General

The term "Miller capacitance" is often seen when reading about guitar amplifier circuit design.  It refers to the effective multiplication of the plate-to-grid capacitance in a triode tube (or transistor) by the gain of the amplifying stage.

When a tube is amplifying a signal, it has to work against the plate-to-grid capacitance, charging and discharging it as the signal changes.  Because the grid is a high impedance, and doesn't sink or source any current, this charging current must be sourced or sinked through the driving source resistance of the previous stage.  This forms a lowpass filter, with a corner frequency determined by the source resistance of the previous stage and the input capacitance.

The Miller capacitance in a triode tube is equal to the plate-to-grid capacitance multiplied by a factor equal to the stage gain plus one.  Pentode and tetrode tubes don't suffer as much from the effects of Miller capacitance because of the shielding effect of the screen grid, which drastically lowers the plate-to-grid capacitance.

The majority of the input capacitance of a triode stage is made up of the combination of the grid-to-cathode capacitance, plus the Miller capacitance formed by the grid-to-plate capacitance multiplied by the stage gain plus one.  The formula for determining the total input capacitance of a triode stage is as follows:

Cin = Cgk + Cgp*(A+1)

where: Cin = input capacitance
           Cgk = grid-to-cathode capacitance, composed of the internal tube capacitance plus the stray capacitance
           Cgp = grid-to-plate capacitance, composed of the internal tube capacitance plus the stray capacitance
             A  = stage gain

The typical interelement capacitances are very small, but, as can be seen from the above equation, the grid-to-plate capacitance is multiplied by the gain of the tube stage plus one, so if the gain is large, the capacitance can very easily become significant, resulting in audible rolloff in frequency response.

For example, a typical 12AX7 stage has the following capacitances and gain:

Cgk = 1.6pF + 0.7pF stray = 2.3pF
Cgp = 1.7pF + 0.7pF stray = 2.4pF
A = 61

Therefore, the total input capacitance would be:

Cin = 2.3pF + (61+1)* 2.4pF = 151.1pF

The input capacitance of the tube, in conjunction with the source impedance of the previous stage, forms a simple, single-pole RC lowpass filter with a -6dB/octave (-20dB/decade) slope, and an  upper -3dB cutoff frequency equal to:

f = 1/(2*pi*R*C)

The -3dB point with a 68K resistor (such as at the input stage of an amplifier) would be:

f = 1/(2*pi*68K*151.1pF) = 15.5kHz

This is not too bad, considering the frequencies involved in guitar amplification.  However, if the source resistance were increased to 470K, the cutoff frequency  would be:

f = 1/(2*pi*470K*151.1pF) = 2.2kHz

This would result in considerable rolloff in the upper part of the passband at guitar frequencies.  Note that the Miller effect applies to any triode, not just small signal triodes.  Pentode output tubes operated in triode mode will exhibit less high frequency response because of the higher input capacitance of the tube in triode mode.  This, in combination with the normally lower level of higher order harmonics in triode mode, will cause the overall tone to be less bright than pentode mode.

Miller capacitance can kill the frequency response rather easily, so there are a few things to consider when designing guitar amplifiers.  If you desire to minimize the effect of the Miller capacitance on frequency response, you can do the following things:

• Reduce the output impedance of the previous stage.    This can be accomplished by lowering the value of the plate load resistor, using a tube with a lower internal plate resistance, or lowering the value of any series or shunt attenuation resistors.  Obviously, all of these things will affect the gain of the stages, so this must be taken into account as well.

• Reduce the gain of the stage.  The Miller capacitance is proportional to the gain of the amplifying stage, so using a lower stage gain will reduce the Miller capacitance, thereby increasing the frequency response.

• Use a pentode or cascode stage instead of a triode stage.  Both the pentode and cascode configuration suffer very little from the effects of Miller capacitance because of the AC grounding of the screen grid in the pentode, and the AC grounding of the upper tube grid in the cascode, which drastically lowers the plate-to-grid capacitance in both configurations.  The cascode configuration has the added advantage of lower noise, when compared to the pentode, because it does not have the "division" noise created by the screen grid of the pentode.

• Use the Miller capacitance to your advantage.  Since the Miller capacitance forms a lowpass filter in conjunction with the output resistance of the previous stage, it can be used as a "free" lowpass filter, thus saving the cost and trouble of adding a capacitor or RC network in the amplifier when a tailored frequency response is desired.  This is commonly used in a not-so-apparent manner in the typical input stage of most guitar amplifiers.  The input resistor, in conjunction with the input capacitance, forms a lowpass filter that reduces the susceptibility of the tube to parasitic oscillations, such as those that can occur when the input stage is being driven by a long guitar cable.  As shown in the example above, the typical cutoff point of this lowpass filter, when using a 68K resistor, is around 15.5kHz.  Also, some high-gain guitar amplifiers have a distortion channel that uses large value series resistors in front of the grids of the preamp tubes.  These resistors not only aid in minimizing blocking distortion, they also act as lowpass filters to reduce some of the high frequency content of the distortion signal, in order to reduce some of the "buzziness" in the tone.  As shown in the above example, the typical cutoff frequency using a series 470K resistor would be 2.2kHz, which would roll off most of the upper harmonics, producing a subjectively smoother distortion tone.