What is Miller Capacitance?

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.

Why is it important?

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.

Example

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

Effect on frequency response

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.