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A D.C. amplifier and reference voltage supply suitable for use in a magnet current regulator 1964

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AD.C. AMPLIFIER AND REFERENCE VOLTAGE SUPPLY SUITABLE FOR USE IN A MAGNET CURRENT REGULATOR by Edward Murray Edwards B . S c , U n i v e r s i t y of B r i t i s h C olumbia, 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of P h y s i c s THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1964 In presenting 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 for 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 f o r reference and study. I further agree that per- mission f o r extensive copying of t h i s t h esis f o r s c h o l a r l y purposes may be granted by the Head of my Department or by hi s representatives. I t i s understood that.copying or p u b l i - c a t i o n of t h i s t h e s i s for 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 if shvo-<i*^~ The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date ABSTRACT The r e q u i r e m e n t s imposed upon a d.c. a m p l i f i e r and r e f e r e n c e v o l t a g e s u p p l y * s u i t a b l e f o r measurement and c o n t r o l of s i g n a l s of the o r d e r of 1 v o l t t o an a c c u r a c y of 10 p a r t s per m i l l i o n are i n v e s t i g a t e d . A b a s i c b l o c k d e s i g n i s proposed t o a c h i e v e t h i s g o a l u s i n g a t r a n s i s t o r d i f f e r e n t i a l a m p l i f i e r of new d e s i g n and a h i g h l y r e g u l a t e d r e f e r e n c e v o l t a g e s u p p l y . The d e t a i l e d d e s i g n procedure f o r th e s e b u i l d i n g b l o c k s i s then p r e s e n t e d . E x a c t a n a l y s i s of i m p o r t a n t c h a r a c t e r i s t i c s of s e v e r a l r e l e v a n t c i r c u i t c o n f i g u r a t i o n s i s p r e s e n t e d * The u n i t under d i s c u s s i o n was c o n s t r u c t e d and the measured performance was found to be i n good agreement w i t h the p r e d i c t e d performance * i ACKNOWLEDGEMENT The au t h o r wishes t o thank Dr. B. B a r r i e , the s u p e r v i s o r of t h i s p r o j e c t , f o r h i s encouragement and support throughout the course of t h i s work. He would a l s o l i k e t o thank Dr. J . B i c h a r d f o r h i s encouragement and i n t e r e s t . D u r i n g the l a t e r stages of t h i s work, the author has been employed by Dr« L. Young of the E l e c t r i c a l E n g i n e e r i n g Department of the U n i v e r s i t y . The author wishes t o thank Dr. Young f o r h i s u n d e r s t a n d i n g and encouragement. The a u t h o r i s i n d e b t e d t o the Defence Research Board f o r t h e i r f i n a n c i a l support and to the E l e c t r i c a l E n g i n e e r i n g Department f o r t y p i n g and p r i n t i n g the f i n a l t h e s i s * L a s t but by no means l e a s t , the auth o r wishes t o thank h i s w i f e f o r g r e a t p a t i e n c e , u n d e r s t a n d i n g and h e l p . v i i TABLE OF CONTENTS Page L i S "b Of F i gUT G S ttooDO**o«oe« S » O » « O S O A O O « « • 1^ L X S ~b O f Tctb X © S « » o o 0 » o o o o o o o o o e e a o o o s o o « » * o * O # * « * * V3l Acknowledgements « o a o . * o » * « o » 0 a * o e o « e » « « * v n X « X H~b 3? O (Xll C "b X O II Ooo<>9oooec<>o«o»««fic«o0«e«*ft« •*•<»« X 2* OutXine of System 2.1 G*eneZ*al e . o ° B » e e o » . o « » . . . o . o o e « » . . . . . . . * . 3 2.2 The D.C. Ampl i f i 6r< • I I > > I I < I I O I > I > < > < I > < < 3 2.3 The Temperature C o n t r o l System 3 2.4 R e g u l a t e d Power Supply ........ 4 2.5 Reference V o l t a g e Supply ..........*..••• 4 3. Module D e s i g n 3*1 The T w o - t r a n s i s t o r Compound B l o c k ....... 6 3*2 The Common—emitter Compound C i r c u i t ••«•• 8 3.3 C o n s i d e r a t i o n s i n the A m p l i f i e r D e s i g n •« 9 , 3*4 The Input Stage of the D.C. A m p l i f i e r ... 10 3*5 The Second Stage of the D.C. A m p l i f i e r 17 3*6 The E m i t t e r - f o l l o w e r T h i r d Stage of the D C . A m p i i f i e r . o o . e o a e . e o o e e . o . . . . . ^ * * . . 21 3»7 The Oven Temperature C o n t r o l System .««• • 22 3*8 The Common—collector R e g u l a t o r .... ...... 26 3«9 The Common-emitter R e g u l a t o r .....»«.»••• 29 3*10 M o d i f i c a t i o n of R e g u l a t o r Output Re S X S "b 3/11C e « o e « o o « » e » e © « t © o o e © o e * » * 30 3.11 Comparison of Common-emitter and Common- c o l l e c t o r R e g u l a t o r s ...<> ..<.........**.. . 30 3.12 The Temperature C o e f f i c i e n t of Breakdown- 3.13'General Power Supply C o n s i d e r a t i o n s *..*• 35 i i I Page 3.14 The R e g u l a t e d Power Supply f o r the D.C. 3*15 The Reference Supply 38 3*16 The P r e - r e g u l a t o r s • 39 3•17 The OV en e f i o o o . o o e o o e « . . o e o . . o « e . « • •»•*'•• 39 4. Performance of Equipment 4.1 The D.C. A m p l i f i e r 44 4*2 The T e m p e r a t u r e - r e g u l a t o r System ........ 44 4„3 Power Supply Performance „............... 46 4»4 P r e — r e g u l a t o r Performance 48 Appendix 1» L i n e and Load R e g u l a t i o n Performance of R e g u l a t e d Power S u p p l i e s ......... 50 Appendix 2* D e r i v a t i o n of E q u a t i o n s P r e d i c t i n g Power Supply Performance 53 Appendix 3« D e r i v a t i o n of the Exact Parameters f o r an E m i t t e r - f o l l o w e r ................. 60 Appendix 4* D e r i v a t i o n of the Ex a c t Parameters f o r a Common-emitter A m p l i f i e r ...... 65 Appendix 5* D e r i v a t i o n of the Exact Parameters f o r a Common-emitter A m p l i f i e r w i t h E x t e r n a l E m i t t e r Impedance . . . . . . . . . . . . . 6 9 Appendix 6. D e r i v a t i o n of the Parameters of the Compound Stage Used i n the D i f f e r e n t i a l A m p l i f i e r « .:«. 74 Ref e r e n c e s . • • • o e o o o © « . o © « o o e o . « . . . . o . . . . . . . . « . * * • i i i LIST OP FIGURES Page 2- 1 B l o c k Diagram of D.C. A m p l i f i e r and Reference Supply ........ 5 3- 1 The Two T r a n s i s t o r Compound «.•••« 8 3-2 The B a s i c T w o - t r a n s i s t o r Compound Common—emitter A m p l i f i e r .............. 8 3-3 The Input Stage of the D.C. A m p l i f i e r . 11 3-4 T r a n s i s t o r w i t h B a s e - e m i t t e r R e s i s t o r * 12 3-5 The Second Stage of the D.C. A m p l i f i e r 18 3-6 The E m i t t e r - f o l l o w e r T h i r d Stage ...... 21 3-7 The Oven Temperature C o n t r o l Servo .... 23 3-8 The Common-collector R e g u l a t o r ........ 26 3-9 The Common-emitter R e g u l a t o r 29 3-10 Breakdown-diode V o l t a g e a t 10 m i l l i a m p s v s . Diode C u r r e n t f o r Zero Temperature C o e f f i c i e n t ...... 34 3-11 The R e g u l a t e d Power Supply f o r the D.C, Allipl 1 f 1 © I* » f r e o o o a o * 0 6 B B * o o e o e o o * 35 3-12 Complete Schematic of D.C. A m p l i f i e r . . 41 3-13 Complete Schematic of Power S u p p l i e s .. 42 3- 14 P a r t s L i s t f o r Temperature C o n t r o l l e r . 43 4- 1 V a l u e of R 302 v s . Oven Temperature ... 45 A l r r l Four T e r m i n a l Network R e p r e s e n t a t i o n of R e g u l a t o r 50 A2—1 P r e — r e g u l a t o r 53 A2-2 S i m p l i f i e d C i r c u i t of P r e - r e g u l a t o r ... 53 A3-1 E m i t t e r - F o l l o w e r 60 A4-1 Common—Emitter A m p l i f i e r . c . . . . . . . . . . 6 5 A5-1 Common—emitter A m p l i f i e r w i t h E x t e r n a l E m i t t e r Impedance o o . . . . . . . . . . . . 69 i v Page A5-2 S m a l l - s i g n a l H y b r i d E q u i v a l e n t C i r c u i t of Common-base T r a n s i s t o r w i t h E x t e r n a l E m i t t e r Impedance 69 A6-1 The T w o — t r a n s i s t o r Compound i n the Common- e m i t t e r C o n f i g u r a t i o n .................... 74 v LIST OP TABLES Page 4—1 P r e d i c t e d and Measured Parameters of the R e g u l a t e d Power Supply •.*.••..... 