A P P L I E D T O
E L E C T R I C A L N E T W O R K S
PROEFSCHRIFTTER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGE-SCHOOL TE DELFT, OP GEZAG VAN DE RECTOR MAGNI-FICUS DR O. BOTTEMA, HOOGLERAAR IN DE AFDELING DER ALGEMENE WETENSCHAPPEN, VOOR EEN COM-MISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENS-DAG 27 JUNI 1956 DES NAMIDWOENS-DAGS TE 4 UUR DOOR
JAN LOURENS BORDEWIJK
Electrotechnisch ingenieur GEBOREN TE MEPPEL 'S-GRAVENHAGE MARTINUS NIJHOFF 1956 / - * ^ \ ƒ i D«elenstr.l01 > 1
\i Delft J I
den heiligen Name van God!"
G u i D o G E Z E L L E
A A N MIJN OUDERS A A N MIJN VROUW
SUMMARY 1 INTRODUCTION 1 CHAPTER I. CONCEPT OF RECIPROCITY 4
§ 1 The reciprocity relation 4 §1.1 Definition of a reciprocal 2n-pole 4
§ 1.2 Examples of reciprocal 2n-poles 4
1.2.1 Two-pole 4 1.2.2 Ideal transformer 4
1.2.3 Coupled coils 5 § 2 Relation between external properties and internal
struc-ture 5 § 2.1 The topological theorem 5
§ 2.2 2n-pole composed of reciprocal 2v-poles 6
§ 3 Matrix symmetry 7 § 3.1 General reciprocity relation in matrix notation . . . . 7
§ 3.2 Admittance (impedance) matrix 7 3.2.1. Symmetry properties with a reciprocal 2n-pole . 7
3.2.2 Conditions for reciprocity 8
§ 3.3 Mixed matrix 9 3.3.1 Symmetry properties with a reciprocal 2n-pole . 9
3.3.2 Conditions for reciprocity 11 3.3.3 Note on sign convention 11
§ 3.4 Scattering matrix 12 CHAPTER I I . CONCEPT OF I N T E R - R E C I P R O C I T Y 14
§ 1 The inter-reciprocity relation 14 § 1.1 Definition of two inter-reciprocal 2n-poles 14
§1.2 Examples of inter-reciprocal 2n-poles 15
1.2.1 Gyrator 15 1.2.2 Triode 15 1.2.3 Reciprocal 2»'-pole 16
§ 2 Relation between inter-reciprocity and internal structure . 17 § 2.1 The topological theorem for two networks of
topologic-ally identical structure 17 §2.2 Two 2n-poles composed of inter-reciprocal 2j'-poles . . 18
§ 3 Matrix relationship 19 §3.1 General inter-reciprocity relation in matrix notation . 19
§ 3.2 Admittance (impedance) matrix 19 3.2.1 Relationship with inter-reciprocal 2n-poles . . . 19
3.2.2 Conditions for inter-reciprocity 19
§ 3.3 Mixed matrix 20 3.3.1 Relationship, with inter-reciprocal 2n-poles . . . 20
3.3.2 Conditions for inter-reciprocity 20
§ 3.4 Scattering matrix 20 CHAPTER I I I . TRANSPOSITION 20
§ 1 Definition of transposition 20 § 2 Transposition of the reciprocal network elements . . . . 21
§ 3 Transposition of non-reciprocal network elements . . . 21
§3.1 Choice of symbols f or triode and transistor 21
§ 3.2 Execution of the transposition 25
3.2.1 Gyrator 25 3.2.2 Triode 26 3.2.3 Dualtriode 28 3.2.4 Transistor 28 3.2.5 Dualtransistor 28 3.2.6 Summary for §§ 3.2.2-3.2.5 29 3.2.7 Isolator 29 3.2.8 Combination of transposition and duality
con-version 30 3.2.9 "Black box" 2r-pole 31
§ 4 Some transposition theorems 31 CHAPTER IV. APPLICATIONS TO AMPLIFIER CIRCUITS 33
§ I Introduction 33 § 2 Grounded-anode and grounded-grid connection 34
§ 3 Influence of cathode feedback on the input and the output
impedance 36 § 4 "Transparent" amplifier 36
§ 5 Transistor negative-resistance converters 38 § 6 Classification of feedback amplifiers 39
§ 7 Symmetry properties of a three-stage repeater 41 § 8 Cathode hybrid-coil arrangements with wide-band
re-peaters 43 § 9 Reactance-tube circuits 44
CHAPTER V. APPLICATION TO PROBLEMS ARISING FROM THE
THEORY OF FEEDBACK AMPLIFIERS 45 § 1 Invariance of return difference 45
§ 1.1 Objective 45 §1.2 Return difference of a triode 45
§ 1.3 Return difference of a unilateral four-pole 47 §1.4 Some practical consequences of theorems V-1 and V-2 . 50
§ 2 A transmission problem 53 CHAPTER VI. T W O - P O L E NOISE-FORMULAE FOR NETWORKS WITH
NON-RECIPROCAL E L E M E N T S 54
§ 1 Definitions 54 § 2 Noise of a gyrator two-pole 55
§ 3 Noise of a triode two-pole 57 § 3.1 Equivalent noise resistance of a triode two-pole . . . 57
§3.2 Internal resistance of the physical triode 60
§3,3 Illustrations 61 3.3.1 Parallel connection 61
3.3.2 Cascade connection 62 § 4 Noise of a transistor two-pole 63 § 5 Correlated noise-sources; resistors with unequal
tempera-tures 65 § 5.1 Definitions 65 § 5.2 Physical representation of correlation 65
CHAPTER VII. TRANSPOSITION AS AN A I D IN THE CALCULATION OF
SIGNAL-TO-NOISE RATIOS 67 § 1 Noise factor for a given frequency 67
§ 2 Application of transposition 68
§ 3 Illustration 70 § 4 Cascade formula 71
NETWORKS
S u m m a r y
As an extension of the concept of reciprocity a new concept called
inter-reciprocity is introduced. The latter concept applies to both reciprocal and
non-reciprocal structures. On behalf of the use of inter-reciprocity for electric-al network investigations a unique topologicelectric-al network operation, celectric-alled
transposition, has been devised in such a way that a network and its transpose
are inter-reciprocal. This transposition is applied in a number of direct calculations as well as in considerations of a more general character concerning transmission and noise properties of gyrator, triode and transistor networks.
INTRODUCTION
It is a matter of common knowledge that the reciprocity relation for electrical networks holds good for only a limited class of linear networks, viz. the class of so-called reciprocal networks, to which belong inter alia networks composed of resistors, capacitors and inductors. To non-reciprocal networks, to which belong, amongst others, the majority of circuits with gyrators, triodes, transistors and the like, the reciprocity relation does not apply, as is evident by definition.Thus it seems as if the advantages obtained in applying the reciprocity relation in the study of the properties of reciprocal networks must be dispensed with in the study of non-reciprocal net-works.
In the present work an inquiry is made into a relation, which is called the inter-reciprocity relation, between two linear electrical net-works, which in themselves may be non-reciprocal, and which are associated one with the other in a certain manner. In the study of non-reciprocal networks the advantages obtained from the inter-reciprocity relation roughly compensate for the loss of validity of the reciprocity relation.
In general there is more than one way of associating one network with another in such a manner t h a t the two networks are inter-reci-procal. However, to deduce a number of rules that are of practical use it is important to introduce a topological operation on a linear electrical network that leads in a unique manner to another network, so that the two networks are inter-reciprocal. It is possible to define such an operation which is reversible, and which will be indicated as transposition. The literal meaning of the word transposition (from the latin trans and ponere) indicates clearly the topological operation to be carried out on a number of network elements in the case of transposition of the network. For example, with transposition of an arrangement in which are included gyrators and triodes, each gyra-tor will be "turned over" and for each triode the grid and anode will be interchanged *). Transistors are also subjected to a transposi-tion, which is specified in chapter I I I § 3.2.4. and § 3.2.8, and which is physically significant in combination with a duality operation.
