The classic gain equations for triodes and pentodes are based on the assumption that amplification factor, plate resistance, and transconductance are constants based on the DC operating point. In other words, we assmume that the tube is operating on small signals within its linear range near that point. This assumption breaks down completely when the tube is overdriven. So how does screen voltage affect gain under large-signal, overdrive conditions?
The issue boils down to our basic definition of gain: output voltage swing divided by input voltage swing. For a pentode preamp the output signal voltage, which is represented by the plate voltage, swings between between a minimum of nearly zero and a maximum equal to the plate supply voltage. The maximum output swing is thus defined by the plate supply voltage. This, it turns out, is independent of the screen voltage.
Maximum plate voltage is at cutoff. Minimum is when the grid reaches 0V. So the voltage gain of a pentode preamp is roughly these two extremes divided by the grid voltage swing needed to create them. This is where screen voltage plays a key role.
Here is a plot of plate current versus grid voltage for an EF86 pentode.
The plot corresponds to a plate voltage of 250 volts, but it doesn't change much for other plate voltages. The screen voltage, which has a much greater influence on plate current, is represented by 4 curves. With the screen at 180 volts the grid voltage swings from a minimum of -6 volts to a maximum of zero volts. At a screen voltage of 60 volts the grid swings only between -2.4 volts and zero. A lower screen voltage enables a substantially lower grid voltage swing to create the same plate voltage swing. This is the essence of our definition of "gain."
For the large, non-linear signal swings often sought by guitar players, where transconductance, plate resistance, and other tube parameters are far from constant, a lower screen voltage can be used to increase the voltage gain of the resulting design. This is achieved by reducing input headroom. With 180-volt screens the input signal can have an amplitude of up to 3 volts without overdriving the EF86 pentode. With 60-volt screens the amplitude can only reach 1.2 volts before the pentode preamp goes into overdrive.
Our thanks to Paul Reid of Rutgers University for his technical assistance with this article.