Miller effect

The Miller effect was named after John Milton Miller.[1] When Miller published his work in 1920, he was working on vacuum tube triodes. The same analysis applies to modern devices such as bipolar junction and field-effect transistors.

Figure 1: An ideal voltage inverting amplifier with an impedance connecting output to input.

Consider an ideal inverting voltage amplifier of gain ${\displaystyle -A_{v}}$ with an impedance ${\displaystyle Z}$ connected between its input and output nodes. The output voltage is therefore ${\displaystyle V_{o}=-A_{v}V_{i}}$. Assuming that the amplifier input draws no current, all of the input current flows through ${\displaystyle Z}$, and is therefore given by

${\displaystyle I_{i}={\frac {V_{i}-V_{o}}{Z}}={\frac {V_{i}(1+A_{v})}{Z}}}$.

The input impedance of the circuit is

${\displaystyle Z_{in}={\frac {V_{i}}{I_{i}}}={\frac {Z}{1+A_{v}}}}$.

If ${\displaystyle Z}$ represents a capacitor with impedance ${\displaystyle Z={\frac {1}{sC}}}$, the resulting input impedance is

${\displaystyle Z_{in}={\frac {1}{sC_{M}}}\quad \mathrm {where} \quad C_{M}=C(1+A_{v})}$.

Thus the effective or Miller capacitance C_M is the physical C multiplied by the factor ${\displaystyle (1+A_{v})}$.[2]

As most amplifiers are inverting (${\displaystyle A_{v}}$ as defined above is positive), the effective capacitance at their inputs is increased due to the Miller effect. This can reduce the bandwidth of the amplifier, restricting its range of operation to lower frequencies. The tiny junction and stray capacitances between the base and collector terminals of a Darlington transistor, for example, may be drastically increased by the Miller effects due to its high gain, lowering the high frequency response of the device.

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 ${\displaystyle {C}({1+1/A_{v}})}$ 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 Miller effect may also 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 circuits, where capacitors can consume significant area, increasing costs.

Figure 2: Amplifier with feedback capacitor C_C.

Figure 2A 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. The output impedance of the amplifier is considered low enough that the relationship V_o= -A_vV_i is presumed to hold. At the output Z_L serves as the load. (The load is irrelevant to this discussion: it just provides a path for the current to leave the circuit.) In Figure 2A, the coupling capacitor delivers a current jωC_C(V_i − V_o) to the output node.

Figure 2B shows a circuit electrically identical to Figure 2A 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 2A. Therefore, the driver sees exactly the same loading in both circuits. On the output side, a capacitor C_Mo = (1 + 1/A_v)C_C draws the same current from the output as does the coupling capacitor in Figure 2A.

In order for the Miller capacitance to draw the same current in Figure 2B as the coupling capacitor in Figure 2A, 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

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

or, rearranging this equation

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

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 2B shows the large Miller capacitance appears at the input of the circuit. The voltage output of the circuit now becomes

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

and rolls off with frequency once frequency is high enough that ωC_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.

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.[3] 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 theorem
• CMOS Amplifiers