In addition to its apphcation to the general theory of non-recipro-cal networks, transposition often affords a good insight, and t h a t surprisingly quickly, into the properties of electric networks, for example, involved feed-back arrangements which are being more and more applied in modern electronics. Furthermore transposition tends to minimize and simplify calculation and also leads to methods t h a t might hardly be found without its aid, as for example in its applications to the noise problems of chapters VI and VII.
In the first three chapters the theoretical basis is laid for the con-cepts inter-reciprocity and transposition. We assume that the charac-teristic parameters of the elements of which the networks are com-posed are independent of both current and voltage, and also inde-pendent of time. We therefore avail ourselves of the term "constant linear networks". The following four chapters include a number of applications.
Chapter I deals with the classical concept of reciprocity. With a view to the concept of inter-reciprocity to be considered in chapter I I , the classical reciprocity concept is dealt with in such a general manner that the parallel between the reciprocity of a network and the inter-reciprocity of two networks is established as clearly as
*) Cf. '). There the term "transposition" is not used, but the term "reversion" is applied to triode networks. In applying the latter word to other unilateral network elements, its scope of meaning is not large enough.
possible. Thus various derivations could be very summarily dealt with in chapter I I . Conversely, the study of the inter-reciprocity concept has advantageously influenced the treatment of the classical reciprocity concept and contributed to a more general description for it. With regard to the definition of a reciprocal network, and t h a t of two inter-reciprocal networks, use is made in chapters I and II of a formulation of the reciprocity relation as was already applied by R a y 1 e i g h ^). Furthermore a general network theorem as applied by T e l l e g e n ^ ) is utihzed. The latter theorem enables us to co-ordinate the properties of reciprocity of a network and those of the elements of which the network is composed. In chapter I, finally, the properties of various matrices of a reciprocal network are considered, and in analogy therewith in chapter II those of two inter-reciprocal networks are considered.
In chapter I I I the operation of transposition is introduced and carried out with reference to a number of network elements. To achieve a logical arrangement a new proposal is made for a choice of symbols for triodes and transistors, a proposal that is to a large extent in line with the existing symbols. Thereafter the combination of transposition and duality conversion is discussed. Chapter I I I then concludes with a number of transposition theorems which are of importance with regard to the applications that are considered in the subsequent chapters.
In chapter IV the transposition concept is used for obtaining a ready insight into a number of triode amplifier arrangements, both as to the internal properties of reciprocity and as to the mutual relationship of various circuits. As an example of the change in a transistor circuit brought about by means of a combination of transposition and duality conversion, a "negative-resistance con-verter" is discussed.
In chapter V transposition is applied to some problems in the theory of feed-back amplifiers. Special attention is given to the behaviour of the return difference as defined by B o d e i^).
In chapter VI with the aid of transposition two-pole noise formulae are derived for gyrator, triode and transistor networks.
In chapter VII it is shown how noise-factor calculation can be replaced by a calculation of the power attenuation of the transposed network, which often leads to a quicker insight and moreover to a simpler calculation.
CHAPTER I. CONCEPT OF RECIPROCITY
§ 1. The reciprocity relation.
1.1. D e f i n i t i o n o f a r e c i p r o c a l 2n-p o 1 e. Let us consider an electrical network with n terminal pairs. Such a network is usually referred to as a 2w-pole network or briefly 2n-pole. In the present investigation the latter term is always used to indicate a constant linear 2n-pole, so that use can be made of the so-called com-plex notation for currents and voltages and their ratios.
- o - ^ = —
'k
Fig. I-l. Symbols and sign convention at the ft'* terminal pair of a 2n-pole. Let /(. and V^ be the complex effective quantities of current and voltage at the ^'^ terminal pair of the 2M-pole, with the sign conven-tion as indicated in fig. I-l. Now the following is defined:
DEFINITION I - 1 . ^ 2n-pole is called reciprocal if
S 7 X = S / X , (I-l)
the summations being taken over all the terminal pairs. The unprimedquantities refer to an arbitrary state 7j., V^ of the 2n-pole, whereas the primed quantities refer to a second arbitrary state l'^, V'y. of the same 2n-pole.
A 2«-pole to which (I-l) does not apply will be called non-reci-procal.
1.2. E x a m p l e s of r e c i p r o c a l 2w-p o 1 e s 1.2.1. Two-pole. For a two-pole for which
V = ZI (1-2) we obviously have IV' = VjZ • ZI' = VI', so that (I-l) holds for
every two-pole. Hence the following theorem can be given: THEOREM I - 1 : Every two-pole is reciprocal.
1.2.2. Ideal transformer. For an ideal transformer with equations
I i = - TI„ (1-3) V, = TV, (1-4)
it is true that I^V', -f I.^V'^ = — TI^V'j_ -f I2TV1 = 0 and also / j F j -\- I;iV2 = 0. Thus the ideal transformer satisfies (I-l).
THEOREM 1-2: Every ideal transformer is reciprocal.
1.2.3. Coupled coils. For a set of n coupled coils with equations
F , = 2 " z , , / , , k= ],2, ...,n, (1-5) i = l
with Z^j = Zjj., we find:
fc=n fc= n j= n j = n h = n j ^n
* = 1 l- = l j - l ; = 1 A: = l 3 = 1
so that (I-l) is also satisfied by a set of n coupled coils. THEOREM 1-3: Every set of n coupled coils is reciprocal. § 2. Relation between external properties and internal structure.
2.1. T h e t o p o l o g i c a l t h e o r e m . In what follows a general network theorem, discussed by T e 11 e g e n 2) and referred to in the introduction, will be indicated as the topological theorem. In the assumed complex notation it reads:
THEOREM 1-4: In a network configuration, imagine branch currents I such that for every node Sƒ = 0, imagine branch voltages V such that for every mesh 2 F = 0 and for every branch associate the positive senses of current and voltage in the same manner (e.g. as in fig. 1-2). Then
1,1V = 0 (1-6) where the summation is over all branches.
If the terminal pairs of the network are interpreted as branches as
/
>1
Fig. 1-2. Symbols and sign conventions for a branch of a network used in formulating the topological theorem.
well, and the same sign convention is used as in fig. I-l and fig. 1-2, the theorem can be re-written as follows:
2 7 , F , = 2 / , F , . (1-7) The first summation in (1-7) is over all the terminal pairs k and the
second over all the internal branches b. The formulation of the theorem is such that the currents and voltages (1-7) may be associated with two different arbitrary states of the network, which will be in-dicated b y primed and unprimed quantities, respectively. Thus
2 / , F ; = 2 7 , F ; . (1-8) but also
2 / ; F , = 2 / ; F , . (1-9) Networks of which the n'g/i^hand members of (1-8) and (1-9) equal
one another,
2 / , F ; = 2 / ; F „ (I-10) satisfy (I-l), for (I-l) is identical with the equality of the /e/^hand
members of (1-8) and (1-9). Under the definition of § 1 networks of this kind are reciprocal. With the aid of condition (I-10) deduced from the topological theorem it can thus be concluded from certain properties of the internal branches whether the network is reciprocal or not. This property has been defined according to (I-l), on the basis of properties of the external terminal pairs.
2.2. 2«-p o l e c o m p o s e d of r e c i p r o c a l 2»'-p o 1 e s. Wc shall now consider a 2n-pole t h a t is composed of reciprocal 2v-poles. To these 2i'-poles (I-10) applies in virtue of the definition of a reciprocal 2«-pole (I-l). For the 2i'-poles the index b has the same function as the index k for the external terminals of the entire net-work.
