Miller effect

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In electronics, the Miller effect accounts for an increase in the equivalent input capacitance of an inverting voltage amplifier due to amplification of capacitance between the input and output terminals. Although Miller effect normally refers to capacitance, any impedance connected between the input and another node exhibiting high gain can modify the amplifier input impedance via the Miller effect.

This increase in input capacitance is given by

C_{M}=C (1-A_v)\,

where Av is the gain of the amplifier and C is the feedback capacitance.

The Miller effect is a special case of Miller's theorem.

Contents

[edit] History

The Miller effect was named after John Milton Miller. When Miller published his work in 1919 or 1920, he was working on vacuum tube triodes, however the same theory applies to more modern devices such as bipolar and MOS transistors.

[edit] Derivation

Figure 1: Circuit for deriving the Miller effect, with ideal voltage amplifier
Figure 1: Circuit for deriving the Miller effect, with ideal voltage amplifier

Consider an ideal voltage amplifier of gain Av with an impedance Z connected between its input and output nodes. The output voltage is therefore Vo = AvVi and the input current is

I_i = \frac{V_i - V_o}{Z} = \frac{V_i (1 - A_v)}{Z}.

As this current flows through the impedance Z, this equation shows that because of the gain of the amplifier a huge current flows in Z; in effect Z behaves as though it were much smaller than it is. The input impedance of the circuit is

Z_{in} = \frac{V_i}{I_i} = \frac{V_i Z}{V_i (1-A_v)} = \frac{Z}{1-A_v}.

If Z represents a capacitor, then

Z = \frac{1}{j \omega C}

and the resulting input impedance is

Z_{in} = \frac{1}{j \omega C (1-A_v)} = \frac{1}{j \omega C_{M}} \quad \mathrm{where} \quad C_{M}=C (1-A_v).

Thus the effective or Miller capacitance CM is the physical C multiplied by the factor (1 − Av)[1].

[edit] Notes

As most amplifiers are inverting amplifiers (i.e. Av < 0) the effective capacitance at the input is larger. For noninverting amplifiers, the Miller effect results in a negative capacitor at the input of the amplifier (compare Negative impedance converter).

Naturally, this increased capacitance can wreak havoc with high frequency response. For example, the tiny junction and stray capacitances in a Darlington transistor drastically reduce the high frequency response through the Miller effect and the Darlington's high gain.

The Miller effect applies to any impedance, not just a capacitance. A pure resistance or pure inductance will be divided by 1 − Av. In addition if the amplifier is non-inverting then a negative resistance or inductance can be created using the Miller effect.

It is also important to note that the Miller capacitance is the capacitance seen looking into the input. If looking for all of the RC time constants (poles) it is important to include as well the capacitance seen by the output. The capacitance on the output is often neglected since it sees \frac{C A_v}{A_v-1} and amplifier outputs are typically low impedance. However if the amplifier has a high impedance output, such as if a gain stage is also the output stage, then this RC can have a significant impact on the performance of the amplifier. This is when pole splitting techniques are used.

The impact of the Miller effect is often reduced by using a cascode or cascade amplifier rather than a common emitter. For feedback amplifiers the Miller effect can actually be very beneficial since stabilizing the amplifier may require a capacitor too large to practically include in the circuit, typically a concern for an integrated circuit where capacitors consume significant area.

[edit] Impact on frequency response

Figure 2: Operational amplifier with feedback capacitor CC.
Figure 2: Operational amplifier with feedback capacitor CC.
Figure 3: Circuit of Figure 2 transformed using Miller's theorem, introducing the Miller capacitance on the input side of the circuit.
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 CC. A Thévenin voltage source VA drives the circuit with Thévenin resistance RA. 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 jωCC( vi - vo ) 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 CM, which draws the same current from the driver as the coupling capacitor in Figure 2. Therefore, the 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 CM to CC. 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 CM of the Derivation Section.

The present example with Av frequency independent shows the implications of the Miller effect, and therefore of CC, upon the frequency response of this circuit, and is typical of the impact of the Miller effect (see, for example, common source). If CC = 0 F, the output voltage of the circuit is simply Av vA, independent of frequency. However, when CC 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

\frac {v_o} {v_A} = A_v \frac {v_i} {v_A} = A_v \frac {1} {1+j \omega C_M R_A},

and rolls off with frequency once frequency is high enough that ω CMRA ≥ 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 ω3dB such that ω3dB CMRA = 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 CM upon the amplifier bandwidth is greatly reduced for low impedance drivers (CM RA is small if RA 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 Av vi. 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. [2] 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.

[edit] Miller approximation

This example also assumes Av is frequency independent, but more generally there is frequency dependence of the amplifier contained implicitly in Av. Such frequency dependence of Av also makes the Miller capacitance frequency dependent, so interpretation of CM as a capacitance becomes a stretch of imagination. However, ordinarily any frequency dependence of Av 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, Av is accurately approximated by its low-frequency value. Determination of CM using Av at low frequencies is the so-called Miller approximation.[1] With the Miller approximation, CM becomes frequency independent, and its interpretation as a capacitance at low frequencies is secure.

[edit] References and notes

  1. ^ a b R.R. Spencer and M.S. Ghausi (2003). Introduction to electronic circuit design.. Upper Saddle River NJ: Prentice Hall/Pearson Education, Inc., p. 533. ISBN 0-201-36183-3. 
  2. ^ See article on pole splitting.

[edit] Further reading

[edit] See also