47 4—2 P r e d i c t e d and Measured Parameters of the Reference Supply 47 v i 1. INTRODUCTION A d.c. a m p l i f i e r and r e f e r e n c e v o l t a g e s u p p l y s u i t a b l e f o r use i n a magnet c u r r e n t c o n t r o l , or p r e c i s i o n p o t e n t i o m e t e r , was d e s i g n e d and b u i l t . I n the p a s t ^ " ^ ' ^ ^ such systems have u s u a l l y employed a m p l i f i e r s based on some m o d u l a t i o n t e c h n i q u e , such as a chopper- s t a b i l i z e d a m p l i f i e r . T h i s t e c h n i q u e u s u a l l y r e q u i r e s an a u x i l i a r y a.c. a m p l i f i e r t o extend the pass-band out to the r e q u i r e d few k i l o c y c l e s . The m o d u l a t i o n system i s v e r y s u s c e p t i b l e t o p i c k u p or n o i s e a t the modulator f r e q u e n c y . E l e c t r o n i c choppers can work a t much h i g h e r f r e q u e n c i e s than m e c h a n i c a l or magnetic choppers or m o d u l a t o r s . However z e r o - d r i f t a g a i n becomes a problem. Magnetic modulators and a m p l i f i e r s were i n v e s t i g a t e d . I t appeared h i g h l y u n l i k e l y t h a t the s t a b i l i t y r e q u i r e d c o u l d be a c h i e v e d w i t h these d e v i c e s . Consequently, a new t r a n s i s t o r d i f f e r e n t i a l a m p l i f i e r was d e s i g n e d which o f f e r e d s u p e r i o r performance. The new a m p l i f i e r has good z e r o - s t a b i l i t y , a low n o i s e f i g u r e and p r o v i d e s a d.c. to 20 k i l o c y c l e s per second band- w i d t h w i t h i n e x p e n s i v e d e v i c e s . S i n c e o n l y one a m p l i f i e r i s used i n s t e a d of sep a r a t e low and h i g h f r e q u e n c y a m p l i f i e r s , the g a i n i s c o n s t a n t and the phase s h i f t s m a l l up to the upper c u t o f f f r e q u e n c y . The new d e v i c e has o p e r a t e d f o r s e v e r a l days w i t h i t s i n p u t s h o r t e d . Under t h i s c o n d i t i o n the peak-to-peak voTtage d r i f t s were of the o r d e r of t h r e e m i c r o v o l t s , r e f e r r e d to the i n p u t . 2 An independent r e f e r e n c e v o l t a g e s u p p l y (which i s e x t r e m e l y s t a b l e under v a r y i n g l i n e v o l t a g e c o n d i t i o n s ) i s i n c l u d e d i n the system. I n p r a c t i c e , the s t a b i l i t y of the r e f e r e n c e v o l t a g e i s l i m i t e d o n l y by t h a t of the break-down- diode used as a r e f e r e n c e * 2. OUTLINE OF SYSTEM 3 2.1 G e n e r a l The b l o c k diagram of the system i s shown i n F i g u r e 2-1. The system c o n s i s t s of a d.c. a m p l i f i e r , temperature c o n t r o l and power a m p l i f i e r , a r e g u l a t e d power su p p l y w i t h p r e - r e g u l a t o r and a r e f e r e n c e s u p p l y w i t h p r e - r e g u l a t o r . The s p e c i f i c a t i o n s r e q u i r e d f o r measurement and c o n t r o l of s i g n a l s of about 1 v o l t to + 10 ppm f o l l o w . 2.2 The D.C. A m p l i f i e r The d.c. a m p l i f i e r must have a zero s t a b i l i t y of + 10 m i c r o v o l t s or b e t t e r . I t s h o u l d have an a v a i l a b l e output of s e v e r a l v o l t s . The g a i n s h o u l d be g r e a t enough, t h a t any . subsequent a m p l i f i e r need not be an u l t r a - l o w d r i f t t y p e . Input impedance s h o u l d be of the order of 100K ohms or g r e a t e r to a v o i d l o a d i n g the s o u r c e . S i n c e such a u n i t would be e x t r e m l y s e n s i t i v e , a h i g h degree of common—mode r e j e c t i o n i s e s s e n t i a l t o reduce the e f f e c t s of s t r a y p i c k u p . The output impedance s h o u l d be low enough to d r i v e a medium impedance meter or r e c o r d e r d i r e c t l y . A w e l l d e f i n e d s t a b l e v o l t a g e , g a i n would a l l o w d i r e c t use f o r low v o l t a g e l e v e l measurements. 2.3 The Temperature C o n t r o l System A u n i t w i t h the s e n s i t i v i t y and s t a b i l i t y of t h a t d e s c r i b e d above would be extemely d i f f i c u l t t o temperature compensate over a temperature range of any a p p r e c i a b l e e x t e n t . 4 I t was t h e r e f o r e f e l t t h a t the more d i r e c t approach of e n c l o s i n g a l l c r i t i c a l components i n an oven at c o n s t a n t temperature would be more f r u i t f u l . I t was f e l t t h a t the oven temperature s h o u l d be h e l d to + 0.1C° or b e t t e r i f p r a c t i c a b l e . 2.4 R e g u l a t e d Power Supply I t i s r e a s o n a b l e t o expect equipment of the n a t u r e of t h a t d e s c r i b e d above to be s e n s i t i v e to power s u p p l y f l u c t u - a t i o n s . ¥hile i t might be p o s s i b l e to compensate the u n i t s f o r s u p p l y f l u c t u a t i o n s , the temperature changes caused by changing power d i s s i p a t i o n and the changes of component parameters w i t h v o l t a g e would be e x p e cted t o cause problems more severe t h a n t h a t of the c o n s t r u c t i o n of a w e l l r e g u l a t e d power s u p p l y . I t was t h e r e f o r e d e c i d e d to d e s i g n a power s u p p l y w i t h r e g u l a t i o n a g a i n s t l i n e v o l t a g e changes of 0.001$ or b e t t e r . T h i s degree of r e g u l a t i o n p r e s e n t s s i m i l a r , but much l e s s severe^ problems t h a n mentioned above i n c o n n e c t i o n w i t h the a m p l i f i e r . I t was t h e r e f o r e d e c i d e d to d i v i d e the r e g u l a t o r i n t o two u n i t s ; a rough p r e - r e g u l a t o r (ifo) and a main r e g u l a t o r i n s i d e the oven. The power s u p p l i e s were d e s i g n e d to t o l e r a t e l i n e v o l t a g e v a r i a t i o n s from 90 to 135 v o l t s rms. 2.5 Reference V o l t a g e Supply A r e f e r e n c e v o l t a g e s u p p l y was i n c o r p o r a t e d to a l l o w d i f f e r e n t i a l measurements or c o n t r o l . The d e s i g n i s v e r y s i m i l a r to the s u p p l y d e s c r i b e d above. To a v o i d l i m i t i n g the a c c u r a c y of the system, t h i s r e f e r e n c e s u p p l y s h o u l d have a s t a b i l i t y of 0.0001$ or b e t t e r . 5 Input o- o- Reference o- Voltoge o- Reference Supply D.C. Amplifier Pre-regulator 2 Power Supply Temperature Sensor and Amplifier Pre-regulator I Heater Power Amplifier Output — — o Heater Transformer Rectifier Filter Transformer Rectifier Filter Transformer Rectifier Filter 115 V. A.C. F i g u r e 2-1* B l o c k Diagram of D.C. A m p l i f i e r and Reference Supply 3. MODULE DESIGN 6 3*1 The Two T r a n s i s t o r Compound B l o c k Appendix 5 g i v e s the g a i n of a common-emitter t r a n s i s t o r a m p l i f i e r w i t h an e x t e r n a l e m i t t e r impedance, Zg, and l o a d impedance as h-p h , Z-p 1-h , A v = -G (-IS 9b_E ) rb 1 + h f e ^ r b 1 + * where h., h „ h , h., h , Z E o b E l+h„ 1-h , f e rb In the u s u a l d e s i g n of common-emitter a m p l i f i e r s the major d e p a r t u r e of the g a i n from the i d e a l v a l u e , -G, i s due to the terms h^^/(l+h^^) and n-L^/Zg« The major source of n o n - l i n e a r i t y i s the term h-j^/Zg. s i n c e hu^ i s i n v e r s e l y p r o p o r t i o n a l to the e m i t t e r c u r r e n t . A g r e a t improvement i n b o t h the p r e d i c t a b i l i t y and s t a b i l i t y of the g a i n can be a c h i e v e d by l o w e r i n g the v a l u e of t u ^ and r a i s i n g the v a l u e of h^ e» The w e l l known D a r l i n g t o n compound r a i s e s h f g to the p r o d u c t of h f e l and h ^ g 2 of the two t r a n s i s t o r s used. U n f o r t u n a t e l y , however, the e f f e c t i v e v a l u e of t u ^ i s a c t u a l l y i n c r e a s e d somewhat and i s s t i l l p r o p o r t i o n a l to the r e c i p r o c a l of the e m i t t e r c u r r e n t . The two t r a n s i s t o r compound suggested h e r e , and shown 7 i n F i g u r e 3-1 a l s o r a i s e s the c u r r e n t g a i n to a p p r o x i m a t e l y the pr o d u c t of the c u r r e n t g a i n s of the i n d i v i d u a l u n i t s . However, h ^ i s reduced g r e a t l y ( t y p i c a l l y one o r d e r of magnitude) over e i t h e r a s i n g l e t r a n s i s t o r or the D a r l i n g t o n compound. I n a d d i t i o n , the " b a s e - e m i t t e r " v o l t a g e drop of the D a r l i n g t o n compound i s the sum of the two b a s e - e m i t t e r v o l t a g e drops w i t h a t t e n d a n t temperature c o e f f i c i e n t s . The " b a s e - e m i t t e r " v o l t a g e drop of the compound suggested here i s o n l y t h a t a s s o c i a t e d w i t h one t r a n s i s t o r . I n the c i r c u i t used i n t h i s work the c o l l e c t o r c u r r e n t of the f i r s t t r a n s i s t o r i s s e v e r a l t i m e s l a r g e r than the c o l l e c t o r c u r r e n t of the second t r a n s i s t o r d i v i d e d by i t s c u r r e n t g a i n . Thus the c o l l e c t o r c u r r e n t of the f i r s t t r a n s i s t o r v a r i e s o n l y s l i g h t l y over the o p e r a t i n g range of the compound. Since h ^ , f ° r the compound, i s p r i m a r i l y determined by h - j ^ j * h ^ , as w e l l as b e i n g s m a l l , w i l l be s u b j e c t t o much l e s s percentage v a r i a t i o n oyer the o p e r a t i n g range. The compound c i r c u i t i s a n a l y z e d i n d e t a i l i n Appendix 6 and o n l y the approximate v a l u e s of the parameters w i l l be reproduced h e r e . , h i b l whpre h» - — h f e 2 3„2 l b - h" 0 w n e r e n f e 2 - h„ _ ...3-2 f e 2 f e 2 i b 2 R B 2 h , ^ h , n + h., , h , 0 rb r b l l b l ob2 «v !