As proved in 2.1 a network, for the internal branches of which (I-10) holds, is reciprocal, so t h a t the following conclusion can be given:
T H E O R E M 1-5: A 2n-pole composed of reciprocal 2v-poles is recipro-cal.
In 1.2 a two-pole, an ideal transformer and a set of coupled coils were quoted as special examples of reciprocal 2j'-poles. Thus the following theorem can be given:
THEOREM l-5a: A 2n-pole composed of two-poles, ideal transformers and coupled coils is reciprocal.
§ 3. Matrix symmetry.
3.1. G e n e r a l r e c i p r o c i t y r e l a t i o n i n m a t r i x n o t a t i o n . In the present § a few symmetry properties of certain matrices of reciprocal 2n-poles will be deduced. Therefore we first re-write the general reciprocity relation (I-1) with the aid of a matrix notation. We then make use of the rule in matrix algebra, relating to the product of two vector-matrices A and B with elements ^ 1 , ^ 2 , ...,A„ and S^, Bg, . . .,B„:
A B = BA = A,B, + A2B2 + . . . + A,B,, (I-l 1) By the --^ - sign used in matrix algebra for indicating transposed matrices it is denoted that the vectors defined as one-column matri-ces are transposed to one-row matrimatri-ces for multiplication purpose (cf. e.g. % Introducing
h
h
In and V = V^v^
Vnthe relation (I-l) can be written, from (I-l 1), as.
I V ' = r V . (1-13) 3.2. A d m i t t a n c e ( i m p e d a n c e ) m a t r i x .
3.2.1. Symmetry properties with a reciprocal 2n-pole. A 2w-pole can be described by n independent linear relations with constant coeffi-cients between the currents and voltages at the terminal pairs. If at each terminal pair the voltage is chosen as the independent variable, the so-called admittance equations are obtained. If the notations of fig. I-1 are used, the n relations in question read:
ƒ, = 2 V , , F , , ^ = 1,2, . . . , « . (1-14)
3 = 1In matrix notation:
I = YV, (1-15) where Y is the admittance matrix.
of a second arbitrary state of the same network can be written as
r = YV'. (1-16)
If the 2M-pole considered is reciprocal, (1-13) establishes a relation between (1-15) and (1-16), from which a property of Y can be derived.
Elimination of I and I' from (1-13) with the aid of (1-15) and (1-16) results in
(YV),V' = (YV'),V (1-17) in which the index t has the same meaning as a ~ ' -sign over the
whole bracketed form. If a few well-known matrix-algebraical rules
are applied *), the left-hand member of (I-17) can be converted into
(YV),V' = VYV' (1-18) and the right-hand member can be made to read:
(YV'),V = V(YV') = VYV'. (1-19) Thus (1-17) becomes
VYV' = VYV'. (1-20)
Now (1-20) must be valid for all values of V and V', because in (I-l) and (1-13) we consider two arbitrary states of the network. Thus
Y = Y, (1-21) or
y,i = yi,- (1-22) If in the foregoing we replace all quantities and operations b y their
duals, we find in a similar manner
Z» = ^ « . (1-23) where Z denotes the impedance matrix of the 2«-pole network
consi-dered. Thus we can conclude that the admittance (impedance) matrix of a reciprocal 2w-pole is symmetric.
3.2.2. Conditions for reciprocity. Once more we consider two arbi-trary states of the 2«-pole with eqs. (1-15) and (1-16). Assuming (1-21) to be given, (1-16) can be written as
l'^urthermore:
(YV),V' = V'(YV) = (V'Y)V = (YV'),V. (1-25) Substitution of (1-15) and (1-24) into (1-25) leads to (1-13), so that availing ourselves also of the dual of this proof, it can be concluded that a 2»-pole with a symmetric admittance (impedance) matrix is reciprocal. The results of 3.2.2 and 3.2.1 can be combined in one theorem:
THEOREM 1-6: A necessary and sufficient condition for a 2n-pole to be reciprocal is that its admittance (impedance) matrix is symmetric. 3.3. M i x e d m a t r i x
3.3.1. Symmetry properties with a reciprocal 2n-pole. For a part of the n terminal pairs, say m pairs, the voltage can be chosen as in-dependent variable, and for the remaining (n — m) terminal pairs the current can be so chosen. This leads to the so-called mixed equa-tions. For the sake of clarity in the notations, the currents and voltages at the first type of terminals are denoted by / j . and F^, respectively, and those at the second type by H^ and E,, respectively (cf. fig. 1-3).
'i
Fig. 1-3. Symbols and sign conventions u.sed with mixed etpiations.
The n relations in question then read as follows: j = m ) = n ƒ , = 2 Y^,V, + 2 M „ //,., k^\,2, ..., m, (1-26) }=\ y — m -f 1 } = m j = n £ , = 2 L,,.F,. + 2 Z „ . i / j , I = m + \, m + 2, . . ., n. (1-27) J = l j ^ m + l
If matrix notation is used, (1-26) and (1-27) can be written as
I = YV -I- M H , (1-28) E = LV -K Z H . (1-29)
An even more concise notation reads I E = D V
H 1
(1-30)in which the mixed matrix D is given by Y M D L Z (1-31) I'
1 ^' 1
= DV' 1
H'Here again it is possible to compare a certain current-voltage con-dition, as e.g. denoted by (1-30), with a second arbitrary state in which the current and voltage quantities are primed:
(1-32)
If the 2w-pole is reciprocal, it is again possible to deduce properties of matrix D from (1-30) and (1-32) with the aid of the general reci-procity relation (I-l), which, in the case of two different symbols for the current and two different symbols for the voltage, must be written as:
i v ' + H E ' = I'V + H ' E . (1-33) Elimination of I, E', I' and E from (1-33) with (1-30) and (1-32) yields
(YV + MH),V' + H(LV' + ZH') =
= ( Y V + MH'),V + H'(LV -h ZH). (1-34) With the aid of the rules used in (1-18) and (1-19), (1-34) can be converted into
V'(YV + MH) -f (LV'-K Z H ' ) ; H = ( V ' Y + H'M)V-h H'(LV-f ZH), V Y V + V M H + V ' L H + H Z H = V Y V + H ' M V + H ' L V + H'ZH), or, after re-arrangement,
V Y V + V ' M H - H'LV - H ZH =
= V YV - V LH + H MV - H ' Z H . (I 35) Now (1-35) must be valid for all possible values of the vectors V,
^kj — ^ik ' M — — T " ki ^ik > ^ki = '^ik • (1-40) (1-41) (1-42) V', H and H'. This can only be so, if the terms of similar composition arranged in corresponding places on the left-hand and right-hand sides of (1-35) are always equal, in other words:
V'YV = V'YV, (1-36) V ' M H = - V'LH, (1-37) - H'LV = H'MV, (1-38) H ' Z H = H ' Z H . (1-39) Because also (1-36) —(1-39) must apply to all values of V, V', H and
H', it follows that
Y = Y, which means that Y M = - L. „
Z = Z, „
3.3.2. Conditions for reciprocity. Conversely, it is now easily proved in the same manner as in 3.2,2. that a 2«-pole to which (1-40) — (1-42) apply is reciprocal. The results of 3.3.1 and 3.3.2 can again be expressed by one theorem:
THEOREM 1-7: A necessary and sufficient condition for a 2n-pole to be reciprocal is that its mixed matrix is symmetric in the matrix elements that determine a transfer admittance or a transfer impedance and is antisymmetric in the matrix elements that determine a transfer current or voltage ratio.
