BIBLIOTHEEK TU Delft P 1294 2293
T H E S Y N T H E S I S O F C O U P L E D T R A N S M I S S I O N - L I N E NETWORKS R E A L I Z I N G ALL-PASS CHARACTERISTICS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS. Ir. H. J. DE WIJS, HOOGLERAAR IN DE AFDELING DER MIJNBOUWKUNDE,
VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 28 APRIL 1965,
DES NAMIDDAGS TE 4 UUR
DOOR
WILLEM J O H A N DIDERIK STEENAART ELEKTROTECHNISCH INGENIEUR
GEBOREN TE BATAVIA
DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. Ir. B. D. H. TELLEGEN
Aan Ellen
Aan Henri, Nancy, Claire en Vivien
CONTENTS
1. INTRODUCTION 7 2. A CONFORMAL REPRESENTATION RELATING
TRANSMISSION-LINE NETWORKS TO LUMPED
NETWORKS 15 2 . 1 Introduction 15 2. 2 The Conformal Mapping 15
2 . 3 The Influence of the Conformal Mapping on the
Network C h a r a c t e r i s t i c s 19 3 . REPRESENTATION OF THE TRANSMISSION LINE
AS A NETWORK ELEMENT 27
3 . 1 Introduction 27 3. 2 The T r a n s m i s s i o n Line a s a Network Element;
Representation in the Complex Frequency Plane 28 4 . THE SYNTHESIS OF CASCADED
TRANSMISSION-LINE NETWORKS 35 4 . 1 Introduction 35 4 . 2 T r a n s m i s s i o n - L i n e Networks Realizing Z e r o s
of T r a n s m i s s i o n on the jw -Axis 35 4 . 3 The Cascaded T r a n s m i s s i o n - L i n e Network;
Analysis and Synthesis 37 4 . 4 The Cascaded T r a n s m i s s i o n - L i n e Network,
Constrained to have all C h a r a c t e r i s t i c Impedances
G r e a t e r than One 44 4 . 5 Synthesis from given T r a n s m i s s i o n Coefficient 51
4 . 6 Synthesis from given T r a n s f e r Impedance; Example 53
5. COUPLED TRANSMISSION LINES 60
5 . 1 Introduction 60 5 . 2 Coupled T r a n s m i s s i o n Lines; the T r a n s f e r Matrix 60
5.3 The Coupled-Line Element as A l l - P a s s Network 67 6. SYNTHESIS OF COUPLED-LINE ALL-PASS NETWORKS 71
6 . 1 Introduction 71 6. 2 Synthesis Based on the Equivalence with a
Cascade of n T r a n s m i s s i o n Lines 72 6 . 3 Another Set of Boimdary Conditions Leading to
a Coupled-Line A l l - P a s s Network 75 6.4 Synthesis using Combinations of Lower O r d e r
Elements 77 7. THE APPROXIMATION OF WIDEBAND
GROUP-DELAY FUNCTIONS 81
7 . 1 Introduction 81 7. 2 Review of Group-Delay Approximation P r o c e d u r e s
for Lumped Realization 81 7.3 Approximation of Wideband Group-Delay Functions
to be Realized by a Cascade of n Coupled-Line
Elements 83 7.4 Example 85 REFERENCES 92 SUMMARY 96 SAMENVATTING 98 ACKNOWLEDGEMENT - CURRICULUM VITAE 100
1. INTRODUCTION
This t h e s i s i s to be a contribution towards the synthesis of n e t works realizing given p h a s e o r groupdelay functions for b r o a d -band application in the u l t r a - h i g h frequency r a n g e . The realization will be by m e a n s of coupled t r a n s m i s s i o n l i n e networks with a l l -p a s s c h a r a c t e r i s t i c s .
Broadband p u l s e - t r a n s m i s s i o n s y s t e m s , broadband antenna s y s t e m s , r a d a r s y s t e m s using pulse compression all need b r o a d -band group-delay e q u a l i z e r s and networks that r e a l i z e a given
( e . g . linear) delay c h a r a c t e r i s t i c .
The t r a n s m i s s i o n - l i n e network consisting of line elements of equal length has network c h a r a c t e r i s t i c s that a r e periodic v e r s u s frequency, and has d i s c r e t e (sampled) functions ( s e r i e s of d e l t a -functions) in time as its i m p u l s e - r e s p o n s e function. It should be emphasized that the r e s t r i c t i o n to equal length i s essential to obtain frequency c h a r a c t e r i s t i c s with a single periodicity. The length of the line elements is not a p a r a m e t e r to be v a r i e d , since this will r e s u l t in frequency c h a r a c t e r i s t i c s with multiple p e r i o d -icity, which cannot be compared with the frequency c h a r a c t e r i s t i c s of a l u m p e d - p a r a m e t e r network, a s is done for elements of equal length.
The l u m p e d - p a r a m e t e r a l l - p a s s network " ', in its lattice or b r i d g e d T realization, is used extensively where phase or g r o u p -delay fvinctions a r e to be r e a l i z e d o r c o r r e c t e d in the lower frequency r a n g e s . We a r e h e r e concerned with the u l t r a - h i g h frequency range where distributed networks a r e usually employed
and subsequently a realization of an a l l - p a s s network for this frequency range will be d i s c u s s e d . Lumped networks fail to have predictable c h a r a c t e r i s t i c s approximately at the beginning of the UHF range (300 M c / s ) , unless v e r y special precautions a r e taken, due to p a r a s i t i c effects and the v e r y small inductance and c a p a c i -tance values of the lumped e l e m e n t s .
Microwave networks, usually thought to c o m p r i s e only the w a v e guide applications, have in recent y e a r s also included t r a n s m i s sionline n e t w o r k s . These a r e r e a l i z e d in coaxialline or s t r i p -line ' form, which, in contrast to the waveguide applications, allow for a v e r y wideband network realization in the UHF r a n g e , due to the absence of a lower cutoff frequency for the only p r o p a -gating mode, the TEM mode. This also allows for the application of Kirchhoffs l a w s . Higher modes occur only at much higher frequencies. It is therefore possible to apply lumped-network realization and approximation methods, with certain adjustments, to the design of t r a n s m i s s i o n - l i n e n e t w o r k s . The analogy with r e a c t i v e networks is v e r y distinct, since the t r a n s m i s s i o n - l i n e elements a r e a s s u m e d to be l o s s l e s s , which i s t r u e to a g r e a t e r extent than the assumed freedom of dissipation of lumped
r e a c t a n c e s .
The realization and approximation p r o b l e m s for distributed n e t -w o r k s , although related to those of lumped net-works, have to be considered s e p a r a t e l y , due to the special configurations possible with s i n g l e - t r a n s m i s s i o n - l i n e and c o u p l e d - t r a n s m i s s i o n - l i n e e l e m e n t s , depending upon the terminating conditions, and due to the periodic frequency v a r i a b l e .
It is thereforff^)ossible to r e a l i z e periodic attenuation or phase, respectively group-delay c u r v e s a s functions of frequency. As
the frequency band of i n t e r e s t usually c o n s i s t s of one period or p a r t of a period of the total function, the remaining periods will not be of influence if the signal h a s been previously bandlimited, so that it falls only within the required frequency r a n g e .
The single-line element is well known in its shorted or open-end v e r s i o n as a o n e - p o r t . The two-port application, consisting of one o r n equal-length lines in cascade, with different c h a r a c t e r i s t i c impedances, is an often encountered network which has been used in filter realization ^ " " '. We will show in the following chapters that it may even b e t t e r be used as a phase equalizer, if one can t o l e r a t e the associated l o s s function, or as a model for a coupled-line a l l - p a s s equalizer network.
