Miller Effect - Impact On Frequency Response

Impact On Frequency Response

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 node.

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

or, rearranging this equation

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

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.