_ . ^ob2 h , s h , , + , • ob o b i h„ , f e l h - h h" fe ~ f e l f e 2 1 + h„ n h " 0 h ..h., _ f e l f e 2 o b i i b 2 T — T 4- ^cbo2 v _ v cbo ~ c b o l + h f x ' BE ~ B E l 8 O C Q2 BO E F i g u r e 3-1. The Two T r a n s i s t o r Compound 3.2 A P r a c t i c a l Common—Emitter A m p l i f i e r S u i t a b l e F o r Use I n a D i f f e r e n t i a l A m p l i f i e r Employing the Two T r a n s i s t o r Compound e m i t t e r a m p l i f i e r to s t a b i l i z e g a i n , r a i s e i n p u t impedance, and make the a m p l i f i e r p r o p e r t i e s l e s s dependent on t r a n s i s t o r parameters i s w e l l known. T h i s t e c h n i q u e i s e s p e c i a l l y e f f e c t i v e w i t h the two t r a n s i s t o r compound d e s c r i b e d above. The a m p l i f i e r h e r e i n d e s c r i b e d employs d i f f e r e n t i a l l y - c o n n e c t e d common-emitter u n i t s i n c o r p o r a t i n g the t w o - t r a n s i s t o r compound w i t h e m i t t e r r e s i s t o r s . T h i s b a s i c b u i l d i n g b l o c k i s shown i n F i g u r e 3-2. F i g u r e 3-2. The B a s i c T w o — T r a n s i s t o r Compound Common-Emitter The t e c h n i q u e of u s i n g an e m i t t e r r e s i s t o r i n a common— <>-°v0 V, tr A m p l i f i e r 9 Appendix 5 g i v e s the v o l t a g e g a i n and i n p u t impedance of a common e m i t t e r a m p l i f i e r w i t h e m i t t e r r e s i s t o r as A = - G ( 5 l L - - 5 s A ) ...3-3 l + h f e l - h r b 1+q where A v g = common e m i t t e r v o l t a g e g a i n and R L G = —• h., B„ B,-,,, h„ h , h.,h , l b i E , EO \ . n i f e r b , , _ i b o b \ B — + ~W~ ' + ^ \ + ) *E0 "EO "E l+h„ 1-h , f e rb ^ 0 l b Gh , ob Z T™ = ( l + h f o ) Bp ^ ...3-4 rb +• ( l + G ) ( l + h f e ) h Q b B E l + h f e E 3.3 C o n s i d e r a t i o n s i n the A m p l i f i e r Design I t was d e c i d e d to d e s i g n a t h r e e stage a m p l i f i e r w i t h an o v e r a l l g a i n of lj , 0 0 0 o and a,n i n p u t impedance of about 100K ohms« The g a i n f i g u r e of 1*000 was chosen so t h a t an a u x i l i a r y a m p l i f i e r used t o i n c r e a s e the g a i n f u r t h e r c o u l d be 10 r e l a t i v e l y s imple w i t h o u t a d v e r s e l y e f f e c t i n g the zero s t a b i l i t y . The t h r e e stages were to c o n s i s t o f ; 1. A h i g h - g a i n v e r y l o w - d r i f t i n p u t stage w i t h good common-mode r e j e c t i o n 2. A medium-gain, l o w - d r i f t second stage 3. An e m i t t e r - f o l l o w e r output stage w i t h p r o v i s i o n s f o r common—mode feedback to the c u r r e n t - s o u r c e t r a n s - i s t o r of the i n p u t s t a g e . 3.4 The Input Stage of the D#C. A m p l i f i e r The i n p u t stage c o n s i s t s of two a m p l i f i e r s of the type shown i n F i g u r e 3—2 connected d i f f e r e n t i a l l y w i t h a t r a n s i s t o r as a high-impedance c u r r e n t source i n the e m i t t e r l e a d s . To reduce problems a s s o c i a t e d w i t h leakage c u r r e n t , the t r a n s i s t o r s used i n the two a m p l i f i e r u n i t s were a l l s i l i c o n . The c i r c u i t i s shown i n F i g u r e 3-3. Q401 and Q402 are bonded t o g e t h e r , cap—to-cap w i t h Dow C o r n i n g 731 S i l i c o n e r u b b e r . T h i s assembly i s t h e n wrapped w i t h a l a y e r of t h i n i n s u l a t i n g t a p e , f o l l o w e d by two l a y e r s of copper tape s e p a r a t e d by i n s u l a t i n g t a p e . The whole assembly i s th e n wrapped w i t h another l a y e r of i n s u l a t i n g t a pe* The wrappings a c t as a th e r m a l f i l t e r t o keep the two t r a n s i s t o r s v e r y n e a r l y a t the same temperature even when the ambient temperature i s changing* The o t h e r p a i r s of t r a n s - i s t o r s i n the a m p l i f i e r are t r e a t e d s i m i l a r i l y . The compound b l o c k c o n s i s t i n g of Q402 and Q404 w i l l now be examined. The b a s e — e m i t t e r v o l t a g e of a s i l i c o n t r a n s i s t o r o p e r a t i n g a t 1 ma a t room temperature i s about 0.6 v o l t s . The common-emitter i n p u t impedance^ h. of such a t r a n s i s t o r i s 11 + I3V F i g u r e 3-3. The Input Stage of the D.C. A m p l i f i e r a p p r o x i m a t e l y ^ f g ^ 1 ^ ^ * where h ^ g i s the common-emitter c u r r e n t g a i n and, h ^ , the common-base i n p u t impedance, i s a p p r o x i m a t e l y 30 ohms. I f a r e s i s t o r Bg, i s p l a c e d between the base and the e m i t t e r , the i n p u t c u r r e n t t o the c o m b i n a t i o n (see F i g u r e 3-4) w i l l be I " = I B x V BE B + I c ^BE h FE Thus h " 0 PE ~ IB" h " — h 3-5 FE - n P E V B 1 ? . . . J D I c R b h P E "where I g " , Ig> IQ« Vgg? hpjj are d.c. l a r g e - s i g n a l parameters, However, HJJ + h. R„ + h„ h., i „ ^ i e B f e I D e l " = l B D b D h„ Rg Rg f e X c 1 f e ~ i " - n f e h h * " •̂B i + f e i b where ig"> ig» i c« hf e» a r e s m a l l - s i g n a l parameters, F i g u r e 3-4. T r a n s i s t o r W ith B a s e - E m i t t e r R e s i s t o r I f the above mentioned parameters are 3-5 and 3-6, 3-7 and 3-8 are o b t a i n e d . s u b s t i t u t e d i n t o h — v» 2 FE _ FE ' n * * * ~ F E h f e " = h f e " ~ 3 - 8 *B" h f e I f h ^ e and h^^, are assumed to be a p p r o x i m a t e l y e q u a l , as i s u s u a l l y the case, i t i s seen t h a t h ^ may be c o n s i d e r a b l y reduced w i t h o u t s e r i o u s r e d u c t i o n of T h i s p e r m i t s l a r g e r c u r r e n t i n the i n p u t t r a n s i s t o r of a compound f o r a g i v e n compound c o l l e c t o r c u r r e n t . I f the c u r r e n t g a i n s of Q404 are t a k e n as h p g = h f g = 40 and R408 i s ta k e n as 5.6K ohms, the n hpg" =7.5 and h^ " = 33. Q404 i s a c t u a l l y o p e r a t e d a t a l i t t l e l e s s than 1 ma. However, the r e s u l t s quoted above are s t i l l a p p r o x i m a t e l y c o r r e c t o OC—201's were chosen f o r Q403 and Q404. Q402 i s oper a t e d a t a p p r o x i m a t e l y 100 jxa. From the d a t a sheet f o r a 2N697 a t I c=100 |j,a» h i b ~ 3 0 0 o h m s h f e - 50 h * 3: X 1 0 = 5 . ..3-9 rb h , ? 2 X 1 0 ~ 7 mhos, ob From the d a t a f o r the OC—201, h^ i s as quoted above, 40 and h o b = .5 \i mho. The presence of R408 l e a d s t o h^ " = 33 as s t a t e d above. Combining t h i s w i t h 3-9 and 3-2 g i v e s as the parameters of the compound Q402, Q404 and R408% h i b ~ 9 o h m h r b 1.8 X 1 0 ~ 4 h Q b * 2.1 X 1 0 ~ 7 , ...3-10 h_ « 2000 i e From 3-3 and 3-10 and t a k i n g G * 60 and and R E O = Jisr— - 850 ohms ob i b „ q * .011 + .01 + ^ - ) h f e -3 # 1 - .5 x 10 ^ 1 1 + h f e RT A V F = -G ((1 - .011 - .01 (-^2. + ^ - ) ) ) ...3-11 V h R E R E o 1 + .01 (^2. + 5L.) + . o n Z t t ;, = — — ° — — — X 10^ ohms lb -n + -021 (G + 1) K E .3-12 15 I t was d e c i d e d to use Rg = 100 ohms because a l a r g e r v a l u e would r e q u i r e a c o r r e s p o n d i n g l y h i g h e r v a l u e of R^ f o r the same g a i n and the stage would t h e n r e q u i r e a h i g h e r v o l t a g e to main- t a i n the d e s i r e d o p e r a t i n g c u r r e n t . Lower o p e r a t i n g c u r r e n t s would g i v e a h i g h e r v a l u e of h and lower v a l u e s of h ^ g and thus degrade the s t a g e . H i g h e r o p e r a t i n g v o l t a g e would r a i s e power d i s s i p a t i o n and aggravate the therm a l d r i f t s . V i t h Rg = 100 ohms, Ayg = -.9G and ZJE = 100K ohms/(.45 + .01 9 ( G + l ) ) . I f Rg was s e t equal t o R g Q , o n l y about 6 per cent more g a i n would be obtained,, I f i t i s assumed t h a t the second stage i n p u t impedance i s v e r y h i g h compared t o R^, the g a i n of the f i r s t stage w i l l be R406 p R406 R404 - ° y 100 Sin c e Ayg = 60 was d e s i r e d ^ R406 was chosen as 6.8K ohms to g i v e G = 68 and Ayg j u s t over 60. These v a l u e s g i v e Zj-g = 57K ohms. I f R401 and R402 are chosen as 150K ohms, the i n p u t impedance would be a p p r o x i m a t e l y 82K ohms between d i f f e r e n t i a l i n p u t s . R401 s u p p l i e s base c u r r e n t to the compound t r a n s i s t o r . I f R410 i s c o r r e c t l y chosen the v o l t a g e drop a c r o s s R402 w i l l be zero a t the d e s i r e d o p e r a t i n g p o i n t and zero i n p u t - v o l t a g e . Two of the above d e s c r i b e d elements are connected t o g e t h e r to form a d i f f e r e n t i a l a m p l i f i e r . Q403 i s the same type as Q404, s e l e c t e d f o r matched • Q401 i s the same type as Q402, s e l e c t e d f o r matched h^ and IQ^Q* The e f f e c t of the c o l l e c t o r - b a s e leakage c u r r e n t , IQ^Q? °^ the compound i s as f o l l o w s . The