To obtain a l l - p a s s c h a r a c t e r i s t i c s , a p a i r of coupled l i n e s , con-sisting of a s y m m e t r i c a l s t r u c t u r e of two lines surrounded by a third (ground) conductor, is needed. The four-port network thus obtained displays a l l - p a s s c h a r a c t e r i s t i c s if the two p o r t s at one end a r e shorted and the two p o r t s at the other end s e r v e a s input, respectively output t e r m i n a l s , and a r e each terminated in the c h a r a c -t e r i s -t i c impedance of one line in -the p r e s e n c e of -the o-ther (fig. 1). The other p a r a m e t e r of the coupled-line element, the coupling factor, is determined by the proximity of the conductors. The a l l p a s s c h a r a c t e r i s t i c s of this s t r u c t u r e have first been r e p o r t
-(7) ed by Jones and Bolljahn
1 3 -o
o-©
i
oo
ffnTt (cross dttctiont)F\g. 1.1 Coupied-I'me all-pass network
When s e v e r a l coupled-line f o u r - p o r t s a r e cascaded, each having
i t s own coupling factor, but retaining the s a m e value of c h a r a c -t e r i s -t i c impedance, and -the s a m e -termina-ting condi-tions apply -to the end sections (fig. 2), an a l l - p a s s c h a r a c t e r i s t i c r e s u l t s that i s of higher o r d e r ; i . e . whereas the two-port of fig. 1 may be d e s c r i b e d by a polynomial of degree one, the one of fig. 2 may be described by a polynomial of degree n .
n-1
fffrff
fig. 1.2 A cascade of n coupled-line elements, terminated to give all-pass characteristics
A cascade of two coupled-line f o u r - p o r t s was f i r s t shown by Schiffman ^ ' , using lines of vinequal length, however. A detailed analysis of the f i r s t - and s e c o n d - o r d e r networks of equal length by the author ', shows that an n - o r d e r delay network may be built, just a s i s done in the lumped c a s e , by using combinations of f i r s t - and s e c o n d - o r d e r networks only.
Another example of an a l l - p a s s s t r u c t u r e i s the meander line ' , which i s illustrated in fig. 3 .
ivhii
Fig. 1.3 Meander line
O 0 - | O O --O 0-1 O 0 -•9 <? I I Ó 6-| o o-untn
It i s comparable to the coupled-line element of fig. 1, but c o n s i s t s of 2n conductors, each of which i s considered t o be coupled to the
two adjacent ones only. The m e a n d e r line h a s been used in a r e c e n t t h e s i s by Dunn ^ ' to r e a l i z e approximations to l i n e a r delay
functions. The s t r u c t u r e of fig. 2, however, h a s g r e a t e r flexibility since each coupled-line element may be assigned a different
coupling factor, which leads to a wide choice of possible s i n g u l a r -ity p a t t e r n s in the complex-frequency plane r e p r e s e n t a t i o n . The nimiber of s h o r t s r e q u i r e d i s reduced to the minimum p o s s i b l e , namely one, which i s an advantage over the meander line, since t h e s e s h o r t s interrupt the field p a t t e r n and p r e s e n t an impedance discontinuity of small but finite length.
Another advantage of the cascade of n coupled-line e l e m e n t s i s that the synthesis p r o b l e m , like that of a c a s c a d e of n directional c o u p l e r s , may be reduced to the synthesis of a c a s c a d e of n t r a n s
-(14)
mission lines ^ ' . It should be noted that a cascade of n coupled-line e l e m e n t s , depending on the boimdary conditions applied to the f o u r - p o r t , may function a s a directional coupler o r an a l l - p a s s network. The possibility of relating the synthesis p r o c e d u r e of both types of networks to that of ihe t r a n s m i s s i o n - l i n e network
(12) h a s been found recently; for the directional coupler by Levy ' and Young ', while the equivalence for the a l l - p a s s network i s shown in chapter 6 of this t h e s i s .
In contrast to the lumped realization of an n - o r d e r a l l - p a s s network, which i s usually avoided due to alignment difficulties, the n o r d e r a l l p a s s coupledline s t r u c t u r e , like any t r a n s m i s -sion-line network r e a l i z e d in s t r i p - l i n e or coaxial-line, may be made to close t o l e r a n c e s in dimension, so t h e r e i s no r e a s o n to favor a combination of f i r s t - and s e c o n d - o r d e r sections from this viewpoint.
In the following c h a p t e r s we will mainly investigate the synthesis
methods and in addition consider the approximation problem. The relation between lumped and distributed l o s s l e s s networks of equal length i s given by a conformal mapping. In other words: after the
(15) conformal mapping function, orginally given by Richards , i s applied, the trigonometrical distributed-network functions become rational lumped-network functions.
This conformal mapping is examined in detail in chapter 2, e s p e -cially the relation between the group-delay function of a distributed network and the one obtained after transformation into a rational function. It is not possible to u s e existing groupdelay a p p r o x i m a -tions computed for lumped networks directly for the approximation
of group-delay in distributed networks; but a recomputation of a modified lumped-delay function will lead to a r e a l i z a b l e c a s e . This i s further discussed in chapter 7.
A general introduction to the representation of the l o s s l e s s t r a n s -m i s s i o n line as a two-port and its lu-mped equivalent: a reactance lattice, is given in chapter 3 .
The synthesis of a c a s c a d e of n t r a n s m i s s i o n lines i s considered in chapter 4, whereby we emphasize the realization of the t r a n s f e r impedance which we need due to the l a t e r to be exposed equivalence with the cascade of n coupled-line e l e m e n t s . Also d i s c u s s e d in chapter 4 i s the synthesis of a c a s c a d e of n lines under the constraint that all c h a r a c t e r i s t i c impedances a r e to be g r e a t e r than one, which i s the value of the n o r m a l i z e d load r e s i s t a n c e . This is n e c e s s a r y because the equivalent coupled-line section cannot be r e a l i z e d if this condition i s not fulfilled.
Chapter 5 d i s c u s s e s the basic equations of the coupledline e l e ment a s derived from the t e l e g r a p h e r s equations for coupled t r a n s -mission lines ' , and the conditions leading to an a l l - p a s s
network and the subsequent reduction in the m a t r i x equations d e -scribing the network, for c a s c a d e s of 1 to n coupled-line e l e m e n t s . The basic relationship that allows for the reduction of the synthesis of four-port cascaded a l l - p a s s networks to the synthesis of two-port cascaded t r a n s m i s s i o n lines is derived in chapter 6. Examples of e x p r e s s i o n s for the t r a n s f e r functions of both networks a r e shown for n = 1 up to n = 3 .
The concluding chapter 7 c o n s i d e r s the n e c e s s a r y approximation methods, which a r e adapted from the existing limipeddelay a p p r o x i -mation methods.
Summing up, this t h e s i s gives the following additions to the p r e s e n t state of the a r t of broadband delay function approximation and r e a l i -zation:
1. It shows the relationship of the approximation procedure for
distributed-network delay functions to the approximation procedure for lumped-network delay functions, so that existing methods may be used to t h e i r fullest extent.
2. It shows the relationship of the sjmthesis method of a cascade of n sections of coupled l o s s l e s s t r a n s m i s s i o n lines, terminated such that a twoport with a l l p a s s c h a r a c t e r i s t i c s r e s u l t s , to the a l -ready existing synthesis method for cascaded t r a n s m i s s i o n l i n e s , so that a given pole configuration, representing a t r a n s f e r function
to be realized, may be t r a n s l a t e d into the c h a r a c t e r i s t i c i m p e dances of the line sections, which subsequently may be t r a n s -lated into the coupling factors of the coupled-line sections. 3. For l o w - o r d e r c a s e s it gives the limitation to be imposed upon
the pole configuration of an a l l - p a s s function that is to be realized by a cascade of n coupled-line e l e m e n t s , which is due to the r e q u i r e m e n t that all coupling factors a r e to be g r e a t e r than
one. For the cascaded-line section serving as a model to the coupled-line synthesis, this leads to the requirement that all characteristic impedances are to be greater than one.
2. A CONFORMAL REPRESENTATION RELATING TRANSMISSION-LINE NETWORKS TO LUMPED NETWORKS
2 . 1 INTRODUCTION
In the preceding introductory chapter it has been mentioned, that approximation p r o b l e m s o c c u r r i n g in connection with distributed networks a r e often comparable to s i m i l a r p r o b l e m s for lumped n e t w o r k s . Thus the available solutions to approximation p r o b l e m s for lumped networks may be useful for distributed networks too, with the exception of approximations to group-delay functions, a s will be examined in this c h a p t e r .
The conformal mapping needed in the c a s e of l o s s l e s s t r a n s m i s s i o n -line networks will be considered f i r s t , followed by a comparison of the frequency c h a r a c t e r i s t i c s of distributed and lumped n e t w o r k s . 2.2 THE CONFORMAL MAPPING
A distributed-network function, represented by its singularities in one complex plane, i s mapped into another complex plane where its singularity pattern may show properties familiar to a lumped-network function. This forms the basis of comparison between the two types of function.
The lumped-network function i s usually represented by its singu-larities: its zero's of transmission and its natural modes. These form a finite pattern in the complex p-plane, ccxistrained by certain well established rules.
The l o s s l e s s transmission-line network consisting of line elements or coupled-line elements may be characterized by an infinite num-ber of singularities in the complex plane, which, by way of contrast
with the p-plane representation of the lumped network, i s named the s-plane. If all the line e l e m e n t s a r e of equal lenght, as will be a s s u m e d , the infinite singularity pattern in the s-plane consists of a repeating finite p a t t e r n . The s-plane may then be divided into horizontal s t r i p s that will each contain one period of the p a t t e r n .
