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The application of Lie derivatives in Lagrangian mechanics for the development of a general holonomic… Gustafson, Ture Kenneth 1964

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T H E ; A P P L I C A T I O N , O P L I E D E R I V A T I V E S I N L A G R A N G I A N M E C H A N I C S F O R T H E D E V E L O P M E N T O F A G E N E R A L H O L O N O M I C T H E O R Y O P E L E C T R I C M A C H I N E S by T U R E K E N N E T H G U S T A F S O N B«AoSeo, U n i v e r s i t y o f B r i t i s h C o l u m b i a ? 1 9 6 3 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F -M A S T E R O F A P P L I E D S C I E N C E i n t h e D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g ¥ e a c c e p t t h i s t h e s i s a-.s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A u g u s t , 1 9 6 4 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of • B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for' reference and study, I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t r c o p y i n g or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department of ^ J J L ^ J A J L ^ ^ & /^o^J'/nccJ\JUv\ ^ The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date ffi j f t f c f r ABSTRACT A general approach to the treatment of e l e c t r i c a l machine systems i s developed. Tensor concepts are adopted; however, m e t r i c a l ideas are avoided i n favour of Hamilton's P r i n c i p l e . Using L i e d e r i v a t i v e s and choosing a holonomic reference system, the equations r e s u l t i n g are g e n e r a l , and thus apply to any p h y s i c a l system of machines. These equations are Faraday's Law f o r the e l e c t r i c a l p o r t i o n and a g r a d i e n t equation f o r the mechanical p o r t i o n . Transformation c h a r a c t e r i s t i c s , which are found to be of two independent types, c a l l e d the v-type and the i - t y p e , a r e i n v e s t i -gated. This leads to tensor c h a r a c t e r and i n v a r i a n c e p r o p e r t i e s a s s o c i a t e d with the t r a n s f o r m a t i o n s . The equations of small o s c i l l a t i o n , which are based on the general equations of motion obtained i n the t h e s i s , are d e r i v e d f o r any p h y s i c a l system. In the f i n a l chapter two examples of a p p l i c a t i o n are given; the power s e l s y n system, and the synchronous machine. i i A C K N O W L E D G E M E N T M a n y o f t h e i d e a s p r e s e n t e d i n t h e t h e s i s a r o s e o u t o f d i s c u s s i o n s w i t h t h e s u p e r v i s i n g p r o f e s s o r , D r . l u . T h e a u t h o r e x p r e s s e s h i s t h a n k s t o h i m a n d o t h e r m e m b e r s o f t h e E l e c t r i c a l E n g i n e e r i n g D e p a r t m e n t a t T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , w h o s e e n c o u r a g e m e n t w a s g r a t e f u l l y a p p r e c i a t e d . T h e a u t h o r i s a l s o g r a t e f u l f o r h i s s t i m u l a t i n g a s s o c i a t i o n w i t h M r , H , G e o r g e . T h a n k s a r e d u e t o t h e N a t i o n a l R e s e a r c h C o u n c i l f o r t h e i r a w a r d o f a b u r s a r y i n 1 9 6 3 . v i i TABLE OP CONTENTS Page L i s t of I l l u s t r a t i o n s v Acknowledgement v i i i 1. I n t r o d u c t i o n 1 2a The Basic Machine Equations f o r Any P h y s i c a l Coordinate System 3 2.1 The Transformation of the Machine Equations from a S l i p - R i n g Coordinate System to Any General P h y s i c a l Coordinate System 3 2.2 The E x c i t a t i o n and Induction Angles of a C o i l Winding 10 2.3 The Lagrangian F u n c t i o n and the Equations of Motion 17 2.4 P e r t i n e n t Equations, D e f i n i t i o n s , and Notations 22 3. The Equations of Motion i n Machine System A n a l y s i s 25 3.1 Loop Equations i n True Coordinate Systems 25 3.2 Transformation Theory 27 3.3 Tensors i n the Equations of Motion 33 3.4 Extension of the Equations of Motion to Machines with R o t a t i n g S a l i e n c y . 36 4. The Equations of Small O s c i l l a t i o n s 39 4.1 The Voltage Equations 39 4.2 The Mechanical Equations of O s c i l l a t i o n 42 4„3 The Combined Equations of Motion 46 5. A p p l i c a t i o n s to the D e r i v a t i o n of Machine System Equations 48 5.1 The Power Selsyn System 48 5.2 Synchronous Machines 56 i i i 6 . C o n c l u s i o n A p p e n d i x I C o n d i t i o n s o n t h e F l u x T e n s o r G u a r a n t e e a n E x t r e m u m A p p e n d i x I I T h e H a m i l t o n i a n ; E n e r g y a n d C o E n e r g y R e f e r e n c e s i v LIST OP ILLUSTRATIONS Figure Page 2A The Determination of the Voltage i n an m - c o i l by a Transformation from a F i x e d a-(i System, F i x e d on the Rotor 4 2B Incremental F l u x Changes 10 2C V i r t u a l R o t a t i o n 11 2D Two General Machine C o i l Windings 15 2E The Induction Angle 17 2F V i r t u a l Rotations 21 3A P o l a r i t y Conventions 26 4A Two Phase Induction Motor 44 5A Selsyn System 48 5B Two-Phase Sel s y n U n i t 52 5C Synchronous Machine 56 5D (d-q) - Reference C o n f i g u r a t i o n 57 IIA Single C o i l B-H Curve 71 v LIST OP TABLES Pages 1 Tensor Q u a n t i t i e s i n the Equations of Motion 34 2 Comparison of Conventional Ideas with Those of the Present Thesis 68 v i THE APPLICATION OP LIE DERIVATIVES IN LAGRANGIAN MECHANICS FOR THE DEVELOPMENT OF A GENERAL HOLONOMIC THEORY OF ELECTRIC MACHINES 1. INTRODUCTION There s t i l l remain . many important unsolved problems i n the theory of e l e c t r i c machines and power systems; the a n a l y s i s of machine and power system s t a b i l i t y , the optimal c o n t r o l c r i t e r i a , and e s p e c i a l l y those analyses i n v o l v i n g n o n - l i n e a r e f f e c t s due to s a t u r a t i o n i n the magnetic c i r c u i t s or due to cross-product terms of cu r r e n t s or of cur r e n t and speed. An extension of s t a t i c c i r c u i t theory to r o t a t i n g (T) " ' machinery by G« Kron, employing matrix and tensor a n a l y s i s , provides a means f o r a general a n a l y s i s approach. This extension deals p r i n c i p a l l y w i t h the a d d i t i o n of the r o t a t i n g c o i l as a g e n e r a l i z e d i n d u c t i v e c i r c u i t element,, However, the i n c l u s i o n "of mechanical motion i n an e l e c t r i c a l c i r c u i t i ntroduces an u n p r o p o r t i o n a l amount of d i f f i c u l t y , mainly because of the n o n - l i n e a r i t y resulting,, Kron's adoption of tensors helped because of the g e n e r a l i z a t i o n and u n i f i c a t i o n achieved thereby* Even though h i s method of attack was complicated by the i n t r o d u c t i o n of no n - h o l o n o m i t i c i t y , i t allowed him to deal with a general c l a s s of problems i n a set manner,. I t appeared as though an extension to power system dynamics would r e a d i l y f o l l o w as we l l as an i n c l u s i o n of n o n - l i n e a r i t i e s caused by s a t u r a t i o n and h y s t e r e s i s * This$ however, d i d not occur* The main b a r r i e r to f u r t h e r development l i e s i n the 2 concept of n o n - h o l o n o m i t i c i t y and i t s i m p l i c a t i o n s . According to Kron, Faraday's Law i s a p p l i c a b l e s o l e l y to s l i p r i n g (6) machines. For commutator machines an a d d i t i o n a l term enters because of the independence of the reference frame and c o i l v e l o c i t i e s . This a l s o i m p l i e s that Lagrange's equations are a p p l i c a b l e only f o r these s p e c i a l cases ( s l i p - r i n g frames), the more general Boltzmann-Hamel equations being needed f o r commutator machines. This i s a very l i m i t i n g r e s t r i c t i o n f o r i t means that n e i t h e r Lagrangian nor Hamiltonian mechanics can be a p p l i e d d i r e c t l y to s t a b i l i t y or o p t i m i z a t i o n problems i n v o l v i n g other than s l i p r i n g machines. In the f o l l o w i n g chapter, Kron's equations are i n i t i a l l y employed to obt a i n the equations of motion i n a s l i p - r i n g frame» By d e f i n i n g a t r a n s f o r m a t i o n from t h i s frame to any general p h y s i c a l frame, i t i s shown that the equations of motion i n the l a t t e r remain of the holonomic form. This i s followed by a d e r i v a t i o n of the general equations of motion using Hamilton's P r i n c i p l e . The g e n e r a l i z e d coordinates s e l e c t e d are r e a d i l y seen to be independent proving that any p h y s i c a l machine i s a holonomic electro-mechanical system. In the succeeding chapters, loop equations i n machine a n a l y s i s , L i e deriva/tive concepts of t r a n s f o r m a t i o n theory, the concept of te n s o r s , and the hunting equations are considered. Examples of a p p l i c a t i o n are i n c l u d e d i n the f i n a l chapter. THE BASIC MACHINE EQUATIONS FOR ANT PHYSICAL COORDINATE SYSTEM 2.1 The Transformation of the Machine Equations From a S l i p -Ring Coordinate System to Any General P h y s i c a l Coordinate  System In the general case Kron obtained as the equations of motion f o r an e l e c t r i c a l machine, a v a r i a t i o n of the Boltzmann— (9) Hamel equation, where R^t* i s the r e s i s t a n c e and i n e r t i a l damping tensor a y o c i s the inductance and i n e r t i a tensor . i s the C h r i s t o f e l Symbol i n a non-Riemannian space and C , the coordinate t r a n s f o r m a t i o n matrix from a holonomic frame to a general frame of re f e r e n c e . The l a s t term of (2-1) takes i n t o c o n s i d e r a t i o n such e f f e c t s as the r o t a t i o n of conductors and coordinate reference frames. The mechanical equations are i n c l u d e d i n (2-1). Consider the case of a c o i l i n which a vol t a g e i s induced by the i n f l u e n c e of an e x c i t a t i o n a p p l i e d to the " e x c i t a t i o n " c o i l as shown i n F i g . 2A. Let m be the i n d u c t i o n c o i l ( c o i l i n 4 which a voltage i s induced) and n the e x c i t a t i o n c o i l . Also l e t p © m , P© n» and p©, r e s p e c t i v e l y be, the speed of the m-coil commutator a x i s , the speed of the n - c o i l commutator a x i s , and the speed of the r o t o r , a l l being independent. I t i s proposed to determine the v o l t a g e i n the m—coil by a t r a n s f o r m a t i o n from the a—p system, which i s f i x e d to the r o t o r . I 4 F i g . 