d i f f e r e n c e between the IQ-^Q of Q401 and t h a t of Q402 f l o w s t h r o u g h the source r e s i s t a n c e and c r e a t e s an o f f s e t v o l t a g e * AI^g^Rg. I t i s t h e r e f o r e d e s i r a b l e to have these IQBO' s a s s m a , H a s p o s s i b l e and as c l o s e l y matched as p o s s i b l e . The e f f e c t of the IQ^Q of Q403 i s s i m i l a r except t h a t the e f f e c t i v e v a l u e i s the a c t u a l v a l u e d i v i d e d by the c u r r e n t g a i n of the i n p u t t r a n s i s t o r , , Thus, i f these l e a k a g e s are of the same or d e r as the leakage of the i n p u t t r a n s i s t o r t h e i r e f f e c t w i l l be n e g l i g i b l e * The I Q B 0 o f Q40l a n d Q402 was -9 a p p r o x i m a t e l y 10 amp. R403 and R404 were hand wound w i t h advance r e s i s t a n c e w i r e on a s i l i c o n e r u b b e r form and matched to w i t h i n 0.05$. R405 and R406 were s e l e c t e d from a v a i l a b l e 6.8K ohm, itfo d e p o s i t e d carbon r e s i s t o r s . A p a i r was found t h a t matched to w i t h i n 0.1$. With the i n p u t s h o r t e d and the l o a d r e s i s t o r s matched, zero o u tput v o l t a g e w i l l be o b t a i n e d from the d i f f e r e n t i a l a m p l i f i e r i f the c o l l e c t o r c u r r e n t of the two compound " t r a n s i s t o r s " are e x a c t l y equals The e m i t t e r c u r r e n t s w i l l t h e n be v e r y n e a r l y e q u a l s i n c e the compound t r a n s i s t o r s have v e r y l a r g e , n e a r l y matched, c u r r e n t g a i n s . T h i s i m p l i e s t h a t the v o l t a g e drops a c r o s s the e m i t t e r r e s i s t o r s w i l l be v e r y n e a r l y e q u a l . T h i s i s o n l y p o s s i b l e i f the b a s e — e m i t t e r v o l t a g e s of the two compound " t r a n s i s t o r s " are equal f o r e q u a l c o l l e c t o r c u r r e n t . F o r t u n a t e l y t h i s parameter i s v a r i a b l e i n the c o m p o u n d - t r a n s i s t o r c i r c u i t u s e d . I f Rg ( F i g u r e 3-4) i s v a r i e d w h i l e I^, i s h e l d c o n s t a n t , I g " ( i n the compound c i r c u i t I g " i s c o l l e c t o r c u r r e n t of the i n p u t t r a n s i s t o r , F i g u r e 3 — l ) must be v a r i e d . When I g " i s v a r i e d the base e m i t t e r v o l t a g e w i l l v a r y . Thus Rg may be v a r i e d t o change the b a s e - e m i t t e r v d l t a g e of the compound a t c o n s t a n t 17 c o l l e c t o r c u r r e n t . S i n c e the b a s e - e m i t t e r v o l t a g e of a modern s i l i c o n t r a n s i s t o r i s a l r e a d y a q u i t e p r e d i c t a b l e parameter, o n l y a s m a l l c o r r e c t i o n i s n e c e s s a r y . T h i s can be e a s i l y a c c o m p l i s h e d w i t h o u t u p s e t t i n g o t h e r parameters of the c i r c u i t . l a r g e r (6.8K ohms i n s t e a d of 5.6K ohms) tha n R408 and shunted by a f i n e - a d j u s t network t h a t c o u l d v a r y the r e s u l t i n g r e s i s t a n c e from somewhat above to somewhat below 5.6K ohms. To a c c o m p l i s h t h i s R415 was made 10K ohms and R416 e x p e r i m e n t a l l y s e l e c t e d t o p l a c e the zero near the c e n t e r of the ad j u s t m e n t . T h i s a d j u s t - ment s h o u l d be made w i t h the i n p u t s h o r t - c i r c u i t e d . R409, R402j R413 and R414 are a s i m i l a r arrangement used to compensate f o r d i f f e r e n c e s i n base c u r r e n t r e q u i r e d t o e s t a b l i s h the d e s i r e d o p e r a t i n g p o i n t . The adjustment of R412 s h o u l d be made 9 a f t e r the adjustment of R415 9 w i t h the i n p u t o p e n — c i r c u i t e d . The zero adjustment s h o u l d t h e n be independent of source impedance. Q411 i s a high-impedance c u r r e n t source f o r the e m i t t e r c i r c u i t . The common-mode feedback s i g n a l , B, (see F i g u r e 3-3) i s g e n e r a t e d from a source of low impedance compared w i t h R411. Q411 i s thus o p e r a t i n g i n the common-base mode, and has an t h a t f l o w s w i l l be a p p r o x i m a t e l y e q u a l to. the v o l t a g e a t the base of Q411 d i v i d e d by R411„ 3.5 The Second Stage of the D.C« A m p l i f i e r For the reasons s e t out above R407 was made somewhat output conductance a p p r o x i m a t e l y e q u a l t o h ob° The a c t u a l c u r r e n t a p p r o x i m a t e l y 18 g r e a t e r g a i n which c o u l d be a d j u s t e d . The c i r c u i t i s shown i n F i g u r e 3-5. • + I3V < R424 Input I O- F i g u r e 3.-5. The Second Stage of the D.C. A m p l i f i e r The t r a n s i s t o r s used i n the second stage are germanium t r a n s i s t o r s . The b a s e - e m i t t e r v o l t a g e of a germanium t r a n s i s t o r a t 1 ma c o l l e c t o r c u r r e n t and room temperature i s t y p i c a l l y 0*1 v o l t compared w i t h 0.6 v o l t f o r s i l i c o n . I n o r d e r t h a t the t e c h n i q u e of u s i n g a r e s i s t o r between the base and the e m i t t e r , t o reduce h^g w i t h o u t s e r i o u s l y e f f e c t i n g h ^ . work e f f e c t i v e l y , i t i s n e c e s s a r y to i n c r e a s e the r a t i o of V-g-g/tUvj• A s i l i c o n diode has a v o l t a g e - c u r r e n t c h a r a c t e r i s t i c v e r y s i m i l a r to the base— v o l t a g e — e m i t t e r - c u r r e n t c h a r a c t e r i s t i c of a s i l i c o n t r a n s i s t o r . Thus the f o r w a r d v o l t a g e drop a t 1 ma c u r r e n t i s about 0.6 v o l t , and the dynamic impedance i s about 30 ohms. Q406, Q408, D402 and B418 form \a p-n-p compound t r a n s i s t o r complimentary t o t h a t shown i n F i g u r e 3-1. D402 r a i s e s the e f f e c t i v e b a s e - e m i t t e r v o l t a g e of Q406 from 0.1 v o l t to 0.7 v o l t a t 1 ma c o l l e c t o r c u r r e n t , w h i l e r a i s i n g the e f f e c t i v e h., from 30 ohms t o 60 ohms. U s i n g a 2N1305 f o r Q408, a 2N1304 f o r Q406, and a 1N465A f o r D402 g i v e s the f o l l o w i n g parameters f o r the compound t r a n s i s t o r thus formed* ¥ith the f o l l o w i n g v a l u e s f o r t r a n s i s t o r p arameters, x f e l 60 h f e 2 - 120 h., , = 300 ohms l b l h r M =" 7 X 10 -4 h =" .12 X 10 o b i -6 h., „ =* 30 ohms l b 2 h , 0 = 5 X 10 rb2 -4 h , . =• .34 X 10 obz -6 h "- 80 n f e 2 _ O U I , =0.1 ma c l I = 1 ma c 2 E q u a t i o n s 3-2 g i v e h., =2.5 ohms l b h , = 8 X 10 rb -4 «© * 3"~ X 3 h « .13 I 10" ob h„ *? 4700 i e Prom 3-3 and 3-13 t a k i n g G = 20 Eo I K ohm R R q = 0.016 + 2*5 X 10" 3 + R E o R E h f e 1 + h = 1 - 2 X 10  4 * 1 f e V E - G f l - *016 - 2.5 X l O " 3 ! ^ - + ^§°) 20 1 + .016 + 2.5 X 1 0 _ 3 ( ^ - + ) 'E J I E 10 -5 .. .3-15 .047 Bg + (l+G) .13 X 10 -6 Fo r reasons s i m i l a r t o those o u t l i n e d r e g a r d i n g the c h o i c e of R404, R420 was s e t equal to 100 ohms. V i t h R420 = 100 ohms, e q u a t i o n s 3-14 and 3-15 become and A v E = - * 9 6 G 'IE 100K .20 + .013 (1+G) Ta k i n g S e t t i n g and G *? 'IE 20, 220K R422 = 3.3K R423 = 10K Thus G = 20 and R L = ^7p- // R422 ~ 2K A v E = ~ 1 9 An i d e n t i c a l u n i t i s used f o r the o t h e r h a l f of the second stage of the d i f f e r e n t i a l a m p l i f i e r w i t h a 2K ohm r e s i s t o r (R424) as an e m i t t e r c u r r e n t s o u r c e . 21 3.6 The E m i t t e r - F o l l o w e r T h i r d Stage of the D.C. A m p l i f i e r The t h i r d stage of the d i f f e r e n t i a l a m p l i f i e r i s a d i f f e r e n t i a l e m i t t e r - f o l l o w e r . I t s purpose i s to p r o v i d e a low output-impedance and thus make the g a i n more independant of l o a d i n g a t the o u t p u t . The c i r c u i t i s shown i n F i g u r e 3-6. e Output I o Output 2 + 13 V F i g u r e 3-6. The E m i t t e r F o l l o w e r T h i r d Stage Q4Q9 and Q410 are 2N1309 h i g h — g a i n germanium t r a n s i s t o r s . E425 and R426 are 8.2K ohms r e s i s t o r s ^ used to e s t a b l i s h the d e s i r e d 1 ma o p e r a t i n g c u r r e n t w i t h the common-mode A output a t +4 v o l t s , The output impedance of an e m i t t e r — f o l l o w e r i s g i v e n by R (Appendix 3) Z = oc l b h f e 1 + h , R ob g ...3-16 22 F o r a 2N1309 at I = 1 ma: c h f e - 1 9 0 h' * .4 X 1 0 " 6 mhos ob h., = 30 ohms l b R v a r i e s , depending on the s e t t i n g of R423, between 0 and 2K ohms* Thus the output impedance w i l l be between 60 ohms and 82 ohms« 3.7 The Oven Tem p e r a t u r e - C o n t r o l Servo The b a s e - e m i t t e r v o l t a g e of a s i l i c o n or germanium t r a n s i s t o r o p e r a t e d a t c o n s t a n t c o l l e c t o r c u r r e n t d ecreases a p p r o x i m a t e l y 2 M i l l i v o l t s f o r each c e n t i g r a d