Each s t r i p of the s-plane contains all the n e c e s s a r y information r e g a r d i n g the network function.
The representation of l o s s l e s s t r a n s m i s s i o n - l i n e functions in t e r m s of transcendental functions and transformation of these into
(15) rational fimctions, has been given by Richards '. The transformation '
-^ =tQnh sT ( 2 . 1 )
changes a transcendental function, r e p r e s e n t e d by an infinite s i n -gularity pattern in the s-plane, into a finite pattern in the p-plane which may be interpreted a s pertaining to a network of lumped elements only. The p-plane normalizing frequency 9. will be taken to be equal to one in the following discussion.
The complex v a r i a b l e s T = crT + j oj T may be r e l a t e d to the e l e c -t r i c a l leng-th 0 of -the -t r a n s m i s s i o n lines: -the imaginary p a r -t of
sT equals B . Subsequently an expression for the p a r a m e t e r T, which d e t e r m i n e s the period on the frequency axis of the s-plane may be obtained, since:
\
or: T= _ 2-n-t - I
w X ''
where t i s the physical length of the line and c is the propagation velocity: c= —^
C-ü
Ol Ol O) OlN
n
N
H nH
Ol Ol Ol a> 01 c Q. I l/lfig. 2.) Comporison ofp-, s- ond z-planes
E x p r e s s e d in r e a l and imaginary p a r t s (1) b e c o m e s : ^ + j n = t a n h ( a + j u » ) T (2.2) which m a y be s e p a r a t e d into:
^= 2 T ^r\^lr 2 T <2.3)
COS* CO I + t a n n CT [. e i n OÜ I and: -0.= i o n u i l (2,4) cosh^ CrT + tan^ (jjT.sinh 0"TEquation (3) shows that v = 0 leads to ^ = 0, which d e m o n s t r a t e s that the jw-axis of the s-plane m a p s into the jfi-axis of the p - p l a n e , and since the signs of corresponding ^ and a values a r e equal, it can be seen that functions positive r e a l in the p - p l a n e , a r e positive r e a l in the s-plane a l s o , and v i c e - v e r s a .
F o r a = 0, (4) i s reduced to
i l = t Q n oüT (2.5) which may be considered a s a substitution of frequency v a r i a b l e
that transforn[is frequency c h a r a c t e r i s t i c s of the transcendental functions into those of the rational functions.
The Inverse of (3) and (4) a r e given by: T A ( i - t ) 2 + n 2
and: 0 0 = ^
2 V(i-^^-n^)^An^
(2.6)
(2.7)
The equations (3) - (7) d e s c r i b e in m o r e detail the relation (1), and allow for the mapping of singularity p a t t e r n s from one plane to the o t h e r .
We win now investigate in how f a r frequency c h a r a c t e r i s t i c s , such a s attenuation and phase o r group-delay, may be r e l a t e d by the substitution of the frequency v a r i a b l e s according to (5).
The transformation (5) may be used cm r e a l and imaginary p a r t s of network functions, so that the frequency c h a r a c t e r i s t i c s in one r e p r e s e n t a t i o n may be redrawn m e r e l y by a point for point r e -scaling p r o c e s s . This applies also to magnitude and phase functions. An exception o c c u r s for the group-delay function, which i s defined a s the derivative of the phase fimction. This will be examined further in section 2 . 3 .
The transformation (1) i s closely associated with two other t r a n s formations encountered in distributed s y s t e m s and in s a m p l e d -data s y s t e m s ' . I t may be obtained by applicaticai of the follow-ing two conformal mappfollow-ings in sequence:
z = e^*^ (2.8) and P " ^ <^-^) Equation (8) i s the z - t r a n s f o r m . The imaginary axis of the s-plane
m a p s onto the u n i t - c i r c l e on the z - p l a n e ,
Equation (9) i s a b i l i n e a r relation which r e l a t e s a rational function r e p r e s e n t e d in the z - p l a n e , with the u n i t - c i r c l e a s frequency a x i s , to a rational function r e p r e s e n t e d in the p p l a n e , with the i m a g -i n a r y ax-is a s frequency a x -i s . Deta-ils about the t r a n s f o r m a t -i o n s (8) and (9) a r e included in fig. 1.
2 . 3 THE INFLUENCE OF THE CONFORMAL MAPPING ON THE NETWORK CHARACTERISTICS
This investigation will show in what r e s p e c t the frequency c h a r a c -t e r i s -t i c s of e i -t h e r lumped o r dis-tribu-ted ne-tworks change, when the transformation (1) o r i t s i n v e r s e i s applied. In the following section s e v e r a l types of function will be used which, in the p-plane r e p r e s e n t a t i o n , fall in t h r e e c a t e g o r i e s , namely:
a. fimctions with z e r o ' s at oo only, b . functions with z e r o ' s at p = + 1 only, c . a l l - p a s s functions. a . F u n c t i o n s w i t h z e r o ' s a t oo o n l y These a r e r e p r e s e n t e d by: F(p) 1 (2.10) i=i with a. > 0; /3. > 0. 1 1
The function c o n s i s t s of n / 2 complex p o l e - p a i r s .
F o r simplicity we will consider the c a s e n = 2, which gives the following r e p r e s e n t a t i o n for the function, i t s magnitude and group-delay c h a r a c t e r i s t i c s respectively, a s function of the r e a l
frequency Ü:
F(p) =
p^ + a p + S lF(jn)l j f c n ) for p = jS2 The magnitude function:
And the delay fimction:
a'
(2.11) (2.12) T ( n ) = - R e F'(p) F(p) (19) p=jn 1 2 rF'(p), F'(-p)' F(p) F(-p) ; or a ( n ^ 6 ) ,^a\ioL^-2(i)n^+(i^ (2.13) p = j jThe t r a n s f o r m a t i o n of F(p) to the complex s-plane, by m e a n s of equation (1), r e s u l t s in the following function:
Ffs)= 1
F r o m (14) the magnitude and group-delay functions a r e derived in the s a m e m a n n e r a s has been done for the lumped-element network function:
| F ( j u , ) | 2 = ^ ( 2 . 1 5 ) t a n * U)T+(a2-26)tQn^ a)T+6
a n d T ( u ) ) = a T t a n ' coT-t-O ^ 2 . 1 6 ) C0s2 ojT [-tan^ ü)T+(Ot^-2fl)+Qn2a}T+(Ï^J
Equation (15) may be obtained from (12) by m e a n s of the frequency substitution (5), a s we have seen in the preceding discussion. The delay function (16) cannot be obtained by frequency substitution only; it also h a s to be multiplied by a factor:
±^ = ^ — (2.17) i—
An example with p a r a m e t e r values a =V2, and ^ = 1, i s shown in fig. 2. It i s evident that for the magnitude functions (figs. 2A and 2C) the mapping i s equal to a r e s c a l i n g of the frequency axis by m e a n s of (5). The groupdelay functions (figs. 2B and 2D) a r e r e -lated by (5) and (17). W h e r e a s the delay function of fig. 2B r e a c h e s z e r o at the frequency fi = oo , the delay function of fig. 2D r e t a i n s a finite value at the corresponding frequencies. F o r comparison the function obtained by frequency r e s c a l i n g alone i s also shown in fig. 2D.
Delay functions for n poles may be e x p r e s s e d a s a sum of t e r m s like (13) or (16).
The network function of o r d e r n i s :
F(P)=7^ 2 J n (2.18)
•• 1 p ^ - i - a - p + ( 3 ^
I
with a j > 0; /3. > 0 .
.m
+ -e c 15 II G V H / ( \ \1
/ / / 1 / / ''1 1
/ 1-The corresponding delay function is:
T(X1)=Z a^ — ^ ' (2.19) '=^ n +(0^1 -2(3;)/! +(3i
It should be noted that, for n = odd, a single pole at - a on the negative real axis contributes a term:
T ( X 1 ) - ^ ^ (2.20)
n +a
to the total delay expression, which has to be added to (19).