2A The Determination of the Voltage i n an m-Coil by a Transformation From a F i x e d oc-g System, F i x e d on the  Rotor The instantaneous e x c i t a t i o n e f f e c t i n the m-coil due to the n — c o i l i s independent of the coordinate frame to which the n - c o i l i s r e f e r r e d . I t depends only on the angular p o s i t i o n and angular speed of the e x c i t a t i o n axis ( n - c o i l commutator a x i s ) . This i m p l i e s t h a t the e x c i t a t i o n c o i l can be assumed to be s i t u a t e d i n an instantaneous holonomic reference frame. T h e e q u a t i o n s o f m o t i o n i n t h e oc-|3 f r a m e , w h i c h i s h o l o n o m i c ^ a r e e , = R t t t i * + a , * ! ! * + [ocf3,Tf] i ~ i * ( 2 - 2 ) F r o m t h i s t h e v o l t a g e e q u a t i o n s i n t h e a a n d p a x e s a r e r> -"t , T • r x - . t . U e S = R S t 1 + L s t d t + L t u » s ] i i = R s n i + L S n ^ + Q-Q— p O l + - g - ^ p O l , ( s = c c , p ) ( 2 - 3 ) s i n c e t h e v o l t a g e i n e i t h e r a x i s i s d u e t o t h e e x c i t a t i o n o f (n) t h e n - c o i l . A p p l y i n g Y u ' s g e n e r a l i n d u c t a n c e f o r m u l a f o r s i n u s o i d a l f l u x v a r i a t i o n , t h a t i s LJL^ = I / i i c o s Q X c o s O 1 " + i A j ^ s i n 0 * s i n 9 4 ( 2 - 4 ) g i v e s L K n = L * ^ c o s 0 c o s 0 n + i A ^ s i n 0 s i n © n L p n = L i ^ n c o s ( © + 9 0 ° ) c o s 0 n + I A p n s i n ( 9 + 9 0 ° ) s i n 9 n . A s s u m i n g y « n = L * j s n = L d a n d L s . ^ „ = I f l . ^ = L ^ , f r o m ( 2 - 4 ) e o t = ^ « r v : i ^ + ( L j c o s 9 c o s 9 " + s i n 9 s i n 9 n ) + ( - L j s i n 9 c o s 9 n + L<^ c o s 9 s i n 9 n ) p 9 + ( - L j c o s 9 s i n 9 n + L ^ s i n 9 c o s 9 l r , ) p 9 n d i n e p = R ^ n i t t + ( - L j s i n 9 c o s 9 n + L ^ c o s 9 s i n © n ) ^ ~ + ( - L j c o s 9 c o s 9 n - L q _ s i n 9 s i n 9 n ) p 9 + ( L j s i n 9 s i n 9 n + L ^ c o s 9 c o s 9 n ) p 9 n i ( 2 - 5 ) The v o l t a g e i n the m - c o i l can be obtained from the t r a n s f o r m a t i o n e w = [ c o s ( Q m - 0 ) , s i n (©*""- ©f| e P Using (2=5), t h i s g i v e s d i n e m = + ) i m + ( L 4 cos © " cos © m + L,, s i n © m s i n ©*)£-{— + (-La s i n © m cos © " + cos © m s i n © n )p© i n + (-Ld cos e w s i n © * + L a s i n © m c o s © n ) p © n i r i . This i s simply e m - ttmni + Lmndt P d e n " I f 0 m i s considered as the angle of the i n d u c t i o n coil» p © ™ the speed of the space i n which t h i s c o i l i s immersed t h i s becomes the holonomic equation of motion. a J) i n . em — a rr»n 1 ^ d t» This r e s u l t suggests the p o s s i b i l i t y of a s s o c i a t i n g two mechanical v a r i a b l e s with each coiL?Sf:. thereby reducing the general machine equations to a holonomic form. The consequence of t h i s would be a s i m p l i f i c a t i o n of the tedious work t h a t was p r e v i o u s l y c a r r i e d out i n the form of t r a n s f o r m a t i o n s from p r i m i t i v e machines. Indeed, the b a s i c p r i m i t i v e machines could be d i s c a r d e d or r e p l a c e d by a more u s e f u l model as p o i n t e d out by Dro I U « A s i m i l a r t r a n s f o r m a t i o n of the torque equation to general 7 c o o r d i n a t e s w i l l now be c a r r i e d o u t . F i r s t , c o n s i d e r t h e t o r q u e e q u a t i o n i n t h e ot-|3 f r a m e o f F i g . 2A. F r o m (2-3) ( j x p , * ] i * i * = [ap,u] i ^ i " w h e r e a a n d p r e f e r t o e l e c t r i c a l c o o r d i n a t e s a n d u r e f e r s t o a m e c h a n i c a l c o o r d i n a t e . S i g n i f y i n g t h i s t e r m b y - t u , t h e n e g a t i v e o f t h e e l e c t r o m a g n e t i c t o r q u e a l o n g t h e u a x i s ; t u - 2 5 x u 1 1 e I n m o s t c a s e s i t i s d e s i r e d t o f i n d t h e t o r q u e a s s o c i a t e d w i t h t h e r o t o r . T h u s x u = 0 i n a n y r e f e r e n c e f r a m e a n d E x p a n d i n g t h e two s u m m a t i o n s i n t f o r F i g . 2A 2 ' d 6 o e a d e (2-6) s i n c e a n d 2 i ^ A k * * d e R e s o l v i n g b o t h i ™ a n d i n ( F i g . 2A) i n t o oc-p c o m p o n e n t s g i v e s i c o s i s i n _ s ( 0 M - ©) j c o s ( © " - ©) .(©*" - ©) , s i n ( © n - ©) (2-7) 8 The inductances and t h e i r d e r i v a t i v e s are L = L <) cos © + s i n © = -L«j cos 9 s i n © + L<^ s i n © cos © Lpjj = L,j s m 0 + L<^cos 0 3 ^ * = -L*(cos*© - sin 2©) + L < L(cos*© - sin*©) °^^=- 2 ( L d s i n © cos 0 - L ^  cos © s i n © ) d e = 2 (-La cos © s i n © + L^ _ s i n 0 cos 0 ) The torque t i n the oc-(3 frame i s t = (-L d cos 0 s i n © + s i n 0 cos ©) i ^ i " * + (-La cos 2© + L ^ o s 20) i K i * + ( L a s i n 20 - L«^ s i n 20) i p i * Using the t r a n s f o r m a t i o n equations (2-7) i«*i« = cos a(© - © m)i ,"i w+ 2 cos(0 - 0 m)cos(©"- © ) i w i n + c o s a ( © © h ) i n i n i*i* = cos(© - O m) sin(© w - © ) i m i m + ( c o s ( © m - ©)sin(© h- ©) + cos(©" - © ) s i n ( 0 w - 0 ) ) i m i * + + cos(0* - 0 ) s i n ( 0 h - 0) i n i * i * i * = s i n £ ( © ^ - ©H^i™ + 2 s i n ( © m - ©)sin(0 n - ©)i wi" + sin a(©" - © ) i h i h ' 0 Hence 9 t = ( L a s i n 0 cos 0 - L < l s i n 0 c o s 0 ) ( i ' i B - i - i * ) + (cos*© -' sin*©) ( L a - L^) i ^ i * B u t i*** = (sin*{Qm- © ) - c o s * ( 9 w - Q))i*i* i ^ - I I -2(cos(2© - © * " - © h ) ) i * i h + ( s i n 2 ( © h - ©) - c o s a ( 9 n - © ) ) i h i » Thus t = ( L d - L < l ) ( s i n 2 2 G - i - i - - ) - cos 2©(i a ei*)) = ( L 4 - L a ) s i n 2 2 ° ( - c o s 2 ( 0 r n - © ) i m i n - 2cos(2©-© r n-© r ,)i h li'' - cos 2 ( © n - © ) i n i n ) - C O S 2 2 Q ( s i n 2 ( © r n - ©) + 2 s i n (©•"" + © * - 2 © ) i r n i n + s i n 2 ( © n - ©) i n i B ) E x p a n d i n g the d o u b l e a n g l e arguments and c o l l e c t i n g terms r e s u l t s i n t = ( L ^ - L^) [ sin© cos© i m i " + (sin© m c o s © h + cos© m s i n © n ) x i r n i n + sin© n c o s © n i n i f " ] = G t n n i w i " where G m n =, (2 - 8 ) Two i m p o r t a n t r e s u l t s a r e o b s e r v e d : l ) The C h r i s t o f e l t r a n s f o r m a t i o n u s e d by K r o n was no more t h a n a m a t h e m a t i c a l t o o l f o r s e l e c t i n g © " "for d i f f e r e n t i a t i o n , 2) & m n i s s i m p l y t h e c o e f f i c i e n t o f c o i l a n g u l a r speed p © m i n t h e i n d u c t i v e impedance m a t r i x . 10 2.2 The E x c i t a t i o n and Induction Angles of a C o i l V i n d i n g A c o i l i s an array of i n t e r c o n n e c t e d wires wound i n some space, which can be r o t a t i n g with r e s p e c t to a chosen i n e r t i a l frame. A c o i l winding c o n s i s t s of the c o i l and the commutator brushes from which e f f e c t s are observed or in t r o d u c e d . A s l i p r i n g machine can be regarded as a s p e c i a l case o c c u r r i n g when the angular speed of the commutator brushes i s equal to the angular speed of the r o t o r . Consider a r o t o r w i t h e x c i t a t i o n s a p p l i e d d i r e c t l y to i t s c o i l s ( i . e . , a p p l i e d through s l i p r i n g s ) . This sets up a f l u x p a t t e r n and i n p a r t i c u l a r a f l u x <P1^ yJc} threading the i - t h c o i l winding, the r o t o r being at s t a n d s t i l l , y * i s the angular p o s i t i o n of the i — t h c o i l winding commutator axes with r e s p e c t to a f i x e d r e f e r e n c e , y * i s allowed to r o t a t e over the e n t i r e r o t o r ( F i g . 2B)„ 0* i s the reference angle of the r o t o r j and pO the r o t o r speed. (a) Rotor at S t a n d s t i l l (b) V i r t u a l R o t a t i o n F i g . 2B Incremental F l u x Changes 11 L e t a v i r t u a l r o t a t i o n b e g i v e n t o t h e r o t o r d u e t o a v i r t ' u a l a n g u l a r v e l o c i t y a c t i n g f o r d t s e c o n d s . S i n c e t h e e x c i t a t i o n s a r e c o n n e c t e d d i r e c t l y t o t h e r o t o r , t h e f l u x p a t t e r n r o t a t e s b y p © ' d t a n d t h e f l u x v e c t o r f o u n d a t y x f o r t h e s t i l l r o t o r w i l l n o w b e f o u n d a t ( y x + t\i p © ^ d t ) j q 4 = 1 . M o r e o v e r , t h e f l u x m e a s u r e d a t t h e l a t t e r p o i n t t r a n s f o r m e d t o t h e c o o r d i n a t e s y s t e m g i v e n b y s 1 = h%(sx - p © * d t ) • ( 2 - 9 ) w h e r e s 1 = y 1 + r| ^ p © ^  d t ( 2 - 1 0 ) w i l l b e e q u a l t o t h e f l u x a t y i n t h e s t i l l m a c h i n e s i n c e • s Z = 8 J y 1 ( 2 - 1 1 ) REFERENCE FRAME FOR \X oYSTEM R E F E R E N C E F R A M E F O ^ X S Y S T E M ' P i g . 2C V i r t u a l R o t a t i o n D e n o t i n g fl)^ a s t h e f l u x i n t h e m o v i n g m a c h i n e ( 2 - 1 2 ) P r o m ( 2 - 9 ) , t h e c o o r d i n a t e t r a n s f o r m a t i o n , 1 2 A t =1§J = S l u t - & < l i p Q A ) * t = S i A f = 6 £ & t = ( 2 - 1 3 ) a n d t h u s ^ { y 1 + C ^ p O ^ d t } = C p ^ y - } ( 2 - 1 4 ) T a k i n g a l i n e a r a p p r o x i m a t i o n t o t h e T a y l o r e x p a n s i o n o f t h e l e f t h a n d s i d e T h e i n c r e m e n t a l c h a n g e i n t h e o b s e r v e d f l u x d u e t o r o t o r r o t a t i o n i s t h u s +• (ck*yA} - Wy 1}) = - | ? i ( r l t p e i ) d t ( 2 _ 1 6 ) w h i c h i s t h e L i e d i f f e r e n t i a l o f C p ^ y k, ) o v e r p 9 * . ' ^ 2 . ^ T h e c o o r d i n a t e s y s t e m T i s d r a g g e d a l o n g b y t h e c o m m u t a t o r s y s t e m i u n d e r t h e p o i n t t r a n s f o r m a t i o n si= yx+ ( » \ ^ p © l ) d t ( 2 - 1 7 ) B y w r i t i n g ( 2 - 1 2 ) a s t h i s i s a l s o e q u a l t o 13 the negative of the L i e d i f f e r e n t i a l of CP^over r^pQ-\ The f l u x w i l l s i m i l a r l y be caused to r o t a t e i f the r o t o r i s s t a t i o n a r y and the axes of e x c i t a t i o n are allowed to r o t a t e . Using standard n o t a t i o n t h i s " .contribution i s Z.