Copyright © 2013 by Wayne Stegall
Created September 17, 2013. See Document History at end for
details.
Square Law
How the squarelaw
of fieldeffect devices arises from physics.
Introduction
I thought it important to know whether the square law transfer
characteristic of FETs and vacuum tubes is arbitrary or concretely
derived from some exact scientific principle. If it were
arbitrary then different devices could vary in their adherence to the
arbitrary model leading to unpredictable results. If based on a
principle, more certain expectations could be held about their
operation. At first thought, it seems possible that principle
could be involved because the capacitor energy equation has a
squarelaw characteristic as well.
Then you might think that device current would be related to the energy
in the gate or grid capacitor.
The Principle
Further inquiry has revealed the actual principle. That is that
the control mechanism does not depend on the control voltage
alone. Rather the voltages throughout the electric field all
contribute to the control mechanism in a distributed way. In the
case of a MOSFET the incremental contribution of voltages at each point
along the electric field to the total drain current could be
represented by the following equation.
(2)

di
dv

= 2k_{n}(v  V_{T})

Then the entire current response could be found by integrating across
the range of voltages distributed across the electric field.
(3)

i_{ds} =

∫ 
v_{gs}
V_{T}

2k_{n}(v  V_{T})
dv 
Now solve the integral
(4)

i_{ds} =


v_{gs}
V_{T}

k_{n}(v  V_{T})^{2} 
(5)

i_{ds} = k_{n}(v_{gs}
 V_{T})^{2} – k_{n}(V_{T}  V_{T})^{2}

The result is the familiar squarelaw MOSFET equation.
(6)

i_{ds} = k_{n}(v_{gs}
 V_{T})^{2} 
Abstractly, this result would have been anticipated without detailed
math from recognizing that the integral of a simple linear relation is
always a second order polynomial. Note also that the original
thought that the controlling factor might be the energy in the gate or
grid capacitor is a parallel coincidence. This is because the
energy in a capacitor is calculated by similarly structured math.
The effect of a nonlinear electric field
All this presumes that the electric field is linear with physical
distance (x), that is that v(x) = ax + b. If a device had a
nonlinear field the calculations would be complicated by adding a
field voltage function v(x) = f(x) and the calculation would have to be
integrated with respect to x. The JFET creates such a
dilemma. Its gate diode capacitance is a function of voltage,
leading to a nonlinear electric field. The capacitance of the
reversed biased pn junction operates as if plate separation increased
with voltage so that the capacitance has an inverse square root
relationship to the control voltage something like:
After the transfer curve is computed from the nonlinear electric
field, the JFET has a 3/2 power law transfer characteristic in a form
close to a squarelaw.
^{1}
(8)

i_{ds} = I_{p0}


1 +

3v_{gs}
V_{p0}

+ 2


– 
v_{gs}
V_{p0} 

^{3/2}


This result is not so bad in comparison as shown below in
figure 1; at least it is not
exponential.
Figure
1:
Comparative
plots of JFET and depletion MOSFET transfer curves


Legend:
———— Depletion MOSFET transfer curve
———— JFET
transfer curve

Comparisons
Vacuum tubes derive their squarelaw transfer curve from the same field
effect. Perhaps they could be regarded as Vacuum FETs for the
sake of comparison. To some like me, this gives hope that MOSFETs
can produce tubelike results without the extra design complexities of
tubes: high voltages, heater circuits, and output
transformers. However there is reason to believe tubes'
squarelaw relationships are more pure. First the dielectric
properties of a vacuum are considered perfect and that of the inert
gases usually filling tubes them nearly so. The silicon dioxide
or glass dielectric of the MOSFET is not as linear with a dissipation
factor of 0.001,
^{2} although a
nonlinearity so small could hardly be expected to produce a bad
result. Also, the MOSFET tends to divide its gate capacitance –
and therefore is electric field – between source and drain at the pinch
off point in dynamic way as the the device operates. The vacuum
tube does not seem to have any of these other possible fielddistorting
affects, its grid to cathode electric field unimpeded from pure
operation.
Finally
These distinctions only apply with certainty if the compared devices
are not under feedback in any way. As soon as local feedback is
applied, the resulting circuits – even the JFET ones – no longer have a
squarelaw transfer curve although they still show squarelaw
tendencies by showing dominance of the second harmonic in their
distortion profiles.
^{3}
^{1}Donald L. Schilling and Charles Belove, Electronic Circuits:
Discrete and Integrated (New York, 1979), p. 138 equation 3.15.
^{2}Arthur B. Williams and Fred J. Taylor, Electronic Design Handbook
(New York, 1981), p. 103
^{3}See article Feedback Distortion Analysis.
Document History
September 17, 2013 Created.