In the s-plane representation, the network function comparable to (18) is:
n/2
F(s)=7r ; (2.21) i»1 tanh' sT+a^tonh sT + S^
The delay function, which compares to (19), is:
T ( a ) ) = - J — ^ a; ^°"'^"^"^i (2.22) cos*OüT i=1 tQn*ü)T+(ai^-2(3i) ton*u)T+0f
To use existing delay approximations directly by means of fre-quency substitution is not possible. We can, however, approxi-mate a certain delay function in the p-domain, using known
techniques, usually iterating computing procedures, if the function is distorted beforehand to allow for later multiplication with the factor (17), which in terms of the p-plane becomes:
iOi =T(1+I1) (2.23)
d u j
Thus the delay function to be approximated should be multiplied 1 2 - 1
by 7p (1 "•" fi ) , to be followed by the approximation procedure that would have been used on the initial function if only the lumped version were of interest. The delay characteristic of the distrib-uted network may then be derived by a substitution of the frequency variable, according to (5).
b . F u n c t i o n s w i t h z e r o ' s a t p = + 1 o n l y In t r a n s m i s s i o n - l i n e networks a function 6(p) = _ i l : P j (2.24) "/^^ IT (p2 + a i P + 6 i ) 1=1
i s often encountered, since the t r a n s f e r impedance of a cascade of t r a n s m i s s i o n lines i s of this form, which will be used in c h a p t e r s 3 and 4 .
The introduction of z e r o ' s of t r a n s m i s s i o n at p = + 1 on the r e a l axis changes the. magnitude of the function, as compared to equation (10). Equation (12) becomes for n = 2:
IGCJH)!' = - ^ V ^ ^ H 2 <2.25)
n +(a - 2 ö ) n + 6
The introduction of the z e r o ' s at + 1 h a s changed the magnitude fimction (fig. 3) considerably, a s compared with the previous function (12), shown in fig. 2 for the s a m e ex. and /3 values, namely a = \ 2 , / 3 = 1 .
The delay v e r s u s frequency function, however, i s unchanged and equal to (13) respectively (16). Introduction of z e r o ' s s y m m e t r i c -ally located in right and left halves of the p - p l a n e , i s of no influ-ence on the phase, respectively the group-delay function. This m a k e s that a cascaded t r a n s m i s s i o n - l i n e network, r e p r e s e n t e d by (24), has a delay function that in i t s p-plane r e p r e s e n t a t i o n i s equal to that of a function with all z e r o ' s at infinity and the s a m e pole configuration (10).
c . A l l - p a s s f u n c t i o n s The a l l - p a s s function
i.i p ' + aip+fl-j
Fig. 2.3 Magnitude characteristics of functions with zeros o t p = ± 1 o r s = ± o o
has a magnitude equal to one, due to the fact that the zero pattern in the right halfplane is symmetrical to the pole pattern in the left halfplane:
|FQ ( j n ) | ' = i
The corresponding delay function is:
T ( n ) . I , ^°'.'^'-°^', , ,2.27,
The increase in delay by a factor 2 as compared to (19) is due to the contribution of the zero configuration.
The delay expression for the distributed case is equal to (22), after multiplication by a factor 2.
We see that all group-delay functions under (a), (b) and (c) are equal, except for a factor 2, while the corresponding magnitude functions are not. The delay-function approximation procedure may thus be the same for all three cases.
3. REPRESENTATION OF THE TRANSMISSION LINE AS A NETWORK ELEMENT
3 . 1 INTRODUCTION
In the preceding chapter the conformal mapping was p r e s e n t e d that r e l a t e s the transcendental t r a n s m i s s i o n - l i n e functions m the s-plane to rational (lumped-element) functions in the p - p l a n e .
H e r e , e x p r e s s i o n s will be derived for the single t r a n s m i s s i o n - l i n e element, its t r a n s f e r m a t r i x and its equivalent lumped-network r e p r e s e n t a t i o n . The system functions will be given in both the s -plane and p--plane r e p r e s e n t a t i o n , which a r e related through con-formal transformation ( 2 . 1 ) .
The s y s t e m functions needed h e r e a r e :
a. Immittance functions, positive r e a l functions in both p l a n e s , transcendental in the s-plane and rational in the p - p l a n e . b . Reflection coefficients, functions of bound one. These a r e
derived from immittance functions by m e a n s of the bilinear relation: R_Z.
I R+Z;
with p.: the reflection coefficient at port i, Z. : the impedance seen at p o r t i, and R : the r e f e r e n c e r e s i s t a n c e .
T r a n s m i s s i o n functions, pertaining to the t r a n s f e r between two p o r t s of the network. Usually the t r a n s m i s s i o n coefficient will be used, which, for a l o s s l e s s two-port t e r m i n a t e d at both ends in r e s i s t a n c e , is defined a s :
1. /. N|2 _ Power delivered to load ' ' Available power
The relation to the reflection factor i s :
l t ( j w ) | ^ = 1 - | p ( j w ) | ^ (3.2) with the reflection factor a s defined under b .
F o r a two-port network between source r e s i s t a n c e R.. and load r e s i s t a n c e R„ the t r a n s m i s s i o n coefficient may also be defined
where V i s the s o u r c e voltage and V„ i s the voltage a c r o s s the load.
Another t r a n s m i s s i o n function that will also be frequently e n -countered i s the t r a n s f e r impedance:
defined in the usual m a n n e r for a network between a c u r r e n t s o u r c e , with infinite shunt r e s i s t a n c e , and a load r e s i s t a n c e commonly taken to be 1 ohm.
3 . 2 THE TRANSMISSION LINE AS A NETWORK ELEMENT; REPRESENTATION IN THE COMPLEX FREQUENCY PLANE F o r a single l o s s l e s s t r a n s m i s s i o n line, the following t r a n s f e r m a t r i x i s well known: " c o s e jZg sine ^ sine o COS
e
(3.3)where Z i s the c h a r a c t e r i s t i c impedance of the line, and 27r
Ö = /3t = ^r-f i s the e l e c t r i c a l length.
The concept of the quarter-wavelength line should also be mention-ed h e r e : it i s a line of physical length i that equals the q u a r t e r w a v e l e n g t h - 7 ^ at a c e r t a i n frequency f . The e x p r e s s i o n for the e l e c t r i c a l length then becomes: 0 = 6 _ O = 2TI _ O = 5 U J .
A X A 2 IOQ
The relation to t h e complex v a r i a b l e S = CT+JU) i s : jQ=ju)T
or: sT=T[+j9=T(cr+jüü)
E x p r e s s i n g the line p a r a m e t e r s a s functions of s, by m e a n s of substitution of: s T = i 0 in (3), we obtain: A C B D cosh sT 1 sinh «T .^0 2g sinh sT cosh sT (3.4)
O r with the v a r i a b l e p , introduced according t o t h e t r a n s f o r m a -tion (2.1):
A B C D
1
W7
1 pZc (3.5)In t e r m s of t h e m a t r i x elements of (4) the input impedance of a line t e r m i n a t e d in R = 1 i s :
y/^\ _ A + B _ cosh sT+ ZQ sinh sT
C+Ï5 1 sinh sT+ cosh sT
^ 0
o r , e x p r e s s e d in the complex v a r i a b l e p , we obtain:
l+pZ^
( 3 . 6 )
Z ( p )
P +1 ( 3 . 7 )
If the line i s t e r m i n a t e d in R = 0 o r R = oo , the input impedance b e c o m e s respectively:
B
Z L ( s ) = - | = Z ^ l Q n h sT or Z L ( P ) = P Z O ( 3 . 8 )
and
'-'^'-Z-ïéLf - ^cCp)4°
(3.9)The input impedance in the p-domain then b e c o m e s the equivalent of an inductive r e a c t a n c e (8) r e s p e c t i v e l y a capacitive r e a c t a n c e (9)
It should be noted that t h i s equivalence f o r m s the b a s i s of the comparison with l u m p e d - e l e m e n t n e t w o r k s , the mapping function (2.1) is based upon it.