$>Ay'} = i™ ( a t > py &. dt 0 — jftw* ( 2 - 1 8 ) Besides angles the f l u x i s a f u n c t i o n of c u r r e n t . As a consequence,the sum of the time and L i e v a r i a t i o n s must be balanced by current v a r i a t i o n s * That i s 3 # x d i A Thus (2-19) dt: =ldP ) d T + a y i d t + a y ^ ^ d t ( 2 - 2 0 ) y k , © k _ a ^ i i l • + M i d i i + M A n f i i i (2-21) This can also be w r i t t e n as The l a s t terms, due to the speed of the space i n which the i - t h c o i l i s wound} i n v o l v e s d i f f e r e n t i a t i o n with respect to the i - t h commutator ax i s angle only and i s independent of a l l 14 commutator axes speeds. I t has the c h a r a c t e r i s t i c s of an i n d u c t i v e type of volta'ge. The commutator a x i s angle y x used i n t h i s sense w i l l be r e f e r r e d to as the i n d u c t i o n angle of the i - t h c o i l winding and w i l l be r e l a b e l l e d x i n any equations i n which i t o c c u r s 0 The c o n t r i b u t i o n - : can then be w r i t t e n + | f i p x % where px ^  ^ k rtf (2-23) The second term on the r i g h t hand side of (2-22) i n v o l v e s d i f f e r e n t i a t i o n over a l l commutator axes angles and depends on commutator axes angular speedso I t i s independent of a l l c o i l speeds. "y-*-" used i n t h i s sense w i l l be r e f e r r e d to as the e x c i t a t i o n angle of the i - t h c o i l winding since d i f f e r e n t i a t i o n w ith respect to i t gives r i s e to the angular speed of the e x c i t a t i o n a p p l i e d to the i ' t h c o i l winding. " y A " w i l l not be r e l a b e l l e d under these c o n d i t i o n s . Using (2-23) , equation (2-22) can be w r i t t e n i n a chain r u l e form d&i. aCPjLdi* dflkdy * 8(Pi.Sxj , v d t ~ 3 i s d t dy»dt ax Jd.t K Z (7 7 1 8 ^ Ve s h a l l now apply the above r e s u l t s to Yu's formula, ' which i s of b a s i c importance i n machine a n a l y s i s . In P i g . 2D l e t the m-coil be' wound i n a space with angular speed pQ**. I f the n c o i l i s e x c i t e d the f l u x i n the m-coil i s given by (P m= L r a r , i p = ( L d w n c o s y m c o s y n + l A m n s i n y m s i n y " ) i n (2-25) 15 R E F E R E N C E m - c o i L F i g . 2D Two General Machine C o i l Windings where y 1™ and y n are the commutator a x i s angles of the m and n-c o i l s r e s p e c t i v e l y . Prom (2-21) d - ^ = ( L ^ r c o s y ' c o s y " + i A ^ s i n y ^  s i n y") d i n dt dt + . ( - L i m n c o s y** s i n y * + l A m n s i n y w c o s y") i n + ( - U ^ s i n y^'cos y " + l A ^ c o s y s i n y ") ( | 2 _ ) ± (2-26) By the previous d i s c u s s i o n , y n i s the e x c i t a t i o n angle of the n - c o i l and y ^ t h e i n d u c t i o n angle of the m - c o i l , which i s denoted by x m r a t h e r than y™, (2-25) and (2-26) then become ou= L ^ o 1 " = ( U ^ c o s x m c o s y ^ + U m r ; s i n x ™sin y n ) i * (2-27) d t ~ a i n d - t + Wyn4.t 8 u d t ; (2-28) As a s p e c i a l case the n - c o i l can c o i n c i d e with the m - c o i l , y becoming y m w h i c h i s n u m e r i c a l l y equal to x*". However, the dy^ a Sx 1^ a operators -^^: mand J*^ - ^ ^ a r e equal only f o r a s l i p r i n g machine, From (2-23) f o r a number of c o i l windings P x = d t U dt (2-29) S x m Here - j ^ - i s the speed of the c o i l comprising the m-th c o i l d© ^  winding and that of the j - t h space. The q u a n t i t y connecting these two speeds w i l l be c a l l e d the i n d u c t i o n angle incidence matrix. For the example of F i g . 2E i t i s given by px3- 1 0 px b 1 0 p x c 0 1 px* 0 1 p©* <1> 1 0 1 0 0 1 0 1 <1>, 1 1 0 0 (1>R = 0 0 1 1 17 ^ R E F E R E N C E \ F i g - 2E The Induction Angle 2.3 The Lagrangian F u n c t i o n and The Equations of Motion The Lagrangian f o r any e l e c t r i c a l machine i s given by L = T = ( K i n e t i c Energy Function) (2-30) o m i t t i n g c a p a c i t i v e e f f e c t s w i t h i n the machine. The k i n e t i c energy i s the sum of the mechanical energy and the stor e d magnetic energy. . In a l l subsequent equations the f o l l o w i n g n o t a t i o n w i l l be used; a) Greek l e t t e r s r e f e r to mechanical q u a n t i t i e s Jotet? the i n e r t i a of the oc'th space Dcteo the damping c o e f f i c i e n t of the a' th space p©**, the angular speed of the a'th space 18 b) Roman l e t t e r s r e f e r to e l e c t r i c a l q u a n t i t i e s L m n , the mutual inductance from the n - c o i l to the m-coil R " k h e r e s i s t a n c e of the m-coil i m , the cur r e n t i n the m-coil c) x and y angles, which a r i s e i n both mechanical and e l e c t r i c a l equations, w i l l be indexed with i , j , or k unless they take s p e c i f i c i n d i c e s . In order to determine the stored magnetic energy the c o i l currents are r a i s e d i n succ e s s i o n , the r e s u l t i n g energy c o n t r i b u t i o n s are c a l c u l a t e d and the i n d i v i d u a l c o n t r i b u t i o n s summed. The order of the cur r e n t s can be chosen a r b i t r a r i l y and enumerated as ( i 1 , i a ... i n ) o The t o t a l s t o r e d energy i s then .cc' E e = Z ( A E f t J L ) = \ Cp.(a') d« J L = 1 J 0 JL + ^ cca) da* + • JL ' + C Qji* ... i n " ' , oc")doc" o = J C P i d ' . . . i 1 " * ,a- ,)da i (2-31) o using the summation convention and a * as the dummy v a r i a b l e of i n t e g r a t i o n . The mechanical energy i n the system i s E w = | JocccPO^P© 0" ' (2-32) 19 and hence the Lagrangian i s L = <Ji(i' i 1 " ' , a^da 1 + | p O * p© (2-33) o The state of any p h y s i c a l l y r e a l i z a b l e machine i s completely d e s c r i b e d by ( q x , © A y y X ) * q"*" being the t o t a l charge pa s s i n g a reference p o i n t an the i H h c o i l winding l e a d ; ©"*", the angular p o s i t i o n of the i T t . h space w i t h re s p e c t to a chosen r e f e r e n c e ; and y *~9 the angular p o s i t i o n of the i ' t h c o i l winding with res p e c t to a chosen referenceo Moreover any "displacement" ( A q x ? A©"*";, Ay"1*) i s _.a p o s s i b l e "displacement" of, the stat e and hence the d e s c r i p t i o n i s holonomic and ( q x , 0 X 9 y X ) i s a true (14) " c o o r d inate" system v „ Consequently, any p h y s i c a l l y r e a l i z a b l e machine i s a true e l e c t r o - m e c h a n i c a l system and Hamilton's P r i n c i p l e a p p l i e s without s u b s i d i a r y c onditionso Since Ee, = Ee {i 1 « o o i *% y 1 °«<> y A 9 x 1 „ 0 „ x A } and •b E m .-= E ^  {p©*}$> the v a r i a t i o n i n the a c t i o n 9 A = ^ ( L ) d t S ) due to the independent v a r i a t i o n s (Aq/% A© , Ay x) i s AA - ^ ( A E e + A E « v ) d t t . | E - A q " + | f > A i - H - 2 E e + ^ + |ff A©* + |f§v A( P©*))dt ^ E The two L i e v a r i a t i o n s enter through the f l u x f u n c t i o n (Eqn 0 (2-=3l)) and hence from Eqn c (2-16) and (2-18) are equal to 2 E e = A © 0 6 (2-35) A V 20 S u b s t i t u t i n g these i n t o (2—34) and i n t e g r a t i n g the f i r s t and l a s t terms by parts maintaining f i x e d end p o i n t s these r e s u l t s : ^ = 5<- ft $h*) + <- it l b - - sp» (2-36) By Hamilton's P r i n c i p l e t h i s i s zero. Since Aq h, A©^ and Ay x are independent the equations of motion i n c l u d i n g d r i v i n g f o r c e s and d i s s i p a t i o n are: e n ft | i * + R » « i n ( 2 " 3 7 )  S»-tt|faf + DMPO*+|f*^ (2-38) ^ =ffi (2-39) The torque r- a r i s i n g from the dependence of E e on the reference axis angle y "** acts on the i ' t h c o i l and hence on the space (3 i f does not equal zero, t ^ the e x t e r n a l l y a p p l i e d torque on the (3-space i s S = s » + ^ l , ( 2 ~ 4 0 ) Since i n (2-31) the n - c o i l can be chosen as l a s t r e f e r r e d to i n the summation M i J. M $ U i ' ... i " - a " } d a ' 3pqn Q i ' = & n{i' ... i n } (2-41) The three equations (2-34), (2-38) and (2-39) are thus reduced to a system of voltage equations and a system of torque equations. f ^ = R - i n • (2-42) 21. t B = D „ pO* + J ^ p f p O ' ) + (2-43) The torque equation, (2-43), can be s i m p l i f i e d as f o l l o w s , a) Before R o t a t i o n b) Rotor V i r t u a l c) R o t a t i o n E x c i t a t i o n V i r t u a l R o t a t i o n Fig„ 2F V i r t u a l Rotations R e f e r r i n g to s e c t i o n 2o2j, i f the r o t o r i s given a v i r t u a l r o t a t i o n A9 from the steady s t a t e value the L i e v a r i a t i o n of f l u x threading the m c o i l winding a x i s i s BCD, , - * - A ^ ° < -m 3 x M * A ° r v A G (2-44) the f l u x having r o t a t e d by ah a d d i t i o n a l A9° I f on the other hand t h e . r o t o r e x c i t a t i o n axes alone are given a v i r t u a l r o t a t i o n , the i ' t h a x i s being r o t a t e d by Ay A, the f l u x change i s 2 0 (2-45) 22 By making Ay x = - r|^A©°% t h a t i s , r o t a t i n g a l l r o t o r e x c i t a t i o n coordinates by -A©°^ t h i s change i s + i j c A©1*. From Fig« 4F(c) the sum of these two f l u x changes i s zero and thus _ ,n x. _ d o Since the d e r i v a t i v e s of (2—43) pass under the i n t e g r a l s (2-33) f o r s u i t a b l e c o n d i t i o n s , The f i n a l equations of motion are then ^ w em - "mm1 + ^ — R rr»m • m . 3$,ndi n ^ . t x ^ , d^dy_^ 1 + d i " dt + §x* dt T d y A dt (2-48) t „ = D „ p©^ + J „ p(p©») + 2 | f l r^j (2-49) 2«4 P e r t i n e n t Equations, D e f i n i t i o n s , and Notations The f o l l o w i n g p e r t i n e n t equations, d e f i n i t i o n s , and n o t a t i o n s w i l l be used i n subsequent chapters, a) Voltage equations : M 1) e - R i n + d - ^ n  1 J err\- amnx + d t R^^ i K V + a i n d t T dt T S y » dt c)CD ^di h a CD* &X_^  a (Pry, dy * n d+, x d t 3v» d t For the l i n e a r case (pm= L r r i n i r i and D n « j . T D I " , 3 L M h Sx n OL Wn d y n 23 2 ) 3 ) 4) 8 L , = G 8 y n 6 x " d t mi, = v i r i n n I n d u c t i o n C o e f f i c i e n t E x c i t a t i o n ' C o e f f i c i e n t n m p O r ' ; H ! T = I n d u c t i o n A n g l e I n c i d e n t 1 v M a t r i x 5) i^ -j.= = I n d u c t i o n A n g l e G r a d i e n t ( i ' t h C o m p o n e n t ) 6 ) 4~».