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Sensitivities 105
obeying at least one Kirchhoff law, the respective branch voltage and current
sets (vectors) have null inner products; i.e.,
(4.91 )
This also applies in time or frequency domains. The second network, N, is
called the adjoint network; it mayor may not be different. First, observe that
an inner product is defined in terms of two vectors such as (4.88) and (4.89);
suppose that they are the n-element vectors x and y. Then the inner product of
x and y is
(4.92)
Example 4.5. Apply Tellegen's theorem to the networks from Figure 4.3b
and c; they are reproduced in Figure 4.18b and c. The branch-4 current arrow
has been reversed so that each branch has its current entering its positive
voltage, consistent with each branch in the common topology shown in Figure
4.18a. All branch voltage and current values are shown in Figure 4.18 as
found by Programs B4-1 and A2-1. Tellegen's theorem says that
(4.93)
In fact, using Program A2-1, y T i=O.OO85+jO.0116, which is as close to zero
as might be expected for the digits carried. There are three more such inner
products that should be equal to zero: yTI, yTi and YTI. Evaluate them using
the data from Figure 4.18. The reader should write a brief program that calls
Program A2-1 subroutines in order to calculate the inner products of complex
numbers; it is much easier, and a lot of time will be saved and errors avoided.
Penfield et al. (1970) generalize the Tellegen theorem statement to include
the conjugation and any linear operator; for the partial derivative operator
with respect to any variable
(4.94)
Consider the network in Figure 4.18b to be its own adjoint network Nand N.
The port input impedance is
v.
Zin=-I-'
(4.95)
- .
and its partial derivative with respect to L yields
-AV.= AZ'n' I.,
(4.96)
where currents are the independent variables. The branch-2 equation in terms
of independent current 1 2
is
V 2 = 1 2 ·wL(d+jl), (4.97)