## Letters to WTF: is tube linearity just a matter of lower gain and higher operating voltage? Or it is inherited with the medium – vacuum vs semiconductor?

Generalizations are difficult to make, but you can look at the theoretical mechanisms of operation for some interesting nuggets. In a tube, current is transferred between anode and cathode by the space charge in a vacuum. Child’s Law states that current in a vacuum is directly proportional to anode voltage (to three halves power) and inversely proportional to the distance between electrodes (squared). The speed of electrons depends solely on the applied voltage. In semiconductors constructed of doped sandwiches of semiconductor material, Child’s Law doesn’t apply. Here we use the Mott-Gurney law. This states that the current density in a semiconductor is directly proportional to anode voltage (squared) and inversely proportional to the thickness of said material (cubed). The speed of the electrons depends on both the electron mobility of the semiconductor (assumed to be constant) and the applied voltage.

There are extra constants for calculations in either case, but notice the similarities. In both cases, we can assume distance between electrodes (whether separated by a semiconductor or a vacuum) doesn’t change. The difference here is the power for the voltage term. The generalization is that current density through a vacuum is less affected by changes in anode voltage than is current through a semiconductor.

Is this the theoretical mechanism to explain the heuristic that vacuum tubes are more linear than transistors? Maybe…at the very least it’s an interesting observation. In reality, geometry, application, and other factors matter, aside from just materials. Some tubes are more linear than others just as some transistors are more linear than others. There are bad ways to bias and operate tubes just as there are good ways to bias and operate transistors.