= W - — E x c i t a t i o n A n g l e G r a d i e n t ( i ' t h 0 y V x C o m p o n e n t ) 7 ) ^p. + = = A n g u l a r G r a d i e n t b ) T o r q u e e q u a t i o n s : v 1 ) t , = D^pp©" + J p p p ( p O P ) + 2 ( ^ 7 E f t ) . ( r ^ ) p F o r t h e l i n e a r c a s e * P = D p p p e * + J „ p ( p Q P ) + ( ^ L ^ ) , ^ i " , i » 2 ) (^7 L n J . O - ^ = ^ £ 1 ^ = G ^ l j ^ T m r > , = T o r q u e T e n s o r f o r i t h e (3 s p a c e 3 ) t * = + 2 ( s y E e ) ( r ^ ) ? = E l e c t r o m a g n e t i c E n e r g y C o n v e r s i o n T o r q u e o n t h e |3 s p a c e T h e d o t p r o d u c t , a s a n e x a m p l e , f o r F i g . « 2 E i s +2 U dE* dE., c)E a 3 E » Qx*' 3 x b ' d x c ' ax 7 t r - +2 "8E e 9Ee. 5 E a 3 E a dx«-' d x b ' d x " ©x 7 24 o r 4-2 1 1 0 0 0 0 1 1 8 E * d x a d E e 3x c 9 E A where t s i s t h e e l e c t r o m a g n e t i c e n e r g y c o n v e r s i o n t o r q u e on the s t a t o r and t r i s t h e e l e c t r o m a g n e t i c e n e r g y c o n v e r s i o n t o r q u e on t h e r o t o r . 3. THE EQUATIONS OP MOTION IN MACHINE SYSTEM ANALYSIS 3.1 Loop Equations i n True Coordinate Systems I t has been proven that Faraday's Induction Lav i s v a l i d f o r any e l e c t r i c a l machine r e f e r r e d to a true coordinate system. Consequently methods of s t a t i c c i r c u i t a n a l y s i s can be extended to i n c l u d e a system c o n t a i n i n g e l e c t r i c a l machines. For an unsaturated system i n c l u d i n g c a p a c i t i v e elements, the general loop equations are e m — R m where = R ^ i " + L w n p i " +1^ ^ 0 * 1 * H - I ^ p y " ^ + = z m n ( p ) i n (3-1) (3-2) a r e t h e impedance m a t r i x e l e m e n t s . N o d a l e q u a t i o n s - c o u l d a l s o be u s e d ; h o w e v e r , s i n c e an e l e c t r i c a l m a c h i n e i s an i n d u c t i v e s y s t e m , and s i n c e t h e d r i v i n g f o r c e s a r e v o l t a g e s , l o o p a n a l y s i s i s more n a t u r a l . 3L m The t o r q u e t e n s o r e l e m e n t T .= Q x ^ n H . i-s " t n e c o e f f i c i e n t o f p© ^ i n z r n n ( p ) « i T h e r e f o r e once t h e l o o p e q u a t i o n have b e e n d e t e r m i n e d , t h e t o r q u e e q u a t i o n s c a n be o b t a i n e d b y i n s p e c t i o n . V = D ^ p © ^ + J^ptp© 5*;) + l ^ r ^ i ^ i " (3-3) 26 Using Yu's formula a standard p o l a r i t y convention can be adopted to determine the signs of a l l terms i n the equations of motion. The c o i l c u r r e n t s are a r b i t r a r i l y chosen to flow towards the center of the machine i f p o s i t i v e , whereas the M*M.F i s r e f e r e n c e d i n an outward d i r e c t i o n , as i s the flux<. Since the v o l t a g e i s a d r i v i n g f o r c e the c u r r e n t flows i n t o the c o i l at i t s plus r e f e r e n c e . From Yu's formula the x and y angles are measured p o s i t i v e l y i n a clockwise d i r e c t i o n from the d - a x i s . P o s i t i v e torque acts i n the d i r e c t i o n of i n c r e a s i n g since i t i s a d r i v i n g f o r c e . The p o s i t i v e r e f e r e n c e sense f o r 9 p can be chosen a r b i t r a r i l y , since the s i g n s o c c u r r i n g i n the equations are given by r j ^ . These r e f e r e n c e s are shown i n Fig» 3A(a) f o r three c o i l s . a) P o s i t i v e p o l a r i t y b) Standard P o l a r i t y Con-v e n t i o n F i g . 3A P o l a r i t y Convention 27 I f the MM/F.direction i n the n - c o i l i s opposite to the chosen r e f e r e n c e d i r e c t i o n a negative s i g n appears i n f r o n t of a l l L m n « Since i n Yu's formula L * m n = N,„ N n P j , t h i s i s e q u i v a l e n t to s e t t i n g N w n e g a t i v e , P o l a r i t y can thus be i n t e r p r e t e d as the presence of a p o s i t i v e or negative number of turns i n a winding. This w i l l be noted by p l a c i n g a dot at the plus end of the n - c o i l i f N n i s p o s i t i v e and at the negative end i f N n i s negative* This gives the standard p o l a r i t y convention of F i g . 3B(b)« 3*2 Transformation Theory Many tran s f o r m a t i o n s used i n machine a n a l y s i s are h y p o t h e t i c a l ; t h a t i s , they are manipulative i n v e n t i o n s which ease the task of d e a l i n g w i t h complicated algebra or cumbersome d i f f e r e n t i a l equations. Since they are h y p o t h e t i c a l , they can be d e f i n e d i n an a r b i t r a r y manner. For i n s t a n c e , a c u r r e n t t r a n s f o r m a t i o n , i n = C" i n f can i n v o l v e the x-angles, y—angles, or any other parameters, z say. The v o l t a g e t r a n s f o r m a t i o n e ^  =-€^"e n can be -defined independently of the c u r r e n t t r a n s -formation, i n v o l v i n g an e n t i r e l y d i f f e r e n t set of parameters. These tra n s f o r m a t i o n s can a l s o i n v o l v e complex numbers. I f the t r a n s f o r m a t i o n s are from one true coordinate system to a new true coordinate system, the equations of motion i n t h i s new system are d e r i v a b l e d i r e c t l y from Hamilton's P r i n c i p l e . The transformed equations must then be r e d u c i b l e to Lagrange's e q u a t i o n s t which r e s t r i c t s the t r a n s -f o r m a t i o n s . To o b t a i n these r e s t r i c t i o n s l e t the c u r r e n t and v o l t a g e transformations from the o l d (unbarred) to the new 28 (barred) system be r e s p e c t i v e l y i " = C £ i n (3-4) e ^ ^ e ^ (3-5) D e a l i n g w i t h the v o l t a g e equations o . n , d_ I d E \ em = 1 + dt \ 3 i m Under the above transformations these become em = *€ m ( R m n C R ± K + f t ^51" C ^ = («£ B^O i " + - e £ ( C i l f r ) (3-6) In order f o r the new system to be a true system ( p h y s i c a l system) we must have where E i s the energy f u n c t i o n i n the new system. This can be s a t i s f i e d only i f •€ ^ or C m or both can be moved through the d i f f e r e n t i a t i o n and then combined to give a Kronecker d e l t a . The f o l l o w i n g c o n d i t i o n s must be s a t i s f i e d . 1) C = (3-8) 2) &|r£ = 0 and/or | | ™ = 0 (3-9) The f i r s t c o n d i t i o n r e s t r i c t s the transformations to those which maintain the power (energy) i n v a r i a n t since ' e = i m = e m « K C „ l = b ^ e _ i n = e ^ i E = ^ e w i * dt = ^ e m i m d t = E o o The Kronecker d e l t a obtained i n (3-8) i s not s t r i c t l y a 29 constant even though i t i s a u n i t matrix since the time d e r i v a t i v e , which i s L i e , i s not zero g e n e r a l l y . The c o n d i t i o n = 0 allows Crn to be taken i n f r o n t of |j dt I t i s then combined w i t h - e ^ to give (3-8). Since no d i f f e r e n t -i a t i p n of (3-8) occurs, the v o l t a g e equations remain e s s e n t i a l l y unchanged and t h e r e f o r e these t r a n s f o r m a t i o n s are of l i t t l e i n t e r e s t . Ve are l e f t with ^ = 0; ^ £ 0 and € = S ^ to be s a t i s f i e d by tran s f o r m a t i o n s from one true system to another true system. These allow j£ to be taken i n s i d e ^ i n (3-6) and then combined with cjj, to give S J R . I n t h i s case, since S ^ i s — u n d e r the d i f f e r e n t i a t i o n , i t s L i e d e r i v a t i v e s e n t e r " t p o s s i b l y changing the v o l t a g e equation quite r a d i c a l l y . rLC" ' Since ^ 0, the L i e d e r i v a t i v e o c c u r r i n g must be over a non-zero speed, Moreover not a l l the p a r t i a l d e r i v a t i v e s S|^ 3u and g^:T c a n v a n i s h . By (3—8) •€ ^  must be the inv e r s e of with a p o s s i b l e interchange of e x c i t a t i o n and i n d u c t i o n angles o c c u r r i n g . However, since ~^~rm = 0 and not a l l d C St and need be zero, the L i e d e r i v a t i v e must be over a zero speed. This i m p l i e s that the L i e d e r i v a t i v e reduces to one over the d i f f e r e n c e between c o i l v e l o c i t i e s i n the new system and the o l d system since these are the only i n v a r i a n t speeds. Consequently •€^ i s a f u n c t i o n of the d i f f e r e n c e between new and o l d i n d u c t i o n angles r e f e r r i n g to c o i l s i n the same space. rlC n -i. R Since there are no p r e f e r r e d angles and d ^ r n ' 0, C m must be a f u n c t i o n of the d i f f e r e n c e between new and o l d e x c i t a t i o n angles o Examples of some such transformations are given on the f o l l o w i n g page» a) mn-dq Transformation, Id. m (y«\ x 4 ) = (0,0) . (py*, px3") - (o, Po r) ( p y m , p x m ) = ( P© r, P o r ) A l l coordinates are r e f e r r e d to the r o t o r space since the t r a n s f o r m a t i o n i s on the r o t o r . 1 costy*" - y4) , sin(y*" - y V r i i n ~sin(.y*" - y4) t c o s ( y * - y4) ~ c o s ( x m - x*) , +sin(x" " - X* )~ - s i n ( x w - x d ) , c o s ( x r " - x d ) _ _ -In these dt 2 c £ * o ; If =2« = 0 b) Synchronous Machine Transformations. (py*, p_xm) - (pO r, pQ s) ( p y d , px a) = (s, p© s) ( y \ x 1 ) = (0,0) ( p y a , p x 1) = (0, p © S ) 31 A l l coordinates are t i e d to the s t a t o r f o r s t a t o r t r a n s -formations. (Note t h a t p x w = p©t)„ s i s the synchronous speed. c o s ( y * - y m ) , s i n ( y d - y w ) ' - s i n ( y d - y w ) , c o s ( y d - y m ) • m 1 _ i - - c o s ( x d - x m ) t s i n ( x d - x1^) - s i n ( x t I - x m ) , c o s ( x 4 - x m ) e m c o s ( y a - y w ) , c o s ( l 2 0 + y 3" - ym), cos(240 + y s i n ( y a " - y " ) , s i n ( l 2 0 + y a - y m ) , sih(.240 + y a. a. y") l 1 4 ? 1 eg. e -IT c o s ( x a - x m ) c o s ( l 2 0 + x a cos(240 + x a s i n ( x a - x m ) x m ) s i n ( l 2 0 •+ x a x m ) s i n (240 + x a x m ) x w ) 1 1 1 e » i -J? cos s i n ( y 1 -( y 1 -1 y*) , cos (120 + y 3-y 3 ) s i n (120 + . y a 1 y * ) , cos(240 + y a - y*) y a) sin(240 + y a - y d ) 1 c o s ( x a - x d ) , sin ( x a " - x 3 ) cos(120 + x a - x 4 ) , s i n ( l 2 0 + x a -cos (240 + x9- - x J ) , sin(240 + x a -x ) x ) 1 1 I e 0 i 1 In a l l of these the L i e d e r i v a t i v e of the vo l t a g e t r a n s -formation i s e q u i v a l e n t to zero whereas that of the c u r r e n t t r a n s f o r m a t i o n i s not. That i s | | w = 0; sL£"» £ 0. 32 c) fb-dq Transformations 1 - J 1 -j(x*°x a) - j ( x m - x 3 ) . j U * " - x*) , j ( x w - x a ) Once again d-6 ™ dt m The torque equation i n t r o d u c e s no a d d i t i o n a l c o n d i t i o n s on transformations from one true system to another true, system, since f o r E = E, the- torque equation r e t a i n s i t s holonomic form. where t * = D ^ p Q + p(pO^) + 2 ^ E D pO»= fclp©* 3 '2 S i t (3-10) (3-11) (3-12) (3-13) (3-14) (3-15) E = E i n the new system, = ^ x , x"1" being the i ' t h i n d u c t i o n angle i n the new system and r^T = r ^ , the incid e n c e matrix i n the new system. 