3.3.3. Note on sign convention. If, instead of the sign convention of fig. 1-3,we had chosen a sign convention in which the positive senses of currents and voltages I^, V^ were all associated in the same man-ner, but in a manner opposite to that of £(, Hi, then the formulation of theorem 7 would have become much simpler. As can easily be verified, (1-40) —(1-42) then change into
D = D, which means that D^^ = D^^. (1-43) Against the advantage of a simpler result, the following objections
hold:
(i) for practical purposes the latter type of sign convention is more involved;
(ii) the driving-point impedance at the /^, V^. terminals or at the El, Hi terminals become the opposite of the voltage-current ratios. 3.4. S c a t t e r i n g m a t r i x . In the n independent relations describing the 2M-pole we have so far chosen the terminal currents and voltages as variables. In the scattering method the terminal voltages are split into two components. The first component is the partial voltage F which is caused by a "wave" that enters the net-work at the terminal pair considered. The second component is the partial voltage T which is caused by a " w a v e " emerging at the same terminal pair. In order to enable this split-up, which is analogous to "long line" practice, the voltages are connected via certain impedan-ces to the various terminal pairs. Usually, impedanimpedan-ces characteristic of the network considered are chosen for the purpose.
<'\.*h'\
Fig. 1-4. Situation at A'* terminal pair.
To investigate the symmetry properties of the scattering matrix occurring in the scattering method, the equation (I-l) of the defini-tion of a reciprocal 2M-pole must be transformed to an expression in the new variables F and T. In fig. 1-4 the situation at the k*'' terminal pair of a 2M-pole has been outlined for the old variables /^ and Fj. as well as for the new F^ and T^. In connection with the "long line" analogy, a generator with voltage 2Fj, and an internal impe-dance Z^. has been chosen. The impeimpe-dances Z^ are to be considered as known. The voltage F^ is now imagined to be split into the following two components:
F,^, being the voltage t h a t would arise at the k*" terminal pair if all the voltages 2Fj at the terminal pairs j ^ k were zero and the generator with e.m.f. 2Fj. and internal impedance Z^. would be terminated with Zj. by the network. This is clearly an imaginary ideal case. Fj, is therefore the contribution to F^, of the so-called "incident wave".
-^* = ^fc ^ ^ i . being the remaining part of the voltage Fj., i.e. the contribution to F^. of the "emerging wave". This contribution is
composed of n parts, viz. S^^F,^, the "reflection terminal voltage" of Fj.; i.e. the supplementary voltage of F^ to the actual voltage Fj^ in the case that all the voltages F , at the terminal pairs f =/= k are zero, and furthermore (n — 1) parts Sj^jF^, which are the contributions of the voltages Fj with ƒ ^ ^ to the voltage F^^.
We then find
T, ='^S,jFj or T = S F , (1-44)
where S is the scattering matrix, which is determined by the 2«-pole and the Zj.'s. The matrix elements S^j. we call the reflection coeffi-cients, S,,j with k ^ j the transfer coefficients. Evolving these n rela-tions, we immediately find the transformation relations (cf. fig. 1-4), which read, after introducing the restriction Z,^ = Z, as
V, = F, + r , or V = F -f T, (1-45)
Ik = ^ or I = - ^ — . (1-46) Substitution into (13) yields
( F - T ) , ( F ' - T ' ) ,
—~~^ (F' -t- T') = ^ j - ^ (F + T ) ; or, in view of (I-l 1)
F T ' = F ' T . (1-47) The surprising result is t h a t (1-47) and (1-13) are of the same form.
This implies that all the symmetry properties deduced in 3.2 for the admittance matrix (impedance matrix) also hold for the scattering matrix if Z^. = Z. And because, conversely, elimination of F and T from (1-45), (1-46) and (1-47) yields (I-13) again, the following theorem holds:
THEOREM 1-8: A necessary and sufficient condition for a 2n-pole, terminated at all the terminal pairs with equal impedances, to be recipro-cal is that its scattering matrix is symmetric.
Note. If the terminal pairs are not all terminated with equal im-pedances, it can easily be deduced that
2 — | — = 2 - ^ (1-48)
k^\ -^A- t = l Z j .
and, in general
CHAPTER I I . CONCEPT OF INTER-RECIPROCITY § 1. The inter-reciprocity relation.
1.1. D e f i n i t i o n o f t w o i n t e r - r e c i p r o c a l 2n-p o l e s . If in addition to the elements enumerated in Cha2n-pter I [two-poles, coupled coils and ideal transformers) a network includes gyrators, triodes or transistors, relation (I-l) will, in general, no longer
apply. As stated in I-1.1 such a network is called non-reciprocal. By means of a modified interpretation it is possible, however, to make the relation (I-l) apply to non-reciprocal networks, in such a manner that the modified interpretation yields for a reciprocal net-work results that, to a certain extent, correspond to the results under the "old" interpretation. The "new" interpretation consists in t h a t the primed and unprimed quantities / and V are used in rela-tion, not to two states of the same network, but to two different net-works, having the same number of terminal pairs. If we do maintain the relation (I-l), these two networks show a certain relationship t h a t we will call "inter-reciprocity". Its formulation requires the following auxiliary definition.
D E F I N I T I O N I I - 1 : If a one-one correspondence is established between the terminal pairs of two 2n-poles and, moreover, those terminals of corresponding terminal pairs are made to correspond that under the sign convention of fig. 7-1 have the same reference polarity, a one-one correspondence is said to be established between the terminals of the two 2n-poles.
With the aid of definition II-1 "inter-reciprocity" can now be defined.
DEFINITION 11-2: Two 2n-poles, between whose terminals a one-one correspondence is established, are called inter-reciprocal if
2 / , F ; = 2 / ; F „ (ii-i)
the summations being taken over all the terminal pairs. The unprimed quantities refer to an arbitrary state I^, F,^ of one 2n-pole, whereas the primed quantities refer to an arbitrary state I^., F». of the other 2n-pole.
Remark. Unless otherwise stated, a one-one correspondence between the terminals is always assumed when inter-reciprocity properties of two 2M-poles are being discussed. In practical examples this one-one correspondence is given by an identical numbering and sign convention for corresponding terminal pairs.
in Chapter I, we shall now (i) by means of the topological theorem couple the inter-reciprocity of two networks with the inter-recipro-city of corresponding elements in them, and (ii) show that, when the above definition is used, the properties of a network that is inter-reciprocal with another network follow from the properties of the other network.
First, to elucidate the new notion and its definition, a few examples will be given.
1.2. E x a m p l e s of i n t e r - r e c i p r o c a l 2n-p o 1 e s. In the present paragraph three simple examples will be given by way of illustration.
1.2.1 Gyrator. The simplest non-reciprocal passive network ele-ment is the gyrator conceived by T e l l e g e n ^ ) .
) d
Fig. II-1. Symbol of the gyrator with equations
In fig. I-l a gyrator is drawn with equations
Ii = fV„
h-(11-2) (11-3) in which p is the gyration admittance. It is easy to prove t h a t two gyrators with opposite gyration admittance are inter-reciprocal. Then the equations of the second gyrator are
/; = - pvi (11-4)
I'z = PV[, (II-5) After multiplying (II-2) by (II-5) and (II-3) by (II-4) the following
is immediately derived:
hV[ = nV,, {11-6) hV'z = I[V„ (II-7) so that (II-I) is satisfied.
1.2.2. Triode. Another simple example of inter-reciprocal net-works is provided by two triodes which are derived from one another
by interchanging their anodes and grids. The triode will be charac-terized by a single, generally complex, parameter, the transadmit-tance S. The other properties of a physical triode, such as tube capacitances and internal resistance, can be accounted for by means of passive elements in the circuit.
_
P"ig. 11-2a. Symbol of the triode with equations
7i = 0, I., = SV\.
