[BUG] multivariable chain rule

[Tevian Dray]

The following comments were sent to me by Larry Tankersley (tank at usna.edu),
who has given me permission to post them to the list.  These comments are
in response to my request back in April for examples of the use of the
multivariable chain rule.

Tevian

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    A design study of a circuit can use the concepts of a multivariable chain
rule. The circuit under consideration might be an amplifier or an
oscillator-timer consisting of a few external components (resistors and
capacitors) and an integrated circuit. The design goal is to set a gain for a
particular frequency or to set the oscillator frequency. Let's focus on gain.

The gain (G) of the amplifier is a function of the values of the several
external resistances and capacitances and the physics of the internal device
for which there is an idealized model and an actual behavior. Each of the
external components has a temperature (T) dependence that is 'known'. Each
type component is available in several families, each with a different known
temperature dependence.

For the idealized model, the problem can be reduced to an equation in the form
of a rational function of rational functions. A few of the values are
complex. Your comment holds. The temperature dependent forms for the
resistances and capacitance can be substituted into the expression for the
gain as it depends on those resistances and capacitances in order that the
temperature dependence be computed directly. In practice the all-at-once
approach is barbaric. Using the multivariable chain rule partitions the task
into computing the partials of the gain with respect to the external
resistances and capacitances and then folding in the temperature dependences
separately. This procedure allows one to examine substituting devices with
different thermal characteristics component by component without requiring a
full recalculation. The resistances often are adequately characterized by a
linear fit although some devices have an exponential variation with
temperature. Capacitance varies more strongly with temperature.

In the actual behavior case, the amplifier is set to operate in the middle of
its expected temperature range. At fixed temperature, the partial derivatives
with respect to the various resistances and capacitances are experimentally
determined. Next the total derivative of the gain with respect to temperature
is measured. Using your expression for the multivariable chain rule and the
known temperature dependences of the external components, one computes the
partial derivative of the gain with respect to temperature.  This last
derivative contains information about the inner workings of the integrated
circuit.

The typical design goal is to achieve a gain that is temperature
independent. The process is to substitute resistors and capacitors of
different types with different characteristic temperature behaviors to
decrease sensitivity of the gain to temperature fluctuations. This search
requires that the intermediate derivatives, the partials with respect to the
values of the external components be known.

The result may not be sensitive to the value of resistor four => buy a cheap
one.  Spend more on the components that have a greater impact.  Components
with small thermal coefficients are usually more expensive.  If good fortune
befriends you, a cheap component with a strong thermal dependence may be
advantageous if it acts to counter the shift due to another component.

I propose this as an application for the multivariable chain rule. Your local
EE department should be able to provide a more practical and current
example. My electronics experiences are quite dated.