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104 Ladder Network Analy.i.
Table 4.6.
First-Order Finite Differences for the Network in Example b in
Figure 4.3
k
L
L
C
C
0.0001
0.01
0.0001
0.01
- 0.001799 + jO.003402
-0.001764+jO.003363
- 0.001999+jO.03691
-0.001968 +jO.0365I
-0.001800+jO.003405
- 0.001836 + jO.003445
- 0.002000 + jO.03692
- 0.002032 + jO.03733
derivatives. How much is x k perturbed? Computers with 7 to 10 decimal-digit
mantissas require x k
to be increased by about 0.01% (a 1.0001 factor). If it is
much less than that, the change in Z;n may fall off the end of the mantissa's
digits, and no change is seen. If it is much more than that, this linear
approximation of slope is too crude. It is easier to talk about the latter
Htruncation" problem in terms of the Taylor series, which will be discussed in
the next chapter.
The network in Example b in Figure 4.3 was analyzed by Program B4-1 ; its
input impedance was calculated for 0.01 and 1% changes in each variable,
namely Land C. The perturbation was tried as an increase and as a decrease.
The input impedances were employed in (4.90), which was evaluated using
Program A2-1. The results are shown in Table 4.6. These values differ from
exact results in the third significant figure.
4.7.3. Exact Partial Derivatives by Tellegen's ·Theorem. There are several
exact means for finding derivatives of complex network functions. It will be
shown in Section 7.1 that the coefficients of bilinear functions, which have the
form of (2.1) or (4.18), can be determined by only three independent function
evaluations. Because the derivative of the bilinear function can be written
easily, its exact value is also available with respect to one of the n variables.
Fidler (\976) has given a means to obtain the exact partial derivatives of a
bilinear function with respect to n variables in just 2n + I function evaluations.
However, Tellegen's theorem enables the calculation of exact partial derivatives
of complex responses with respect to alJ n variables in just one or two
network analyses, depending on whether the response is at only one end of the
network or is a transfer function, respectively. This is a spectacular result, and
the computer memory requirements fOf variables and code are not too severe
for desktop computers. Branin (\973) and others have observed that the same
result is available algebraically with a slight savings in computation; so
Tellegen's theorem is not really necessary. Even so, it is worth knowing for its
general enlightenment and compactness. Penfield et al. (\970) have neatly
derived 101 fundamental theorems in electrical engineering using Tellegen's
theorem. They correctly claim that no circuit designer should be without it.
Tellegen's theorem states that for any two entirely different (or identical)
linear or nonlinear networks (N and N) having the same branch topology and