General properties of frequency-converting networks

FREQUENCY CONVERTING NETWORKS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETEN-SCHAP AAN DE TECHNISCHE HOGE-SCHOOL TE DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS. DR. O. BOTTEMA. HOOGLERAAR IN DE AFDELING DER ALGEMENE WETENSCHAPPEN. VOOR EEN COMMISSIE UIT DE SENAAT TE VER-DEDIGEN OP WOENSDAG 19 JUNI 1057.

DES NAMIDDAGS TE 4 UUR

DOOR

SIMON DUINKER

-r/' H ^-

DOOR DE PROMOTOR PROF IR. B. D. H. TELLEGEN

I. I N T R O D U C T I O N

1.1. F r e q u e n c y conversion; general aspects a n d notions 4 1.2. P u r p o s e a n d outline of t h e p r e s e n t investigation . 4 1.3. S u r v e y of t h e l i t e r a t u r e 6 References 7 I I . N O N L I N E A R E L E M E N T S A N D N E T W O R K S 11.1. General r e m a r k s 8 11.2. N o n l i n e a r two-pole elements 8 11.3. N o n l i n e a r coupling elements 9

11.3.1. Sets of nonlinearly coupled coils 11

11.3.1.1. Sets of nonlinearly coupled coils

w i t h o u t leakage 14

11.3.1.2. Necessity for t h e i n t r o d u c t i o n of

nonlinear t r a n s f o r m e r s 17

11.3.2. Sets of nonlinearly i n s u l a t e d conductors . . 17

11.3.3. Nonlinearly c o n d u c t i n g b o d y 20

11.4. N e t w o r k e q u a t i o n s 21

References 22

III. PERTURBATIONAL EQUATIONS FOR NONLINEAR

NETWORKS

111.1. S t a t e m e n t of t h e p r o b l e m 23

111.2. Description of t h e s y s t e m considered 23

111.3. F i r s t - o r d e r p e r t u r b a t i o n a l e q u a t i o n s 25 111.4. Reciprocity for p e r t u r b a t i o n a l e q u a t i o n s 27 111.5. Conditions for linear v a r i a b l e e q u a t i o n s 28

I V . E Q U A T I O N S F O R S M A L L P E R I O D I C S I G N A L S I N C O M P L E X F O R M

I V . 1 . A s s u m p t i o n s for t h e s y s t e m considered 29 I V . 2 . I n t r o d u c t i o n of c o m p l e x q u a n t i t i e s 29 I V . 3 . Complex expressions for t h e voltages across

indivi-d u a l elements in a mesh 31 I V . 3 . 1 . Capacitive voltages 32 I V . 3 . 2 . Resistive voltages 34 I Y . 3 . 3 . I n d u c t i v e voltages 35

I V . 4 . M a t r i x r e p r e s e n t a t i o n of m e s h e q u a t i o n s 36 I V . 5 . M a t r i x r e p r e s e n t a t i o n of n o d a l e q u a t i o n s . . . . 38 I V . 6 . Multipole e q u a t i o n s 4 0 I V . 7 . Discussion of t h e l i t e r a t u r e 40 References 4 1 V. D I S C U S S I O N O F M A T R I X E Q U A T I O N S V . l . P r o p e r t i e s of t h e m a t r i x e q u a t i o n s 42 V.2. D e p e n d e n c y of t h e m a t r i x e q u a t i o n s on t h e t i m e origin 42 V . 3 . D e p e n d e n c y of t h e m a t r i x e q u a t i o n s on t h e signal frequency 44 V . 4 . Effect of a s y m m e t r i c f u n d a m e n t a l s t a t e 4 5

V . 5 . Effect of t h e presence of linear i m p e d a n c e s 48 V.6. Infinite m a t r i x s y s t e m s a n d t h e i r r e d u c t i o n in p r a c t i c e 49 References 51 V I . E Q U I V A L E N T C I R C U I T S F O R C O N V E R S I O N N E T -W O R K S V I . 1 . I n t r o d u c t i o n 52 V I . 2 . Symbolical e q u i v a l e n t circuits 53 V I . 2 . 1 . Symbolical r e p r e s e n t a t i o n of resistive t e r m s 54 V I . 2 . 2 . Symbolical r e p r e s e n t a t i o n of reactive t e r m s 55 V I . 3 . Symbolical r e p r e s e n t a t i o n of t h e coupling b e t w e e n t w o fictitious meshes 57 V I . 4 . Symbolical r e p r e s e n t a t i o n of t e r m s in t h e n o d a l e q u a t i o n s 57 V I . 5 . Discussion of t h e l i t e r a t u r e concerning e q u i v a l e n t circuits 59 V I . 6 . Discussion of n o t i o n s of reciprocity from l i t e r a t u r e 60

References 62

VII. ENERGY RELATIONS FOR PURELY REACTIVE

CONVERSION NETWORKS

V I I . 1 . I n t r o d u c t i o n 63 V I I . 2 . E n e r g y t h e o r e m 63 V I I . 3 . H a r t l e y eflfect 67 V I I . 4 . Conversion gain or loss 69

V I I . 5 . Condition for t h e absence of r e a c t i o n 71

V I I . 6 . Discussion of t h e l i t e r a t u r e 73 References 'i 5

VIII. MATRIX EQUATIONS FOR BALANCED NETWORKS

V I I I . 1 . Balancing of elements of the same kind 76 VII1.2. Balanced magnetic series modulator 76 VIII.3. Balanced dielectric modulator 81 VIII.4. Balanced diode modulator 83 VIII.5. Balanced magnetic polyphase modulator 85

VIII.6. Discussion of the literature 89

References 89

Summary 90 Resume 90 Zusammenfassung 91

I.l. Frequency conversion; general aspects and notions

Telecommunication engineering comprises the field of the transmission of information by means of electrical signals. An electrical signal can be decomposed into simple harmonic components, each of which is character-ized by a certain frequency, amplitude and phase. Depending upon the nature of the information to be transmitted, the frequencies of the signal components cover a certain interval, the frequency band. For various reasons it may be desirable to shift a given frequency band by adding a constant amount to the frequency of each component in the band without changing the mutual amplitude and phase relations of the components.

In the literature the process of shifting a given frequency band has been given widely divergent names, such as modulation, demodulation, detec-tion, mixing, transposidetec-tion, transladetec-tion, changing, shifting, conversion, depending upon the way it is achieved and upon the object pursued. Here, the expression frequency conversion will be used throughout.

To obtain frequency conversion it is necessary to use systems that contain at least one element whose parameter varies with the time in a particular, for instance, periodic way. The manner in which this time dependency is obtained is of no importance for the process of conversion as such. It is possible, although difficult to physically realize in the case of rapid varia-tions, to vary the parameter of a linear element by mechanical means. However, it is preferable to introduce the desired time dependency by using a nonlinear element that is subjected to an auxiliary signal of sufficient strength. If the signals to be converted are small, the two methods are equivalent so far as the process of conversion is concerned, since both lead to linear differential equations with variable, i.e., time-dependent coefficients.

1.2. Purpose and outline of the present investigation

Conversion elements can be classified into two categories: electron tubes and transistors on the one hand, and nonlinear coils (which may or may not be coupled), resistors and condensers on the other. Elements of the first category, possessing by their very nature a certain preference for the direction of the transmission, will preserve this preferred direction if used for conversion purposes. This implies that for these elements only a small, often negligible, reaction of the output circuit on the input circuit occurs, in contrast to the elements of the second category where this reaction is strong. In the first case a conversion system can often be analyzed i.ilo parts

separated by conversion elements. In the second case the system has to be analyzed as a whole. It is clear that the analysis of the latter case will be more complicated than that of the first case, especially if it is realized that generally in the process of conversion, apart from the shifting desired for the frequency band, there will appear a series of other shiftings, necessitat-ing a large number of frequency components to be taken into account.

It is the purpose of the following considerations to present, by means of complex quantities, an exposition and an extension of the theory of net-works comprising time-dependent inductances, resistances and capacitan-ces, possibly combined with constant elements. The theory can be extended in an obvious way so as to comprise converting electron tubes and transistors. The considerations are given for purely electrical systems but the results obtained may also be used for, e.g., electro-mechanical systems, which are described by similar differential equations.

Since, generally, nonlinear elements are used to obtain conversion, these elements will be defined first in chap. I I . So-called coupling elements, such as sets of coupled coils, insulated conductors and conducting bodies supplied with terminals, will also be considered. It appears that in the nonlinear case these structures are generally not equivalent to combinations of simple two-pole elements and ideal transformers.