e degree r i s e i n temperature over a wide temperature range. T h i s v o l t a g e and i t s temperature c o e f f i c i e n t appear s t a b l e w i t h t i m e . T h i s phenomenum i s u t i l i z e d t o c o n t r o l the temperature of the oven. The c i r c u i t f o r the temperature c o n t r o l servomechanism i s shown i n F i g u r e 3-7. B a s i c a l l y the u n i t i s a d.c. a m p l i f i e r which i s not compensated f o r the e f f e c t of the change of the b a s e - e m i t t e r v o l t a g e of the i n p u t t a n s i s t o r w i t h t e m p e r a t u r e . I n f a c t the a m p l i f i e r i s d e s i g n e d t o emphasize t h i s e f f e c t . A germanium t r a n s i s t o r i s used f o r Q301. A s i l i c o n t r a n s i s t o r would have about s i x times as l a r g e a b a s e - e m i t t e r v o l t a g e w i t h about the same v a l u e of O BE of a p p r o x i m a t e l y b T —2 mv/C°. The b a s e - c i r c u i t v o l t a g e d i v i d e r , R301 and R302, 23 o + Heater 24V c - OVEN F i g u r e 3-7. The Oven Temperature-Control System R 302 —1 would then have to have about s i x times the r a t i o ( l + g~302") and the s e n s i t i v i t y of the c i r c u i t t o power s u p p l y v o l t a g e changes would be i n c r e a s e d by the same f a c t o r . T r a n s i s t o r Q301 i s o p e r a t e d i n the common e m i t t e r mode. The impedance of the b a s e - c i r c u i t v o l t a g e d i v i d e r must be much l e s s t h a n the common-emitter i n p u t impedance to maximize the v o l t a g e s e n s i t i v i t y of Q301* The performance of the c i r c u i t may be a n a l y s e d by r e p l a c i n g Q301 w i t h a h y p o t h e t i c a l t r a n s i s t o r w i t h q BE _ Q o T and h a v i n g a v o l t a g e g e n e r a t o r of 2 mv ( T Q - T) i n s e r i e s w i t h i t s base l e a d . U s i n g e q u a t i o n s A5—97 A5—12, and A5-13 of Appendix 5* and A4-9 of Appendix 4 the g a i n o f the a m p l i f i e r c o n s i s t i n g of Q301, Q302 and Q303 may be shown t o be A y E = 1.5 X 10 24 The e m i t t e r - f o l l o w e r , Q304 w i l l p r o v i d e an o u t p u t - impedance of a p p r o x i m a t e l y 250 ohms. R310 i n the c o l l e c t o r c i r c u i t of Q304 se r v e s to l i m i t the c u r r e n t through Q304 i n the event of o v e r l o a d or s h o r t c i r c u i t . The v o l t a g e a t the p o i n t marked V̂ , i n F i g u r e 3-7 w i l l have a temperature c o e f f i c i e n t of — - = 1.5 X 1CT X (-2) mv/C° ...3-17 dT = 30 v o l t s / C }T and w i l l be generated by a source of a p p r o x i m a t e l y 250 ohms impedance. As w e l l as b e i n g s e n s i t i v e t o temperature changes, the u n i t i s s e n s i t i v e t o s u p p l y v o l t a g e v a r i a t i o n s . The u n i t t h a t was c o n s t r u c t e d gave an o p e r a t i n g p o i n t of 35°C w i t h R302 = 15K and R301 = 100 ohms. The d i v i d e r r a t i o f o r R302 and -3 R301 i s t h e n 6.7 X 10 . When t h i s i s m u l t i p l i e d by the 4 a m p l i f i e r g a i n , 1.5 X 10 , the r e s u l t i s 2 VT — - = 100 v o l t s / v o l t where = v o l t a g e a t t e s t p o i n t ( f i g u r e 3-7) Vg = s u p p l y v o l t a g e 2 5 <^T / <^T o / The r a t i o / = 3.1 C / v o l t i s the e r r o r i n o p e r a t i n g a v s ^T p o i n t t h a t would be caused by a change of 1 v o l t i n s u p p l y v o l t a g e * The power s u p p l i e s used i n t h i s equipment are capable of more t h a n s u f f i c i e n t r e g u l a t i o n t o keep t h i s source of e r r o r below a few m i l l i d e g r e e s f o r l i n e v o l t a g e s between 9 0 and 1 3 5 v p l t s r.m.s. The e f f e c t of the change of b a s e - e m i t t e r v o l t a g e of Q 3 0 2 w i t h temperature i s suppressed by the pr o d u c t of the v o l t a g e g a i n of Q 3 0 1 and one p l u s the r a t i o of output impedance of Q 3 0 1 t o the i n p u t impedance of Q 3 0 2 . T h i s q u a n t i t y i s a p p r o x i m a t e l y 3 3 0 . The temperature s e n s i t i v i t y of the base- e m i t t e r v o l t a g e of Q 3 0 2 o n l y changes the o v e r a l l s e n s i t i v i t y by about l / 3 $ . T r a n s i s t o r s f u r t h e r a l o n g i n the u n i t w i l l have even l e s s e f f e c t . The e f f e c t of the leakage c u r r e n t of Q 3 0 2 i s much more s e r i o u s . I f the IQ^Q of Q 3 0 2 changes by 1 jj,a, t h i s w i l l appear as a 1 aa change i n c u r r e n t through the output impedance of Q 3 0 1 w h i c h , i n c l u d i n g the e f f e c t of i t s l o a d r e s i s t o r , R 3 0 3 , i s 3 6 K ohms. The 1 \xa, change i n I Q B Q of Q 3 0 2 then appears as a 3 6 m i l l i v o l t change a t the output of Q 3 0 1 which i s e q u i v a l e n t to about l / 3 mv a t the i n p u t or about l / 6 C°. I t would t h e r e - f o r e be d e s i r a b l e to r e p l a c e Q 3 0 2 w i t h a low leakage s i l i c o n t r a n s i s t o r . S e l f h e a t i n g of Q 3 0 1 i s another p o t e n t i a l source of e r r o r and i n s t a b i l i t y . Q 3 0 1 , which i s r e q u i r e d t o d i s s i p a t e about \ mw, i s mounted i n a f i n n e d h e a t - s i n k which g i v e s an o v e r a l l o / c o l l e c t o r j u n c t i o n - t o - a m b i e n t t h e r m a l r e s i s t a n c e of about .2 C /mw. 26 The j u n c t i o n of Q301 would t h e n be e x p e cted to be a t about 0.05C 0 above ambient. S i n c e t h i s q u a n t i t y s h o u l d be s t a b l e to w i t h i n about 10$, the i n s t a b i l i t y caused s h o u l d be n e g l i g i b l e . The h e a t e r power a m p l i f i e r i s a common-emitter power s t a g e , w i t h a v o l t a g e g a i n of about 5, d r i v e n by an e m i t t e r - f o l l o w e r * B311, R312 and B 315 p r o v i d e adequate t h e r m a l . s t a b i l i t y . R313 and R314 l i m i t the power d i s s i p a t i o n i n the c o l l e c t o r c i r c u i t of Q305. The output v o l t a g e to the h e a t e r i s t h e n about 150 volts/C°. The oven w i l l t h e n be t u r n e d from f u l l on to f u l l o f f by an i n t e r n a l temperature change of about .15 C°. For the u n i t c o n s t r u c t e d , the v a l u e of R302 r e q u i r e d f o r 35°C o p e r a t i o n was a p p r o x i m a t e l y 15K ohms and dR302 w a g 3 T found t o be 250 ohms/C°. 3«8 The Common-Collector R e g u l a t o r A v e r y common r e g u l a t o r i s the s o — c a l l e d e m i t t e r - f o l l o w e r , or common—collector, r e g u l a t o r shown s c h e m a t i c a l l y i n F i g u r e 3-8. +Vj o- 01 I — W V «i R — ° v 0 :(l-a)R. f ° V R S a R , F i g u r e 3-8. Common-Collector R e g u l a t o r The performance of t h i s r e g u l a t o r w i l l be a n a l y s e d by e v a l u a t i n g g and r as d i s c u s s e d i n Appendix 1, where g a i V = const o L e t V ^ be changed by A"V\ w h i l e V q i s h e l d c o n s t a n t by changing the l o a d as much as i s n e c e s s a r y . The i n p u t t o Q2 remains c o n s t a n t s i n c e V , and thus the output of a m p l i f i e r , A, remains constant.. The b a s e - e m i t t e r v o l t a g e of Ql w i l l change s l i g h t l y when V ^ i s changed. T h i s s m a l l change w i l l cause a n e g l i g i b l e change i n the c o l l e c t o r c u r r e n t of Q2. Thus the c o l l e c t o r c u r r e n t of Q2 may be regarde d as c o n s t a n t . Any change i n V-g-ĝ w i l l be much l e s s than AV.« As V. i s i n c r e a s e d I w i l l i n c r e a s e 1 1 due t o the i n c r e a s e i n c o l l e c t o r - e m i t t e r v o l t a g e of Ql and due to the i n c r e a s e of base c u r r e n t t h r o u g h R« A I = A V. I + h AV. l ...3-19 R o e l f o r e V. - V R - h. PE1 l mm I max o 3-20 Sin c e a s m a l l v a l u e of R w i l l degrade the performance of the s u p p l y , we choose the l a r g e s t a c c e p t a b l e v a l u e and s u b s t i t u t e i t i n t o E q u a t i o n 3.19 f o r A I 28 I h„ A I = ( E M ^ e i h . J AV. ...3-21 V - V h P E l f e l o b l 1 i min o Tf h <v v. ± T . f e l ~ FE1' ° _ AV V - V i i min o To f i n d r the output r e s i s t a n c e , l e t V. be h e l d o r ' i c o n s t a n t and examine the change i n I , A I , due to a s m a l l change i n YQf ^ V q . The change i n v o l t a g e a t the base of Q2, A V ^ ? < l u e to A V w i l l be: o A V ^ = aA A V ...3-23 £3 Z O T h i s w i l l cause a change i n base c u r r e n t of Q2 which w i l l be a m p l i f i e d by Q2 and Ql t o g i v e aA A V A I = (1.+ h f e l ) h f e 2 2. ...3-24 i e 2 aA A V " h f e l h f e 2 i f h f e l » 1 h i e 2 u s i n g A V Q h. 0 = h „ ~ h . , 0 and r - — — i e 2 f e 2 i b 2 o A V h., . r = - 2. = i b 2 _ 3 _ 2 5 0 A I a (-A) h f e l 29 E q u a t i o n 3-25 n e g l e c t s the f a c t t h a t r e s i s t o r , R, shunts the i n p u t t o Q l . r Q w i l l i n p r a c t i c e be somewhat h i g h e r than p r e d i c t e d by 3-25. 