The t r a n s m i s s i o n l i n e , t e r m i n a t e d in an a r b i t r a r y r e s i s t a n c e R, h a s a t r a n s f e r impedance: R _ R ^ 2 1 ^ ^ ^ = C R ^ cosh sT + ^ sinh sT ''o (3.10) o r Z 2 , C p ) = R [ I - P ^ 1 ^ 1 + p j - (3.11) The p o l e - z e r o p a t t e r n of (10) and (11) i s shown in figs. 1 and 2,
7 7
r e s p e c t i v e l y for the c a s e s that « ^ > 1 and -^ < 1. The locations of the poles in the s-plane corresponding to t h e s e two c a s e s can b e found from fig. 2 . 1 and from (2.T). The single line section i s unusual in sofar that the z e r o s of t r a n s m i s s i o n in the p-plane a r e of o r d e r 5. When the line i s t e r m i n a t e d in i t s c h a r a c t e r i s t i c impedance (Z = R), the r e s u l t i n g singularities a r e , in the p -plane, a pole of o r d e r 5 at p = - 1 , and a z e r o of o r d e r g at p = "H. F o r the c a s c a d e of two t r a n s m i s s i o n l i n e s , of equal lengths and with c h a r a c t e r i s t i c impedance values Z , and Z.., the overall t r a n s f e r m a t r i x b e c o m e s : cosh sT Z j sinh sT 4 - sinh sT sh sT sh sT sin h s T Z l ••h sT cosh sT c o s h ^ s T + - ^ sinh sT ( Z . | + Z 2 ) s i n h sT cosh sT ( J - + 4 " ) sinh sT cosh sT cosh^ sT+—1 s i n h ^ sT
J O
0/2)
- Z o / R -1 o -(V2) Zo/R J W +1 o-0/2)
p-plane
- | . „ | ^ ^ :
fig. 3.1 Singularities of (3.10) and (3.11) for ^ > 1
j3Tr/2T JTT/ 2T s-plane -JT1/2T -J3TT/2T j n Jw p-plane (1/2) . 1 l n R l | o - ^ T R-Zo
fig. 3.2 Singularities of (3.10) and (3.11) for '^'' < )
Jïï/T
s-plane
—
In the p-plane r e p r e s e n t a t i o n (12) becomes:
( J _ + ^ ) p u | l _ p 2
The t r a n s f e r impedance in the s-plane i s :
(3.13) Z21 i n (s) the co6h2sT+R(J-p-plane: Z21 (p) = - + Zl Zz =l-)sinh sTc ^2 R(1-p2) p 2 + R ( ± . ^1 Dsh sT
f)p
' ^ +1 sinhZ sT (3.14) (3.15)The singularities corresponding to (14) and (15) a r e shown in figs. 3 and 4 .
The poles can b e anywhere in the left halfplane.
When the two c h a r a c t e r i s t i c impedances a r e equal, the t r a n s f e r m a t r i x (12) b e c o m e s :
cosh^ sT+ sinh^ sT 2Z- sinh sT cosh sT
y- Sinh sT cosh sT cosh^sT+ sinh^ sT
(3.16) o r in the p-plane: 1 1-o2 1 + p2 2p 2pZo (3.17)
which may be r e p r e s e n t e d by the lattice network of fig. 5, which, when ternainated
c h a r a c t e r i s t i c s .
X
\ J3ïï/2T
XI jT,y2T
X Xi Jïï/2T
X XI j3Tr/2T
X s-plane Fig. 3.3 Singularities of (3.14) X -1 X +1 p-plane fig. 3.4 Singularities of (3.15) 33L = Z,
- ^
k ^ p-^lant
/
fig. 3.S Lattice representation of loss/ess transmission lines Fig 3.6 Singu/orities of (3.18)»
(15)
This r e p r e s e n t a t i o n h a s been given by R i c h a r d s ^ ', without derivation, however. The singularities of the t r a n s f e r impedance, for a r b i t r a r y termination R, a r e shown in fig. 6, since (15) b e c o m e s : R(1-p2) Z,, (p) = 2 4 . 2 R , P + 7 - P+1 '-o (3.18)
If R = Z the pole p a i r coincides with the z e r o at - 1 and the line behaves a s an a l l - p a s s section with ideal termination:
1-P 2.. (p) = R,
1-P (3.19)
The pole pair can not be imaginary for finite values of Z^ and nonzero values of R.
A transformation different from (2.1) can be found to r e p r e s e n t a t r a n s m i s s i o n - l i n e two-port, thought to be consisting of two lines in cascade of half the length each, by a lattice network of lumped e l e m e n t s , namely „ • - + „ „ ! , sT (3 20^ p»= lonh ^
sT
When tanh sT i s written as a function of tanh -^ and this i s sub-stituted in (4), the matrix (16) i s obtained. In this p*-plane representation a cascaded line network i s equivalent to a cascade of lattices of the type of fig. 5, each with its own Z value. The resulting network does not have the all-pass property, as follows from the preceding discussion.
Using the transformation (2.1), instead of (20), we do not have a readily available representation for a single-line two-port in the p-plane.
A cascade of two transmission lines with unequal characteristic impedances is described by the matrix (12); the resulting transfer impedance is formulated in (14) and (15), having a singularity pattern as in figs. 3 and 4. Similarly, the transfer impedance of a cascade of n lines has n poles in the left half of the p-plane, forming — pairs, or —r— pairs and a single pole on the negative real axis, and zeros of order — at p = + 1.
The input impedance of a cascade of lines terminated in an open circuit or a short circuit in its p-plane representation is a reactance function. If the termination is resistive, the input impedance is a positive real function. These properties of the input impedance reflect in the matrix elements that pertain to a cascade of n transmission lines. This will be further discussed in the next chapter.
The transfer impedance of a cascade of n transmission lines falls into category (b) of the function types mentioned in section 2.3, namely functions with transmission zeros at p = + 1 only.
The group delay of the cascaded-line network is therefore equal to the group delay of the functions of category (a), fimctions with all zeros of transmission at infinity, and an equal pole pattern, As pointed out in section 2.3, the group delay is also one half of the group delay of the all-pass function (category c).
4. THE SYNTHESIS OF CASCADED TRANSMISSION-LINE NETWORKS
4 . 1 INTRODUCTION
The synthesis of coupled t r a n s m i s s i o n - l i n e a l l - p a s s networks will be based upon an analogy with the synthesis of cascaded t r a n s -m i s s i o n - l i n e n e t w o r k s . This analogy, to be derived in c h a p t e r s 5 and 6, may in general be explained by noting that in using the coupled-line a l l - p a s s network, we r e s t r i c t ourselves to the
variation of only one (out of two available) p a r a m e t e r s p e r section; while in the c a s e of cascaded t r a n s m i s s i o n lines t h e r e i s also only one available p a r a m e t e r to be v a r i e d p e r section, the c h a r a c t e r -istic impedance.
The analogy r e d u c e s the four-port coupled-line synthesis problem to the m o r e familiar two-port cascaded-line synthesis p r o b l e m . It i s therefore of I n t e r e s t to review h e r e the essential points of c a s c a d e d - t r a n s m i s s i o n - l i n e synthesis in general, followed by the synthesis method needed especially for our application.
4 . 2 TRANSMISSION-LINE NETWORKS REALIZING ZEROS OF TRANSMISSION ON THE jw-AXlS
Using the transformation (2.1), the input impedance of the shorted r e s p e c t i v e l y open-circuited l o s s l e s s t r a n s m i s s i o n line b e c o m e s equivalent to the impedance of an inductor respectively c a p a c i t o r . This equivalence f o r m s the b a s i s of the comparison between l u m p ed and t r a n s m i s s i o n l i n e n e t w o r k s . And so, it i s accepted p r a c -tice to look for equivalent lumped approximation and synthesis p r o c e d u r e s to obtain distributed networks with s i m i l a r behavior.
This behavior, however, i s not exactly the s a m e due to the p e r i -odicity of the frequency c h a r a c t e r i s t i c , of which one period m a y be compared with the total c h a r a c t e r i s t i c of the lumped network. A lumped network function, determined by natural modes and without finite z e r o s of t r a n s m i s s i o n , i s prototype for many filter
(21)
functions . F o r the l o w - p a s s filter a realization a s a l o s s l e s s ladder network consisting of s e r i e s inductors and shunt c a p a c i t o r s i s u s u a l .
The t r a n s m i s s i o n - l i n e equivalent would consist of a ladder con-figuration of shorted l i n e s , replacing the inductors, and opencircuited l i n e s , replacing the c a p a c i t o r s . To e a s e the c o n s t r u c -tion difficulties that this s t r u c t u r e p r e s e n t s , a cascade of two-port line elements of unit impedance i s introduced between the network and the (unit) load r e s i s t a n c e . The line e l e m e n t s a r e then t n t e r
-(4) changed with the stub lines according to the identities of Kuroda^ ' and a network r e s u l t s that i s realizable in p r a c t i c e . The p o l e - z e r o pattern i s unchanged by the addition of the line elements; in the
s-plane r e p r e s e n t a t i o n the z e r o s occur at + (2k -i- l ) i r / T ; k = 0, 1, 2 00 , and a r e equal in o r d e r to the number of poles in a horizontal s t r i p of the s - p l a n e .
It has also been t r i e d to r e a l i z e filter c h a r a c t e r i s t i c s by using cascaded line elements only ' , without any stub l i n e s . As we have seen in chapter 3 , this r e s u l t s in a t r a n s f e r impedance that c o n s i s t s of n poles in the left half of the p-plane and z e r o s of o r d e r n / 2 at p = + 1. In the s-plane representation, however, no finite z e r o s occur on the jw - a x i s , and this c a u s e s the filter c h a r a c t e r i s t i c s to have little attenuation between p a s s b a n d s at the frequencies oi = (2k "i- 1) T T / T .