Chapter I I I deals with the linear differential equations with variable coefficients pertaining to small signals if the general network is made to depend on time by auxiliary signals of sufficient strength, that is, if a specific fundamental state is established. From the treatment in chap. II it then follows that the variable coefficients satisfy a condition of reciprocity. The following considerations, in which the interconnection of the various conversion components are analyzed more closely, will be based upon the small-signal equations.

In chap. IV the case of a periodic fundamental state and the conversion of a simple harmonic signal will be considered. This relatively simple case already gives rise to an infinite number of frequencies. If complex quantities are used, an infinite system of equations is obtained, each relating to a distinct frequency. This system of equations, which adequately can be written in matrix form, fixes the relationship of amplitudes and phase angles of the various components and, therefore, completely characterizes the conversion network.

In chap. V the matrix equations are further analyzed as to their depend-ency on the chosen origin of time and on the signal frequdepend-ency. In addition, the effect of a symmetrical fundamental state and of the presence of linear impedances is investigated.

Equivalent circuits for conversion networks are derived in chap. VI. In contrast to the case of constant linear networks these equivalent circuits

generally do not represent a physically realizable system but present only a symbolical picture of the various mutual reactions. The considerations in this chapter show that the process of conversion may give rise to power gain. This power gain is more closely analyzed in chap. VII, where a general theorem is derived for the mean power supplied to a purely reactive con-version network. Based on this theorem several aspects of power gain are investigated, among other things the possibility of spontaneous oscillations, the so-called Hartley effect.

Finally, in chap. VIII equations are derived for several symmetrical (so-called balanced) circuits of interest for practical applications. These circuits exclusively contain elements of the same kind. It then appears that the balancing results in a simplification of the system of equations. 1.3. Survey of the literature

The frequency-converting properties of linear networks with time-dependent parameters have been frequently investigated in the literature. Without exception the considerations involved arc valid only under certain restrictions. Often only a single conversion element is considered or a net-work comprising a special configuration of elements of the same kind. An-other restriction often introduced is the consideration of only a very limited number of frequencies, ignoring all of the rest. The methods followed in these analyses are widely divergent and, in many cases, are directed too much towards the solution of specific problems so as to be of little use for dealing with general circuits.

In this investigation the theory will be developed with reference to a general conversion network which enables the results of other authors to be interrelated or to be generalized. In order not to interrupt the discussion too much, the comparison of the results given in the sequel with those obtained by other workers is concentrated as much as possible into separate sections appended to each chapter. A more general survey of the literature will now be presented.

One of the first attempts to arrive at a general theory of networks with time-dependent parameters was made by Nichols ^) as far back as 1917. He showed that under special conditions power gain for small signals is possible, the power required being derived from the sources maintaining the fundamental state. However, he did not arrive at general expressions. That gain may be obtained from magnetic modulators had already been discovered by Alexanderson ^) who made use of this fact for modulation in long-wave transmitters ^). Almost simultaneously Hartley seems *) to have investigated the properties of the variable inductance resulting in a discovery of the possibility of negative-resistance effects which may give rise to spontaneous oscillations.

The application of complex Fourier expressions to these problems seems to have been given for the first time by Guillemin ^) in connection with variable resistances. In an extensive analysis of the so-called ring modulator, Kruse ^) obtained an infinite set of equations by using complex quantities. However, not only does he postulate a symmetric network configuration but also his final results are made to depend upon the assumption of the existence of a symmetric fundamental state. Stieltjes ' ) , mainly following Kruse's reasoning, considered among other things switching-modulators and a special type of magnetic modulator. He also considered the significance of negative frequencies which may occur in applying complex quantities. Haantjes and Tellegen **) also introduced complex quantities in their analysis of the diode, although following another line of thought. They avoided the occurrence of negative frequencies by introducing conjugate complex quantities.

Analyses of conversion networks without using complex symbolism are numerous. A survey of recent contributions to the theory of linear-vari-able networks is to be found in ref. ^).

R E F E R E N C E S

1) H. W. N i c h o l s , Phys. Rev. 10, 171-193, 1917; Elect. Comm. 1, 11-14, 1922. 2) E. F. W. A l e x a n d e r s o n , U.S. Pats 1.206.643 (11-28-1916), 1.328.473 (1-20-1920),

1.328.797 (1-20-1920).

' ) E. F. W. A l e x a n d e r s o n and S. P. N i x d o r f f , Free. Inst. Radio Engrs 4, 101-129, 1916.

*) E. P e t e r s o n , Bell Labs Rec. 7, 231, 1929.

^) E. A. G u i l l e m i n , "Com:nunication Networks" Vol I, Wiley, New York, 1935, Chap. X . «) S. K r u s e , Ericsson Technics no. 2, 17-54, 1939.

' ) F. H. S t i e l t j e s , Tijdschr. Ned. Radiogen. 11, 221-270, 1946.

») J. H a a n t j e s and B. D. H. T e l l e g e n , Tijdschr. Ned. Radiogen. 10, 237-260, 1943; Philips Res. Rep. 2, 401-419, 1947.

II. NONLINEAR ELEMENTS AND NETWORKS

I I . l . General remarks

To include certain practical cases it is necessary to widen t h e conventional concept of n e t w o r k c l e m e n t . H e r e , a n e t w o r k will be u n d e r s t o o d t o c o n t a i n not only resistances, coils, condensers and sources of energy, b u t also, as will be specified l a t e r on, sets of coupled coils e m b e d d e d in a nonlinear ferromagnetic m e d i u m , sets of insulated conductors e m b e d d e d in a non-linear dielectric, a n d a restricted class of nonnon-linearly conducting bodies supplied with t e r m i n a l s . T h e t w o categories of elements will be design-ated as two-pole a n d coupling elements and will be discussed s e p a r a t e l y in sees 2 a n d 3 . E l e c t r o n t u b e s a n d t r a n s i s t o r s are excluded from t h e considerations.

F o r a rigorous m a t h e m a t i c a l t r e a t m e n t of t h e problems t o be possible, t h e usual idealizations of n e t w o r k t h e o r y will be i n t r o d u c e d . This implies t h a t t h e elements are free from p a r a s i t i c p h e n o m e n a .

I I . 2 . Nonlinear two-pole elements

A two-pole element is defined by a (generally nonlinear) characteristic, i.e., a relationship between electrical q u a n t i t i e s , viz., fluxes q) and charges q, a n d their time d e r i v a t i v e s , voltages e = ^ a n d c u r r e n t s i = g. I n accord-ance with linear t h e o r y nonlinear two-pole elements, depending upon t h e n a t u r e of their defining q u a n t i t i e s {x,y), can be classified as resistances (^,e), coils (95,1) a n d condensers {q,e). T h e positive senses of i a n d e will be chosen such t h a t t h e energy supplied to t h e element in a n y i n t e r v a l of t i m e d« is d I F = ci dt. F u r t h e r , t h e positive senses of q and 99 are chosen in accordance w i t h t h a t of i a n d e, respectively.

F o r convenience t h e restriction will be i n t r o d u c e d t h a t t h e nonlinear ele-m e n t will b e h a v e like a passive linear s y s t e ele-m for sele-mall v a r i a t i o n s a b o u t a n y equilibrium p o i n t . This will be characterized by saying t h a t t h e element possesses t h e p r o p e r t y of ^Hocal passivity". Consequently, (i) t h e c h a r a c t e r -istics h a v e e v e r y w h e r e a positive slope. F o r resistances t h e passivity, i.e., ei ^ 0, implies t h a t (ii) t h e characteristics contain t h e origin. This l a t t e r p r o p e r t y will be assumed t o hold equally well for coils and condensers, b u t it does n o t impose restrictions u p o n t h e generality since it has only t h e effect of defining a certain zero p o i n t of q> a n d q. P r o p e r t i e s (i) and (ii) imply t h a t t h e characteristics are s i t u a t e d in t h e first a n d t h i r d q u a d r a n t s . F u r t h e r it will be assumed t h a t (Hi) t h e characteristics h a v e no hysteresis, no d i s c o n t i n u i t i e s a n d no k i n k s and therefore are single-valued a n d

s m o o t h a n d t h a t (iv) t h e y are not subjected t o dispersion, so t h a t t h e s t a t i c characteristics also define t h e relationship b e t w e e n i n s t a n t a n e o u s values of t h e q u a n t i t i e s for a r b i t r a r y t i m e d e p e n d e n c y .