33 3«3 Tensors In the Equations of Motion A tensor index can transform by the volt a g e t r a n s f o r m a t i o n matrix •€, i n which case i t w i l l be c a l l e d a v-index, or i t can transform'by the c u r r e n t t r a n s f o r m a t i o n matrix C, i n which case i t w i l l be c a l l e d an i - i n d e x . Any tensor q u a n t i t y can have; a l l i - i n d i c e s ( i - t e n s o r ) , a l l v — i n d i c i e s (v-tensor) or a combination of v and i i n d i c e s ( v - i tensor),, I f (3-8) i s s a t i s f i e d , so -€y£ i s the inv e r s e of except f o r a p o s s i b l e interchange of i n d u c t i o n and e x c i t a t i o n angles, a l l tensor i n d i c e s transform i n the same manner except f o r a p o s s i b l e interchange of i n d u c t i o n and e x c i t a t i o n angles* I f i t i s necessary to show e x p l i c i t l y how an index t r a n s -forms a ' V'or a n l ( i " w i l l be p l a c e d e i t h e r above or below i t . Tensor q u a n t i t i e s o c c u r r i n g i n the equations of motion can be determined from the expansion of (3-6), and from (3-10) through (3-15). They are l i s t e d i n Table 1 on the f o l l o w i n g page. Since the e x c i t a t i o n and i n d u c t i o n angles are not tensor q u a n t i t i e s n e i t h e r are the i n d u c t i o n or e x c i t a t i o n c o e f f i c i e n t s , I t can be seen from the t a b l e that i n general the torque tensor elements transform as a v — i tensor i n the vo l t a g e equation whereas they transform as an i - i tensor i n the torque equation. The d e t a i l e d tensor n o t a t i o n used above i s u s e f u l f o r checking tensor c h a r a c t e r or f o r i n v e s t i g a t i o n g d e t a i l s , however, f o r computational purposes matrix n o t a t i o n i s b e t t e r . In t h i s form the vo l t a g e equation i n a true reference frame are •(e) = ( (R) + p (L) ) ( i ) , p = | T (3-16) 34 T e n s o r Q u a n t i t i e s T r a n s f o r m a t i o n P r o p e r t i e s a ) V o l t a g e E q u a ; t i o n V 1 — ' w m 3 i m T dj&m = o c m I—i tn p V ^ ( 3<Dwi U"» ^rp <T> m C n C B T b ) T o r q u e E q u a t i o n P * » t ? ( a < P , r n H w T T a b l e I T e n s o r Q u a n t i t i e s i n t h e E q u a t i o n s o f M o t i o n A m a t r i x m u l t i p l i c a t i o n . o c c u r s f o r e a c h |3 i n t h e c a l c u l a t i o n o f t h e e l e c t r o m a g n e t i c t o r q u e s i n c e T m n p i s a t r i a d o R e t a i n i n g t h i s i n d e x f o r c l a r i t y t h e t o r q u e e q u a t i o n s i n a t r u e f r a m e a r e ( t p ) = ( D „ ) ( p © * ) + ( J ^ p M p © * ) + 2 ( ( ^ E ) 0 ( r ^ ) p ) (3-17) F o r t h e u n s a t u r a t e d c a s e t h i s i s 35 (t») = ( D „ ) ( p © 3 ) + (J wp)(.p©*) + 2 ( ( i f ( T ) p ( i ) ) (3-18) The t r a n s f o r m a t i o n equations (3-4) and (3-5) are ( i ) = ( C ) ( i ) (3-19) ( e ) = (-ef (e) (3-20) For these transformations (3-16) and (3-18) become • * (e) = (€)_*'(R) CC) (I) + ( € ) * p ( L ) ( C ) ( T ) (3-31) ( t , ) = ( D & p ) ( p © P ) + ' (J Mp)(p0 1 1 ) + ( ( D t ( C ) t ( T ) ? ( C ) ( i ) ) (3-22) I f p© ^ does not equal zero and (3-8) holds so that (C) i s r e p l a c e a b l e by("€!) except f o r an interchange of i n d u c t i o n and e x c i t a t i o n a n g l e s p the transformed torque tensor (C)*' ( T ) B ( C ) i s the matrix of c o e f f i c i e n t s of p© ^ in' '(3.-21) before p o s s i b l e di.( C) (•e) (L) Q^f- ( i ) c o n t r i b u t ions a-.re added 0 In t h i s case then. once the v o l t a g e equations have been transformed, the transformed torque equations can be obtained by i n s p e c t i o n a f t e r a p o s s i b l e interchange of i n d u c t i o n and e x c i t a t i o n coordinates„ Invariance., a property u s u a l l y a s s o c i a t e d with the con-v e c t i o n of two tensors of opposite types' 9 i s no longer a general r e s u l t . As an example under (3—4) and (3-5) e ^ i ^ e . - e ^ C ^ i ' ^ - e J c S e ^ . ' (3-23) (3-23) i s understandable i f i t i s r e a l i z e d t h a t power and energy can be d e f i n e d only i n a 'true system f o r which e m and i m are p h y s i c a l l y measurable. For a t r a n s f o r m a t i o n from one true system to another (3-8) i s s a t i s f i e d and the i n v a r i a n c e property f o l l o w s . 36 For an untrue system e R i m and ^ ( e ^ i ^ d t are a b s t r a c t q u a n t i t i e s ; they are not the true power and. energy. Power i n the (m) system i s • € ™ C ™ e ^ i * r e f e r e n c e d to the r e a l (m) system. The energy i s l i k e w i s e j ( C ™ C ™ e ^ i ^ ) d t r e f e r e n c e d to the (m) system. In t h i s sense both power and energy are i n v a r i a n t s of any t r a n s f o r m a t i o n . Another form of i n v a r i a n e e occurs i n the torque equation. The torque t p i s a c o v a r i a n t tensor; however } i t s t r a n s f o r m a t i o n ma,irix. i s always the Kronecker d e l t a and i t i s thus a c t u a l l y i n v a r i a n t . Mechanical angles are transformed s i m i l a r l y a,nd hence the e n t i r e torque equation i s e s s e n t i a l l y i n v a r i a n t . T h i s a p p l i e s i n p a r t i c u l a r to the electromagnetic energy conversion torque which i s c o v a r i a n t i n the mechanical c o o r d i n a t e s . 3.4 E x t e n s i o n of the.Equations of Motion to a Machine With  Rota t i n g S a l i e n c y S a l i e n c y i n the equations of motion i s expressed by a dependence of the f l u x upon angular displacements, 0' 9 of the machine spaces. In the equations of motion (2-48) and (2—49) any s a l i e n c y was assumed f i x e d w i t h r e s p e c t to the chosen refere n c e and hence © dependencies were i r r e l e v e n t and could be n e g l e c t e d . In general then the f l u x and thus the s t o r e d e l e c t r o -magnetic energy are i n f l u e n c e d by mechanical r o t a t i o n s as w e l l as the mechanical energy./ These added v a r i a t i o n s are L i e 9 the g e n e r a l i z a t i o n s of (2-22) and (2-=34) being ^^m&^Sfr+ZVx (3-24) _ CLS-L d i * , S CD.JL dy 1 a CD JL d © ? . 3 C D ^ J d© ? ~ di*. dt + 3 p L dt + B©^ a t + ^ d t (3-25) 37 •t AA= 5<^ » ^ i - +»R. •a^ o Proceeding as i n 2«3, the equations of motion are easily-found to be M „ _ u -j n , 3.(Pun e tr> - "mn1 + at (3-27) t , = ' D - ^ P © * + J „ p ( p O * ) + 2 ^ E e ^ + | | | (3-28) As a s p e c i a l case to be used l a t e r , c o n s i d e r Yu's formula f o r a r o t a t i n g d - a x i s . I t can be w r i t t e n i n the form L = L * m n c o s X m c o s Y n + iAfn^sin X m s i n Y n (3-29) where X m = x*" - S " > (3-30) YN- = y* - S * (3-31) x a n d y n are the i n d u c t i o n and e x c i t a t i o n abg3.es of the m and n c o i l s r e s p e c t i v e l y . 6 X > which w i l l be c a l l e d the s a l i e n c y angle of the i - t h c o i l , i s d e f i n e d by a s 1 d s x d Q » ft^-ao*- (T\O\ dt - 8 0 ? dt _ " » dt U - ^ J " I p " ' ^ e sa-*--'-ency anS-'-e i n c i d e n c e matrix, has i n t h i s case, elements which are one's i f (3 r e f e r s to the space i n which the d-axis i s l o c a t e d . Otherwise the elements are zeroes. (3-27) and (3-28) are then Here (3-33) (3-34) £ J. T ^ ^ O ^ p©^ + J?? p(p©*) •+ 2\y,E e <r p X m = ^ (x* 1 -s*) = ( r^p©* - fjp© 5*) = <T^p© ? (3-35) p y h = f t ( y * ~ s h ) = ( p y n ~ ^ p q P ) ( 3 " 3 6 ) The farm of these equations i s i d e n t i c a l to (2-48) and (2-49) w i t h (Tp&s the g e n e r a l i z e d i n d u c t i o n angle i n c i d e n c e matrix. The torque tensor f o r the (3 spa.ce, as before, i s the array of c o e f f i c i e n t s of p© e x c l u d i n g those from pT -.<, 39 4. THE EQUATIONS OP SMALL OSCILLATIONS To d e r i v e the hunting equations, the general network loop equations and the torque equations i n the f o l l o w i n g form w i l l , be usedo e m ~ R m r, . w , aCPm d i n cXPn.Sx1 a(p wdy ^  x l C -n,, 1 + d i " dt + d x - dt + a y - dt C m " J (4-1) t p - p©* + J p j l p(p©*) + 2( ^ E ) . ( r | ) ^ (4-2) 4 o l The Voltage Equations I f . s m a l l v a r i a t i o n s i n v o l t a g e are a p p l i e d to a system of machines s e v e r a l c o n d i t i o n s change s l i g h t l y , a l l of which c o n t r i b u t e to the "hunting" of the system, a) The angular p o s i t i o n s of the c o i l spaces and/or of the r e f e r e n c e angles can be changed s l i g h t l y from the steady s t a t e v a l u e s ( i , e , , the instantaneous n o n - o s c i l l a t i n g v a l u e s ) . These v a r i a t i o n s are analogous to f l u x v a r i a t i o n s w i t h © ^ and y and hence are L i e v a r i a t i o n s given by 1) tC e - = ! f S *{P^* (4-3). 2) SCe^^l^C Ay - V (4-4) The a p p l i e d v o l t a g e e m c a n only be a f u n c t i o n of the r e f e r e n c e angle of the c o i l winding to which i t i s a p p l i e d . Therefore "GI^^ and = - The above v a r i a t i o n i s as a r e s u l t |f£ ( I ? ^ + Ay-) (4-5) b) The steady sta t e v e l o c i t i e s of the c o i l spaces and/or of the c o i l winding commutator axes can change s l i g h t l y . The r e s u l t i n g v a r i a t i o n s are obtainable from (4-1) c) The c u r r e n t s can be o s c i l l a t i n g about t h e i r steady s t a t e values g i v i n g a v a r i a t i o n c)e»v> * ; i /-a . ,, d f tCD m E x 1 d aCpm d y K -A S i * A l = ( R w , A + C>i*di w P + d i ^ d x ^ dt + SJL'dy-*. d t } A l (4-7) d) The r a t e s of change of currents can be o s c i l l a t i n g about t h e i r steady sta t e v a l u e s to give 3 ( d i ^ ) d t d l dt ' e) From the presence of capacitance, the i n t e g r a l of the c u r r e n t can be o s c i l l a t i n g A( $ i"dt) = ^ A ( i i h ) = c % A i n (4-9) C> ( $ i n d t ) 0 In analogy with (2—22) the f i r s t approximation of the v o l t a g e o s c i l l a t i o n equations i s 41 Ae m = - e | n o ) = §p ApO> + Apy - + | ^ Ai" e_j» A • 4 o + e*, + £ £ e m (4-10) A e m i s *the v o l t a g e v a r i a t i o n i n the m' th loop, e m o b e i n g the steady stat e value and e ^ t h e a c t u a l v a l u e . S u b s t i t u t i n g (4-2) through. (4-9) i n t o (4-10) , A e w = (R m„ + ^  p + ^ i h 8 > . J L J l — - ^ f + 5i^< S T * ' ~ d t +c5iF'lcVC,-1dT c ^ i ) A 1" +H^APX- -h^TApy^ + | ^ ( n - A © ? + Ay-) (4-11) There are three d i s t i n c t c o n t r i b u t i o n s i n the above equation. •H\ (T, • _ . ^.