costs.

Mitigation

The Miller effect may be undesired in many cases, and approaches may be sought to lower its impact. Several such techniques are used in the design of amplifiers.

A current buffer stage may be added at the output to lower the gain $A_{v}$ between the input and output terminals of the amplifier (though not necessarily the overall gain). For example, a common base may be used as a current buffer at the output of a common emitter stage, forming a cascode. This will typically reduce the Miller effect and increase the bandwidth of the amplifier.

Alternatively, a voltage buffer may be used before the amplifier input, reducing the effective source impedance seen by the input terminals. This lowers the $RC$ time constant of the circuit and typically increases the bandwidth.

Impact on frequency response

Figure 2: Operational amplifier with feedback capacitor C_C.

Figure 3: Circuit of Figure 2 transformed using Miller's theorem, introducing the Miller capacitance on the input side of the circuit.

Figure 2 shows an example of Figure 1 where the impedance coupling the input to the output is the coupling capacitor C_C. A Thévenin voltage source V_A drives the circuit with Thévenin resistance R_A. At the output a parallel RC-circuit serves as load. (The load is irrelevant to this discussion: it just provides a path for the current to leave the circuit.) In Figure 2, the coupling capacitor delivers a current jωC_C( v_i - v_o ) to the output circuit.

Figure 3 shows a circuit electrically identical to Figure 2 using Miller's theorem. The coupling capacitor is replaced on the input side of the circuit by the Miller capacitance C_M, which draws the same current from the driver as the coupling capacitor in Figure 2. Therefore, the driver sees exactly the same loading in both circuits. On the output side, a dependent current source in Figure 3 delivers the same current to the output as does the coupling capacitor in Figure 2. That is, the R-C-load sees the same current in Figure 3 that it does in Figure 2.

In order that the Miller capacitance draw the same current in Figure 3 as the coupling capacitor in Figure 2, the Miller transformation is used to relate C_M to C_C. In this example, this transformation is equivalent to setting the currents equal, that is

$\ j\omega C_{C}(v_{i}-v_{O})=j\omega C_{M}v_{i},$

or, rearranging this equation

$C_{M}=C_{C}\left(1+{\frac {v_{o}}{v_{i}}}\right)=C_{C}(1+A_{v}).$

This result is the same as C_M of the Derivation Section.

The present example with A_v frequency independent shows the implications of the Miller effect, and therefore of C_C, upon the frequency response of this circuit, and is typical of the impact of the Miller effect (see, for example, common source). If C_C = 0 F, the output voltage of the circuit is simply A_v v_A, independent of frequency. However, when C_C is not zero, Figure 3 shows the large Miller capacitance appears at the input of the circuit. The voltage output of the circuit now becomes

$v_{o}=-A_{v}v_{i}=A_{v}{\frac {v_{A}}{1+j\omega C_{M}R_{A}}},$

and rolls off with frequency once frequency is high enough that ω C_MR_A ≥ 1. It is a low-pass filter. In analog amplifiers this curtailment of frequency response is a major implication of the Miller effect. In this example, the frequency ω_3dB such that ω_3dB C_MR_A = 1 marks the end of the low-frequency response region and sets the bandwidth or cutoff frequency of the amplifier.

It is important to notice that the effect of C_M upon the amplifier bandwidth is greatly reduced for low impedance drivers (C_M R_A is small if R_A 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, which reduces the apparent driver impedance seen by the amplifier.

The output voltage of this simple circuit is always A_v v_i. However, real amplifiers have output resistance. If the amplifier output resistance is included in the analysis, the output voltage exhibits a more complex frequency response and the impact of the frequency-dependent current source on the output side must be taken into account. Ordinarily these effects show up only at frequencies much higher than the roll-off due to the Miller capacitance, so the analysis presented here is adequate to determine the useful frequency range of an amplifier dominated by the Miller effect.

Miller approximation

This example also assumes A_v is frequency independent, but more generally there is frequency dependence of the amplifier contained implicitly in A_v. Such frequency dependence of A_v also makes the Miller capacitance frequency dependent, so interpretation of C_M as a capacitance becomes more difficult. However, ordinarily any frequency dependence of A_v arises only at frequencies much higher than the roll-off with frequency caused by the Miller effect, so for frequencies up to the Miller-effect roll-off of the gain, A_v is accurately approximated by its low-frequency value. Determination of C_M using A_v at low frequencies is the so-called Miller approximation. With the Miller approximation, C_M becomes frequency independent, and its interpretation as a capacitance at low frequencies is secure.