3.9 The Common-Emitter R e g u l a t o r An a l t e r n a t i v e r e g u l a t o r c i r c u i t , the common-emitter r e g u l a t o r , i s shown i n F i g u r e 3-9. +V: o- I -o + V 0 ( l - a ) R , *R > a R i F i g u r e 3-9. The Common-Emitter R e g u l a t o r The parameters g and r Q are e v a l u a t e d as i n 3-8. I f V. i s i n c r e a s e d by AV. w h i l e V i s h e l d c o n s t a n t , I w i l l 1 J 1 o 7 i n c r e a s e due t o the i n c r e a s e d b a s e - e m i t t e r v o l t a g e on Q l . I w i l l i n c r e a s e f u r t h e r due to i n c r e a s e d c o l l e c t o r e m i t t e r v o l t a g e on Q2 c a u s i n g more base c u r r e n t t o f l o w i n t o Q l . I f i t i s assumed t h a t s A VBE1 « A V i and — » h . , g 02 1 6 1 where g Q2 = output conductance of Q2 30 t h e n „ _ L L _ > l f e l v " o b l " s o 2 g = A - i - = h , o l ( h r t K 1 + g„ 0) ...3-26 A V. 1 I f V q i s h e l d c o n s t a n t and the change i n I , A I , due to a change, A V q , i n V q i s examined, then the same argument used to o b t a i n E q u a t i o n 3-25 y i e l d s : r = ^ ...3-27 a(-A) h f e l where A i s the g a i n of the a m p l i f i e r , A, i n F i g u r e 3-9. 3.10 M o d i f i c a t i o n of R e g u l a t o r O u t p u t — R e s i s t a n c e by I n t e r - a c t i o n of Output V o l t a g e and Reference V o l t a g e I f a p a s s i v e r e f e r e n c e element ( e . g . a breakdown d i o d e ) i s d r i v e n by a r e g u l a t o r , the r e f e r e n c e v o l t a g e , V^^ w i l l , i n g e n e r a l * change w i t h a change of output v o l t a g e , V q . The a n a l y s i s f o r r must be m o d i f i e d to account f o r t h i s e f f e c t . 0 a v R The i n p u t t o the a m p l i f i e r u n i t becomes ( a — ) A V dY 0 o and t h u s * k j b 2 1 _ _ r _ ...3-2o A( a — ) * V o 3»11 Comparison of Common-Emitter and Common—Collector R e g u l a t o r s I t i s seen t h a t , f o r e q u i v a l e n t s u p p l i e s , the output r e s i s t a n c e s are the same. The i n p u t conductances d i f f e r . 31 g c c = y I m a I v + h f e l h o b l • ' ' 3 " 2 2 1 min o and >cE = h f e l ( S 0 2 + h o b l } ...3-26 I t may be seen from Appendix 1, Al»6 t h a t the l i n e r e g u l a t i o n f a c t o r , K, i s g i v e n by K = g r Q ...3-29 S i n c e r and r are a p p r o x i m a t e l y equal i f l o a d i n g K g due t o the d r i v e r e s i s t o r , R, may be n e g l e c t e d , j r ^ - = — — . cc ^cc The o p e r a t i n g c o n d i t i o n s of the r e g u l a t e d s u p p l y f o r the d*c* a m p l i f i e r w i l l be t a k e n as an example* I = 50 ma V T - V = 5V I mm o h f e l = 7 0 h = 3 X 1 0 ~ 6 mho ob i g Q 2 i s the output conductance of the common-emitter a m p l i f i e r , Q2, and i s g i v e n by A-4-15 i n Appendix 4 as 1 _ h f e 2 h i b 2 , , Z >o2 ~ 2 v l lob2 i lfe2 T h.^J + 2 U o b 2 n f e 2 " h. ~ 1 i b 2 i b 2 h f e 2 h l b 2 \ ..*3-30 32 i f h f e 2 » 1 and h r b 2 « 1 T y p i c a l v a l u e s f o r Q2 h , , = .4 X 10 6 mho ob2 h r b 2 = 7 X I D " 4 h ^ b 2 = 30 ohms h f e 2 = 5 0 These w i t h 3-29 g i v e g o 2 * 22 X I D " 6 U s i n g t h i s v a l u e i n 3-21 and 3-25 y e i l d s 1 - 1500 Z -6 -6 g g 0 = 22 X 10 ° - 1.5 X 10 ° o 2 1 + Z g ...3-31 g C E ^ 1.8 X 10 3 C l e a r l y the common-emitter r e g u l a t o r w i l l p e r f o r m more than f i v e t imes as w e l l as the common-collector r e g u l a t o r i n t h i s a p p l i c a t i o n . 33 3.12 The Temperature C o e f f i c i e n t of Breakdown-Diodes I t i s w e l l known t h a t l o w - v o l t a g e breakdown-diodes have a n e g a t i v e temperature c o e f f i c i e n t of v o l t a g e and h i g h - v o l t a g e d i o d e s have a p o s i t i v e temperature c o e f f i c i e n t . A v e r y s m a l l temperature c o e f f i c i e n t i s observed i n d i o d e s which break down a t about 5.5 v o l t s . I t i s a l s o w e l l known t h a t the temperature c o e f f i c i e n t i s a f u n c t i o n of c u r r e n t , becoming more p o s i t i v e a t h i g h e r c u r r e n t . These phenomena were i n v e s t i g a t e d and s e v e r a l s a l i e n t f e a t u r e s n o t e d . The breakdown v o l t a g e a t 10 ma and the o p e r a t i n g c u r r e n t t h a t produced zero average temperature c o e f f i c i e n t between 0°C o and 100 C were measured on f o u r t e e n d i o d e s . The r e s u l t i n g d i s t r i b u t i o n i s shown i n F i g u r e 3-10* The v o l t a g e - t e m p e r a t u r e c h a r a c t e r i s t i c s between -50 C and +100 C of these d i o d e s were measured a t the c u r r e n t which gave zero average temperature c o e f f i c i e n t . The r e s u l t s were f i t t e d t o the e q u a t i o n V = V (50°C) 1 + a(T - 50) + b(T - 5 0 ) 2 + c(T - 5 0 ) 3 + d(T - 5 0 ) 4 ...3-32 V i t h i n the a c c u r a c y of a v a i l a b l e t e s t equipment (.01$), i t was found t h a t a = 0 (by c h o i c e of o p e r a t i n g p o i n t ) ...3-33 b = .7 X 1 0 " 6 34 Diode V o l t a g e at lOma I 6.3 Diode C u r r e n t f o r Zero Temperature C o e f f i c i e n t (ma) F i g u r e 3-10. Breakdown-diode V o l t a g e a t lOma v s . Diode C u r r e n t f o r Zero Temperature C u r r e n t 35 3.13 General Power Supply C o n s i d e r a t i o n s There are f o u r r e g u l a t o r s i n the system. Two serve as rough p r e - r e g u l a t o r s and two as r e g u l a t o r s . These u n i t s are a l l common-emitter r e g u l a t o r s . The main r e g u l a t o r s have a d i f f e r e n t i a l a m p l i f i e r s t a g e . The p r e - r e g u l a t o r s do not have t h e i r own r e f e r e n c e d i o d e s but use the output of the main r e g u l a t o r s as r e f e r e n c e s . The p r e - r e g u l a t o r s a l s o use s m a l l amounts of open- l o o p feedback to improve t h e i r r e g u l a t i o n . 3.14 The R e g u l a t e d Power Supply f o r the D.C. A m p l i f i e r The c i r c u i t of the r e g u l a t e d power su p p l y f o r the d.c. a m p l i f i e r i s shown i n F i g u r e 3—11. I t i s of the common-emitter type w i t h a d i f f e r e n t i a l a m p l i f i e r w i t h a g a i n of about 5. F i g u r e 3.11 The R e g u l a t e d Power Supply f o r the D.C. A m p l i f i e r R208 i s p r e s e n t t o improve the t h e r m a l s t a b i l i t y of Q204 and a l s o 36 to r a i s e the g a i n of Q203 by i n c r e a s i n g the c o l l e c t o r c u r r e n t of Q203 and thus r e d u c i n g i t s R207 and R205 are p r o t e c t i o n r e s i s t o r s t o l i m i t the c u r r e n t s i n paths t h a t would o t h e r w i s e c o n t a i n o n l y f o r w a r d b i a s e d d i o d e s and t r a n s i s t o r s , w h i c h c o u l d be momentarily o v e r l o a d e d by t r a n s i e n t s . R206 serve s t o i n c r e a s e the c o l l e c t o r c u r r e n t of Q202 and improve the therm a l s t a b i l i t y of Q203. (See 3-5 and 3-6). D202 r a i s e s the b a s e - e m i t t e r v o l t a g e of Q203 as d i s c u s s e d i n 3.5. Q201, Q202j and R204 form a c o n v e n t i o n a l " l o n g - t a i l e d - p a i r " d i f f e r e n t i a l a m p l i f i e r . D201 i s the r e f e r e n c e diode s u p p l i e d w i t h c u r r e n t from the r e g u l a t e d output through R203. R202 and R201 form the sampl i n g d i v i d e r . The r e s i s t a n c e of R201 and R202 i n p a r a l l e l s h o u l d be much l e s s t h a n the i n p u t impedance of Q201 to a v o i d l o s s of g a i n . R203 i s chosen such t h a t i t passes the c o r r e c t c u r r e n t f o r zero temperature c o e f f i c i e n t t o the p a r t i c u l a r r e f e r e n c e diode used. The q u a n t i t y 1 + R202/R201 i s made equal to the d e s i r e d v a l u e of Vout/V r e f . and R204 i s a d j u s t e d t o make the base v o l t a g e of Q201 equal to the base v o l t a g e of Q202. C201 keeps the s u p p l y impedance low a t h i g h f r e q u e n c i e s and a l s o reduces the l o o p g a i n of the r e g u l a t o r t o below u n i t y a t f r e q u e n c i e s below which phase s h i f t s become e x c e s s i v e . R209 i s n e c e s s a r y to t u r n the u n i t on and reduce the power d i s s i p a t i o n i n Q204. The performance of the s u p p l y was e s t i m a t e d on the b a s i s of the f o l l o w i n g t y p i c a l p a r a m e t e r s . 