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102 Ladder Network Analysis
which can be viewed as a normalized partial derivative. Recalling the derivative
formula for natural logarithms (In), (4.82) can also be written as
z __ Ak In Z 4 83
S (. )
Xk A k
In x ' k
which shows that the normalized derivatives are describing relative changes in
logarithmic space. Optimization (automatic component adjustment) in logarithmic
space often is better behaved because of the normalization of partial
derivatives that would otherwise be badly scaled (grossly different magnitudes).
Suppose that Z= IZle j '. Then differentiation of Z in the right-hand term in
(4.82) follows the rule for differentiation of a product, namely d(uv) =v du+
u dv. It follows that
which reduces to a useful identity:
S~ = x,, [(AkIZI)ej'+jIZI(AkO)ej'J, (4.84)
, IZle J
SZ = Sizi +J'°S' . (4.85)
Xk x~ x~
This says that when the complex sensitivity of a complex response function is
obtained, the real sensitivities of both the magnitude and angle (phase) are
immediately available.
First-order prediction of response behavior for small changes in several
independent variables may be derived by recalling the total differential
Table 4.5.
I.
2.
3.
4.
5.
6.
7.
8.
9,
10.
Useful Identities for Partial Derivative Applications
Z=U+jW:
liZ = i£ = IIIZle j '.
aXk
IIZ=IIU+jIlW.
Z*~U-jW.
lI(aV+fl[)~a(IIV)+jl(II[);
For Z ~ IZle j ¢:
Z*(IIZ)
IIIZI ~ Re IZI ~ IIIZlcos(O- ,p),
a and fl are scalars.
(liZ)
II~lm-Z seconds. Multiply by 360E6 to get degrees/MHz.
IIIZI' ~ 2[U(IIU) + W(II W)]~ 2 Re[Z*(IIZ)].
az,
az, az,
~ = ~Z - if Xk is only in domain of Zl .
uX k u I 3Xk
!oglOe
II log IOU = --U(IIU).
!og,oe=0.434 294 482.
. . ,(R,+R,)'
InsertIOn-loss ratlO=ls2d 4R R .
j 2