(Dvn d i * , 3 [.cXPmxfix* , S i " p • d i ^ d ' i * dt d i ^ d x 1 " V dt a i a o d±n cyp a ( a <pm\ d y A _ i \ •. i±n \ I d t + c . m r , p J A l A aw*n (p ) A i * 3 a v a r i a t i o n a r i s i n g from the e f f e c t of the o s c i l l a t i o n impedance (-gmJ a c t i n g oh o s c i l l a t i n g currents,, Here s m„(p) = z W n ( p ) -ip g>j!^A 0 T H E l a s t term vanishes f o r unsaturated systems. b) (gpT P + Ay ng v o l t a g e s a r i s i n g from o s c i l l a t i n g r eference axes. c ) (|y^ + ^ " P ) 1?Ap© pi v o l t a g e s a r i s i n g due to o s c i l l a t i n g eo.il spa,ees. 42 For the unsaturated case Cp^= L ^ i " * , (4-10) becomes A e ^ = z m n ( p ) A i n + V ^ ^ i ' 1 Apy" + A y *» + 1? Ap©* + f f ^ 1? AO > (4-12) where ( P ) = ( R * r m + L ^ h p + G ^ ^ J f J p O * + V ^ p y * * 7 ^ As an example consider the i n d u c t i o n motor of F i g . 4A with f i x e d commutator axes* py = 0 and gj^X = 0 since the a p p l i e d v o l t a g e s are a f u n c t i o n of time only,, The general equation reduces to A e * = (Rv««+ L m « P + G ^ ^ J p G ^ A i " + 0 m w „ i n l $ ( A p Q * ) (4-13) 4.2 The Mechanical Equations of O s c i l l a t i o n For the v o l t a g e o s c i l l a t i o n e q u a t i o n s t a v a r i a t i o n of the loop equations was takenj equations c o n t a i n i n g r e l a t i o n s h i p s between d i f f e r e n t machines. Torque v a r i a t i o n s , i n c o n t r a s t f must deal w i t h i n d i v i d u a l machine spaces and i n t h i s sense the equations are separateda This s e p a r a t i o n a r i s e s computationally when one c a l c u l a t e s the torque tensor f o r the component spaces (|L^° f p r the p »-.th space). Since f o r each 0, a matrix r e s u l t s 9 the torque tensor i s a t r i a d i c . Operations (transpose, i n v e r s e , etc.) on the torque tensor a p p l y ' : f o r each p i n d i v i d u a l l y . Proceeding as i n the d e r i v a t i o n of the volt a g e o s c i l l a t i o n equations c o n t r i b u t i o n s ti» targue o s c i l l a t i o n s a r i s e from 43 a ) V a r i a t i o n s due to o s c i l l a t i n g mechanical angles ft Q © , mechanical angular speeds, pO } and mechanical angular a c c e l e r a t i o n s , p (p©*) <> From (4—2) the sum of these c o n t r i -b u t i o n s i s f t * A ( P ( P ^ ) ) + |§fr A(pO>) = §g* A©* + D^ApO' + J w A p ( p © * ) (4-14) b) A v a r i a t i o n due to o s c i l l a t i n g c u r r e n t s given by | | | A i A = | | ^ A i A = 2 ( ^ | | x ) a ( > l V A i : 2( C0A).(r|)^A (4-15) c) A v a r i a t i o n due to o s c i l l a t i n g r eference axes i ^ + dx<> x-3x> \ = 2 ^ r ( ( ^E)„ ( ^ ) p ) A y 1 (4-16) To a f i r s t approximation the sum of these v a r i a t i o n s i s equal to the a p p l i e d torque v a r i a t i o n A t ^ = t ^ - t p o <, The mechanical o s c i l l a t i o n equations are thus Atf* = ! © * + + J w p ) A p © } i + 2y{( ^ E ) . ( ^ ) A y ' A -'+ 2( ^ - C P ^ P . . ^ ) ^ A i V (4-17) 3t, $©* | A©* + ( D w + J^p p)Ap© 2 + VS(V^E A y ^ + C F U A i ^ r ^ (4-18) 44 For unsaturated systems (p^ss L £ ji. i •** and E = ^ L ^ This gives A i (4-20) In matrix n o t a t i o n r e t a i n i n g the mechanical index these c o n t r i b u t i o n s are ( V | j i ) ( A y ) and ( ( T ) ^ ( i ) + ( T ) J ( i ) ) * ( A i ) where t i s the transpose operator. As an example consider F i g . 4A? the two phase i n d u c t i o n motor,, (t) the electromagnetic a b c d S 1 1 0 0 r 0 0 r l 1 F i g . 4A Two Phase Induction Motor energy conversion torque v e c t o r i s on the s t a t o r and t K , that a c t i n g on t where t 8 i s that a c t i n g le r o t o r . 45 V J* A y a Ay c Ay d a ^ _a_ a x * + a y ' o o o o B x c + D y c V ' Sxi+ax*^ The zeroes are present since t s i s independent of c or d angles = 0 f o r 3 = S and i = c,d) and s i m i l a r l y f o r For the sec ond c o n t r i b u t i o n , denote ( T ) g as the torque tensor f o r the s t a t o r ; ( T ) R , the torque tensor the r o t o r ; and ( i ) as the column v e c t o r i c ( ( T ) j j ( i ) +- ( T ) * ( i j )* (Ai) i s then equal to "(T) s ( i ) ( i ) + ( T ) r ( T ) * (Ai) The torque o s c i l l a t i o n equation i n matrix fornv-for t n e unsaturated case i s ( A t 3 ) = ( § | | A 9 * ) + ((T + T * ) ( i ) * ) ( M ) + (D + Jp) (ApO) + ( v J j O(Ay) (4-21) 4.3 The Combined Equations of Motion 46 The system equations Ae * ^ ( p ) A i " + ($fep +^^?A0^+(^r+|y*«5^ Z ^ ( p ) A p O * + ||f A©* + V i f t r t f A i * + VAM^ y ^  (4-22) 3 3 c) c) where Z = (D & o t + J ^ p ) and V u = C § ^ C + g p ^ ^ T + QJ^) form a set of coupled matrix d i f f e r e n t i a l equations which determine the steady s t a t e and hunting sta t e of a machine system once the i n i t i a l c o n d i t i o n s are known. The o s c i l l a t i o n equations can be c o n v e n i e n t l y combined to give a s i n g l e matrix equation of o s c i l l a t i o n "(Ae)" (At)_ ( A i H ) (Ap© p) 3< 3y* LEQ 3y* ( A y " ) ¥hen s a t u r a t i o n i s absent t h i s s i m p l i f i e s to (4-23) 4 7 (Ae) (At)_ ( ( T + T * ) ( i ) ) * , (D + Jp) + ^ S x m V + ^2>yw p S o * p (Ai") (ApO*) 3 y m i + U y » 1 ' P (Ay") ^ ' ( 4 - 2 4 ) In the general hunting equations (Ay) and (Apy) r e f e r to commutator axes and hence are c o n s t r a i n e d e x t e r n a l l y -5. APPLICATIONS TO THE DERIVATION OF MACHINE SYSTEM EQUATIONS 5.1 The Power Selsyn System Two i n d u c t i o n motors i n t e r c o n n e c t e d as shown i n F i g . 5A form a s e l s y n system. The t r a n s m i t t e r i s d r i v e n e x t e r n a l l y . The r e c i e v e r runs at the same speed with an angle of l a g 6 . F i g . 5A Selsyn System The loop equations f o r a s i n g l e phase connection are e - 0 n 0 — _j o p t _ r n R 35 -P ^ t t i i + p L 35 (5.1) 49 The inductances are obtained from Yu's formula (2-27) p k m i = L m J i ( c o s y a p - s i n y*py 4) (5-2) pt, A m= L ^ A ( c o s x*-p - s i n x Apx J-) (5^3) pt, n r= L n r ( c o s y r p - s i n y r p y r ) (5-4) p-L r n= L r n ( c o s x r p - s i n x r p x r ) (5-5) p^xi= P LJL1= L i X P ( i = m,l fn,r) (5-6)" pt,33= P ( U i a + L r r ) The incid e n c e matrix i s given by "px-= 0 " 1 0 0 0 ~ " p © ^ 0 px x 0 1 0 0 p © 1 px r . — 0. 0 1 0 p'e' px h = 0 0 0 0 1 _ p ° n = 0 (5-7) S u b s t i t u t i n g (5-6) and' (5-7) i n t o (5-1), 0 R r r , r » + P L r Y , r n 0 - L r n A ( cos x p - s m x Ape^) , 0 f ,^hn + P^r.n , L h r ( c o s x r p - s i n x rpe r') The torque tensor f o r the two r o t o r spaces i s , L m J l ( c o s y A p - s i n . y l p y < ) , - L n r ( c o s y r p - s i n y r p y r ) ,B 3 3 + p L 3 3 0 , o, 6 o 0 L^&sin x % 0j_ 0 0; 0 ,0 0, 0 ,o o, s i n x r ,0 (5-8) (5-9) 50 The corresponding torque equations f o r the system are t A = ( J a a P + D i a ) pO 2 + L ^ s i n x*i™i* (5-10) t r = ( J r r p + D r r ) pO - L n r s i n x r i ° i 3 Assuming that a l l angles are measured i n terms of e l e c t r i c a l angles, and that the machine o p e r a t i o n i s balanced, an analogous set of equations f o r a system o p e r a t i o n on a g r e a t e r number of phases can be obtained., as f o l l o w s s For a two phase arrangement, the currents i n the second phase l a g those of the f i r s t by an angle of (TC/2). In matrix form i*(D~ i n ( l ) i 3 ( l ) (5-11) i w ( 2 ) i n ( l ) i n ( 2 ) i 3 ( D i 3 ( 2 ) •-] where the bracketed numbers r e f e r to the phase number. S i m i l a r l y f o r a polyphase machine, adding the f a c t o r — one obtains TP i l ( o ) i'-(o) i X ( o ) i £ ( o ) « i h(o) a 1 6 i h ( D i x ( p - D o e a e p e 3 e - j a n g - i -qanp—a. Ln(o-) (5-12) 51 i"(*>) =1 e _ i P i , ( 5 - 1 3 ) where <J) i s the phase number, the lowest being zero p i s the number of phases present, n r e f e r s to the n'th c o i l . Note t h a t a two phase system i s a c t u a l l y a semi-quarter phase system (p=4). Taking a tensor approach, t h i s matrix equation can be thought of as a t r a n s f o r m a t i o n of the form i n M = C ™ £ i R W (5-14) i n which the unbracketed indic£v(| r e f e r to the c o i l number and the bracketed indicie^Vj (dead) to the phase number. Here3.= Os and < S = e _ i p 6^co) (5-15) Since the t r a n s f o r m a t i o n i s assumed to be from one true m u l t i -phase r e p r e s e n t a t i o n to a true s i n g l e phase r e p r e s e n t a t i o n e*k)= C £ s > e r u o o ' (5-16) The transformed impedance tensor i s then Zmn(5e>tp — fits.) Z m r n * x p ^ r i ^ ~ . Z f * F \ <£)(?>= b("Ho> O m e P Zrrn « ) h t ? ) ^S_e ±> 0(B)(0) (5-18) /P~ /P" Dropping the (o) indic.e*-J z _ _ z F f U o o ^ y , g j ^ 1 ^ - ^ ) (5-18) Pnm p Or i n terms of inductances 1^-= Lnwm (ja e ^ ^ ^ (5-19) P 52 In matrix form t h i s i s 1 1 " S r 2 (z>= . « n «e if ( p - i ) .n -3£n e P e -r e p -jaTTP-i 6 p J ( 5 - 2 0 ) These r e s u l t s w i l l now be used to reduce the equations of a two phase system to an e q u i v a l e n t s i n g l e phase system,. The impedance tensor of a s i n g l e two—phase machine i s obtained from l u ' s formulae "s" denotes s t a t o r c o i l s , v 9 r o t o r c o i l s and the braeketed i n d e x 9 phase number ( f i g . 5c) esd); i r0U Fig,, 5B Two-P&ase Sel s y n Unit 53 ( Z m n ) = s ( l ) s(2) r ( l ) T(2) 0 0 R s s +pL ss L 6 r p ( c o s y r ) , - L s r p ( s i n y r ) L s r p ( s i n y r ) , L £ r p ( c o s y r ) L S r p ( c o s x r ) t L s r p ( s i n x r ) , R r r+pL -L<. rp(sin x r ) , L s r p ( c o s x r)» rr 0 0 » ^ r r + P ^ r r (5-21 The balanced s i n g l e phase e q u i v a l e n t i s thus - s ( l ) 1 1 j 0 0 0 0 1 j 1 0 0 1 0 j s ( l ) r ( l ) r ( l ) p R s s + P L s s t L s r P e i y r L s r P e " S X » Brr+P L rr (5-22) jHavi_ng obtained the e q u i v a l e n t impedance matrix of a s i n g l e machine j) loop equations f o r the system can once again be obtained, These are given by ('5-1) wi t h the mutual impedance elements ob-t a i n e d i n ( 5 - 2 2 ) . Using the n o t a t i o n of P i g . (5A) .1 T l 0 , R, - L m A p e ~ i X , 0 + pL ~ L ™ P e L h r P e L n r p e " 5 , R 3 3 + p L 3 3 (5-23) These equations are best handled by r o t a t i n g the reference a x i s of the 1 — c o i l i n t o the n-axis by i ^ ~1 0 = 0 1 i § 0 0 0 V (5-24) 1 0 0 - 0 1 0 ei=°_ 0 0 e • s i x 1 - * * ) e 3=o This g i v e s , using>the i n c i d e n c e matrix of ( 5 - 7 ) , 0 »-LrnAP 0 0 \ , R h n + P L r ^ •L m A ( p - j p © M. ?Lnr e i U \ X ( p - j p Q r ) *R33 +L 3 3 ( p - j p y A ) ? L n r e ^ y - y a ) ( P + j ( p y ^ y M ( 5 - 2 5 J 54 where x i - x r = y x-y d€ These equations are i n a true system since 2~ = 0 f o r both (5-24) and (5-16). The p o s s i b i l i t y of commutator r o t a t i o n i s i n c l u d e d i n these equations through p y r and p x r . As an example assume a steady sta t e c o n d i t i o n : py r= py A=0; p © r = p © A and p=jw. L e t t i n g w-p© .