37 f e ob h rb l b h oe h. l e Q204 70 3 X 10 4 X 10 1 2 X 10 70 -6 -4 Q203 +D202 50 *4 X 10 7 X 1 0 ~ 4 60 -6 Q201 and Q202 50 .2 X 1 0 " 6 mho 7 X 1 0 ~ 4 150 ohms mho ohms D201 250 ohms 6 v o l t s Prom 3-30 g Q of Q202 = 5 X 1 0 ~ 6 mho. These v a l u e s are s u b s t i t u t e d i n t o 3-26 to o b t a i n 1.2 X 10""3 mho ...3-34 To the above v a l u e of g must be added 1/820 mho due to the presence of R209 (820 ohms) to y i e l d g = 2.4 X 1 0 ~ 3 mho The impedance a t the base of Q203 i s a p p r o x i m a t e l y h f ^ h ^ + i„ h , RT f e ob L r ^ ) / ( l + h_pQ ^ T ) i n p a r a l l e l w i t h R206, t h a t i s , about 1.5K, 38 The g a i n of the a m p l i f i e r i s then RT A 1 i-t rv j-= -2 h — = 5 l b The f a c t o r \ appears due to the f a c t t h a t a s i n g l e - e n d e d output i s b e i n g t a k e n from a d i f f e r e n t i a l a m p l i f i e r . The a t t e n u a t i o n of the v o l t a g e d i v i d e r i s a = .45 ^ R R /V >. r- • , v f = = .05 « . a 6 0 r R + R 203 S u b s t i t u t i o n of these v a l u e s i n t o 3-25 y e i l d s Tq = 0.4 ohm ...3-36 and s i n c e K = gr 0 K = 10 ...3-37 3.15 The Reference Supply The d e s i g n of the r e f e r e n c e supply i s s i m i l a r t o t h a t of the r e g u l a t e d s u p p l y f o r the d.c, a m p l i f i e r . The parameter a was chosen to be \» S i m i l a r c a l c u l a t i o n s t o those of 3.14 show K £ 1 0 ~ 3 ...3-37 The r e f e r e n c e v o l t a g e output i s t a k e n d i r e c t l y from the r e f e r e n c e d i o d e . Any change i n output v o l t a g e of the r e g u l a t o r i s t h e n 39 f u r t h e r a t t e n u a t e d by | ^ * 40 X 1 0 ~ 3 * v o Thus K „ =" 40 X 1 0 " 6 ...3-38 r e i 3.16 The P r e - r e g u l a t o r s The p r e - r e g u l a t o r s are s i m p l e r t w o - t r a n s i s t o r r e g u l a t o r s u s i n g the output v o l t a g e of the main r e g u l a t o r s as r e f e r e n c e s * The v o l t a g e d i v i d e r s are i n the e m i t t e r s of the d r i v e r t r a n s i s t o r s . T h i s r e s u l t s i n the d r i v e r t r a n s i s t o r s o p e r a t i n g i n the common—base mode w i t h g _ = h The d e t a i l e d d e s i g n °o2 ob2 c a l c u l a t i o n s are g i v e n i n Appendix 2 and the c i r c u i t diagrams i n F i g u r e 3-13. 3*17 The Oven The oven i s a 6 i n c h by 3 i n c h by 5 i n c h box f a b r i c a t e d of \ i n c h t h i c k aluminium* The t o p and bottom are i n s u l a t e d w i t h about 1 i n c h of s t y r o f o a m . The s i d e s are wrapped w i t h f i b e r g l a s s tape on which i s wound a 25 ohm nichrome h e a t e r . The h e a t e r i s h e l d i n p l a c e w i t h another l a y e r of tape and o u t e r p a n e l s of f i b r e b o a r d . The f i b r e board p a n e l s a l s o p r o v i d e some the r m a l i n s u l a t i o n * The g a i n of the temperature c o n t r o l system i s s e t a t about \ the v a l u e r e q u i r e d t o s u s t a i n o s c i l l a t i o n s * The p e r i o d of the damped o s c i l l a t i o n s which occur when f i r s t 40 s w i t c h e d on i s about 6 minutes* A f t e r about 2 hours of o p e r a t i o n the temperature appears t o have s t e a d i e d down t o w i t h i n a few hundredths of a c e n t i g r a d e degree .from the e q u i l i b r i u m v a l u e . F i g u r e 3-12. Complete Schematic of D.C. A m p l i f i e r F i g u r e 3-13. Complete Schematic of Power S u p p l i e s R 301 100 ohm C301 .1 u f a R 302 see f i g . 4-1 Rr^03 47 K Q 301 2N 1303 R 304 3.3 K Q 302 2N 1305 R 305 330 Q 303 2N 1304 R 306 8.2 K Q 304 2N 1305 R 307 560 Q 305 2N 1305 R 308 1.5 K Q 306 2N 250 R 309 6.8 K R 310 2.7 K R 311 3.3 K R 312 330 R 313 270 R 314 680 R 315 5 ohm/5 watt see F i g u r e 3-7. F i g u r e 3-14. P a r t s L i s t f o r Temperature C o n t r o l l 44 4. PERFORMANCE OF EQUIPMENT 4.1 The D.C. A m p l i f i e r The a m p l i f i e r o utput was connected t o a s t r i p c h a r t r e c o r d e r and the system r u n f o r s e v e r a l days w i t h the i n p u t s h o r t e d * D u r i n g the f i r s t day a stea d y d r i f t of about 1 m i c r o v o l t per hour ( r e f e r r e d t o the i n p u t ) was obse r v e d . A f t e r the f i r s t 24 hours the d r i f t remained below + 2 m i c r o v o l t s . Source r e s i s t a n c e s up to about 1,000 ohms made no a p p r e c i a b l e d i f f e r e n c e . A source r e s i s t a n c e of 10K ohms gave r i s e t o about 6 m i c r o v o l t s p e a k — t o - pe ak of v e r y l o w - f r e q u e n c y n o i s e . 100K ohms source r e s i s t a n c e gave about 40 m i c r o v o l t s . The f r e q u e n c y response of the a m p l i f i e r was down 1 db a t 10 k c , 3 db a t 20 k c , and 6 db a t 40 k c . The broad—band n o i s e f i g u r e was found to be about 3 db w i t h a 10K ohm source impedance and about 6 db w i t h a IK ohm source impedance* The i n p u t impedance was measured as 120K w i t h the v o l t a g e g a i n s e t a t 1,000. 4.2 The Temperature R e g u l a t o r System The temperature c o n t r o l u n i t was p l a c e d i n an oven w i t h a thermometer. The temperature was s l o w l y r a i s e d w h i l e the output of the c o n t r o l u n i t was m a i n t a i n e d c o n s t a n t at 1 v o l t by v a r y i n g R302. The r e s u l t i n g d a t a i s p l o t t e d i n F i g u r e 4-1. From the sl o p e of the p l o t of R302 v s . t e m p e r a t u r e , i t was dete r m i n e d t h a t a 250 ohm change i n R302 corresponds to a change of 1°C. R302 was the n v a r i e d a t c o n s t a n t temperature and i t was found t h a t 10 ohms change i n R302 caused the output of temperature 45 46 c o n t r o l a m p l i f i e r (V^ of F i g u r e 3-7) to change 1 v o l t . I t i s thus seen t h a t |JT = 250 ohms/c° = 2 5 v o l t s / G o m e a s u r e d d i 10 ohms/V This v a l u e compares f a v o r a b l y w i t h the e s t i m a t e d v a l u e (3-17) of 30 volts/C°. The h e a t e r c u r r e n t r e q u i r e d t o r a i s e the oven temperature from 25°C to 35°C was measured and found t o be 0.6A. The h e a t e r c o n t r o l and power a m p l i f i e r can s u p p l y about 1.4 times t h i s amount of c u r r e n t or about t w i c e the power. I t i s t h e r e f o r e e x p e c t ed t h a t the oven and temperature c o n t r o l w i l l f u n c t i o n p r o p e r l y i n ambient temperatures from about 15°C to j u s t under 35°C. 4.3 Power Supply Performance The output impedance and l i n e r e g u l a t i o n f a c t o r of the r e g u l a t e d power s u p p l y were measured and compared to the p r e d i c t e d v a l u e s . A change i n l o a d c u r r e n t of 11 millamps was found to cause a change i n output v o l t a g e of 5 m i l l i v o l t s from which r Q = 0.45 ohm. The output v o l t a g e changed 6 m i l l i v o l t s f o r an i n p u t change of 5 v o l t s . The l i n e r e g u l a t i o n f a c t o r , K, i s th e n -3 1.2 X 10 . P r e d i c t e d and measured v a l u e s are compared i n t a b l e 4-1. 47 P r e d i c t e d Measured r 0,4 ohm 0.45 ohm o K 1 0 " 3 1.2 X 1 0 " 3 Table 4-1. P r e d i c t e d and Measured Parameters of the R e g u l a t e d Power Supply- S i m i l a r t e s t s performed on the r e f e r e n c e s u p p l y gave the r e s u l t s of Table 4-2. P r e d i c t e d Measured K 1 0 ~ 3 1.4 X 1 0 " 3 3 V R 4 X 1 0 ~ 2 2 X 1 0 ~ 2 d V o K .'f 40 X 1 0 " 6 < 50 X 1 0 ~ 6 Table 4-2. d V R The d i s c r e p a n c y i n ̂  y was due to the f a c t t h a t r e f e r e n c e diode used had a dynamic r e s i s t a n c e of o n l y 27 ohms i n s t e a d of the 50 ohms a n t i c i p a t e d from the d a t a s h e e t . The v a l u e o f K ' o b t a i n e d by m u l t i p l y i n g the measured * VR v a l u e s of K and r = - , namely o K „ = 28 X 1 0 ~ 6 r e f i s p r o b a b l y more a c c u r a t e t h a n the v a l u e a c t u a l l y measured, s i n c e the change i n r e f e r e n c e v o l t a g e caused by a 5 v o l t change i n i n p u t v o l t a g e was o n l y a few tim e s the t h r e s h o l d of d e t e c t a b i l i t y on the measuring equipment used. 48 4.4 P r e - r e g u l a t o r Performance I t was found p o s s i b l e t o a d j u s t the p r e - r e g u l a t o r s t o g i v e l e s s t h a n 15 m i l l i v o l t s o u tput change f o r 15 v o l t s i n p u t change^ t h a t i s , K = 10 « & ' ' p r e - r e g The o v e r a l l l i n e r e g u l a t i o n f a c t o r f o r the r e f e r e n c e system should t h e n be K r e f t o t a l = 3 0 X 1 0 " 9 The r e f e r e n c e s u p p l y and p r e - r e g u l a t o r operate a t a nominal i n p u t of 22 v o l t s and a nominal output of 5.6 v o l t s . A 10$ change i n l i n e v o l t a g e would t h e n be expected to produce about 0»07 m i c r o v o l t change i n 5.6 v o l t s or about 0.0013 ppm. A change of a.c. l i n e v o l t a g e from 90 to 135 v o l t s (+ 20$) caused a change i n r e f e r e n c e v o l t a g e of l e s s t h a n the l i m i t of r e s o l u t i o n of the measuring system used (about 2 (ov). The use of h i g h q u a l i t y s h i e l d e d l i n e i s o l a t i o n t r a n s f o r m e r s would g r e a t l y reduce the problems of 60 c y c l e c o u p l i n g when w o r k i n g w i t h h i g h impedance s o u r c e s . A v o l t a g e d i v i d e r , c o n s i s t i n g of a 1000 ohm p r e c i s i o n r e s i s t o r and a decade r e s i s t a n c e box, was p l a c e d a c r o s s the o u t - put of the r e f e r e n c e s u p p l y * A s t a n d a r d c e l l was connected i n s e r i e s - o p p o s i n g t o the v o l t a g e a c r o s s the 1000 ohm r e s i s t o r and the r e s u l t a n t v o l t a g e was f e d i n t o the a m p l i f i e r . The r e s i s t a n c e box was a d j u s t e d f o r zero output v o l t a g e from the a m p l i f i e r and the a m p l i f i e r output was connected to a r e c o r d e r . D u r i n g s e v e r a l days of o p e r a t i o n the d r i f t never exceeded 5 m i c r o v o l t s r e f e r r e d t o the i n p u t , t h a t i s , about p a r t s per m i l l i o n . 