We will continue by considering a network consisting of cascaded t r a n s m i s s i o n lines only and emphasize its usefulness as a model for p h a s e - c o r r e c t i n g n e t w o r k s .
4 . 3 THE CASCADED TRANSMISSION-LINE NETWORK; ANALYSIS AND SYNTHESIS
A cascaded t r a n s m i s s i o n - l i n e network c o n s i s t s of n line e l e m e n t s of equal length, each having a different c h a r a c t e r i s t i c impedance. An analysis of the single t r a n s m i s s i o n line and two t r a n s m i s s i o n lines in cascade has been given in section 3 . 2 . By multiplication of n t r a n s f e r matrices., r e p r e s e n t i n g n t r a n s m i s s i o n lines in c a s c a d e , a t r a n s f e r impedance i s obtained of which all the z e r o s of t r a n s m i s s i o n occur at p = + 1 in the p - p l a n e , o r at <r = + oo in the s - p l a n e . We a r e interested to know what r e s t r i c t i o n , if any, i s to be placed upon the location of the poles in the left half of the p p l a n e . It i s the s y m m e t r i c a l location of the z e r o s of t r a n s m i s -sion that m a k e s the c a s c a d e d - l i n e t r a n s f e r function so i n t e r e s t i n g for the purpose of phase o r group-delay c o r r e c t i o n . Due to t h e i r s y m m e t r i c a l location the z e r o s a r e of no influence on the phase c h a r a c t e r i s t i c of the network.
In publications ^ ^^ '^ ' sofar the c a s c a d e d - t r a n s m i s s i o n - l i n e network h a s been used only a s a filter o r i m p e d a n c e - t r a n s f o r m e r network which, due to its lack of z e r o s of t r a n s m i s s i o n between p a s s b a n d s , h a s need of added stub elements to r e a l i z e some l o s s between p a s s b a n d s . If the cascaded-line filter has to be designed between equal t e r m i n a t i o n s , the only way to r e a l i z e some l o s s in between the passbands is to allow for sharply varying c h a r a c t e r -istic impedances of adjacent line elements '; or if unequal t e r m i n a t i o n s a r e allowed, these contribute towards the l o s s in between the p a s s b a n d s .
It i s pointed out h e r e , however, that the cascaded-line network itself would be ideally suited a s a p h a s e - c o r r e c t i n g device in c a s e s where the associated l o s s function would be t o l e r a b l e . Where this i s not so we have to u s e a coupledline a l l p a s s n e t -work, for which the cascaded-line network may be used a s a prototype in design.
Initially we will consider which pole locations for the t r a n s f e r impedance a r e possible, this to be followed by a discussion of the known realization methods of R i c h a r d s ' ' and Riblet ' ' which lead to equivalent r e s u l t s .
F o r a cascade of n lines the t r a n s f e r impedance has — pole p a i r s in the p-plane r e p r e s e n t a t i o n , for n = even (or has ï^zL pole p a i r s and a negative r e a l pole for n = odd). We will now investigate whether o r not the locations may be assigned freely. As usual the functions will be p r e s e n t e d a s rational functions in the p-plane, r a t h e r than to work with the transcendental s-plane functions. The t r a n s f e r impedance b e c o m e s for n i s even:
^21^^^=-;;^-^^ ; a i > O ; 0 . > O ( 4 . 1 )
77"(p2+a.n+(3i)
o r for n i s odd:
Z2i(p)= "''(„.i)/2 .oc,>0,(i,>0,Y>0 (4.2) (p+y) TT (p2+aip+(3i)
i=l
F o r given poles in the left half of the pplane the t r a n s f e r i m p e -dance may be formed according to (1) o r (2), and subsequently the t r a n s f e r m a t r i x o r the input impedance of a cascade of t r a n s
mission Lines that realizes the transfer impedance has to be derived.
The transfer matrix of n cascaded lines obtained by the multi-plication of n matrices (3. 5) may be written as:
A C B D 1 (1-p2)"^2 A„(P)
k^p)
Bn(P> Dn(p) (4.3) (14) The conditions on the polynomials of (3) are:A (p) and D (p) even in p, C (p) and B (p) odd in p, with
for n = even: A (p) and D (p) of degree n, C (p) and B (p) of degree n - 1 ; and for n = odd:
A (p) and D (p) of degree n - 1 , C (p)and B (p) of degree n. The reciprocity condition is:
AD - BC = 1
or A„(p)D„(p)-B„(p)C„(p)-C1-p»)'^ (4.4)
The input impedance of a cascade of n transmission lines, termi-nated in a unit resistance, i s , in terms of the polynomials in (3):
Z. (p) = A ^ , ^ " ^ ^ " (4,5)
The transfer impedance may also be written in terms of the poly-nomials:
Z„(p) = (1-p2)" /2
C,(p)-.D„(p) (4.6)
A given set of poles determines the polynomials C and D , which are the odd and even parts respectively of the denominator, within
a constant multiplier, which may be determined from the condition Z 2 i ( 0 ) = l .
since for z e r o frequency the cascaded line network b e c o m e s t r a n s p a r e n t .
The polynomials C and D , for even n, may be r e p r e s e n t e d by:
n-1 n-3
Cn=Cn-lP +'^n-3P + + «^l P (4-7)
K-<^r. p " + d , . 2 p " " ' - ^ + do (4-8)
The unknown polynomials A and B a r e of the form:
An = ° n P + an-2P + "^ °o (4-9)
B n = b n - l p " " ' + ^ - 3 P " " ^ + + h P (4.10)
Substitution of equations (7) - (10) into (4) and separation of (4) into n -)- 1 equations according to equal powers of p , r e s u l t s in a set of n "I- 1 l i n e a r equations with n "i- 1 unknowns, namely the coefficients a. and b . . 1 1
We will proceed to prove that, for any given Hurwitz polynomial C -t- D , the resulting input impedance (5) i s a positive r e a l function. Since we s t a r t with a Hurwitz polynomial in the denom-inator of (5), and poles on the jfi-axis a r e excluded, the conditions to be satisfied to make (5) a positive r e a l function a r e :
Z(p) i s r e a l for p r e a l , and Re Z(jJ2) > 0 for p = jü.
The linear set of equations derived from condition (4), when solved, yields r e a l coefficients a. and b . , satisfying the first condition for positive reality of the input impedance. To see
whether o r not the second condition is satisfied we form the r e a l p a r t of (5): ReZ(jn)= A n ^ n - BnC, 2 2 n n p ' j n ( 1 - p 2 ) " D' •C, 2 ' ( 1 + n ) p»jn.
D'(n)+c'(n)
Both n u m e r a t o r and denominator of this expression a r e positive for p = jfi, and the condition for positive reality of (5) i s thus satisfied for any Hurwitz polynomial C + D .
The analysis i s concluded h e r e , since we have shown that from any given configuration of poles in the left half of the p-plane (excluding the jfi-axis) a t r a n s f e r impedance may be formed, that m a y be r e a l i z e d by a cascade of n lines; we will l a t e r on indicate that to form the t r a n s m i s s i o n coefficient of a cascade of n lines from a given pole pattern is not directly p o s s i b l e . Our i n t e r e s t is mainly in the t r a n s f e r impedance, however, since it will be shown in chapter 6 that the comparison with the coupled-line a l l - p a s s network is on the b a s i s of the t r a n s f e r impedance only.
Once the t r a n s f e r m a t r i x or the input impedance of the network is given, the realization procedure follows one of two methods:
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Riblet premultiplies the m a t r i x (2) with the i n v e r s e of the m a t r i x of a single t r a n s m i s s i o n line, of which the c h a r a c t e r i s t i c impedance i s determined by the condition that the elements of the r e s u l t i n g m a t r i x a r e to be of lower d e g r e e . This p r o c e s s i s repeated until all n c h a r a c t e r i s t i c impedances a r e known. In his publication Riblet formulates e x p r e s s i o n s for each of the c h a r a c -t e r i s -t i c impedances of a cascade of n lines in -t e r m s of -the coeffi-cients of the polynomials A and C of the overall t r a n s f e r m a t r i x (2). An example i s given in section 4 . 6 .