T h e m a g n e t i c a n d t h e electric energy stored in t h e elements are k n o w n t o be given b y

a n d

T((p) = ƒ i{(p) d(p,

U{q)= i e(q)dq,

respectively. F r o m t h e local p a s s i t i v i t y follows t h a t

d^r di d^U

~ = 5 : 0 a n d

dq)^ dq) dq^ de

= — 5s 0 ,

so t h a t t h e functions T{(p) a n d U(q) are always non-negative a n d c o n v e x t o w a r d s t h e 99-axis a n d t h e g-axis, respectively.

As t o t h e shape of their c h a r a c t e r i s t i c t h e elements can be divided i n t o symmetrical a n d non-symmetrical elements according as t h e characteristic is radially s y m m e t r i c a b o u t t h e origin (fig. l a ) or n o t (fig. 16). E x a m p l e s of t h e first kind are: iron-cored coils, ferro-electric condensers a n d silicon-carbide resistors, a n d , of course, t h e linear elements. To t h e second category belong t h e various t y p e s of rectifier.

Fig. 1. Characteristics of nonlinear two-pole elements: (a) symmetrical characteristic, (fc) non-symmetrical characteristic.

I I . 3 . Nonlinear coupling elements

A coupling element will be u n d e r s t o o d to be a s y s t e m defined b y m e a n s of t w o series of electrical q u a n t i t i e s x-^,..., x^ a n d J i , . . . , yni b y relations of t h e form j ; = y; (x^,..., « „ ) , ƒ = 1,..., n. H e n c e , i n s t e a d of being c h a r a c t e r

-ized by a number of curves, these elements are determined by n surfaces in an (n -j- l)-dimensional space.

The various couphng elements that can be distinguished by the nature of the defining quantities (x, y) are sets of coupled coils (99, i), sets of insulated conductors (q,e or q,v, where v represents a potential) and conducting bodies supplied with terminals (i,e or i,v). The positive senses of ij and e; are chosen so that the energy supplied to the element in any interval of time dt is d IF = S ejijdt, while the positive senses of qj and (pj are chosen in accordance with those of ij and ey, respectively.

For the coupling elements too the property of local passivity will be assumed to hold. If a two-pole element is derived from a coupling element by keeping all independent variables x^ constant (k =t=y), the assumptions (i), (Hi) and (iv) of sec. 2 can accordingly be introduced for any relation

yj ^= yj(xj). Assumption (ii) can only be introduced for any relation yj = jy(«i, ..., Xn)- It is to be noted that the assumptions (i) and (ii) of

sec. 2 do not follow from the passivity for relations of the form y; := y/(«fe) (/(;=t=y), since these are not describing a two-pole characteristic but a transfer characteristic. Assumptions (üi) and (iv), can be introduced without difficulties.

Similar considerations hold if the inverse set of defining equations, i.e., Xj = xj{yi, ..., j „ ) , j ^ 1, ..., n, is used for the characterization of the coupling network. However, assumption (Hi), the absence of hysteresis, then has to be interpreted more carefully, viz., as implying that the same transfer characteristic Xj(yi() is traversed irrespective of the sense in which

yic is ^'aried.

One may ask whether it is necessary to consider, besides the two-pole elements, the more general class of coupling elements. In the theory of constant linear networks it has been demonstrated ^*^) that a system of coupled coils is equivalent to a combination of non-coupled coils and ideal transformers. Further, a system of n conductors insulated by a linear dielectric can be shown to be equivalent to a complete n-gon of capacitances and, similarly, a linear conducting body with n terminals to a complete n-gon of resistances. In the linear theory, therefore, analysis can be confined to networks containing two-pole elements and ideal transformers only.

It will appear, however, that, in general, nonlinear coupling elements are not similarly equivalent to combinations of two-pole elements and ideal transformers. Since coils coupled by nonlinear magnetic media, conductors insulated by nonlinear dielectrics and nonlinear conducting bodies supplied with more than two terminals cannot be excluded from the following considerations, owing to their practical importance for frequency conversion, the nonlinear coupling elements have to be investigated separately.

I I . 3 . 1 . Sets of nonlinearly coupled coils

F i r s t , t w o coils coupled b y a nonlinear m a g n e t i c m e d i u m will be considered. These are characterized b y t w o relations b e t w e e n c u r r e n t s i a n d fluxes 99,

ii = i i ( ? ' i ' 9 ' 2 ) ' ^ .

According t o t h e law of conservation of energy, after a closed series of v a r i a t i o n s of s t a t e , t h e s y s t e m h a s n e i t h e r absorbed nor delivered energy. F r o m t h i s it can be concluded t h a t , a p a r t from an a d d i t i v e c o n s t a n t , t h e r e exists a uniquely defined function, viz., t h e m a g n e t i c energy T{(pi, 993). T h e c o n s t a n t can be eliminated b y setting T(0,0) = 0.

S t a r t i n g from an a r b i t r a r y s t a t e {(pio, (P20) ^ small i n c r e m e n t of T(cp-^, q)^ is considered, d T = ij d99i + 12 dq>^ (2) which can a l t e r n a t i v e l y be w r i t t e n as d T = - — d99i + - — d992. (3) ii<Pi Ö992 F r o m eqs (2) a n d (3) i m m e d i a t e l y follows t h e r e c i p r o c i t y r e l a t i o n dii Ö ö T Ö ö T öi2 Ö992 Ö992 Ö99j Ö991 ögsg 099^

O t h e r relations can be derived from t h e p r o p e r t y of local passivity pre-supposed. T o this end t h e differential expressions for eqs (1) are considered,

dii =

(;-^) ^<Pi

+

[-—-]

^92^

\Ö99i/„ VÖ992/0

(-^^\ do,. + (' '^' dig = ( -^— ) d9,i + (-^— ) d992,

\Ö99i/„ \Ö992/

which d e t e r m i n e small v a r i a t i o n s a b o u t some specific equilibrium position (ho' ^20' 'P101 9'2o)- Local p a s s i v i t y now implies for a n y equilibrium position

(5)

(6) 0^1 _ 0 ' ^ > 0

Ö991 099^ "

and further, for the discriminant

^ _ dii 0^2 / dii \^_ 09?! Ö992 \dq>2'

where use is made of (4).

012 Ö992 dq>l dq>l 09^2 1 1 "^ 0 V 093^0993 /

I t is t o be n o t e d t h a t it c a n n o t be deduced from relations (5) a n d (6) t h a t t h e transfer q u a n t i t y dii/dqi2 will be positive, since this d e p e n d s on t h e sign c o n v e n t i o n of ij a n d 993.

The energy T{cirj.(f2) can be represented by a surface in a three-dimensional space. The mathematical significance of the expression D from (6) follows from a consideration of

ö^T d^T dT'

--^ (d(pi)^ + 2 ^ -T df^dtp^ + -—^ '

the complete differential of the second order of T. This quadratic form in the differentials

d(p^ and d(p2 determines the direction cosines of the tangents to the intersection curve

situated in the tangent plane, at any point of the surface T{<p-^, cp^). If condition (6) for D, the so-called discriminant of the surface T, is satisfied, the tangents at the point considered are imaginary (or, in the limiting case, real and coincident) so t h a t the tangent plane has only one point in common with T (or a curve at any point of which the tangents are coincident). Hence, if (6) holds, T possesses elliptic (or, in the limiting case, parabolic) points but no hyperbolic points at which D < 0 and at which the tangents are real and distinct. I t has been shown ' ' ) t h a t condition (6) is necessary and sufficient for the ex-clusion of hyperbolic points on T, i.e., for (semi-)definiteness of the form d'^T. On account of (5), d^T is positive (semi-) definite, so t h a t , if one puts T(0,0) = 0, the origin corresponds to a minimum of T, while T is convex towards the {tpi, (p^) plane.

