Design Theory and Data for Electrical Filters

712 Pages·1965·165.988 MB·English

by  J. K. Skwirzynski

Design Theory
and Data
for Electrical
Filters
J. K. SKWIRZYNSKI
MARCONI SERIES
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Fantomas Ping
I\
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Chapter I
LADDER NETWORKS
AND COEFFICIENTS
Ladder Structure
The filters whose design data are tabulated numerically in this book form
a compact and rigorously defined group. It will be convenient to specify pre
cisely in this chapter their particular electrical structure. This is done for two
reasons. Firstly, it is essential for the user of our Tables to develop a close
familiarity with the network configurations which can be obtained by direct
use of the numerical data provided. Secondly, since the possible specifications
of filters tabulated here cover a very extensive range of practical requirements,
the theoretical treatment can be limited to these cases without much ·1oss
of generality; a logical description of this, somewhat restricted, insertion
loss theory is most conveniently started with the structural aspects of
resultant filters. Thus, ignoring for the present the wider aspects of electrical
performance ··of filter networks considered here, these networks consist
of reactive ladder structures operating between two resistances which, in
the practical sense, represent the internal impedances of a generator and
a load.
Ladder structures are built up of successive connections of series and shunt
branches; each branch contains one or more reactive components, capaci
tances or inductances. One end of each of the shunt branches is connected to
the common bas~ line (earth terminal), the other joins one or two neighbour
ing series branches. A quick glance at Figs. 1.1 and 1.2 will explain the situa
tion. The filters considered in this and the following five chapters are low-pass
filters; the extension to band-pass, band-stop and high-pass filters will be
made in Chapter 7. The branches of a low-pass filter network consist of one
or at most two reactive compontnts. The resultant ladder structures, shown
in Figs. 1.1 and 1.2 will be called standard ladders or, more precisely, stan
dard low-pass ladders. Since the branch components of standard ladders,
particularly the inductances, are assumed to be reactive, their effective Q
factors are infinite. This idealization of the physical state of affairs is neces-
. sary so far as the underlying theory is concerned; in fact it is impossible to
develop an exact theoretical treatment of the insertion loss theory with
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$
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Fig. 1.1 The reactive Ladder T structures.
=:;-
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Fig. 1.2 The reactive ladder II structures.
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I.ADDER NETWORKS AND COEFFICIENTS [Ch
sufficient generality and clarity unless the filters are assumed to be purely
reactive. The inclusion of resistances in the ladder branches, in order to
account for finite Q-factors, can only be accomplished under certain special
conditions within the general framework of the theory. The procedure
adopted in such cases is essentially an approximation, but it is one which
retains the expected agreement between imposed initial specifications and the
resultant response; it is called the 'predistortion' technique and is developed
in Chapter 8.
Thus, the main design data in the numerical Tables are for purely reactive
filters. On the other hand, graphical data are also provided for the design of
'predistorted' filters. For reasons of simplicity (see Chapter 8', pp. 156 et
seq.), only filters with open-circuit (or short-circuit) tenninations are treated
in this way. These graphical data may be used to produce reliable filters with
inductances whose Q-factors are realizable in practice. The necessary dissipa
tion resistances may also be utilized to relax the strict theoretical requirements
for infinite or zero resistance termination ratios of these filters and to provide
selective networks with large but finite termination ratios (see for instance
Example 5, for the design of a narrow band-pass filter, not necessarily
equally terminated, pp. 533 et seq.).
In Figs. 1.1. and 1.2, the standard ladders are built up by successive con
nection of series reactances !£ and shunt susceptances [fl: starting from the
input end (from the left), the ladders may commence with a series reactance
!!"1, as in Fig. l.l(a) and in Fig. l.l(b), wherej~1 :::ajwLi. or with a shunt
susceptance !11, as in Figs. 1.2(a) and 1.2(b), wherej111 = jroC1. The ladders
may finish at the output end (to the right) with a series reactance j!E ,+
2 1
= jwL2v+ 1 in Fig. l.l(a) and j!E 2v+z = jruL2v+ 2 in Fig. l.2(b), or with a
shunt susceptancej.112v+2 =j(J)C2v+2 in Fig. l.l(b) andj.912,+1 =jwC2,+1
in Fig. l.2(a).