R i c h a r d s ^ ' h a s derived a method based upon s u c c e s s i v e reduc-tions of the input-impedance function. His theorem (15a) s t a t e s
that, if Z (p) i s a positive real rational function and Z (p)/Z (1) i s not identically equal to p o r l / p *, the function
i s a l s o a positive r e a l rational function. A corollary s t a t e s that a factor (p - 1) always cancels in Z.. (p), and that an additional factor (p "I" 1) will cancel in Z.. (p), provided the original Z (p) conforms to the condition Z (- 1 ) = - Z (1). F o r the proof of both the theorem and the c o r o l l a r y we r e f e r to the work of R i c h a r d s . It can be easily shown that for the input impedance the condition Z (- 1) = - Z (1) i s valid, since according to condition (4) we have
A (1) D (1) - B (1) C (1) = 0, n ' n ' n ^ ' n ^ ' ' from which we may derive
A (1) - B (1) A (1) "H B (1) n ' ^ " ^ n n ' ^ -C (1)+ D (1) C (1) -t- D (1) n' ' n ' n ' n ^ ' or Z „ ( p ) — Z„=Z„(1) Z (- 1) = - Z (1). "n Zn-.tp)
Jj
Fig. 4.0 Reduction of the input impedanceAccording to fig. Q, the impedance Z (p) may be reduced in d e g r e e by removing a t r a n s m i s s i o n line of c h a r a c t e r i s t i c i m p e -dance Z = Z (1) from it. The remaining impe-dance Z _, (p) i s
*) This possibility does not occur for the input impedance of a c a s c a d e of n t r a n s m i s s i o n l i n e s , since, due to condition (4), denominator and n u m e r a t o r both contain factors p and p .
of lower d e g r e e , which can be seen from fig. 1, after writing the e x p r e s s i o n s for Z (p), respectively Z _,(p):
+ 1
and
According to the t h e o r e m of Richards Z _.. (p) i s also a positive r e a l function, of degree one lower than Z (p), since a factor p - 1 cancels in n u m e r a t o r and denominator. The condition (4) i s valid for Z _.. (p) a l s o , with the index and exponent n replaced by n - 1 . The realization technique of Richards c o n s i s t s of a repeated
application of this p r o c e s s , which r e s u l t s in a chain of t r a n s m i s s i o n lines with c h a r a c t e r i s t i c impedance values: Z (1), Z ^(1)
Z..(l). The method yields identical r e s u l t s as the one by Riblet, and this proves that the c h a r a c t e r i s t i c impedances of a cascade of t r a n s m i s s i o n lines all have positive v a l u e s .
4 . 4 THE CASCADED TRANSMISSION-LINE NETWORK, CONSTRAINED TO HAVE ALL CHARACTERISTIC IMPEDANCES GREATER THAN ONE
When a constraint is added to allow for only c h a r a c t e r i s t i c i m p e -d a n c e s t o be g r e a t e r than one, the casca-de-d t r a n s m i s s i o n - l i n e network synthesis will be r e s t r i c t e d in the s e n s e that not all pole configurations of the t r a n s f e r impedance left half of the p-plane will lead to networks that fulfil this condition. However, the condition i s n e c e s s a r y to e n s u r e the existence of a realizable
coupled-line section for each t r a n s m i s s i o n - l i n e section, as will be shown in chapter 6.
The limitations on the choice of the pole configuration of the t r a n s f e r impedance have been found for c a s c a d e s of one to t h r e e lines only; for m o r e lines in cascade the realization p r o c e s s as outlined in section 4 . 3 will have to be t r i e d out in o r d e r to e s t a b -lish whether or not the network fulfils the condition.
We will proceed to derive the limitations on the pole locations for the l o w - o r d e r c a s e s .
F o r n = 1 :
As given in (3.11), the t r a n s f e r impedance of the single t r a n s m i s -sion line, terminated in a. unit r e s i s t a n c e i s :
The pole o c c u r s at ! = - Z , fi = 0, and the condition Z >1 r e q u i r e s ! < - l . It may be located on the negative r e a l axis in the interval ["1. -°°J only, as shown in fig. 1.
p-plan« -1
fig. 4.1 Excluded pan of the negative real axis (0,-1)
for the poles of the first-order function
It is obvious from fig. 2 . 1 that this p a r t of the negative r e a l axis maps onto the lines
a) = ±(2n+1) T r / T , n - 0 . 1 , 2 . . .
of the left half of the s-plane. Poles on the co = + 2nT/T lines cannot be realized by a single t r a n s m i s s i o n line with Z > 1.
F o r n = 2 :
The t r a n s f e r impedance of a cascade of two lines with c h a r a c t e r -istic impedances Z and Z.. respectively, terminated in R = 1, is according to (3.15):
Z2/P)= z ^^-'l\^ (4.15)
P V * P 7 7*^
The r o o t s of the denominator occur at:
P , . , = - ^ ± j n = - ^ ± .
Zz (Zi^Z2)2 Zi 4ZtThis expression may be solved for Z.. and Z in t e r m s of the coordinates ! and fi:
r , . . l ± ^ , ,4.16,
F o r any negative value of ! we have:
and consequently: Z > 1 .
The equation (17), however, may lead to Z„ < 1. We will proceed to find for what values of ^ and Q this is s o .
The condition for which Z = 1, is found from (17):
(X'^+n}) (^•^X,^•^•a'^)+2•s,^o (4.i8)
This curve i s shown in fig. 2. Choice of a pole p a i r in the left halfplane that falls outside or on this curve will lead to a
c h a r a c t e r i s t i c impedance Z^ > !• A pole pair chosen inside the curve will r e s u l t in Z < 1.
1.0 .8 ,6 .A
N^
\ ,2 / /fig. 4.2 Excluded part of the p-plane for the poles of the second-order function
The choice of the pole pattern is thus found to b e limited to p a r t of the left halfplane only. A pole pair falling inside the given curve does not lead to a r e a l i z a b l e two-section coupled-line network, but may be r e a l i z e d a s a t r a n s m i s s i o n - l i n e network with Z < 1.
F o r n = 3 :
The t r a n s f e r impedance of a c a s c a d e of t h r e e lines is c h a r a c t e r -ized by a pole pattern consisting of a conjugate-complex pole p a i r and a pole on the negative r e a l a x i s . We will derive a l i m i -tation for the pole on the negative r e a l a x i s , which will prove to be equal to the limitation in the case of n = 1. A method for deriving the limitation for the complex pole pair will be indicated.
The denominator of the t r a n s f e r impedance e x p r e s s e d in the coordinates of the poles i s , according to (1) and (4):
K(C3 + D 3 ) = ( p - ^ l ) ( p 2 - 2 ^ 2 P + ^ 2 ^ + n 2 ' > =
= p^-CCl-h2-?2)p2+(2?,t2 + ^ 2 ' * . f i 2 ' ) p - ^ l ( ? 2 ^ + ^ 2 ^ ) <4.19) Another expression for K (C„ + D„), using the c h a r a c t e r i s t i c
impedances numbered 1-3 from load to s o u r c e of the c a s c a d e of t h r e e l i n e s , i s obtained by pre-multiplication of the m a t r i x (3.13) with the m a t r i x of a single line (3.5) with Z = Z„ substituted:
K(C3 + D3) =
- P ^ p ^ ^ ( | l ^ | l ^ | l ) . p ^ ( 1 ^ 1 . J . ) . , ^ (4.20) ^2 ^3 ^ 3 '•2 ^2 ^1 ^2 ^3 ^Z
Equating (19) and (20) leads to the following t h r e e equations:
-(^1+21^2) = ^ ( | l ^ | 2 . ^ | l ) , (4.21)
2 ? , t 2 + ? 2 ' + - f i 2 ' = ^ ( ^ + ^ + f - ) , (4.22)
- ^ 1 ( ^ + ^ 2 ' ^ = ^ 7— (4.23)
^2
F r o m equations (21) - (23) e x p r e s s i o n s for the t h r e e impedances in t e r m s of the coordinates of the poles may be obtained.
The expression for Z.. i s :
Zi = - ? ^ - 2 t 2 ^~^' , (4.24) 1 + 2 ? i : ? 2 + - C 2 ' + - a 2 2
which shows that for ^-^ = - 1 we have Z = 1. Assuming Z > 1 we may write (24) a s :
( t i + 1 ) ( l + t 2 ^ + n 2 ^ + 2 t 2 ) < 0 (4.25) Equation (25) i s satisfied if ^ , < - 1 , since always:
1 + t 2 ^ + n 2 ^ + 2 ? 2 > 0
F o r Z < 1 we may also prove: ! . > - ! , from which we may conclude that the sign of (Z - 1) is only dependent upon the p a r a m e t e r f^ of the pole on the negative r e a l a x i s . This condition i s equal to the one for the c a s e of n = 1, which
excludes the interval (0, - 1) a s possible pole location to guarantee the copdition Z.. > 1.