N o w , assuming for t h e m o m e n t t h e defining eqs (1) t o be such as t o p e r m i t t h e e q u a l i t y sign in conditions (5) a n d (6) t o be d r o p p e d , t h e s a m e set of coupled coils can be defined b y relations of t h e form

fj = <Pj{ii, i a ) , j = 1,2 . (7) P a r t i a l differentiation of eqs (7) with respect t o 99^ a n d 932 respectively,

gives four p a r t i a l differential e q u a t i o n s , viz., {j,k = 1,2) dq>j oil dqij di2

oil Ö9^7c öig dqk

_ ^ 1 '

^ 0 ,

; = fe

7 + fc

If dq)Jdi2 is solved from t h e t w o e q u a t i o n s for which 7 = 1 a n d 09,2/öii from those for which j = 2, it follows i m m e d i a t e l y t h a t condition (4) implies t h a t

Ö91 __ ^

öig oil (8)

T h e e q u a l i t y (8) gives a necessary a n d sufficient condition for t h e existence of a function T'{ii,i2) such t h a t

dT' dT'

9^1 = - 7 - ' ?'2 = • — •

O i l OI2

This function was first i n t r o d u c e d by Cherry ^^) a n d labelled m a g n e t i c co-energy. The energy T a n d t h e co-energy T' are connected b y t h e so-called Legendre t r a n s f o r m a t i o n ^^)

T'(ii,i2) = ii99i + i2992 — ^(991, 9-2), assuming T'(0,0) = 0.

Similarly it can be shown t h a t t h e conditions for local p a s s i v i t y i m p l y t h a t

'^^-^0, 5 ^ > 0 and ^ ^ ^ _ _ ( ^ ) % o ,

ÖI d i o o i l Öl2 Ölo (9)

for a n y value of ii a n d

12-Sets comprising m o r e t h a n t w o coupled coils c a n be t r e a t e d i n a similar w a y . If, for e x a m p l e , a s y s t e m is defined b y relations

ij = ijiVi^---^ Vn), y = l , - - - , 1 , (10) t h e n t h e consideration of d T , t h e differential of t h e energy T(q}^,..., 99„) yields

öifc Ö dT d dT di;

(11)

dq>j dq>j dq>k dqJk dq)j d<pk

T h e local passivity condition requires 1*) t h e Hessian H of T, which is equal t o t h e J a c o b i a n , i.e., t h e functional d e t e r m i n a n t ()(ii,..., i„)/ö(99j, ..., 99^), t o satisfy H d^T ö^T Ö991 Ö99iÖ9J„ d^T Ö^T d99„ö9Ji Ö99„ oil Ö991 öi„ oil i>q>n din Ö9J1 Ö99„ ö(ii,...,i„) d{q>^,...,q>„) >0, (12) a n d , in a d d i t i o n , t h e principal minors t o be n o n - n e g a t i v e . On a c c o u n t of (11) t h e d e t e r m i n a n t s in (12) are s y m m e t r i c a l .

A similar reasoning can be given s t a r t i n g from relations Vj = 9'y(h'---' i " ) ' j = I'---' " • One can t h e n deduce from (11)

dq)k _ dqij di; dik

(13)

(14)

from which, in t u r n , t h e existence of t h e co-energy function T ' ( i i , . . . , i „ ) can be concluded. Local p a s s i v i t y now requires

diq>i,...,q>n) >0, ö(ii,...,i„)

a n d , a t t h e same t i m e , all of its principal minors t o be n o n - n e g a t i v e . (15)

I I . 3 . 1 . 1 . Sets of nonlinearly coupled coils without leakage

T h e relations (12) a n d (15) i n d i c a t e t h a t , in general, t h e surface T{q)i,...,q)n) a n d t h e surface T ' ( i i , . . . , i „ ) wUl contain elliptic a n d parabolic p o i n t s . As a limiting case it m a y occur t h a t T or T' possesses only parabolic p o i n t s so t h a t a t a n y p o i n t t h e functional d e t e r m i n a n t ö ( i j , . . . , in)/ö (991,..., 99^) or 0(991, ...,99„)/ö(ii, . . . , i „ ) is e q u a l t o zero. F r o m analysis it is k n o w n t h a t if t h e J a c o b i a n of a s y s t e m of functions vanishes, these functions are n o t i n d e p e n d e n t .

On t h e o t h e r h a n d , it can be concluded '•') from t h e vanishing of t h e Hessian t h a t T(99i, ...,99^) or T ' ( i i , . . . , i „ ) represent developable ruled sur-faces. W i t h t h e aid of (7), t h e physical significance of these m a t h e m a t i c a l relations will be i n v e s t i g a t e d . Similar considerations can be given, of course, s t a r t i n g from eqs (1), (10) or (13).

I f t h e J a c o b i a n 0(991, q>2)ld{ii, 12) is zero for any value of i^ a n d i j , t h e n , as a consequence of t h e d e p e n d e n c y t h a t can be concluded from it, one can p u t

<P2=f{<Pl) (16) P a r t i a l differentiation of (16) w i t h respect t o ii yields

oil ^H 0*2

t h e second e q u a l i t y following from t h e reciprocity relation. F r o m this m a y b e concluded t h e v a n i s h i n g of t h e J a c o b i a n 0(991, ii)/ö(ii,i2) of t h e s y s t e m

*1 ^ *1 T ƒ *2 5

in which ƒ ' m a y b e a n implicit f u n c t i o n of ii a n d i2, i.e., f'\q>x{ii,i2)[-This expresses t h e existence of a r e l a t i o n b e t w e e n 991 a n d ii in w h i c h ij a n d i2 do n o t occur, so t h a t one can write

h = h + f'k = gM • (17)

T h e condition of local p a s s i v i t y imposes some further restrictions u p o n t h e functions ƒ a n d g. B y p a r t i a l differentiation of (17) w i t h respect t o i j , keeping i2 c o n s t a n t , one o b t a i n s

~ = {g'-f"Hy'- (18)

oil

B y requiring t h e r i g h t - h a n d side of eq. (18) t o be non-negative for a n y v a l u e of 9?! a n d i2 on a c c o u n t of (9), it follows t h a t g' ^0 and, in a d d i t i o n , ƒ " = 0.

H e n c e „, , , ,_^^

where fc is a constant; the second constant of integration is zero, since the characteristic goes through the origin.

The system considered can now also be characterized by (19) together with

ii = ii + A:i2 = g(99i), (20) instead of by eqs (7).

The proportionality of primary and secondary fluxes implies that the two coils are linking the same flux, possibly with a different number of turns. In other words, the nonlinear transformer formed by the coils has no leakage. Typical combinations of transformers without leakage will be dealt with in chap. V I I I .

Equation (20) characterizes a single coil carrying a current ij^ = ii -j- fci2. It is easy to see that the system considered is equivalent to a single non-linear coU connected in parallel to the primary side of an ideal transformer defined by eq. (19) and the condition ii = 0, or ii = —feij (see fig. 2a).

The ruled surface T'(ij,i2) is a cylinder, and the (ii,^) plane is tangent to it, cutting it in the generator I'l -feij. The representation (20), which can inversely he written as

<p = h{ii), corresponds to a rotation of the axes by introduction of the new independent

variables, ij from eq. (20), and ij = fei^ — i j , resulting in a transformation of the expression

^'(.h'h) ^"to t h a t for a single coil.

If the coupled coils are defined by equations of the form (1) and the Jacobian is equal to zero, one will find by a similar reasoning that the system is equivalent to a single nonlinear coil in series with an ideal transformer (fig. 26).

Fig. 2. Equivalent circuit for two nonlinearly coupled coils without leakage: (a) defined by relations of the form (7), (6) defined by relations of the form (1).

For more than two coupled coils a similar treatment can be given. For instance, if the second-order minors of the Jacobian of the system (13), for n ^ 3, are equal to zero, then

92 = fiifi) and 9,3 = /2(99i).