Fig. 11-26. Symbol of the triode that is inter-reciprocal with the one of fig. II-2a with corresponding equations
T, = SV'„ I', = 0.
In fig. II-2a this triode is drawn which is a 4-pole defined by
/ i = 0, (II-8) h = SV,. (II-9) The unconventional system of drawing is elucidated in Chapter
I I I . The equations of a second triode (fig. II-2&), resulting from the first one by interchanging anode and grid, read
/ ; = svi (11-10) / ; = o. (11-11) From (II-8) and (II-l 1) it follows that
hV[ = l',V, = 0 (11-12) and from (II-9) and (II-10) that
I,V', = I[V„ (11-13)
so t h a t (II-l) is also satisfied as regards the two triodes.
1.2.3. Reciprocal 2v-pole. As a third example we can give the following theorem resulting directly from definitions I-l and II-2. T H E O R E M I I - 1 : Two identical reciprocal 2v-poles, whose topological-ly similar terminals are one-one corresponding, are inter-reciprocal.
Because every two-pole is reciprocal (cf. theorem I-1), the follow-ing theorem holds:
§ 2. Relation between inter-reciprocity and internal structure. 2.1. T h e t o p o l o g i c a l t h e o r e m f o r t w o n e t -w o r k s o f t o p o l o g i c a l l y i d e n t i c a l s t r u c t u r e . The topological theorem discussed in 1-2.1 can be formulated regard-ing its application to the theory of inter-reciprocity as follows.
THEOREM II-2: In two networks composed in a topologically identical manner of an equal number of branches, imagine branch currents I for the first, I' for the second network, such that for each node 2 / = 0 and 2 7 ' = 0, respectively; branch voltages V for the first, V' for the second network, such that for each mesh 2 F = 0 and 2 F ' = 0, respectively, and for every branch take the same sign convention (e.g. as in fig. II-7). Then
2 7 F = 0 , (11-14) 2 7 ' F ' = 0, (11-15) 2 7 F ' = 0 , (11-16) 2 F F = 0, (11-17) if the latter four summations are over all the branches and each term
always comprises quantities of the same branch or of two corresponding branches.
(11-14) and (11-15) are identical with (1-6). (11-16) and (11-17) can again be applied to networks with external terminal pairs by con-sidering these terminal pairs as branches of the network. Keeping to the sign conventions of fig. 1-3 and fig. 1-4 we get
2 7 , F ; = 2 7 , F ; , (II-18) 2 7 ; F , = 2 7 ; F „ (11-19) each having at the left-hand side a summation over the external
•1 '•I
Fig. II-3. Symbols and sign conventions for two corresponding network branches used in the formulation of the topological theorem for two networks
of the same configuration,
terminal pairs k, at and the right-hand side a summation over the internal branches b. Although (II-18) and (II-19) are formally identical with (1-8) and (1-9), there is a difference in the changed meaning of the prime, which now denotes the quantities of a second network built up in a topologically similar manner. Equations (11-18) and (11-19) enable us to investigate the inter-reciprocity of two 2w-poles on the basis of their internal structure.
vi;
Fig, II-4, Symbols and sign conventions used for two corresponding terminal pairs of two 2w-poles,
2.2. T w o 2w-p o 1 e s c o m p o s e d of i n t e r - r e c i p r o c a l 2v-p o 1 e s. Let us compare two 2M-poIes that in a topologically identical manner are composed of a number of 2v-poles and in which the topologically corresponding 2v-poles are inter-reciprocal. By virtue of (II-l) the following applies to two corresponding 2j'-poles:
2 7 , F ; = 2 7 ; F , , (11-20) the summation extending over all the terminal pairs of the two
corre-sponding 2)'-poles. These terminal pairs act as internal branches in the 2w-pole considered, hence the index b.
If (11-20) is substituted for all the 2v-poles into the relations (11-18) and (11-19), equation (II-l) arises from the equality of the left-hand members of (11-18 )and (11-19), which leads to
THEOREM I I - 3 : Two 2n-poles that in a topologically identical man-ner are composed of a number of 2v-poles, are inter-reciprocal if the topologically corresponding 2v-poles are inter-reciprocal.
As a corollary we may state that two 2M-poles, that are composed of two-poles, coupled coils, ideal transformers, gyrators and triodes, and t h a t result from one another by inverting the signs of the parameters of gyration and by interchanging the anodes and grids of the triodes, are inter-reciprocal.
Remark: Theorem 11-3 contains a sufficient but not necessary
con-+ | 0
dition for two 2«-poles to be inter-reciprocal. One of the triodes in the foregoing example of the two triodes that result from each other by interchanging plate and grid may e.g. be replaced by two triodes in cascade, the cascade circuit having the same four-pole properties as the one triode. This equivalent network and the other triode are still inter-reciprocal. Generally, the relation of inter-reciprocity is not destroyed by replacing one of the 2w-pole networks by an equiva-lent network.
§ 3. Matrix relationship.
3.1. G e n e r a l i n t e r r e c i p r o c i t y r e l a t i o n i n m a -t r i x n o -t a -t i o n . Because -the general in-ter-reciproci-ty rela-tion (II-l) and the general reciprocity relation (I-l) are formally the same, it follows that the matrix notation (1-13) can also be used to represent (II-l), with the proviso of a change in the meaning of the primes.
3.2. A d m i t t a n c e ( i m p e d a n c e ) m a t r i x
3.2.1. Relationship with inter-reciprocal 2n-poles. The admittance matrices (impedance matrices) of two inter-reciprocal 2w-poles are each other's transpose. The proof of this follows 1-3.2.1 almost in its entirety, bearing in mind the changed meaning of the primes.
Because we now discuss two networks of which the admittance matrices are generally not equal and which we shall therefore dis-tinguish as Y and Y', equation (1-16) should now be replaced by
I' = Y'V'. (11-21) This is the equation for an arbitrary state I', V' of the second
net-work. The result (1-21) is then changed into
Y' = Y, which means t h a t Y'^j = Y^,^. (11-22) 3.2.2. Conditions for inter-reciprocity. Two 2n-poles of which the admittance (impedance) matrices are each other's transpose, are inter-reciprocal. To prove this, 1-3.2.2 can again be followed. In this case it is not (1-21), but (11-22) which should be the starting point. The results of 3.2.1 and 3.2.2 can be combined in the following theorem.
THEOREM 11-4: A necessary and sufficient condition for two 2M-poles to be inter-reciprocal is that their admittance [impedance) matrices are each other's transpose.
Y' = M ' = L ' = Z ' = Y, - L , - M , Z,
which means that which means that which means t h a t which means that
in.
M;, E]ci \Z'ki Xz'kk = = = Yik, Ykk, I^ik> - M , , , Zik, Zkk-3.3. M i x e d m a t r i x .3.3.1 Relationship with inter-reciprocal 2n-poles. We can again avail ourselves of the results of 1-3.3.1, if in (1-32) D is replaced by D ' . In analogy with (I-40)-(I-42) the following is found:
(II-23«) (II-23&) (11-24) (11-25) (II-26a) (11-266) Again the primed quantities belong to the second network. 3.3.2. Conditions for inter-reciprocity. These conditions follow from 1-3.3.2 and can be formulated with (II-23)-(II-26) as
T H E O R E M I I - 5 : A necessary and sufficient condition for 'wo 2M-poles to be inter-reciprocal is that their mixed matrices result c::e from the other by transposition and multiplication by — 1 of the ::iatrix elements that determine a transfer ratio of currents or voltages.
3.4. S c a t t e r i n g m a t r i x . When two scattering matrices S and S' are introduced with regard to the two networks considered, 1-3.4 leads to the following
THEOREM II-6: A necessary and sufficient condition for two 2M-poles, terminated at all the terminal pairs with equal impedancf.s, to be inter-reciprocal is that their scattering matrices are each other's trans-pose.
