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formula:
Sensitivities 103
dZ""A,Z tox, + A,Ztox,+ .,. + AnZ toxn. (4.86)
Dividing both sides by Z and placing xk/xk in each term on the right-hand
side yields
(4.87)
This shows that the relative change in a (complex) response is approximately
the sum of relative signed changes of all independent variables weighted by
the (complex) signed sensitivity numbers. Table 4.5 provides some useful
identities for partial derivatives of complex variables.
4.7.2. Approximate Sensitivity. It is essential that the reader feel comfortable
about partial derivatives, especially those that are complex. First-order
finite differences will be explained because it is a practical method and should
convince even the most apprehensive reader that partial derivatives are nice. It
is presumed that the connection between real-function slope and derivative
can be recalled, particularly as it defines an ordinary real first derivative. The
kth variable xk has been discussed; a formal notation of the entire set of
variables needs to be introduced; it is called a column vector:
x,
x,
X= (4.88)
xk
It may be written in row form, using the transpose operator that swaps rows
and columns:
xn
x = (Xl' X 2 ,···, X k ,···, X n ) T. (4.89)
A convenient definition of a finite-difference approximation to a partial
derivative is now possible:
(4.90)
For instance, suppose that there is a ladder network with n L's and C's. For
their nominal values residing in the vector x defined by (4.88), the input
impedance Z;n(x) is computed at a frequency that does not change. Now the
kth component xk is changed by a small amount, tox" and the slightly
different input impedance Zin(x + toxk) is calculated. These three numbers, two
being complex, are used in (4.90) to approximate the partial derivative. It
requires n + I complete analyses of the ladder network to get all n partial