By partial differentiation of these relations one easily verifies that the Jacobian ö(ii, 993,993)/ö(ii,i2,i3) vanishes, where

As a consequence of t h i s , t o g e t h e r w i t h t h e local-passivity condition from which one concludes f^ = k^ and f^ = k^ where k^ a n d fcg are c o n s t a n t s , t h e s y s t e m c a n be characterized b y t h e e q u a t i o n s

H = H + KH + KH = g(9'i)'

which p e r m i t s t h e r e p r e s e n t a t i o n given in fig. 3 . T h e analysis c a n b e e x t e n d e d in a n obvious w a y t o include t h e case of n coupled coils defined b y equations of t h e form (10) or (13).

l.K,.^—-^

Fig. 3. Equivalent circuit for three nonlinearly coupled coils without leakage, defined by relations of the form (13).

i-'2 !j

As a less special case one c a n h a v e t h a t t h e J a c o b i a n of a defining set of equations is zero a n d some, b u t n o t all, of its principal minors zero. I n o t h e r words, t h e r a n k of t h e J a c o b i a n , i.e., t h e highest order of t h e principal m i n o r t h a t is u n e q u a l t o zero, is g r e a t e r t h a n one. F o r example, for n = 3 ,

d{.9\-> 92-. 93) diivii^i^) a n d Q ( y i , 9^2) Ö ( i i , i 2 ) 0 ( ^ 2 , 9s) d{i2,i3)

+ 0,

+ 0,

assuming, dqi^/dii, dq)2ldi2 a n d dq>Jdi^ t o b e u n e q u a l t o zero. This s i t u a t i o n arises if t h e r e exists a relation b e t w e e n t w o of t h e defining e q u a t i o n s , viz., 993 ^ f(q>2)- I n a completely similar w a y as w a s followed for t h e first e x a m p l e , t h e defining e q u a t i o n s c a n n o w b e w r i t t e n as

9 ^ 1 = 951(11,12)»

952 = 92{H'>H) = ' c ^ V s '

considered c a n b e r e p r e s e n t e d b y t h e circuit p i c t u r e d i n fig. 4 , comprising a n i d e a l a n d a n o n l i n e a r t r a n s f o r m e r . v -0 ^ + i ^, 0-• ^

Fig. 4. Equivalent circuit for three nonlinearly coupled coils, two of which are mutually without leak-age, defined by relations of t h e form (13).

II.3.1.2. Necessity for the introduction of nonlinear transformers

Considering a nonlinear transformer defined by eqs (7), one obtains after differentiation, _ d(pj dii dqrj d i j

•' dii dt dia dt

which equations can be easily written in another form, namely dii / d(pi \~^ d(pi I d(pi \~' dig

(21)

, ^ , , , , . \ d(p, óqPo I o f , \ V / Ö9'i \ ' fli2 e, = — ^ — ^ e,

1,2,

dt

^ 0 ^ ld<pi\ 0^2 \ dii ' In the linear case one has c

dig \ oil S dqPi d<p2 ( dii 0^2 d( I d(pi \ dt

bii = L j , öï'j/öia = -L2 ^ " ^ öij^i/öi-j --= M, where L j , Lg and M are constants. I t is then possible to represent the system by an ideal transformer with transformation ratio M / L j , connected in parallel to an inductance L^ at the primary side and in series to an inductance (L1L2 — M^)/Li at the secondary side.

In the nonlinear case, however, the partial derivatives in eqs (21) are not constant b u t functions of i, and ij. Consequently, the transfer terms in (21) cannot be represented by an ideal transformer. Further, the first equation of (21) cannot be interpreted as representing the primary current ij, divided into a part t h a t is transformed to the secondary side and another part t h a t flows through an inductance, since the latter depends upon ij and i^. Similarly, the second equation of (21) cannot be conceived of as a transformed voltage in series with an inductance, since the latter also depends upon ij and ig. Hence, the equivalent circuit for linearly coupled coils does not hold for the nonlinear case, and therefore the nonlinear transformer is an essential element.

I I . 3 . 2 . Sets of nonlinearly insulated conductors

A set of n i n s u l a t e d c o n d u c t o r s e m b e d d e d in a n o n l i n e a r dielectric a n d

c a r r y i n g charges qj(j = ! , . . . , « ) is c h a r a c t e r i z e d by n r e l a t i o n s b e t w e e n t h e p o t e n t i a l s vj a n d charges qj,

j = l , . . . , n . (22) Vj = Vj (gi _^<ln)

Since a small a m o u n t of e n e r g y supplied t o t h e s y s t e m c a n b e identified t o b e t h e t o t a l differential dU of t h e electric energy U s t o r e d in t h e s y s t e m , w h e r e

" dU

one c a n d e d u c e t h a t

d dU dvj dvk d dU

= —i- = = (24)

dqk dqj dqk dqj dqj dqk

T h e c o n d i t i o n for local p a s s i v i t y n o w r e q u i r e s t h e functional d e t e r m i n a n t

d{Vi,...,Vn)

d{qi,...,qn) (25)

a n d all of i t s p r i n c i p a l m i n o r s t o b e n o n - n e g a t i v e .

B y c h a r a c t e r i z i n g t h e s a m e s y s t e m b y a n inverse set of relations of t h e form

9j= Ij i^i^-'-^^n), j = l , . . . , n , (26)

it c a n again b e p r o v e d t h a t c o n d i t i o n (24) implies t h e equalities

M = ^

(27)

dvk dvj

t o h o l d f o r y =1= k, from w h i c h t h e existence of a function {7'(t'i,...,t)„) c a n b e d e d u c e d , such t h a t qj = dU'jdvj,j = l,...,n. This was also first i n t r o -d u c e -d b y C h e r r y '^) u n -d e r t h e n a m e of electric co-energy. Local p a s s i v i t y n o w requires t h e functional d e t e r m i n a n t

^(9v-'in) (28) d{Vi,...,Vn)

a n d all of its p r i n c i p a l m i n o r s t o b e n o n - n e g a t i v e .

The conditions (25) and (28), although sufficient for ensuring local passivity, can be replaced by more stringent ones with t h e aid of some hypothetical experimental reasoning.

Consider the differential expressions for the set (22),

dvj = V ^dqk, j = l,...,n, (29) k=l dqk

for some specific distribution of charge quo- By supplying a small positive charge dqj to conductor J while keeping the other charges constant, i.e., dqk = 0 for k =}; j , one obtains a slightly different pattern for the lines of force. The charge of a conductor is measured by the excess of the number of lines of force which issue from it over those which terminate in it. Hence, new lines of force will emerge only from conductor y and these are terminated in the other conductors and/or ground. Since lines of force are directed from places of greater potential to places of smaller potential, the increment of the potential vj exceeds t h a t of the other potentials Vk, which in their turn are positive. Hence

dvj ^ dvk J> 0 ,

and on account of (29),

^ > ^ 5 = 0 . (30)

dqj dqj

By repeating t h e experiment for all possible initial charge distributions and for any con-ductor separately, the condition will be seen to hold generally.

Starting from the differential expression of the set (26),

dqj= 2 ^dvk, ; = l , . . , n , (31) •^ k=l OVk

for some specific distribution of potential Vkg, a small positive increment dvj in the potential of conductor/, while keeping the other conductors at a constant potential, i.e., dvk = 0 for k :^ j , will result in a slight variation of the charge distribution. Now dqj > 0, while, as a result of electrostatic induction, dqk ^ 0, so that it follows from (31) that

^ • > 0 and ^ < 0 . ovj ovj

Depending upon whether or not part of the new lines of force are going to ground, the total charge variation of conductors k ^ j will be smaller than or equal to dqj. Hence

^ + 2 ^ ^ 0 . (32) dvj fc^j dvj

This condition also holds for any possible initial potential distribution and for any j = 1,..., n. If, as usual, t h e c o n d u c t o r w i t h zero p o t e n t i a l (ground or " i n f i n i t y " ) , for which arbitrarily t h e one labelled n m a y he chosen, is p a r t of t h e s y s t e m considered, t h e n t h e s y s t e m is said t o be closed. F r o m a closed s y s t e m as a whole no lines of force will emerge. I n o t h e r w o r d s , t h e

n - l

t o t a l s u m of t h e charges is zero, i.e., qn = — 2 ij- T h e p o t e n t i a l differ-ences of t h e c o n d u c t o r s w i t h respect t o g r o u n d can be defined as voltages e. As a result of t h e relation b e t w e e n t h e charges, t h e functional d e t e r m i n a n t of t h e set (22) will be equal t o zero. T h e defining e q u a t i o n s (22) can be replaced b y relations b e t w e e n voltages a n d charges,

ej = ej{qi,...,qn^i), y = l , . . . , n — 1 . (33) T h e functional d e t e r m i n a n t of this set, t o g e t h e r with its principal m i n o r s ,

h a s again to be n o n - n e g a t i v e in order t o ensure local p a s s i v i t y . Similarly, it is also possible t o write t h e set (26) in t h e form

qj =^ qj {ei,...,en-i), j=l,...,n—l. (34) H o w e v e r , it c a n n o t be concluded from these equations t h a t t h e closed

s y s t e m is equivalent t o a complete w-gon of nonlinear condensers. If this were so it would be possible t o express t h e charge of a c o n d u c t o r as a s u m , each t e r m of which d e p e n d s only u p o n t h e p o t e n t i a l difference b e t w e e n t h i s c o n d u c t o r and a n o t h e r , including t h e g r o u n d e d reference c o n d u c t o r . F o r i n s t a n c e , for n ^ 3 :

?i = 91 (ei) + q'iiei — «2), etc.