CHAPTER I I I . TRANSPOSITION
§ 1. Definition of transposition. It was already mentioned in § II-2.2 that the concept of inter-reciprocity does not yield a unique and reversible operation on a 2M-pole such t h a t the resulting and the original 2M-poles are inter-reciprocal. As will appear from practical applications, however, it is of importance to introduce such a unique and reversible operation, which we shall call transposition and which will be defined as follows.
DEFINITION 11 I-l. By the transposition of a 2n-pole network is understood the replacing of the separate network elements by network elements that are inter-reciprocal with regard to the aforesaid network elements.
As to the type of network element we shall restrict ourselves to resistors, capacitors, inductors, systems of coupled coils,
transfor-mers, gyrators, triodes, dualtriodes, transistors, dualtransistors, isola-tors (to be introduced below) and the "black b o x " 2j'-pole, for all of which one unique and reversible transposition result will be proved to exist, so t h a t the transposition process of Definition I I I - l is in-deed unique and reversible. In view of Definition I I I - l and Theorem II-3 we are justified in concluding:
THEOREM I I I - l : Two 2n-poles that arise one from the other by trans-position, are inter-reciprocal.
As is apparent from the remark on equivalent networks in § II-2.2 the reverse, i.e. that two inter-reciprocal 2M-poles arise one from the other by transposition, is not true in general.
§ 2. Transposition of the reciprocal network elements. The trans-position oi reciprocal network elements does not yield a new problem. According to II-3.3.1 for a mixed matrix D there holds
D ' Y M L Z Y M L Z
(III-According to (I-40)-(I-42) this can be rewritten for a reciprocal 2M-pole as D ' Y L M Z D. (111-2)
As in view of (III-2) the equations of the transposed reciprocal net-work element are equal to those of the original element, we have
THEOREM I I I - 2 : A reciprocal network element is invariant for transposition.
When use is also made of Definition, I I I - l , we can conclude: THEOREM I I I - 3 : A 2n-pole composed of reciprocal network elements is invariant for transposition.
§ 3. Transposition of non-reciprocal network elements.
3.1. C h o i c e of s y m b o l s f o r t r i o d e a n d t r a n s i s -t o r . In order -to keep -the ac-tual deriva-tion of -the -transposi-tion of non-reciprocal four-pole elements as clear as possible, we shall first deal with the symbols to be chosen for triode and transistor. In studying the transposition of the transistor, the element dual with the transistor automatically manifests itself. We shall call it dual-transistor. To round off the whole set-up it seems desirable to also
consider the element dual with the triode. We shall call that the dualtriode. With regard to the four elements triode, dualtriode, tran-sistor and dualtrantran-sistor, we propose to use the symbolism worked out in fig. I I I - l . Each of the four elements is characterized by one, generally complex, quantity, t h e c h a r a c t e r i s t i c para-meter, as given in the defining equations of column 4 of fig. I I I - l . The other properties of a physical element, such as inter-electrode capacitances, electrode self-inductances, internal resistances, and the like, we shall consider as being accounted for by means of passive elements in the network to which the element concerned belongs. Properly we should speak of ideal triode, ideal transistor, etc.
We shall call these elements unilateral on the basis of the following definition.
D E F I N I T I O N I I I - 2 : A four-pole with terminals kk' and jj' of which the transfer admittance, the transfer impedance, the transfer voltage ratio, the transfer current ratio or the transfer coefficient is finite in the direction from terminal pair kk' to jj', but is zero in the direction from terminal pair jj' to kk', is called unilateral.
As is apparent from column 6 in fig. I I I - l , the possibilities for unilateral four-poles with input and output impedance 0 or co are exhausted by the four elements mentioned.
In figs III-2 and I I I - 3 the duality conversion of triode into dual triode and that of transistor into dual transistor are worked out respectively.
An elaborate elucidation of the construction of dual networks will be found with T e l l e g e n * ) . Within each mesh of the original network a node of the dual network is placed. Each branch of the original network belongs to two meshes. Now, with such a branch, there is associated a branch of the new network connecting the nodes placed in these two meshes. With the current flowing through a branch of the original network is then associated the voltage on the associated branch of the new network, with the following convention. The arrow denoting the positive sense of the current through any branch of the original network, if turned clockwise for about 90°, must follow the direction from the -\- to the — denoting the positive sense of the voltage on the associated branch of the new network. Furthermore in the two networks, with regard to each branch, the positive sense of the voltage is always similarly associated with the positive sense of the current.
a b c d triode dualtriode transistor dualtransistor \ + +
^h.
/ O v i .
1
X
TT
.h- r\i^.
\r
I , ' .= ^ o ^0
w
I, I .^i r\_h-_
y
r
1, I . Vz + . + ^2 + l \ =•- 0 K, = 0 ƒ , - 0 K^ = /SI/, voltage input current output current input voltage output current input current output voltage input voltage output o o 0 0 oo oo 0 oo 0 pentodetransistor with low r^ in grounded emissor arrangement p — n-p and n — p — n transistor triode in cathode follower arrangement
Remark. The sign conventions obtained in this manner at the inputs of the dualtriode in fig. III-2 and of the dualtransistor in fig. I I I - 3 are the opposite of those adhered to in fig. I I I - l . If we choose for the dual triode in fig. III-2 and the dualtransistor in fig. I I I - 3 the same sign convention as in fig. I I I - l , we find, with a duality conversion factor k^, for their characteristic parameters:
F = - k-'S, (III-3) ^ = - a, (III-4) in which all symbols have the same meaning as in fig. I I I - l . The
minus signs arise from the change of sign convention at the inputs of
~^^
ii=^-Fig. III-2. Duality conversion of triode into dualtriode.
- ' ^ ^
a
>^=S-Fig. III-3. Duality conversion of transistor into dualtransistor. the dualtriode and the dualtransistor that are necessary for them to be in accordance with the sign convention of fig. I I I - l .
The sjmibols to be chosen for the four elements a-d involve a diffi-culty, since those for the physical triode and for the physical tran-sistor are well-established. The conventional triode symbol has been maintained in fig. I I I - l ; the symbols for grid and anode, however, have been drawn in a symmetrical position, which is important with a view to the execution of the transposition.
aware that the change is more or less revolutionary as to the existing order. It has such a large number of advantages, however, that it is here suggested for adoption by virtue of the following arguments.
Due to the introduction of the two elements b and d, the dual-triode and the dualtransistor, two new element symbols and six new names for the electrodes are required. Apart from the question whether they could be adequately given, the use of the large number of new symbols and names constitutes a complication that will impede the introduction of the transposition concept.
Now, the proposal as contained in fig. I I I - l offers opportunities to avail ourselves of practically all the existing symbols and names. In a general sense, a grid symbol denotes voltage control between grid and cathode. An anode symbol denotes a current source between anode and cathode. An emissor symbol denotes current control through emissor and cathode (base), and a collector symbol denotes a voltage source between collector and cathode (base).
The suggested symbol convention furthermore provides for easy and quick reading of the driving-point impedances at the inputs and outputs of the four elements. With the suggested symbols the touching of the electrodes (emissor and collector with cathode) always denotes that the driving-point impedance concerned is zero, whereas the isolated condition of the electrode (grid and anode) always means that the driving-point impedance concerned is infinite. Thus it is much easier to bear the four symbols in mind, so much so that in the author's opinion it outweighs a possible advantage yielded by an exchange of the symbols of fig. Ill-16 and fig. I l l - I c when the existing transistor symbol is maintained. Base and cathode have been similarly indica-ted since no ambiguity can arise. In the transposition of the four unilateral elements the symbol convention proves to be such that one type of electrode is always transposed into one other type only (cf. III-3.2.6). This is perhaps the most attractive argument for the convention suggested.
