Miller effect
Encyclopedia
In electronics
, the Miller effect accounts for the increase in the equivalent input capacitance
of an inverting voltage amplifier
due to amplification of the effect of capacitance between the input and output terminals. The virtually increased input capacitance due to the Miller effect is given by
where
is the gain of the amplifier and C is the feedback capacitance.
Although the term Miller effect normally refers to capacitance, any impedance connected between the input and another node exhibiting gain can modify the amplifier input impedance via this effect. These properties of the Miller effect are generalized in the Miller theorem
.
. 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.
of gain
with an impedance
connected between its input and output nodes. The output voltage is therefore 
. Assuming that the amplifier input draws no current, all of the input current flows through 
, and is therefore given by
.
The input impedance of the circuit is
If Z represents a capacitor with impedance
, the resulting input impedance is
Thus the effective or Miller capacitance CM is the physical C multiplied by the factor
.
), the effective capacitance at their inputs is increased due to the Miller effect. This can lower the bandwidth of the amplifier, reducing 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 constant
s (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
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 circuit
, where capacitors can consume significant area, increasing costs.
A current buffer stage may be added at the output to lower the gain
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
time constant of the circuit and typically increases the bandwidth.
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 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 CM, 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 CM to CC. In this example, this transformation is equivalent to setting the currents equal, that is
or, rearranging this equation
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
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. 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.
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
, the Miller effect accounts for the increase in the equivalent input capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
of an inverting voltage amplifier
Amplifier
Generally, an amplifier or simply amp, is a device for increasing the power of a signal.In popular use, the term usually describes an electronic amplifier, in which the input "signal" is usually a voltage or a current. In audio applications, amplifiers drive the loudspeakers used in PA systems to...
due to amplification of the effect of capacitance between the input and output terminals. The virtually increased input capacitance due to the Miller effect is given by
where
is the gain of the amplifier and C is the feedback capacitance.Although the term Miller effect normally refers to capacitance, any impedance connected between the input and another node exhibiting gain can modify the amplifier input impedance via this effect. These properties of the Miller effect are generalized in the Miller theorem
Miller theorem
Miller theorem refers to the process of creating equivalent circuits. It asserts that a floating impedance element supplied by two voltage sources connected in series may be split into two grounded elements with corresponding impedances. There is also a dual Miller theorem with regards to impedance...
.
History
The Miller effect was named after John Milton MillerJohn Milton Miller
John Milton Miller was a noted American electrical engineer, best known for discovering the Miller effect and inventing fundamental circuits for quartz crystal oscillators ....
. When Miller published his work in 1920, he was working on vacuum tube
Vacuum tube
In electronics, a vacuum tube, electron tube , or thermionic valve , reduced to simply "tube" or "valve" in everyday parlance, is a device that relies on the flow of electric current through a vacuum...
triodes, however the same theory applies to more modern devices such as bipolar and MOS transistors.
Derivation
Consider an ideal inverting voltage amplifierAmplifier
Generally, an amplifier or simply amp, is a device for increasing the power of a signal.In popular use, the term usually describes an electronic amplifier, in which the input "signal" is usually a voltage or a current. In audio applications, amplifiers drive the loudspeakers used in PA systems to...
of gain
with an impedanceElectrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
connected between its input and output nodes. The output voltage is therefore . Assuming that the amplifier input draws no current, all of the input current flows through , and is therefore given by.The input impedance of the circuit is
If Z represents a capacitor with impedance
, the resulting input impedance isThus the effective or Miller capacitance CM is the physical C multiplied by the factor
.Effects
As most amplifiers are inverting (i.e.), the effective capacitance at their inputs is increased due to the Miller effect. This can lower the bandwidth of the amplifier, reducing its range of operation to lower frequencies. The tiny junction and stray capacitances between the base and collector terminals of a Darlington transistorDarlington transistor
In electronics, the Darlington transistor is a compound structure consisting of two bipolar transistors connected in such a way that the current amplified by the first transistor is amplified further by the second one...
, 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 constant
RC time constant
In an RC circuit, the value of the time constant is equal to the product of the circuit resistance and the circuit capacitance , i.e. \tau = R × C. It is the time required to charge the capacitor, through the resistor, to 63.2 percent of full charge; or to discharge it to 36.8 percent of its...
s (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
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 impactOpen-circuit time constant method
The open-circuit time constant method is an approximate analysis technique used in electronic circuit design to determine the corner frequency of complex circuits. It also is known as the zero-value time constant technique...
on the performance of the amplifier. This is when pole splitting
Pole splitting
Pole splitting is a phenomenon exploited in some forms of frequency compensation used in an electronic amplifier. When a capacitor is introduced between the input and output sides of the amplifier with the intention of moving the pole lowest in frequency to lower frequencies, pole splitting causes...
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 circuit
Integrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...
, 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
between the input and output terminals of the amplifier (though not necessarily the overall gain). For example, a common baseCommon base
In electronics, a common-base amplifier is one of three basic single-stage bipolar junction transistor amplifier topologies, typically used as a current buffer or voltage amplifier...
may be used as a current buffer at the output of a common emitter
Common emitter
In electronics, a common-emitter amplifier is one of three basic single-stage bipolar-junction-transistor amplifier topologies, typically used as a voltage amplifier...
stage, forming a cascode
Cascode
The cascode is a two-stage amplifier composed of a transconductance amplifier followed by a current buffer. Compared to a single amplifier stage, this combination may have one or more of the following characteristics: higher input-output isolation, higher input impedance, high output impedance,...
. 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
time constant of the circuit and typically increases the bandwidth.Impact on frequency response
Thévenin's theorem
In circuit theory, Thévenin's theorem for linear electrical networks states that any combination of voltage sources, current sources, and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem...
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 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 CM, 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 CM to CC. In this example, this transformation is equivalent to setting the currents equal, that is
or, rearranging this equation
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
Common source
In electronics, a common-source amplifier is one of three basic single-stage field-effect transistor amplifier topologies, typically used as a voltage or transconductance amplifier. The easiest way to tell if a FET is common source, common drain, or common gate is to examine where the signal...
). 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
and rolls off with frequency once frequency is high enough that ω CMRA ≥ 1. It is a low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut 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
Cutoff frequency
In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.Typically in electronic systems such as filters and...
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. Ordinarily these effects show up only at frequencies much higher than the roll-off
Roll-off
Roll-off is a term commonly used to describe the steepness of a transmission function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband...
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.