Generalizing this reasoning in an obvious w a y , one could s t a t e t h a t a necessary condition t h a t a set of i n s u l a t e d conductors be e q u i v a l e n t t o a c o m p l e t e ;i-gon of condensers w o u l d b e

d^q;

— ^ = 0 , for k ^ l ^ j =l,...,n~l. dekdei

Since, however, for a physical nonlinear system this condition is n o t necessary, t h e equivalency does n o t exist in general.

I I . 3 . 3 . Nonlinearly conducting body

I n a c o n d u c t i n g b o d y w i t h n-{~l t e r m i n a l s t h e relations b e t w e e n t h e c u r r e n t s ij supplied t o t h e t e r m i n a l s a n d n of t h e p o t e n t i a l s v; of t h e ter-minals are fixed.

B y a r b i t r a r i l y selecting t h e ( ? i + l ) t h t e r m i n a l as a reference one o b t a i n s

"j = f'j ih^-'-^in), j = 1,..-,«, (35) where ey = Vj — Vn+i.

Since t h e p h e n o m e n o n of energy dissipation is an irreversible process, c o n t r a r y t o t h e reversible energy e x c h a n g e w i t h t h e s u r r o u n d i n g s t a k i n g place in r e a c t i v e s y s t e m s , it c a n n o t be p r o v e d in t h e w a y of t h e foregoing t h a t öe; dek -J-=.-^, k^j. (36) dik dij

Y e t , t h e v a l i d i t y of (36) can be p r o v e d b y using t h e so-called Onsager principle, which is based on t h e reversibility of t h e t i m e in microscopic processes 1^), p r o v i d e d one restricts oneself t o small v a r i a t i o n s a b o u t t h e origin a n d if e x t e r n a l m a g n e t i c fields are a b s e n t . For t h e conducting bodies u n d e r consideration, it will be assumed t h a t (36) holds for any equilibrium position. I t can t h e n be concluded t h a t t h e r e exists, a p a r t from an a d d i t i v e c o n s t a n t , a u n i q u e l y defined function G(ij,...,i„) t h e differential of which is given b y

n öG 1 dG = S ^ di,- = 2 e; di,-.

T h e function G, fixed by p u t t i n g G(0,...,0) = 0, was i n t r o d u c e d b y Millar ^') u n d e r t h e n a m e of content in his investigation of n e t w o r k s containing nonlinear resistances. I n this case t h e relations (36) are satisfied since t h e y t h e n i m p l y t h a t for small v a r i a t i o n s a b o u t a n y equilibrium position t h e differential resistance of t h e b r a n c h in c o m m o n t o t w o adjacent meshes is entirely d e t e r m i n e d b y t h e v a r i a t i o n s of t h e b r a n c h c u r r e n t . B y a s s u m i n g (36) t o hold generally, t h e notion of c o n t e n t is now applicable t o nonlinear c o n d u c t i n g bodies.

T h e same c o n d u c t i n g b o d y can also be defined b y relations of t h e form

V = V (ei,---,e„), j = l,...,n. (37) B y p a r t i a l differentiation of t h e e q u a t i o n s w i t h suffices j and k w i t h respect

is solved from t h e first set a n d öife/öe.- is solved from t h e second, it follows i m m e d i a t e l y t h a t relations (36) i m p l y t h a t

di; dik

^ = - ~ , (38) dek dej

if, as will be assumed, t h e functional d e t e r m i n a n t ö(ei,...,e„)/ö(ii,...,i„) is u n e q u a l to zero.

F r o m (38) t h e existence of a function J{e^,...,en), t h e co-content as it was n a m e d b y Millar, can be d e d u c e d , t h e differential of which is given b y

" dj " dj =•- 2 de,- = V i d e ; . T a k i n g J ( 0 , . . . , 0 ) = 0, it is seen t h a t

d J + dG = S d i,-e, = d P

j = i J •>

so t h a t t h e energy P dissipated per unit t i m e is given b y

j=i J •'

T h e local-passivity condition requires t h e non-negativeness of t h e func-tional d e t e r m i n a n t of t h e set (35) or (37) t o g e t h e r with its principal minors. F r o m t h i s it follows t h a t t h e functions G a n d J are n o n - n e g a t i v e .

F o r nonlinearly c o n d u c t i n g bodies, j u s t as for sets of i n s u l a t e d con-d u c t o r s , it will generally n o t be possible t o give an e q u i v a l e n t circuit consisting of a complete /i-gon of resistances. F o r t h e p a r t i a l d e r i v a t i v e s one can derive again r e l a t i o n s w h i c h are m o r e s t r i n g e n t t h a n t h o s e c o r r e s p o n d i n g t o t h e f u n c t i o n a l d e t e r m i n a n t s .

11.4. Network equations

T h e n e t w o r k s t o be considered contain a r b i t r a r y interconnections of sources of energy, two-pole elements a n d coupling elements. T h e c u r r e n t s flowing t o g e t h e r a t a node a n d t h e voltages across t h e elements in a m e s h are subjected t o Kirchhoff's laws: S i = 0 a n d 2 e = 0. Using t h e relations discussed in t h e foregoing sections, t h e differential e q u a t i o n s for c u r r e n t s a n d voltages m a y be o b t a i n e d from these laws, w h e r e b y t h e b e h a v i o u r of t h e n e t w o r k is d e t e r m i n e d if t h e initial conditions are specified.

F o r a n e t w o r k t o be analyzed on t h e basis of mesh e q u a t i o n s , supposing t h e n e t w o r k t o consist of r i n d e p e n d e n t meshes, one o b t a i n s

9s . c . r s 1 / o n \

- r - + ej + « 5 = eg, s = l , . . . , r . (39) dt

In (39) 99s represents the sum of the fluxes of the inductive elements in mesh s. Further e, and eg represent the total sum in mesh s of the capa-citive voltages and the resistive voltages, respectively, while Cg represents the voltage sources. Substituting the corresponding relations for the elements (for coupling elements relations of the form(13), (33) or (35)), eq. (39) transforms into a set of differential equations for the charges. The nodal equations for a network with r -f 1 nodes, one of which is arbitrarily selected as a reference point, read

i^l + lig + X = 'ig, .s = l,...,r. (40)

dt

In (40), 5s represents the total charge of the capacitive elements attached to node s, while Hg, ig and ^ig represent the total inductive, resistive and impressed currents, respectively, flowing towards node s. Substituting the corresponding relations for the elements (for coupling elements relations of the form (34), (10) or (37)), eq. (40) transforms into a set of differential equations for the fluxes, i.e., the time integrals of the potential differences. The differential equations obtained from (39) and (40) will be nonlinear if the network considered comprises nonlinear elements, and therefore will not be soluble in general. For some categories of problem, particularly those to be discussed in the sequel, the solution is not of primary importance. Therefore, eqs (39) and (40) will be used in the following chapter in the form given. The specification of the characteristics of the individual elements will not be necessary for the purpose of the present investigations.

R E F E R E N C E S

'") W. C a u e r , ,,Theorie der linearen Wechselstromschaltungen", Akademie-Verlag, Berlin, 1954, Ch. IV, App. 3.

^') J. A. S e r r e t and G. S c h e f f e r s , "Lehrbuch der Differential- und IntegralrechnuBg", Teubner, Leipzig, 1915, § 156 ff.

12) E. C. C h e r r y , Phil. Mag. 42, 1161-1177, 1951. 13) J. A. S e r r e t and G. S c h e f f e r s , ref. n ) , § 100.

^^) E. C e s a r o , "Algebraïsche Analysis und Infinitesimahechnung", Teubner, Leipzig, 1904, § 659 ff.

15) E. C e s a r o , ref. " ) , § 675.

i«) H. B. G. C a s i m i r , Philips Res. Rep. 1, 185-196, 1946. 1') W. M i l l a r , Phil. Mag. 42, 1150-1160, 1951.