The parameter v is of of prime importance in the theory; it may be called
the number of sections in a ladder. In Figs. 1.1 and 1.2 the ladders are con
veniently subdivided into sections as indicated by broken lines. Such a divi
sion of the ladder structure recalls the image parameter theory. It should be
observed, however, and well remembered, that the filters considered in this
book are designed as complete, indivisible entities; once the components of
the branches are determined from the Tables of numerical data, the ladders
cannot be broken in the middle (for instance) and a new section included
there; similarly, once the filter is designed, a new section cannot be added at
the input or output end without destroying the matching properties of the
network and its specified response to signals. Such a procedure is perfectly
justified in the image parameter theory where the concept of a section has a
precise physical meaning. In the insertion loss theory the number of sections,
v, indicates the complexity of a filter (e.g. the number of branches) and the
identification of v with the number of sections does not imply in any waythe
6
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I) LADDER COEFFICIENTS
tp-
subdivision of a filter into separate entities matched together. This constitutes
one of the main differences between the conventional image parameter
and the modern insertion loss technique.\( The consequence of this 'l?a er
:r
a ue concept of sections in the present theory is that each reactance and
each susceptance 11 (in Figs. 1.1 and 1.2) is specified by a single parameter,
whereas in the image parameter theory the branch common to two adjacent
sections (i.e. a series branch in the structure in Fig. 1.1 or a shunt branch in
Fig. 1.2) is a combination of two similar re qtanc s. Thus, a reactanc_e _in . ol,.q_
Fig. 1.l,,£,~"a_.s:q.~ceptance in Fig. 1.2 cannot be regar e as split into two h11~e~~ '"
each ~~TOilglrig to one of the two neigKSo'iifufg sections. ·
The terminology of the image parameter theory will be occasionally used
here, without necessarily implying that all the corresponding properties of the
'h'grcr,
given term as it is employed on this theory. This should cause no con
fusion once it is remembered that these terms are used here as a convenient
description of purely structural properties of a standard ladder. Thus, the
. ladders in Figs. 1.1 and 1.2 may be called 'm-derived' structures, since they
'e~Ocfy"""resonant reactances (or susceptances) respectively, in shunt and
series branches; however, no special meaning is attached to the quantity m.
Similarly, a term 'constant-k' half-section may be used when these branches
contain a single reactance only, again without attaching special meaning to
the quantity k; in fact throughout this book, the quantity k denotes the
selectivity factor of a filter which will be defined in the next chapter. To stress
this symbolic use of these two terms they are put in inverted commas.
Ladder Coefficients
The standard low-pass ladder structure shown in Fig. l.l(a) can be con
sidered as consisting of v :&rh6te 'm-derived' T sections, comprising together
+
2v 1 branches. Such a Iiia<ie'r configuration will be called (recalling again the
image parameter theory) a 'mid-series'( T-symmetrical/ structure and will be
shortly denoted as a TS structure.WThe symmetry referred to here is of a
purely structural nature and it does not necessarily iffipfythe symmetry of the
values of components about the central series b~.
The series branches in a TS structure contain inductive reactances
j!£ ls-1,TS = j(J)Lls-1,TS (1.1)
s = 1, 2, ... v+ 1
Ys-1 =1.:\-5,i. · ·-
where oo = 2nf and f is the frequency. The second subscript, namely.TS, is
~ti on (IJ) mere!~ to J,ocate the reactan~es .1!._nd th~correg~onding
inductances in appropriate IadJer configurations; in this case, in Fig. l.l(a).
These suffixes inaybe conveiiieiilly dropped as~'Oif-as a particular structure
7
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LADDER NETWORKS AND COEFFICIENTS [Ch
i:;J~
is considered and no confusion can arise. The branches in a TS struc
ture consist of resonant susceptances
·,a _ jwC2s,TS (1.2)
}l7il 2s,TS - --2---'-'-'-=---
l -W L1s,TsC2s,TS
s = 1, 2, ... v
which resonate at the tuning frequencies
TRt!Q.\lEl'lc.y Of
J l'lftl'll'i c LOSS (1.3)
s = 1, 2, ... v
infinit~~
Each of these frequencies will be called a frequency of since at
such a frequency the corresponding shunt branch presents a short circuit to
the signal, preventing it from appearing at the load terminals. Thus, when
f = f 00s, for each of :J = 1, 2, ... v, the loss characteristic of a filter is said to
have a pole or an infinity.
The ladder configurati911 in Fig. l.l(b) is not symmetrical in the structural
sense; it consists of v wb.'tffe, 'm-derived', 'mid-series' T sections plus one
'constant-.k' half-section. This latter configuration will be called a 'mid-·
series' T-asymmetrical structure and will be shortly denoted as a TA structure.