= w(l-v) = w(s) and wL^= X J L J , the reactance, one obtains Rmm+jXrom ^ r 0 i 0 •dx m 4 - i s J A n r e » S + S m (5-26) CT) ~0 0 0 0 0 0 0_ •0 0 0 0 0 0 e J 6 0 _ and the torque equations are t r = D r r p © r + J r r p (p© r) + j L m J L i m i 5 (5-27) (5-28) t A = D X ApO l + J A A p (p© 1) - j L n r e i £ > i " i ~ 3 The s t a b i l i t y of the s e l s y n system can be s t u d i e d using the equations of small o s c i l l a t i o n , which apply d i r e c t l y since the 3 e, machine i s i n a true reference system. Since dy' = 0, a t d Q P ^ = 0 ( p - r f l ) ' ^ ^ e o s c i H a ' t i o n s e q u a t i o n s a r e (Ae) U m n l p ) » (At) + (T •+ T**) ( i ) ) \ Sx^ 1 M.* V ^ J + 8 y " P BLirTir, - n (-Vi^ ) (Ay) (5^29) In t h i s set of equations z f n n ( p ) i s given by (5-25). D i r e c t l y from these 55 3 x m > 1 r m ~0 0 n 0 0 3 J r r e (5-30) i'Apy" = m 0 n j L n r e 3 6 A(py r -py A ) 3 |_ -jL.rr-Apy 1 ( (T + T t ) ( i ) ) t = (i,) ^  (T + -T*) ( i - , i % i B ) (i<" f i % i 1 ) 0 , 0 , J'L^JL 0 , 0 , 0 _0_,_ 0 _ 0 , 0 0 , 0 (5-31) ' 0 , - j L r r e 68 0 , - j L h r e J , 0 + j L m i i 3 , 0 , * J L m A 0 , - j L h r e ° 6 i 3 , - j L n r e j 6 i 5 (5-32) V 4 P A y i = V i - j L n r e j & i ^ 0 Ay - j L n r i f t i 3 e j £ ( j A y * - j A y r ) (5-33) The hunting equations f o r the s i n g l e phase e q u i v a l e n t of the / • • two—phase selsyn^system'are thus Ae^ A.e„ 0 At A At. 0 f- LmiP , 0 , 0 e" (p+jp(yr-yO o , o -LmxCp-JP^U^e 5 6 (p^jpoO^Baa +L 3 3 (p-jpyO.» j x L i % - r j L n r e 3 6 L 0 ; 3 0 • T j 6 • 3 - j L n r e l • T i 6 • i , - j L n r e l , D J U +JJUL- p, 0 » 0 > D R R + J R R p A i ^ Aifv' A i 5 ApQ1 ApO3 0 j L h r e " j 6 A ( p y r - p y M -3't r rApy A - ; j L h r i n i V 6 ( j A y ^ ; j A y * ) (5-34) 56 The brush a x i s angles and t h e i r speeds as w e l l as t h e i r v a r i a t i o n s are c o n s t r a i n e d e x t e r n a l l y and are thus known i n a l l equations. This example i l l u s t r a t e s a general procedure f o r s e t t i n g up the equations of motion f o r any system, transforming them, and d e r i v i n g the hunting equations. To solve these equations and study t h e i r general beha.vior i s complicated by the n o n - l i n e a r i t y i n v o l v e d . (1-6,10,13,15,16) 5o2 Synchronous Machines In. the conventional a n a l y s i s of synchronous machines the d-q axes are connected to the r o t a t i n g s a l i e n t p o l e s , which are viewed as s t a t i o n a r y . This i s convenient f o r handling s i n g l e machine problems; however, when multi-machine problems are attempted, complicated i n t e r c o n n e c t i o n m a t r i c e s a r i s e . This can be avoided by c o n s i d e r i n g .a common d-axis f o r the complete system and ap p l y i n g the equations of motion with r o t a t i n g s a l i e n c y to i n d i v i d u a l machines. Moreover, each machine can be viewed with i t s ; s t a t o r s t ationary,- r a t h e r than from an arrange-ment i n which the s t a t o r r o t a t e s . F i g . 5C Synchronous Machine I. 57 I-The hew referen c e scheme i s i l l u s t r a t e d i n F i g . 5C. The O L — (3 system i s t i e d to the r o t a r . The d-q system i s r o t a t i n g at the synchronous speed. For a system of in t e r c o n n e c t e d machines-, the d-q systems are p a r a l l e l y d i s p l a c e d , each r o t a t i n g at the syn-chronous speed. Consequently i n t e r c o n n e c t i o n can be made d i r e c t l y along the d-q ax i s and loop a n a l y s i s used to o b t a i n the system equations. The general impedance matrix f o r a s i n g l e machine w i l l now be d e r i v e d u s i n g Yu's general formula (3-29) and (3-33). The conventional theory w i l l be considered as a s p e c i a l case obtained when the (d-q) axes are c o n s t r a i n e d to move with the r o t o r . The standard p o l a r i t y of F i g . 5D.will be used. F i g . 5D (d-q) - (aQ-3) Reference C o n f i g u r a t i o n S i n t h i s case i s x , u s i n g c f o r cosine and s f o r s i n e , the impedance elements are t h e r e f o r e : p L m n = (L» m r icX mct n + l A n w s X w s r ) p 58 RY nn + ( - L ^ s X ^ Y " + L w ^ s Y * ) f f -+ (-L^ neX msY n + L i m , s X m c Y n ) f f ^ ( 5 _ 3 5 ) From F i g . 5D p t ^ * (Ld c X m c Y n + U r n h s X m s Y n ) p + (-L3 m nsX r T ,cY n + U m n c X m s Y n ) px" + (-L* n hcX r*sY n + U ^ s X , n c Y n ) (py"- py~) (5-36) (m,n = d,q) p i = ( L i m n c X m c Y n + . U m | s X w s T n ) p - ( - L i ^ s X W + L ^ p i ^ s T " ) px- (5-37) (m=ds,q; n=f,g,h) P L m n ^ ( L ^ n c X m c T m + L ^ s X - s Y * ) p + (-I/ m ncX msY n + L V n s X m c Y n ) (py* - py*) (5-38) (m=fjg,h; n=q,d) V^mrT L < W P (m,n,=f,g) p t m n = If l - m n p . (m,n=h) (5-39) ptc,h = pLh-P = pLh^= pLfh= 0 (5-40) In a l l these elements p y ^ p x ^ p © r These could a l s o be obtained by using the synchronous ma-chine transformations of 2%2 b)« However, the procedure i s more t e d i o u s . The zero sequence equation i s omitted i n the present a n a l y s i s s i n c e i t i s uncoupled from the remaining set of equations. From s e c t i o n 3;5 the torque tensor elements are the c o e f f i c i e n t s of p@^ excluding c o n t r i b u t i o n s from pY* . For the r o t o r space they are - ( - I / W ^ c Y * + I A ^ C X ^ S I " ) (m=d,q,i,n=d,q,f,g,h) 59 The above procedure can be extended to mult-machine systems using l o o p - a n a l y s i s . The conventional synchronous machine equations are obtained by r e s t r i c t i n g ( x 4 , y d ) such t h a t ; 1* = 0, pX d= -px~= -pe-*" ,pYd=0. P h y s i c a l l y the d-q axi s i s t i e d to the oc -(3« a x i s i n the s t a t o r space. Assuming N d = N<1 and usi n g L^ n= TJnm; 1/^ = I A ^ , "the im-pedance matrix i n standard n o t a t i o n i s R f+pL£ , M f 3 p , 0 Mf3 p , Rg+L 9p , 0 0 G .»,R>±P*!h. M £ ap , M 3 dp f M h q p © M £ dp * 0 _ 2 _ i M H E _ Ri+LiPjM^p© 1^ ~ M d p © r yR^+L^p (5-41) In the l i t e r a t u r e , f o r a generator ( a l t e r n a t o r ) , the p o l a r i t y r e f e r e n c e s of x^, p © r , and t r are opposite to those shown i n Pig« 5D Adopting t h i s r e f e r e n c e scheme and p a r t i t i o n i n g , the equations f o r an a l t e r n a t o r are 3 I e f i ' = i f 0 5 i 9 S>_ i i ed i a = \l 5 (z (.) = -R £-pL £, -M f 3p , -Mrx p , -R a-L 9p, ^ 9 0 0 0 0 ,-TR, —pL • < * 3 > = - M f d P y -^Mgip , M K < l p © - M f i p © , -Mg dp© , -M^V ( z a ) -M £ dp -Msa p o , o , o -M -R 4-L ap ; M c lp© r - M d p © r , -Rq-L^p (5-41) (e_) i s then the d and q axi s generated vol t a g e The f i r s t matrix equation i n the p a r t i t i o n e d set can be used iminate l ! : from the second set g i v i n g = ( z 8 ) ( z j ' M e J - ( z a ) ( z x ) _ : L ( z a ) ( i a ) + (z«)' i a (5-42) 60 Wr i t t e n i n f u l l t h i s i s G(p)pE G(p)p© rE + L^(p)p© r - r d - L 4 ( p ) p i t (5.-43) assuming r<j.=r L d ( p ) = p a ( L a ( M f J f -2M a<iM £ 9M £ d + L £ ( M 9 a ) £ + p ( R f ( M a * f +Rg (Msa f) ft : p a ( L f L a - ( M £ g ) 2 ) + p ( R 3 L f + R f L 9 ) +R gR £ L^(p) = (M h q.f p Rh + LhP G(p) = P ( L 9 M £ < L - M £ q Ma + .BaMja p 2 ( L f L a - (M £ g ) £ ) + p ( R 3 L £ + R^Lg) + R gB £ (5-44) E = e f ¥riting these equations i n the form e a - G(p)pE - G(p)pO rE t-r- L i ( p ) p , L^(p) p © r - L d ( p ) p Q r , -r - L ^ ( p ) p (5-45) i t i s seen that f o r the armature axes, the impedance tensor has e x a c t l y the same form as p r e v i o u s l y , (z.,), the only d i f f e r e n c e being t h a t ' a l l open c i r c u i t q u a n t i t i e s are r e p l a c e d by short c i r c u i t q u a n t i t i e s . In per u n i t n o t a t i o n , adding the zero sequence equation (i° = ^  ( i 3 - + i b + i c ) , where i 3 " , i b 9 i c a r e the phase q u a n t i t i e s ) v 3 which remains unchanged throughout^ Parks equations are obtained. e d= G(p)pE - z d ( p ) i + x ^ p ) 1°L (6) e < L= G(p)p© rE <- x d ( p ) p © r i d - z q ( p ) i ^ (5-46) e 0= -z 0i° The torque equation (3-04) i * e . t r = D r r P © r + J r r P ( p © r ) + ^ " ? i m i n The electromagnetic energy conversion torque t r i s the torque to e be overcome by the prime mover. 61 o , o M- H 0 M - M f i , -M g i » 0 -M d i 4 = (i^-)(-K£i i f - M 3 < i i 9 - M ( 1 i c L ) + i i ( M h c i i V l + M ^ ) (5-47) Using once again the f i r s t p a r t i t i o n of (5-4l) to e l i m i n a t e ( i 1 ) i d i ^ ( x , ( p ) - x d ( p ) ) + i ^ p ) e £ (5-48) i n Park's p e * - u n i t system. A d i f f e r e n t form of the equations (5-46) and (5-48) can be obtained as f o l l o w s . For the unsaturated case, once the impedance matrix' Has been obtained i t s c o e f f i c i e n t s no longer d i s p l a y any angular dependence-(e) = - ( R ) ( i ) where (L) • = ¥e can w r i t e f o r the a l t e r n a t o r £ ^ ( L ) ( i ) + ( T) r(i)p© r (5-49) (TL = +L £ 0 Mfa 0~ 0 M g d 0 0 0 0 +M£d 0 0 0 0 M K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 % -M £ d -M g i 0 -Md 0 (5-50) ( L ) ( i ) , gives the f l u x i n each c o i l due to e x c i t a t i o n of a l l c o i l s i n the same a x i s . ( T ) r ( i ) , g i v e s the f l u x i n each c o i l due to e x c i t a t i o n of a l l c o i l s at r i g h t angles to i t and hence 62 i s a c r o s s - f l u x . A f t e r e l i m i n a t i o n of ( i 1 ) , the r o t o r c u r r e n t s , (5-49) can be (6) w r i t t e n with the zero sequence added — e d = - r i d + p y d - ^ p © e«l-= - r i ^ + p ^ 4 ^ d p © = - r i 6 +pvy0 Here .yd = G(p)E - x d ( p ) i d \ In t h i s form the torque equations (5-48) i s (5-51) (5-52) t r = i ^ - - (5-53) For an a l t e r n a t o r at steady s t a t e p=0 and p© r=l i n the per u n i t system A l s o x d ( p ) = x d and x^(p) = x^, G(p) = I f E = E f p j - j where E f i s the " i n t e r n a l generated v o l t a g e " , then G(p)E = E f . Assuming the a l t e r n a t o r i s connected to an i n f i n i t e bus whose f i e l d angle lags t h a t of the generator by 6 so that (5-54) E = e s i n 6 e cos 6 the equations of motion become e s i n £ = — r x<^  e cos &- E ^ _-x d - r _ i ^ t*= i d i ^ ( x o - X d ) + E £ i ^ (5-55) (5-56) The hunting equations f o r a synchronous machine can be obtained from (4-23) by the f o l l o w i n g changes ( c . f . 