III. PERTURBATIONAL EQUATIONS FOR NONLINEAR NETWORKS

111.1. Statement of the problem

For a general nonlinear network under the influence of sources of energy, the mesh and nodal equations were given in chap. I I . For the following considerations the knowledge of the actual currents and voltages in the network is not required. It suffices to assume that there exists a certain current and voltage distribution, the so-called fundamental state. In the following chapters the fundamental state will be assumed to be periodic but it is not necessary to impose this restriction on the considerations in the present chapter.

The fundamental state will be subjected to a small perturbation. The treatment is restricted to the case where the perturbation is sufficiently small so as to permit the consideration of the first-order terms exclusively. The first-order terms determine a linear network with time-dependent parameters, defining what will be termed the perturbational state. Thus, distortion problems arising with larger signals and necessitating the inclusion of higher-order terms, are excluded.

In this chapter, after a description of the system to be considered, the perturbational equations will be derived.

111.2. Description of the system considered

The system considered contains an arbitrary configuration of coils (which may or may not be coupled), of sets of insulated conductors and of conduct-ing bodies. The nonlinear network together with the sources of energy that produce the fundamental state form a closed system. An example containing a voltage source e, a current source i, resistors, coils, condensers and a nonlinear transformer is given in fig. l a .

In the network a perturbation results from the introduction of small-signal sources s. It is assumed that these small-signal sources are either voltage sources or current sources permitting the perturbational state of the entire network to be described either by mesh of by nodal equations exclusively. This is not an essential restriction, since a signal voltage source together with its internal series impedance, which it will possess in almost any practical case, can also be represented as a current source with an internal admittance in parallel, and vice versa.

The number of independent meshes needed for defining the perturbational state may be different from that required for describing the fundamental state. Some possibilities are represented in fig. 16 which shows the

perturba-tional state of the circuit given in fig. l a generated by the small-signal volt-age sources s. Since the sources s have no effect on the current supplied by the source i in the original circuit, the corresponding branch and, hence, a corresponding mesh in fig. 16 vanishes, while the number of nodes remains

Fig. 1. (a) General nonlinear circuit containing resistors, coils, condensers and a nonlinear transformer, in which the fundamental state is determined by a voltage source e and a current source i. (h) Circuit determining the perturbational state of the network in fig. l a , as a result of the presence of small-signal voltage sources s.

unaltered. Conversely, as a result of the presence of the small-signal source •Sg a new branch and a new node, but not a new mesh, must be added to the original circuit of fig. l a .

In order to discuss the effects of these signals it may be desirable, as will be discussed in later chapters, to regard the network as a 2n-pole rather than as a closed system. Terminal pairs can be introduced by cutting a branch or by selecting two existing nodes. However, since the introduction of terminal pairs must not affect the fundamental state, some additional precautions are necessary.

When a terminal pair is created by cutting a branch, one has to extend the network in the fundamental state at this place by adding a current source in parallel, this source having a strength equal to the actual current in the branch before it was cut. When a terminal pair is created by selection of two existing nodes, one has to include a voltage source in series of a strength equal to the actual voltage between the nodes. In this way the original fundamental state is maintained when the newly created terminal pairs are open or closed.

Examples are given in fig. 2a which represents the network of fig. l a extended by the insertion of a voltage source e' and a current source i'. The corresponding perturbational state is given in fig. 26 which shows — besides the terminals corresponding to the signal sources Sj, Sg and s., — the newly created terminal pairs 4 and 5, which are short-circuited and open-circuited, respectively, for the fundamental state, the short-circuits being indicated by the dashed lines.

The treatment on the basis of nodal equations is similar to that on the basis of mesh equations. In the sequel the case of mesh equations will be presented in detail, while that of the nodal equations will be indicated only briefly.

Fig. 2. (a) General nonlinear circuit of fig. l a , extended by the insertion of a voltage source e' and a current source i' for obtaining terminal pairs, (b) Circuit determining the perturbational state of the network in fig. 2a, as a result of the introduction of small signals at the created terminal pairs.

II 1.3. First-order perturbational equations

In the network defined in sec. 2, the voltage sources «i, Cj, ... will

produce mesh currents q^, q2, It is assumed, that in describing the fundamental state, the meshes have been specified in the way needed for defining the perturbational state. For a branch containing a current source the algebraic sum of the corresponding mesh currents is equal to the current delivered by that source. Then one of the mesh currents can be eliminated, if the voltage across the branch is introduced as the unknown. When the network possesses r' independent meshes it is completely described by eqs (II. 39), viz.,

- v - + eg + eg = eg, s = l , . . . , r , (1) dt

where ^Cg is equal to zero if mesh s does not contain an energy source. Depending upon the initial conditions certain functions of time for the charges (^so fo' mesh s) wUl satisfy (1), determining the fundamental state. The introduction of small disturbances by small-signal voltage sources d^Cg gives rise to variations öqg in the mesh charges, producing a deviation from the fundamental state. If the variations are infinitely small so that a restriction to first-order terms is permitted, this deviation will be described by the system of variational equations

where r m a y differ from r', as was discussed in sec. 2. These e q u a t i o n s also a p p l y t o t h e 2n-pole discussed in sec. 2 if a d e p a r t u r e is m a d e from t h e e x t e n d e d f u n d a m e n t a l s t a t e . I n t h a t case t h e signal voltage ö Cg represents t h e voltage across t h e newly created t e r m i n a l pair lying in m e s h s.

Using t h e dependencies of t h e fluxes, t h e capacitive a n d t h e resistive voltages on t h e various mesh charges a n d m e s h c u r r e n t s , as t r e a t e d in c h a p . I I , eqs (2) can b e w r i t t e n as

>• ( d / dqg , \ Ö «s Ö e„ . ; _ ( = i ( d l \ dqt / dqt dqt ^

T h e p a r t i a l derivatives in (3) are functions of t h e f u n d a m e n t a l s t a t e a n d are therefore t o be considered as given functions of t h e t i m e . P u t t i n g for shortness

dqig d eg Ö Bs

^st = —. ' egt = ^ — , Qst = —^-, (4) dqt dqt dqt

a n d for t h e charge a n d t h e voltage v a r i a t i o n s

ft = öqt, "s = ^^«s, (5) eqs (3) can b e w r i t t e n as

r i d - • )

Y. W {''•St ft) + >'st ft + Ogt ft = Ws, s = l , - . . , r . (6) I' 1 (dt )

A similar reasoning can b e given d e p a r t i n g from t h e nodal e q u a t i o n s (11.40)

-, ^ ''Is + ls= Is, S = l , . . . , r . (1 )

Now, t h e i n d e p e n d e n t variables are t h e fluxes q. Using t h e relations dis-cussed in c h a p . I I a n d i n t r o d u c i n g symbols for t h e p a r t i a l derivatives

dqg dHg d ig Yst = —r'> /'st = ^ ' ^si = — T - ' (4') dq>t dq>t dq)t a n d for t h e flux a n d c u r r e n t v a r i a t i o n s Vt = ^9i^ <s = ^^is, (5') one is led t o t h e v a r i a t i o n a l e q u a t i o n s r ( d . . ) S j ]-r (ïst Vt) + fht nt + ><st Vti^h-, s= l , . . . , r . (6')

111.4. Reciprocity for perturbational equations

The variable coefficients in the differential equations (6) determining the perturbational state are subjected to certain relations as discussed in chap. II. If the notation of sec. 3 is introduced, the equalities (11.14, 24 and 36) read

ht = -^(s, i^st = ets a n d Ogt = Qtg, (7 a,b,c) respectively. Similarly, for the coefficients of the nodal equations (6') one can write, as a consequence of the relations (11.11, 27 and 38),

Pst = fits-, yst=yts and Xgi =^xts-, (7'a,b,c) respectively.

If the network possesses only two-pole elements in the sense of sec. II.2, the relations (7) and (7') are trivial since they then result from the fact that a certain element is common to meshes s and f, or connects nodes .s and t, respectively. However, as soon as the network contains couphng elements in the sense of sec. II.3, the equalities (7a,b) and (7'a,b) are no longer self-evident but follow from the principle of conservation of energy for reactive systems, while (7c) and (7c') were assumed to hold for conducting bodies.

In the theory of constant linear networks the coefficients in the differen-tial equations are constant. In the case of a sinusoidal variation these equa-tions can be converted into algebraic ones by introducing complex quantities for the currents and voltages. After elimination of the equations correspond-ing to internal meshes or nodes a new system of algebraic equations is obtained which represents the 2n-pole considered. In these equations the coefficients of the mutual-coupling terms, usually designated as transfer impedances or admittances, are also equal.

