5.1 The Importance of Choosing the Correct Pair of Variables
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Example 5.1 (Time-Varying Capacitor) Consider the CR
i = C
dv
dt
(5.1)
of a linear time-invariant capacitor C in the (v, i)-domain. Then, consider a timevarying capacitor C(t) obtained by varying the distance between plates via the
application of a mechanical force. We can ask whether there holds the obvious
generalization
i = C(t)
dv
dt
.
Unfortunately, laboratory experiments would show that this generalized CR would
not give correct results for current i for any applied signal v. The problem is that we
have chosen the incorrect pair of variables for the CR of a time-varying capacitor.
Suppose instead to consider the pair of variables (v, q) = (v, i (−1) ) to describe
C, i.e., to use the CR
q = Cv
which is obtained by integrating (5.1) in time. The correct CR of the time-varying
capacitor C(t) is simply given as
q = C(t)v
from which it is possible to derive the right CR in the (v, i)-domain, that is
i = C(t)
dv
dt
+
dC(t)
dt
v.
Note the extra term (dC(t)/dt)v that is needed to predict the correct values of the
time-varying capacitor current.
Example 5.2 (Rocket Launching) An example similar to the time-varying capacitor
can be found in mechanics concerning a rocket launching.
The rocket mass m(t) changes with time due to fuel consumption. We can ask
whether the following generalization of Newton’s Law of Motion holds
f = m(t)
dν
dt
where f is the force and ν = dx/dt is the velocity. Again, such formula turns out
to fail to predict the experimental results, i.e., it has no predicting ability.
Integrating in time Newton’s Law of Motion f = mdν/dt for a constant mass m
yields