Remark: As a matter of possible interest we record here that there is no direct relation between the notions described in the present sections and those used by W a l l a c e and R a i s b e c k ' ' ) . 3.2. E x e c u t i o n o f t h e t r a n s p o s i t i o n .
3.2.1. Gyrator. In II-1.2.1 two gyrators with opposite gyration admittances were quoted as examples of two inter-reciprocal
four-poles. It follows iipmediately from the matrix relationship derived in II-3.2.1 that for the inter-reciprocity of two gyrators it is also a necessary condition that the gyration admittances should be opposite.
Let the admittance matrix of the given gyrator be (cf. fig. III-4a)
0 0
(111-5)
Then, according to II-3.2.1 the admittance matrix Y' of the trans-posed gyrator will be
0 -p Y = Y
0
(III-6) q.e.d.
Physically the transposition of the gyrator four-pole can be repre-sented in two ways (cf. figs III-4a and III-4&). (i) the sign con-ventions of the gyrator are reversed either at terminal pair 1 or at terminal pair 2 {w and x are interchanged or y and z are interchanged); (ii) the terminal pairs 1 and 2 are interchanged (this can be done in two ways: interchanging w with y and x with z, or w with z and x with y). Because the interchange of w with y and that of x with z bears most on the transposition of the triode to be discussed, we shall stress this latter way of interchange more particularly and denote it by "turning over the gyrator". This nomenclature clearly shows that the operation is reversible.
3.2.2. Triode. The triode has already been quoted as an example in II-l.2.2. Let the admittance matrix of a triode (cf. figs I l l - l a and III-4c) be
S 0
Then the matrix Y' of the transposed triode will be 0 S
Y' = Y =
0 0
(III-7)
(III-8) This matrix determines a triode arising from the original one by interchanging grid and anode (fig. IW-Ad). Proceeding from (III-8) as the original matrix, (III-7) is obtained after transposition. The operation is thus reversible.
NAME OF THE ELEMENT QyRATOR TRIODE DUALTRIODE TRANSISTOR O U M . T R ANSISTDR ORIGINAL ELEMENT W V « . \
)C
" X Z " - P 0 a I , I , + V,. ° u; °
+ \ h Y= ° " ' S 0 c H. H . E, 0 ^ O 0° (xJ °
„ ^2 H, H^ Y = ° ° R 0 e H. I ,u
o ~ ( ° { E, ^ • ^ ' ' '^K
i^
0 1 0 (X ; 0 9 I . H, + V,0 r^ 0
u>
+ E2 1, ÏT TRANSPOSED ELEMENT'1
i
S')C
K
P 0 b i; n + ''; —'°
^ "1 "~ 0 J ° -K
fr ïT Y'= ° ^ ° ° d H! H! + E; -^ " ^ ^ -^ 0/> °
„+
E; i^ sr .Y'= " " 0 0 f H; I ; -1-E; - L°
< ^ "l '~ a) °
^ "; HT Ï I h i; H' + W —°
€
„ \ ^ n) °
..+
E; rr ?; 0 1 0 ^ P"ig, 111-4, S u r v e y of t h e t r a n s p o s i t i o n of a n u m b e r of e l e m e n t s ,3.2.3. Dualtriode. The dualtriode with its equations is represented in fig. III-lÖ and also drawn in fig. lll-Ae. Let the impedance matrix of a dualtriode be
(111-9) R 0
Then the matrix Z ' of the transposed dualtriode will be 0 R
Z = Z =
0 0
(III-10) This matrix determines a dualtriode arising from the original one by interchange of emissor and collector (fig. III-4/). This operation is reversible too.
3.2.4. Transistor. The transistor is best described with the aid of a "mixed matrix". Let the mixed matrix for a transistor (figs I I I - l c and 111-4^) be
D Then according to II-3.3
D ' = 0 a 0 0
• °
i 0
1 — a 0 (III-l 1) (III-12) This matrix determines a dualtransistor arising from the given tran-sistor by changing the emissor into a collector and the anode into a grid (cf. fig. III-4A). For the voltage ratio /? of the dualtransistorthus produced ;8 = - a . (Ill-13) Proceeding from (III-12) we get ( I I I - I l ) , so that this operation is
reversible as well.
3.2.5. Dualtransistor. In a manner analogous to t h a t described in 3.2.4. we obtain from a dualtransistor (figs. lll-\d and 111-4/) with matrix D 0 ^ 0 0 by transposition, a transistor with matrix
0 D ' = 0 - ) 3 0 (III-14) (III-15)
as represented in fig. 111-4^, which results from fig. 111-4/ by changing the collector into an emissor and the grid into an anode.
For the current ratio of the transistor so obtained
a = - / 3 . (in-16) This operation is reversible too.
3.2.6. Summary for §§ 3.2.2-3.2.5. It is interesting to note t h a t the topological effect of the transposition for §§ 3.2.2-3.25 can be adequately covered by the following rule
grid '^ anode emissor ^ collector.
3.2.7. Isolator. Let there be an "isolator" with scattering matrix (fig. III-5)
0 0 I 0 then for the transposed "isolator"
0 a
(III-17)
S'
0 0
(III-18)
and this is the matrix of an isolator that may arise from the original one by interchanging terminals in the same manner as described for the gyrator. We wish to lay particular stress on the interchange of w with y and of x with z, i.e. the process of "turning over". This operation is reversible too.
+ c 2F, ^ z 3 — ( = 1 w o
'4
<l
5 y 1 + 1 + 9 z i i c°
Z 0 c z i ^ + 1 T; 1 + 1 "^J — i i +r^
1 +H
X — ^ z t3.2.8. Combination of transposition and duality conversion. For a network composed of the elements enumerated in § III-l it follows from Definition III-l et seq. that the transposed network is obtained by the transposition of each of the unilateral elements provided in the network.
The network dual to another network is found by establishing the dual element of each element separately and by connecting these dual elements to form a network whose configuration is dual to that of the original network.
As is apparent from figs III-2 and I I I - 3 and the summary given in III-3.2.6, a combined operation, i.e. duality conversion followed by transposition, or in reversed order, will obey the following scheme
grid ^ collector anode '^ emissor.
Considering that a gyrator can be replaced by an equivalent work consisting of two triodes and an isolator by an equivalent net-work consisting of a triode and two impedances, the following theo-rem can be deduced:
THEOREM I I I - 4 : When transposition and duality conversion are con-secutively applied to a 2n-pole, the result obtained is independent of the sequence of the two operations.
Let the duality conversion factor be k^ (cf. III-3.1). Then, if the sign convention of fig. I I I - l is used, the following results are ob-tained by a combined operation, i.e. duality conversion followed by transposition, or in reversed order.
before combined operation after combined operation element characteristic parameter element characteristic parameter gyrator P gyrator 1 k'p triode S dual-triode —h'-S dual triode R triode R transis-tor a transis-tor a dualtran-sistor ^ dualtran-sistor
P
isolator a isolator a 1 1 According to the Remark in III-3.1 a dual conversion of each of the four unilateral elements gives rise to a minus sign. Only in the case of a transistor and a dualtransistor does a transposition give rise to a minus sign; cf. fig. III-4 and equations (III-8), (III-IO), (III-13)and (III-16). The combined operation thus leads to a minus sign only in the case of the triode and the dualtriode. The above result is of practical importance for transistor networks. The reason is that the transposition result derived for transistors is, to a certain extent, less attractive than that derived for triodes, since the transposition of a transistor network leads to a network with elements of another type, viz. dualtransistors, and vice versa. Obviously, an approxima-tion to a "dualtransistor" as a physical element can be obtained by means of a "low resistance triode" or, otherwise, by means of a triode in grounded-anode connection. However, in the study of related transistor networks it is desirable to find, after transposition is applied, again a transistor network. This can be achieved by applica-tion of duality conversion to the transposed network.