The addition of the single half-section at the output, rather than at the input
end of the ladder, mermtandardizes the representation of a network; in
!ifxes
particular, it the numerical sequence of the branch reactances fl' zs-t
and susceptances f!J 2 •• The ~l position of the generator and of the load
can be interchanged under certain loading conditions as is explained in Chap
ter 3; thus, the terminals 1-1' and 2-2' neecl not be considered as permanently
tied to the generator and the load terminals respectively. On the other hand,
the single half-section is in a sense degenerate since its shtint branch contains
a single susceptance which does not resonate at any finite frequency; in tltjs
sense it is a 'constant-k' half-section. This particular arrangement of la'tfcief
components is caused by the fact that the numerical design data of filters with
asymmetrical structures tabulated in this book are only for the cases where
the pole of a loss characteristic at infinite frequency is double. This particular
condition is fully explained in Chapters 3 and 5; it is necessitated by the re
quirements of the underlying theory of filter synthesis.
The series reactances of a TA structure [Fig. 1.I(b)] again become
jfl' 2s-1,TA = jWLzs-1,TA (1.4)
s = 1, 2, ... v+ 1
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8
I) LADDER COEFFICIENTS
while the shunt susceptances become
}·;/;1o7) 2s,TA -_ j2wLC 2s,TAC (1.5)
1-w 2s,TA 2s,TA
s = 1, 2, ... v
and the last of these is
jflzv+2,TA = jWC2v+2,TA (1.6)
(J'.,Q,t"""'
It is convenient to consider this last susceptance as consisting of a finite
capacitance in series "'.ith an in~u~tance of zero value, so that the resul~ant
I\ tuning frequency of this branch is mfinite.j
It will nowT e possible to combine equations (1.1) to (1.6) into a general
set of relations valid for any of the T structures considered here, whether TS
or TA. The susceptance of the last shunt branch in the TS structure is as
sumed (for the sake of generality) to be zer£2:£~~tt¥.;g.J.fil:.Toii~p"Qii<ling'
tapacity to zero (open circuit). . Lz.-i
or a genera struc ure ._rrm---o
jfl' 2s-1,T = jWLzs-1,T (1.7)
'fl - -j(w-5C:2!_s,_T) 2 (1.8)
} 2s,T - l
Wcos
s = 1, 2, ... v+ 1
where
(1.9)
and
C2v+2,T = 0, for TS structure (1.10)
L2v+z,T = 0, for TA structure
The low-pass ladder structure shown in Fig. 1.2(a) consists of v whole 'm
derived' II sections, again comprising 2v + 1 branches. Such a symmetrical
configuration will now be called a ITS structure (II-symmetrical structure).
The ladder in Fig. l .2(b) is furthermore terminated by a single 'constant-k'
half-section and thus comprises 2v + 2 branches; hence, this configuration is
called a IIA structure (II-asymmetrical structure). The shunt branches of a II •
structure are capacitive susceptances
jflzs-1,n = jwC2.-1,n (1.11)
s = 1, 2, ... v+ 1
9
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LADDER NETWORKS AND COEFFICIENTS [Ch
for both ITS and IIA structures. The series branches consist of resonant
reactances
·~ _ jwL2.,n
(1.12)
J 2.r,n - -( w )2
1 --
Wcos
s=l,2, ... v+l
where
1
27r/cos = Wcos = -;::====- (1.13)
.J
L2s,nC2s,n
and where, by analogy with the corresponding treatment of TS and TA struc-
tures, i:=:::t I~
L2v+i,n = 0, for ITS structure (1.14)
C2v+2,n = 0, for ITA structure
The resonant reactances in the series branches (1.12) now present open .
circuits at appropriate frequencies / ¥j~~ no power appears in the load
00.,
resistance terminating the filter. The loss characteristic of a II low-pass net-,
work thus h~s Jl.Ol«~~t thei;_e Jreg~ncies.I It is not necessary to provide [
00•
with a special suffix'. T or II, as these fre uencies are iven b numerical de
sign parameters tabulated in this book), a licable to both T and II ladders.
e efimtion of ladder coefficients can be readily established from the ex
pre~sions for reactances and susceptances of both T and II configurations
wh'ct'fier symmetrical or asymmetrical. It is convenient to introduce now a
variable n which is non-dimensional a g,roportional to frequency; it will
e called the normalized frequency and is related to f by some convenient
fre uency fB, which will be chosen on the basis of the properties of the re
\\ quired loss response of a low-pass filter. Let then I
£
n = = ~ (1.15)
fB WB
Analogously
n = foos =
(1)00!_ (1.16)
cos fa (J)B
so that the poles of a filter loss characteristic are related to this normalizing
frequency. For a low-pass filter
~-~n ... > i (1.11)
It will be convenient at this stage to introduce the complex normalized fre
quency variable
p =jil (1.18)
to facilitate further theoretical treatment and to simplify the analysis.
10
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