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280 Direct-Coupled Filters
frequency variable jF with exponents I and O. When W=Wo, F=O, and thus
C(w o ) is a real number. Clearly, the N = 3 loss function has the form
OF)' + U 20F)2 + U ,(jF) + U o
L(w) = U
o
(8.15)
This confirms that the prototype network in Figure 8.1 can produce the exact
polynomial response function of the classical bandpass network in Figure
6.31, because (8.15) is the manifestation of the c1assicallowpass-to-bandpass
mapping in (6.83).
The expressions for chain parameter C, such as (8.11) and (8.12), may he
converted to the loaded-Q parameter using definitions (8.3), (8.4), and (8.9).
For example, the (8.15) loss function for N = 3 is
L=t[(jF)'QuQuQu + (jF)2(Qu Qu + QuQu)
Similarly, the N = 2 loss function is
+(jF)(Qu+Qu+Qu)+2j. (8.16)
L=t[OF)2Qu QL2+(jF)(Qu +Qul+2j. (8.17)
The important conclusion is that prototype network selectivity is a function of
only loaded-Q values. This allows the arbitrary choice of parallel resistance
levels throughout the direct-coupled filter to accommodate convenient element
values.
1,
-
~
15
L
0.3
I\. f 2
- f 1 / ,
F=--
"
'\
,-------
5
L, ~1~;~':'I:d_
o
0.4 0.5
.618
1.618
V
\ - +----- V
Defined
r\idth passband
1\
~f- F,~
r-,\V,\!N
Near
stop
ban':J
2
/
l/
2nd harmoniC
I I
Far stopband
3
4
-2.67 ~1.5 -1.0 Q 1.0 +1.5 +2.B7 +3.15 F
Figure 8.7.
Standard semilogarithmic selectivity plot with passband normalization.