3.2.9. "Black box" 2v-pole. Often it will not be necessary to know the transposition results in detail with regard to a large part of a network. The latter part can then be represented by a "black box" with V terminal pairs. We can characterize this transposed "black box" part by means of a ^^-sign. The properties of the transposed 'black box" follow immediately from the properties derived in II-3. § 4. Some transposition theorems. Confining ourselves to the network elements enumerated in I I I - l , the following practical rules can be given for the transposition of a 2M-pole, on the basis of the results derived in the present chapter.
THEOREM I I I - 5 : Transposition of a network includes the following operations:
(i) "turning over" of the gyrators and isolators; (ii) replacing all the grids by anodes, and vice versa; (iii) replacing all the emissors by collectors, and vice versa.
Availing ourselves of the results of II-3.3 and II-3.4 (cf. also the notations there!) and on account of Theorem I I I - l , we find that for two 2«-poles arising one from the other by transposition
y'kk = Ykk. (in-19) Z'kk = Z,„ (III-20) 5 ; . = S,,. (III-21) This can be expressed in words.
T H E O R E M I I I - 6 : The driving-point impedances, the driving-point admittances and the reflexion coefficients of a 2n-pole are invariant for transposition.
It also follows from II-3.3 and II-3.4 t h a t for 2«-poles arising one from the other by transposition
^ ki — '^ ik' Zki = •^jft > M'^j= - L r t , L ; , = -Mj^, Ski = ^ik-(III-22) (III-23) (III-24) (III-25) (111-26) Adopting the convention t h a t "reciprocally-invariant" denotes the property t h a t two quantities with interchanged indices are equal, and furthermore t h a t "reciprocally-antivariant" denotes the property that two quantities with interchanged indices are opposite, (III-22)-(III-26) can be worded as follows:
THEOREM 111-7: The transfer admittances, the transfer impedances and the transfer coefficients of a 2n-pole are reciprocally-invariant for transposition. The transfer ratios of a 2n-pole are reciprocally-anti-variant for transposition.
With regard to the determinant of the matrices considered we can make the following derivation:
det [Y'] = det [Y] = det [Y], (III-27) det [Z'] = det [Z] = det [Z], (III-28) r Y ' M ' n r Y - L " !
r Y - M ~ l TY M"|
= ^^*L-L J = c i e t [ j ^ J = det[D], (III-29)
det [S'] = det [S] = det [S]. (III-30) Since the proof for the cofactors of the main-diagonal elements is
entirely analogous, we m a y conclude:
THEOREM I I I - 8 : The determinant and the cofactors of diagonal elements of the admittance matrix, the impedance matrix, the mixed
matrices and the scattering matrix of a 2n-pole are invariant for transposition.
With regard to the cofactors of the non-diagonal elements of the various matrices it can be derived that
cof,, [Y'] = cof,, [Y] = cof,, [Y], (III-31) cof,, [Z'] = cof,, [Z] = cof,, [Z], (III-32)
FY'M'n r Y - L " i r Y - M I cof,,[D] = c o f , , L j ^ , ^ , J = c o f , L _ j ^ ~J = cof,|__^ ^ J = rYM"i rYM~i -cof,.;t L L z J = -cof,* [D] (III-33) for elements
be-longing to Y or Z, for elements be-longing to M or L,
cof,, [S'] = cof,, [S] = cof,, [S]. (III-34) The above (III-31)-(III-34) can be expressed in words.
THEOREM III-9: The cofactors of non-diagonal elements of the
admit-tance matrix, the impedance matrix, the scattering matrix and the cofac-tors of the non-diagonal admittance and impedance elements of mixed matrices of a 2n-pole are reciprocally-invariant, the cofactors of ratio elements of the mixed matrices are reciprocally-antivariant for trans-position.
CH.A.PTER I V . A P P L I C A T I O N S TO AMPLIFIER CIRCUITS
§ 1. Introduction. As a first application of the transposition theory dealt with in the preceding chapters a number of simple amplifier circuits will be considered. The term "amplifier circuits" has to be interpreted in such a manner that it also comprises oscillator cir-cuits, negative-impedance converters, and suchlike. For these ampli-fier circuits the transposition theory is of importance because of the fact that by means of it the equivalence or analogue properties of certain variants of a circuit can be traced. So, through an analysis of the variant which is most suitable for the computation of a certain property, an insight can be gained into the properties of other va-riants. With respect to computation and insight the transposition
theory also affords a possibility of making use of certain symmetry properties inherent in some structures, if all the non-reciprocal elements in the circuits would be supplemented with their transpose to reciprocal elements.
The triodes and transistors occurring in the amplifier circuits are, just as in chapter I I I , imagined to be characterized by one complex parameter. The further properties of a physical triode such as inter-electrode capacitances, internal resistance, and suchlike, and those of an actual transistor, such as base resistance, emissor resistance, etc., are considered to belong to the reciprocal network in which the the triodes and transistors have been arranged. This implies that the transposition of an actual amplifier circuit will only lead to a circuit to be built up by means of the same reciprocal and non-reciprocal elements if
(i) the parasitic properties are negligibly small, because e.g. the internal resistance between anode and cathode of a triode is large compared with the anode impedance, or (ii) the parasitic properties show a certain symmetry, because e.g. for the triode C^j. = C„j. But even if none of these conditions should be fulfilled, an approach by means of elements which are somewhat different often remains possible. Moreover, at any rate the computing process is greatly reduced, because the formulae for a network, and for its transpose are of the same kind.Apart from this a double transposition always leads to a network that can be realized physically, viz. to the same network, which often permits interesting conclusions regarding its driving-point quantities.
§ 2. Grounded-anode and grounded-grid connection. As a first example we make a comparison between the properties of two well-known triode connections: the anode and the grounded-grid connection, in itself a rather trivial case.
It is well-known that the output admittance Y of the ideal triode in the grounded-anode connection, as drawn in fig. IV-la, is equal to the transconductance S of the triode. This holds also for the input admittance Y' of the grounded-grid connection shown in fig. IV-16, which has been derived from fig. IV-la by transposition, i.e. by interchanging anode and grid. The equality of Y and Y' illustrates the fact that the driving-point admittance is invariant when anode and grid are interchanged (Theorem III-6).
The input impedance of a grounded-anode circuit with ideal triode is infinite. Thus, on account of Theorem III-6, the output impedance of the grounded-grid circuit is infinite as well.
Fig. IV-la. Triode in grounded-anode connection.
Fig. TV-lè. Transpose of fig, IV-1 a,
Also with regard to the transmission properties of grounded-anode and grounded-grid circuits an illustration of the transposition theory is worth-while. In order not to make this example too simple two additional impedances Z and Z,^ are present in figs IV-2a and IV-2Ö.
-{••\
I .
Fig, IV-2a, Transmission in grounded-anode connection,
— ° t v; Zj, |v,
Fig, \N-2h. Transmission in grounded-grid connection,
A slight acquaintance with the influence of a cathode impedance on the resulting transadmittance of a triode allows one to read quickly from fig. IV-2«:
F ,
hZ
S1 + 5 Z , (IV-1)
It is a little less easy to write down directly the ratio of V[ to T^ in fig. IV-2Ö. As, however, the networks in figs IV-2a and IV-2Ö are each other's transpose, IiV[ = I'^V^ and consequently also
V[ = I'2Z
s
1 + 5 Z ,which can easily be checked by computation. Similar relations observed in the literature, see P e t e r s o n ^ ) and K e e n ^ ) , become directly understandable from the transposition theory.