3 -33; 3-34). 63 r p In the r e p r e s e n t a t i o n used here the reference axes remain t i e d to the f i e l d axes and thus o s c i l l a t e (AY n =0)« The hunting equ-a t i o n s become "(Ae) (z* m(p)) ? (At) p ) ^ d Y ^ ^ ' i " (Ai) (ApO) ( 5 - 5 7 ) For these equations the r e f e r e n c e s of Fig„ 5D w i l l be. used, From (5-54) d e in c>Yn 0 0 0 e co s 8 as 3 Y n |-e s i n (5-58) For a machine connected to an i n f i n i t e -feus S = y'°S y* f c = y*- y 4 - ( y ^ - y 4 ) where y ^ i s the r o t o r angle of the bus with r e s p e c t to the chosen r e f e r e n c e ? Using t h i s 3 Y n Hence 3 e B Y n to _ 0 0 0 -e cos 6 e s i n 6 (5-59) The v o l t a g e v a r i a t i o n Ae i s Ae A -'ef - " — 0 ^Ae f 0 0 0 o 0 0 e s i n £> e cos £ Ae s i n 6 § cos £>_ -e s i n &_ _Ae cos6_ (5-60) 64 where A & i s the v a r i a t i o n of & other than L i e , the l a t t e r having al r e a d y been i n c l u d e d i n the equations of motion. Here A& = L{y~- y*) - My^-y 0 6) = Ay"= A9 r The g e n e r a l i z e d i n c i d e n c e matrix i s given by (p©r) (5-61) pX f 0 PX9 0 pX h 0 px - i ^\ ~ i (5-62) Therefore the remaining terms of (5-57) are j e ^ ^ n 8 O i r y n r c)I n r ~ 0 0 0 0 0 0 0 0 0 —e cos B -1 e c o s t - e sin6•—1 -e s i n 6 0 0 0 0 0 0 0 G 0 0 19 G 0 0 0 0 i * 0 0 0 M a i4 d o -Md 0 i i (5r63) (5-64) 0 0 0 M K i h + - M £ < i i £ - MadLi! 9 _ ^P,^= it)\(T% + ( T ) r ) -M M d i c —t t 3 d i ^ Mhqi 4-M h c i i h + (M^Md) i ^ ( M £ c L i £ + M 3 < j i 2 ) + (M^-Ml) i d (5-65) (5-66) 65 S u b s t i t u t i n g these i n (5-54) the o s c i l l a t i o n s equations with the p o l a r i t y of F i g . 5D are on the f o l l o w i n g page^ Although t h i s d e r i v a t i o n i s f o r a s i n g l e machine, the ap-proach can be extended to a multi-machine system 0 Ae £ -R £ +pli£ J M £ 3 P > 0 > M £ d P 7 0 , 0 r A i f 0 M £ gP 9 Rg+pLg 9 0 9 M g dp ? 0 0 9 A i 9 0 0 o , R n+pL h 0 S MhqP , "0 A i h Ae s i n £ M f dp 1 , MgdP 9 Mhq.P©r o R^+L^p r M^p© 1^ ,M h < Li h A i d A e cos S At ~ M £ d p © r ? ™M£ di^ 9 ~M 9 d p©*\ -=Mc,di^ - , Mh^P 9 - M d P © »• ML, i h R^+L^p -(•M £ di +Mqdi a + (Mq_^ Md) i d , - M £ d i £ -M gdi a - M d i a ), Dp +Jp £ A i ^ A © T ( 5 - 6 7 ) 67 6. CONCLUSIONS By s e l e c t i n g a set of independent c o o r d i n a t e s , i . e . , a holonomie system, and ap p l y i n g Hamilton's P r i n c i p l e , the volt a g e and torque equations a p p l i c a b l e to any p h y s i c a l machine were d e r i v e d * C o n t r i b u t i o n s a r i s i n g from i n t e r a c t i o n s between me-c h a n i c a l r o t a t i o n s and magnetic f l u x e s were shown to be due to L i e v a r i a t i o n s . Once the b a s i c equations were d e r i v e d , the hunting equations and t r a n s f o r m a t i o n p r o p e r t i e s f o l l o w e d r e a d i l y . Of importance i s the independance of the v o l t a g e t r a n s f o r m a t i o n matrix, (€!), and the c u r r e n t t r a n s f o r m a t i o n matrix, (C), f o r a general t r a n s f o r m a t -i o n of the system equations. R e s t r i c t i o n s a r i s i n g when the t r a n s -f o r m a t i o n i s from one true system to another leads to the i n v a r i a n c e p r o p e r t i e s of power.. The t a b l e f o l l o w i n g compares the conventional i d e a s , p r i n c i p a l l y those of Kron, w i t h those presented i n the present t h e s i s . The t h e s i s i n d i c a t e s s e v e r a l p o s s i b l e areas of i n v e s t i -g a tion: s t a b i l i t y and o p t i m i z a t i o n s t u d i e s using the Hamiltonian approach^ the r e a l i z a b i l i t y of a true system, which may not n e c e s s a r i l y be a p h y s i c a l system although the converse i s always t r u e ; and the i n v e s t i g a t i o n of n o n - l i n e a r problems. 68. Conventional Present Thesis Holonomic system S l i p r i n g machines only Any p h y s i c a l machine Lagrange -r,s equations S l i p r i n g machines only Any p h y s i c a l machine t a k i n g i n t o account L i e v a r i a t i o n s Quasi-^holonomic F i x e d commutator a x i s machines No such system Non—holonomic Anything which i s not i n c l u d e d i n the above, hypothetical or p h y s i c a l Any hypotheteifjLl machine f o r which the t r a n s -formation matrices (C). and (•€) from a p h y s i c a l machine s a t i s f y dt r u ' dt ^ D i r e c t a p p l i c a t i o n of loop or Nodal a n a l y s i s S l i p , r i n g machines Any true system; i n c l u d e s a l l p h y s i c a l systems t Holonomic Moving s a l i e n c y Non-holonomic Invarianee of tensor products — — — Follows from (C) and (£) p r o p e r t i e s P o l a r i t y convention No standard r e f e r e n c e s Follows from the b a s i c equations General comment General equations d e r i v e d by Kron apply only to a l i n e a r system, i b e , , System equations obtained are completely g e n e r a l , i n c l u d i n g n o n - l i n e a r i t i e s , and are l e s s compli-cated i n form. Table 2„ Comparison of Conventional Ideas Thesis with those of the Present 69 APPENDIX I CONDITIONS ON THE FLUX TENSOR TO GUARANTEE AN EXTREMUM The existance of an extremum of the a c t i o n i s guaranteed by three t e s t s which can be a p p l i e d to the Lagrangian. These extend the r e s u l t s of r e a l i t y t e s t s f o r mutual inductance din \ \ s t a t i c c i r c u i t theory. -\ \ The three t e s t s w i l l be d e a l t with i n t u r n . \ '» a) The E u l e r Equations are s a t i s f i e d . | * I b) jThe Legendre T e s t e I f S = $ L(t,q ;-,q 1)dt i s an extremum I j ! ' o .;!;' the quadratic forni whose matrix i s ( ) = ( E ^ i , ^ i ) must be/ / / ,/ p o s i t i v e d e f i n i t e . Thus the matrix M L . S i ' ' d i a ' d i ' ' c5ia ' * " i s p o s i t i v e d e f i n i t e . A necessary and s u f f i c i e n t c o n d i t i o n f o r t h i s to be true i s th a t the minor determinants be grea t e r than or equal to zero. That i s c)(P. dO, di' ' di £ > 0 LSi' Si £. For the. unsaturated case t h i s i s L 6 i > ° II » > 0 70 c) The Jacob! Te/st: A l l terms of the form; ( d ~ E u d*E \ / 6 A E \ A 3 i ? , d i r n d i 8 d , £ s ' ^ d i £ d i r V must be.greater than or equal to zero . That i s l " d T F H a i ^ ) " 0 For no s a t u r a t i o n t h i s i s ( L r r ) ( L s s ) - ( L r s ) £ ^ 0 / This i s i n c l u d e d i n the above since L-£s = L a r by energy con-s i d e r a t i o n s or as can be proved d i r e c t l y since d i r G ) £ E _ aor _ _ a £ E _ d^s e ) i s d i r - d i s _ d±Tdis ~ Dir 71 APPENDIX II THE HAMILTONIAN; ENERGY AND CO-ENERGY The'Lagrangian of an elect r o m e c h a n i c a l system i s an i m p l i c i t f u n c t i o n of time and hence i f d i s s i p a t i o n i s neglected ( c l o s e d system) the Hamiltonian i s conserved. I f H i s the Hamiltonian f o r the e l e c t r i c a l p a r t of the system and L e the La,grangian c)L H = cpii- E H i s t h i s case i s not equal to E @ g e n e r a l l y and hence the energy i s g e n e r a l l y not a constant under f r e e o s c i l l a t i o n s of the (14 ) machine. I t w i l l f l u c t u a t e so as to maintain H a constant. An i n t e r p r e t a t i o n of the Hamiltonian can be made by g e n e r a l i z i n g the case of a s i n g l e c o i l . ( P i g . I I A ) . F i g . IIA Si n g l e C o i l B H Curve i In t h i s case the energy i s ^ ( p d i which i s equal to the shaded area, The co-energy i s d e f i n e d as i C =Cpi - E = t p i - \ ( p ( i ) d i which i s the unshaded area of the r e c t a n g l e , and i s the H a m i l t o n i a n 0 Thus as a g e n e r a l i z a t i o n , the co-energy of the system i s d e f i n e d as C E = (PJL I 1 - E e = H and the Hamiltonian i s equal to the g e n e r a l i z e d co-energy. REFERENCES 73 L Adkins,, B., The General Theory of E l e c t r i c a l Machines (Book) s John ¥iley & Sons Inc., New York, 1957. 2. Bewley, L.V. „. Tensor A n a l y s i s of E l e c t r i c a l C i r c u i t s and Machines (Book),, Ronald Press Co., New York, .1961. 3. Concordia, C , Synchronous Machines (Book), John Wiley-& Sons, New York, 1951. 4. Gibbs, W.J., Tensors i n E l e c t r i c a l Machine Theory (Book), Chapman & H a l l , Tendon, 1952. 5. ,K e l l e r , E r n e s t G. , Mathematics of Modern En g i n e e r i n g ', V o l I I Dover P u b l i c a t i o n s Inc. New York. 6. Kron,G„, Tens ors For C i r c u i t s (Book), Dover P u b l i c a t i o n s Inc., New York, 1959. 7. Kron, G.., Non-Rtemannian Dyna.mics of Ro t a t i n g E l e c t r i c a l Machinery, Journal c f Mathematics and Physics', X III 2, May 1934, pp. 103-1.94. 8. Kron, G., S e r i e s of P u b l i c a t i o n s i n the General E l e c t r i c Review, Schenectady, New York, Parts I-XVIII, 1935-38. 9. Kron, G., The A p p l i c a t i o n of Tensors To the A n a l y s i s of Ro t a t i n g E l e c t r i c a l Machinery (Book) T 10. Park, R o H . , Two Reaction Theory of Synchronous Machines, AIEE T r a n s a c t i o n s , Part I , V o l . 48, 1929, pp. 716-73Q, Part I I , V o l . 52, 1.938, pp. 3 52-354. 11. Schouten, J.A., R i c c i C a l c u l u s (Book), 2nd ed., S p r i n g e r - V e r l a g , Berlins, Gottingen, 1.954. 12. Schouten, J . A . T e n s o r s For P h y s i c i s t s (Book), Clarendon Press, Oxford, 1951o ; 13. Takeuchi, T.J., Matrix Theory of E l e c t r i c a l Machinery (Book), Ohm-Sha, T.okyo, 1962. 14. Whittaker, E.T., A n a l y t i c a l Dynamics (Book), Cambridge U n i v e r s i t y Press, 4th ed., 1961. 15. White, D.C., and Woodsen, H 0H 0 , E l e c t r o m e c h a n i c a l Energy Conversion (Bo\ok), John Wil^y^fe Sons, New York, 1959. 16. Wood, W o S o , Theory of E l e c t r i c a l Machines (Book), London, Butterworth, I9\58. 17. Yu, Yao-nan, The Impedance Tensor of the General Machine, Trans. AIEE, p t . I (C ommunication and E l e c t r o n i c s ) , V o l . 75, May, 1956, pp. 181-1.87. 18. Yu, Yao-nan, The Torque Tensor of the General Machine, Power Apparatus and Systems, Feb., 1963, pp. 623-629. 

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