It is easy to see from (6) and (6') that reciprocity in this wider sense will not hold for the linear variable systems considered here. For if the signal voltages Ug (or currents ig) vary sinusoidally, then the charges f( (or fluxes rjt) and their derivatives wUl generally be complicated functions of time, so that it is not possible to introduce complex quantities directly for obtaining algebraic equations. In the next chapter it will be indicated how in the special case of a periodic fundamental state, after splitting up (6) and (6') according to the various frequency components, one can arrive at a generalization of the impedance concept. Thereby the relations (7) and (7') give rise to a more restricted kind of reciprocity.

Networks containing electron tubes or transistors do not satisfy the reciprocity relation. This is the reason for excluding these elements from the considerations. At those places in the following where the reciprocity rela-tion is not used, the method of treatment will also apply to networks containing electron tubes and transistors.

111.5. Conditions for linear variable equations

A p a r t from t h e reciprocity relations given in sec. 4, t h e variable co-efficients in (6) a n d (6') will be subjected t o further conditions following from t h e r e q u i r e m e n t for local p a s s i v i t y .

I n c h a p . I I these conditions were given s e p a r a t e l y for systems containing elements of t h e s a m e kind only. These conditions r e m a i n u n a l t e r e d for a n e t w o r k c o n t a i n i n g elements of different k i n d . Therefore, one o b t a i n s in t h e n o t a t i o n of sec. 3, for t h e q u a n t i t i e s /< of eq. (6'), t h e conditions (see eq. I I . 12) t h a t all principal minors of t h e d e t e r m i n a n t

/ ' l l • • • / ' ' i r

fhi--- I'rr

should be n o n - n e g a t i v e .

Similar conditions will hold for t h e q u a n t i t i e s X, Q, e, y and K.

For capacitances, in chap. I I more stringent relations could be derived. The elastances f are subjected to inequalities of the form (11.30), or

fjj > nj ^ 0 , and the capacitances y, on account of eq. (11.32), satisfy

where the equality signs apply if one of the conductors surrounds the whole structure. Similar relations will hold for the resistances g and the conductances x.

IV. EQUATIONS FOR SMALL PERIODIC SIGNALS IN COMPLEX FORM

IV.I. Assumptions for the system considered

In deriving the perturbational equations in chap. I l l no restrictions were imposed upon the time dependency of the sources of energy present. In order to apply the preceding general considerations to the nonlinear net-works used for frequency conversion, it wUI now be assumed that the sources of energy establishing the fundamental state are periodic with the same fundamental frequency p. This assumption does not exclude the presence of d.c. sources.

Further, it will be assumed that a forced periodic state with fundamental frequency p will finally be attained. The proof of the existence of such a periodic fundamental state for an arbitrary nonlinear network has not yet been given. For a restricted number of degrees of freedom one can find a proof in Stoker ^^). As is known (see, for instance, ref. ^^)), different periodic fundamental states can occur in a nonlinear system, depending solely upon the history of the system. In such a case, one of the stable states, arbitrarily chosen, will be considered to be the fundamental state.

The fundamental state will next be considered to be varied by a small harmonic disturbance with frequency q, which will be further designated as

signal and signal frequency, respectively. In accordance with the usual

nomenclature the currents and voltages in the fundamental state will be called carriers.

As a consequence of the stabiUty assumed for the fundamental state, after the introduction of the disturbance, a slightly different situation will arise in which, as a result of the nonlinearity, new frequencies, which are linear combinations of carrier and signal frequencies: ±mp±nq [m, n = 0,1,2,3,...), will occur.

The signal will be understood to be sufficiently small so as to permit the use of the first-order perturbational equations of sec. III.3. This implies that signal-distortion components (gr-harmonics) are neglected so that the relevant conversion components have frequencies ±mp±q.

In this chapter, after the introduction of complex quantities, the equa-tions for the conversion components at the various frequencies will be derived. These equations can be conveniently represented in matrix form. IV.2. Introduction of complex quantities

Assuming r independent meshes for the signals, the frequency-converting network can be described by the following system of r simultaneous Unear

differential equations (see eqs III.6)

r i d - • )

I.^W i^st h) + Qst h + egt ^ = Ug, s = l,...,r, (1)

confining oneself to first-order terms, and ignoring the fundamental state. In (1), the self and mutual inductances 2, resistances g and elastances e are periodic functions of the time with fundamental frequency p on account of their dependence upon the fundamental state. Therefore, one can put, as was first done by Guillemin ^), for the elastances:

est = S *'e„ cos {mpt + 'Vm) = 2 ^'^m exp {j'^Pt)^ 'Vo = 0, (2)

m = 0 m = — CO

for the resistances:

_ . , S

m — 0 m = — C O

and for the inductances:

est = 2 ''r^n cos {mpt + %„,) = S ' ' P „ exp (jmpt), 'Vo = 0, (3)

>-st = 2 X cos {mpt + ''ênt) = S "A„, exp (ympt), "^o = 0 . (4)

m ^ — C O

In eqs (2), (3) and (4) the time-independent complex quantities "i?m, *'Pm and *'/I„i are introduced, which are determined by

2JI/P

"JEm = TT- / «St exp {—jmpt) dt, etc.

0

Since £«<, ^(s and Xgt are real, these quantities, as is easily verified, have to satisfy the following equalities (m = 1,2,3,...):

'% = \ , , ^ ' £ „ = i ' ' e „ exp (j ^Vm) = "£*m; (5)

^'P„ = ' \ , ^'P,„ = 1 'V„, exp (;• ' ' a „ ) = ''P*m; (6)

''A, = X , ' ' ^ m = i ''«m exp (j ''&„,) = ^ ' / l ! ^ ; (7)

where conjugate complex quantities have been denoted by an asterisk. Furthermore, from the reciprocity conditions (III.7) it follows, by equating terms with equal frequencies and using (5), (6) and (7), that

''E^ = ''Ern, (8)

^'P,„ = " P , „ , (9)

''An, = ' M „ , (10)

for all integral values of m. The complex quantities E, P and A will depend both upon the fundamental state and upon the characteristics of the various elements.

On account of the assumptions in sec. 1, the charge variations ft can always be represented by a series of the form*)

ft = 'q'q cos {qt + %) +

CO

+ ^2 J ['?mp -^ q COS ) (m/) + g) t + '§^p + ,( + Vmp-q COS ) (w/) — g) t + '«?mp-g|],

which can be written in complex form as f t = '<?, exp (jgt) + '<?_, exp (-jgt) +

CC

+ ^2 J'^q-Fmp exp ^j(5f + mp)t( + 'Q^^q+mp) exp ^—/(5 -f mp)t\]^ + ^ j ^ ^

CO

+ _^2 J'^q-mp exp \j{q — mp)t< + 'Q-^q-mp) exp \~j{q — mp)t\] ,

if one defines, for m = 1,2,3,...,

'Qq = Ï%^^VU%) ='Q*q,

Qq+mp = ^ imp+q exp (7 &mp+q) = ^_(g-l-mp), (12a) Vq~mp ^^ 2 imp-q exp (j "mp-q) = y^(q-nip) •

Combination of the first, third and fifth terms on the right of (11a) and, also, of the second, fourth and sixth terms, gives

+ CO

ft = ^ 2 ^ ['Qq+mp exp \j{q + mp)t\ + '(>^(q-Fmp) exp \—j{q + m/))((], (11) while the relations (12a) can be contracted to

'Qq+mp = 'Q--(q+mp)^ m = ..., — 1 , 0, + 1, ... . (12)

The current variations f( are easily derived from (11) by differentiation:

+ 00

ft = 2 ['Iq+mp exp \j{q + mp)t\ + 'l^^q+^p) exp \—i{q + mp)t\], (13)

m —— CO ^

where the current components satisfy, in virtue of (12),

'Iq+mp = j{q + mp) 'Qq+mp = j{q + mp) 'Q*(q+mp) = '/*(q + mp),

m = . . . , - 1 , 0 , + 1 , . . . . (14)

IV.3. Complex expressions for the voltages across individual elements in a mesh

Equations (1) express the voltage equilibrium in the various meshes. The product terms ejtft, gstft and d(Astft)/dt represent the voltages across the terminals of the capacitive, resistive and inductive two-pole and coupling

*) To avoid confusion, the charge amplitudes q have been jirovided with a prime for discriminating them from the signal frequency q.