The expressions for Z and Z a r e :
z
t^(^+t.2^•^n2^)4•2t,J 1+t2'+^2^ + 2l?1^2 (4.26) ^ , ( ^ 2 ' ' + ^ 2 ' K U ^ 2 ' + ^ 2 ^ + 2 ^ 1 ^ 2 ) + ^ i n - K t 2 ' + ^ 2 ' ) + 2 g 2 2 ^ l ' ^ 2 + ^ - ? , ? 2 ^ + 2 t 2 ( ^ 2 ^ + - f ^ 2 ^ ) 49Za = - ^ 1 ( ^ 2 ^ + ^ 2 2 ) .
(4.27)
2%A'^z^^ta2^+2t.zU2^+n2^)
Equations (26) and (27) respectively, reduce to (16) and (17) for ! .. = - 1 , since then Z = 1 and the c a s e of n = 3 i s reduced to n = 2, a cascade of two lines with c h a r a c t e r i s t i c impedances Z„ and Z„ respectively, and t e r m i n a t e d in Z = R = 1. The curve of fig. 2 applies, allowing for a choice of a pole p a i r in the left halfplane outside t h i s curve to a s s u r e a value of Z„ > 1, while Z i s always g r e a t e r than one.
Values of ^^ < - 1 , however, will lead to other conditions for the pole p a i r . Each given value of !.. < -1 may be substituted into
(26) and (27), which, when equated to one, leads to two sixth degree c u r v e s in ^ „ and Ü , comparable to the fourth degree curve (18) for the c a s e of n = 2. F o r a choice of a pole pair such that values of Z_ > 1 and Z > 1 will be obtained, it i s n e c e s s a r y to t e s t which p a r t s of the left halfplane correspond to the r e q u i r e d impedance v a l u e s .
This p r o c e d u r e rapidly b e c o m e s unwieldy for higher values of n, and will therefore not be pursued further. Each combination of poles of the t r a n s f e r impedance will have to be t r i e d with the realization method of section 4 . 3 , without advance knowledge whether o r not the resulting c h a r a c t e r i s t i c impedances will all be g r e a t e r than one.
4 . 5 . SYNTHESIS FROM GIVEN TRANSMISSION COEFFICIENT In the realization of a groupdelay function by a cascaded t r a n s -m i s s i o n - l i n e network itself, with its associated loss function, we have the advantage that the r e s t r i c t i o n to c h a r a c t e r i s t i c impedan-c e s g r e a t e r than one does not apply, sinimpedan-ce any values may be r e a l i z e d .
F o r p r a c t i c a l design, however, finite s o u r c e and load r e s i s t a n c e s have t o be allowed for, which c a l l s for the u s e of the t r a n s m i s -sion coefficient in the synthesis p r o c e d u r e r a t h e r than the t r a n s f e r impedance. The question r e g a r d i n g the allowed location for the pole configuration needs re-examination, since in the c a s e of the t r a n s m i s s i o n coefficient synthesis other conditions apply a s in the c a s e of the t r a n s f e r i m p e d a n c e . F o r equal t e r m i n a t i o n s (R = 1), the t r a n s m i s s i o n coefficient i s :
2(1-P^) "/2
^"^''^-A.Bk.D (4.28)
n n n n
with the denominator polynomials A , B , C and D a s given in equation (3).
E x p r e s s e d in t e r m s of i t s n a t u r a l modes the t r a n s m i s s i o n coeffi-cient becomes:
^•^(p)- 2K(1-p )" w i t h o . > 0 ; 7r(p^ + aiP+0i) /3. >0
Isl 1
(4.29)
Due to the requirement:
t„(0)=1 ,
since for p = 0 the network i s t r a n s p a r e n t , we find n/2
7r/3L=2K. i=i
2 n
F r o m the equality A D - B C = ( l - p ) the following expression i s obtained: n n n n "12 - ( i - p 2 ) " =
\-K
L 2 J
2 Bn-Cnl n ni
J
(4.30)The left half of (30) may be evaluated from the denominator of (29) for any given s e t of a- and /3. values; the right half may be
itten as: A -D„ B„-C" n n , n n
[ 2 ' 2 J
• n n 2Bn-C
2which shows that the z e r o s may be separated in the two halfplanes, since the z e r o s of the reflection coefficient:
, -An-Dn+Bn-C„
^ " " V D n + V C „
(4.31)should fall in the left halfplane.
The polynomials A , B , C and D may be found for any pole configuration in the left halfplane, excluding the jfl-axis and the
2 n
condition A D B C = (1p ) will be satisfied, but the r e a l i z a -n -n -n -n ^ ' ' bility of the network still depends upon the following two conditions:
The input impedance is to be a positive r e a l function in p , and [ t „ ( p ) . t ^ ( - p ) ] p ^ j S 2 l l , f o r a l l f i .
The f i r s t condition is always fulfilled, since:
the sum of n u m e r a t o r and denominator A + B "i-C + D i s n n n n a Hurwitz polynomial, A + B „ . , _ n n ^(P) ~ Q ^ Y) ^^ ^®^^ ^'^^ P r e a l , and, as in section 3, n n
The second condition leads to r e s t r i c t i o n s on the possible pole configurations in the p-plane, and will be examined for the c a s e that n=2:
F r o m (28) and (29) we have:
A2 + D2+t.2+l^2 2 + b p + a p '
^ ^ ^2(p)-^2M=^^^^p2)2_b2p2
o r t2 (j.f^)-*2 f-J^) = , j ! ^ ^ ' ^ ' , (4.33)
To make this s m a l l e r than o r equal to one for all Ü, the coefficients a and b have to fulfil the conditions:
a^^A- and k ^ A s . $ 2 . (4.34)
As an example we take a=2 and b=4; then
1+p
The denominator i s the well-knoAvn Butterworth-filter function. Several other filter functions have been obtained for the denominat o r of denominathis denominatype of funcdenomination , budenominat for denominathe purpose of g r o u p -delay equalization further work on the realizability of pole p a t t e r n s i s r e q u i r e d .
4 . 6 SYNTHESIS FROM GIVEN TRANSFER IMPEDANCE; EXAMPLE
F o r a given pole configuration of the t r a n s f e r impedance the c h a r a c t e r i s t i c - i m p e d a n c e values of the cascaded-line network will be d e r i v e d .
F r o m the coordinates of the pole p a i r s the denominator poly-nomial C + D of the t r a n s f e r impedance i s formed. The other
n n ^ polynomials a r e then derived from condition (4):
A„D„-B„C„=(1-p^)". n n n n
F o r two pole p a i r s (n = 4) at (^ + jfi ) and (^ + jfi ) in the p-plane J. J. ^ Li
we obtain;
K(C4+Dj=(p=-2^iP+?^+nt)(p'-2-^2P+^'2+n'2) =
= p ^ 2 ( t i + ^ 2 ) p ^ ( ^ C l t 2 + t i + ^ l + " ^ 2 + ^ 2 ) p '
-Since Z (0) = 1, the constant t e r m in K (C. "t- D.) should be set equal to one to obtain the denominator polynomial C . H- D . . Taking the odd and even p a r t s , respectively, of the denominator polynomial we obtain e x p r e s s i o n s for C. and D •
C^=c ^ - t j p - f C ^ p = 2 2 2 . ( ^ i + n ^ ) ( ^ 2 + ^ 2 ) -2(?i+t2) 3 2 : P 2?2 2 t i P (4.35) and D^ = d^p +d2P +1 = ^ 1 pA , A t l t 2 + ' g l + - ^ 1 + ' l 2 + - ^ 2 2^., (ti+ni)C^'2-Hn2) (^?+il^)(t2+^2) ^ (4.36)
The unknown polynomials A . and B a r e assumed to be:
^ 4 = 04? + 0 2 P -Hi ,
The coefficients a. and b. may be found from the application of condition (4):
-=i
(4.37) -(ü-Ki^/d^) 0 -C3 6-(d^-h1/d4) -C3 - c i - ( / i + d j ) - c i 0 ^4 d2 1 0 -=3 -=1 - = 3 - ^ 1 0 (4.38) b3 = -(4 + d2/d4) 6-(d^+1/d^) -(-i+dz) - < = 3 0 (4.39) b , = «I. d2 1 0 -=3 -=1 -(4+d2/d4) 6-(d^+1/d4) -(4+d2) (4.40)The derivation of the additional polynomials is n e c e s s a r y where the realization r e q u i r e s the u s e of all four polynomials, a s i s the
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c a s e for the impedance realization of Richards ' . The cascadei line network i s , however, completely determined by two of the
(14)
four poljmomials. The realization method of Riblet ' u s e s only the polynomials A and C, which differ somewhat from ours since