50 APPENDIX 1. LINE AND LOAD REGULATION OP REGULATED POWER SUPPLIES The r e g u l a t o r i s regard e d as a f o u r - t e r m i n a l network ( F i g u r e A l - l ) w i t h c h a r a c t e r i s t i c s d e s c r i b e d by E q u a t i o n s A l - 1 and A l - 2 . AAA Oe—>-t——-'VVV Oe— V S Q ! r s fv F i g u r e Al-1.. F o u r - T e r m i n a l Network R e p r e s e n t a t i o n of R e g u l a t o r where where d l — dV + g„dV. r o B f 1 o . . . A l - 1 3V r = — o I V. i and S4 31. ^ V i g d l — dV + g.dV. r o toi l r . . . A l - 2 3V r = -* ~ 3 1 , 51 and • i av. The l i n e r e g u l a t i o n f a c t o r , K, i s d e f i n e d by E q u a t i o n A l - 3 . K dV o d V i . . * A l - 3 I f the i n p u t v o l t a g e i s changed, b o t h the output v o l t a g e and c u r r e n t w i l l change. These changes are r e l a t e d b y j d i = ±- dV o bL o . . . A l - 4 E q u a t i o n A l - 4 i s s u b s t i t u t e d i n t o E q u a t i o n A l - 1 , and the r e s u l t i n g e q u a t i o n s o l v e d f o r K as d e f i n e d by E q u a t i o n A l - 3 K = g r RT o L f RT + r L o •«.Al-5 S i n c e a s u p p l y w e l l r e g u l a t e d a g a i n s t l o a d v a r i a t i o n s , w i l l have r R T, the a p p r o x i m a t i o n A l - 6 i s j u s t i f i e d . O Li to .* . . A l - 6 The output r e s i s t a n c e , r, i s d e f i n e d by E q u a t i o n A l - 7 . dY c d i . . . A l - 7 52 I f the output c u r r e n t i s changed, b o t h the output v o l t a g and i n p u t v o l t a g e w i l l change. The i n p u t v o l t a g e change and the i n p u t c u r r e n t change are r e l a t e d by E q u a t i o n A l - 8 . 1 d I i — = g g = - ( d e f i n i t i o n of g g and r ) „.*Al-8 s i E q u a t i o n AI—8 i s s u b s t i t u t e d i n t o E q u a t i o n AI—2 and the r e s u l t i s s o l v e d f o r dV. i n terms of dV . 1 o dV. = dV . . . A l - 9 r r ( g i + g s ) E q u a t i o n AI—9 i s s u b s t i t u t e d i n t o E q u a t i o n 1—1 and the r e s u l t i s s o l v e d f o r r u s i n g E q u a t i o n A l - 7 to y i e l d : r = r i ...Al-10 o g f r o ( g i + g s ) r r I n the r e g u l a t o r s c o n s i d e r e d here d l ^ = dI Q« Thus g^ = g = r . I f these approxima" output impedance i s found to be: and r ~ Q  pr o x i m a t i o n s are made i n AI—10, the r = r ( l + g„r ) »•» .AI—11 o tof s Sin c e g ^ r g « l i n the r e g u l a t o r s and power s u p p l i e s c o n s i d e r e d ? 53 APPENDIX 2. DERIVATION OF EQUATIONS PREDICTING POWER SUPPLY PERFORMANCE !• P r e - r e g u l a t o r R5 AAA- Q I R4 R2 \ 0 2 / R3 R7« R 6 R7 A R8 • V A r VR -o A F i g u r e 1. P r e - r e g u l a t o r F i r s t , c o n s i d e r the s i m p l i f i e d c i r c u i t of F i g u r e 2. 01 Z. *" R 7 N 02 R6 V R v 2 F i g u r e 2. S i m p l i f i e d C i r c u i t of P r e - r e g u l a t o r C a l c u l a t i o n of g' and tq the f o r c i r c u i t of F i g u r e 2. We s e t AV'2 = 0, assume A V ' R = 0 and c a l c u l a t e g*• The output c u r r e n t w i l l i n c r e a s e when i s i n c r e a s e d f o r two r e a s o n s ; (a) VQ^i i n c r e a s e s and causes a s m a l l i n c r e a s e i n c o l l e c t o r c u r r e n t of Q l . (b) ^CB2 ^ n c r e a s e s a n d causes a s m a l l i n c r e a s e i n c o l l e c t o r c u r r e n t of Q2 which i s m u l t i p l i e d by the c u r r e n t g a i n of Q l . When a t r a n s i s t o r i s o p e r a t e d w i t h non-zero impedance i n both e m i t t e r and base c i r c u i t s , i t cannot, s t r i c t l y s p e a k i n g , be s a i d t o be o p e r a t i n g e i t h e r common-base or common-emitter. How- ev e r , i f the e m i t t e r — c i r c u i t impedance i s much l a r g e r t h a n the base c i r c u i t impedance d i v i d e d by h ^ e , the c i r c u i t may be re g a r d e d as common-base w i t h n e g l i g i b l e e r r o r . S i m i l a r i l y , i f the e m i t t e r - c i r c u i t impedance i s much l a r g e r t h a n the base- c i r c u i t impedance d i v i d e d by hfes> the c i r c u i t may be s a i d to approximate a common—emitter. The v o l t a g e r e f e r e n c e , V R , w i l l n o r m a l l y be a l o w - impedance s o u r c e . Thus Q2 may be regard e d as a common—base c o n f i g u r a t i o n and i t s output conductance w i l l be h^^* The b a s e - c i r c u i t impedance f o r Ql i s the output impedance of Q2, which i s ̂ / n G> J2* ^ n e e m i " t " t e r - c i r c u i t impedance seen by Ql i s the i n t e r n a l impedance of the power s o u r c e . ^)•0]:)2, ^ o r ^ e s u p p l i e s c o n s i d e r e d here w i l l be ^ 1 micro mho. I f we assume h^.^ — .100, Ql w i l l p e r f o r m as a common-emitter as l o n g as r « 10K ohms which w i l l c e r t a i n l y be the case f o r a source J r e a s o i i ab l e t r a n s f o r m e r — r e c t i f i e r - f i l t e r system. T h e r e f o r e the 55 out p u t admittance of Ql w i l l be b Q e ^ , Thus: d I = h o e l d V l + h f e l h o b 2 d V l — A 2 " 1 = h f e l ( h o b l + h o b 2 > d V l J h t e l ^ ^ b ^ 1 ' = h f e l f i L o b l + h o b 2 ) - ' k 2 ~ 2 I f we now assume a change i n output v o l t a g e AV 2, w i t h h e l d c o n s t a n t , we can c a l c u l a t e AI and thus r . The j u n c t i o n ° R 6 of B7 and R6 may be regard e d as a source of v o l t a g e , V" = =,—•—=— z 5 7 w i t h i n t e r n a l impedance, R' = p p d r i v i n g the e m i t t e r of 6 7 Q2.. AV" 2 AI„ 0 = ...A2-3 R' + h., 0 i b 2 6 A V R 6 + R 7 2 R,R 7 6 7 h R 6+ R ? AV 1 R 7 + h i b 2 R 7 R 6 56 AI c2 / h f b 2 / A I E 2 =• A I E 2 ...A2-4 AI f e l c2 ..A2-5 h f e l A Y ' 2 R 7 + h i b 2 + IT> 6 R 7 R + h (1 + ^L) AV' 7 l b 2 R 6 ••»A2-6 AI f e l K ' = = < hobl + hob2> | R 7 + W 1 + 5^> ] *.*A2—7 R e t u r n i n g to the c i r c u i t of F i g u r e 1, we s e t R^t R^ and Rr- - oo and examine the e f f e c t of R c. 5 o V - V - IR V2 _ V 2 a£ .*.A2-8 V , R - V R + I R 8 • •.•A2-9 From A2-8 ^ i ^ i dV R' + Rc 3" d i "8 R .•.A2-10 and from A2-9 DVR< d l - tt8 ...A2-11 combining A2-10 and A2-11 we have r = r V R - 1) : * V , 2 VR« A2-12 I f I i s h e l d c o n s t a n t , I must remain c o n s t a n t and t h e r e f o r e I g ^ -tB2 a n d VBE2 r e m a i n c o n s t a n t . Summing c u r r e n t s a t the j u n c t i o n o f R̂  and R̂  g i v e s Vt - (v.t _ v ) R, R 'BE2 y T _ V'R ~ VBE2 + ± E 2 - 7 ,*A2-13 Taking d i f f e r e n t i a l s i n A2-13 w i t h I = c o n s t a n t y i e l d s dv-«2 - dV-t.R d V R I R, 7 R, d V * 2 = D V*R R7 (RT + fc) = d V 6 7 R£ + R.-7 ! 6 7 R R̂- .•A2-14 58 thus + R~ — '- •..A2-15 R 6 ; S u b s t i t u t i n g A2-15 i n t o A2-12 y i e l d s R r = R ' — Ro Tr- ...A2-16 O O o xtg L e t us now examine the e f f e c t of R^, R 2 and R^* I f R-̂ and R 2 are chosen such t h a t a t the nominal v a l u e o f R l V, (nom) ± = V„ 0(nom) •**A2-17 1 V R 2 t h e n the c u r r e n t t h r o u g h R^ i s zero a t nominal s u p p l y v o l t a g e . I f i n c r e a s e s ( a t c o n s t a n t I ) c u r r e n t w i l l f l o w t h r o u g h R^ and cause V g 2 to r i s e thus t e n d i n g t o reduce V-g-g-p and t h e r e f o r e ^E2* ^ h ^ s w i l l * i n turn-, tend t o reduce I . I f the v a l u e s are chosen c o r r e c t l y i t s h o u l d be p o s s i b l e t o j u s t o f f s e t the tendency of I t o i n c r e a s e w i t h i n c r e a s i n g V^. The c u r r e n t t h r o u g h R 3 i s AV^/R* A V t l = A V 1 R ^ + R 2 - * A 2 - 1 8 AT i = v i - Vnom) R , 3 = ( R 1 / / R 2 + R 3 + h i b 2 / / R 6 ) T h i s c u r r e n t causes A I c 2 = | h f b 2 | A I E 2 = + A I E 2 = " ZR3 — A 2 - 1 9 R l h f e l AI = h„ , AI „ = -h„ n I„„ = -AV, f e l c2 ~ f e l R3 ~ 1 R±+ R 2 R* 3 •••A2-20 5 I _ R l h f e l _ „ dY± - - R l + R 2 g'T t g' R^ i s n e c e s s a r y t o s t a r t the s u p p l y s i n c e w i t h Ql non- c o n d u c t i n g t h e r e i s no power to the r e f e r e n c e s e c t i o n * R,- lowe r s g t o (g* + ^— ) + g" and R^ improves the t h e r m a l 5 s t a b i l i t y of Ql but l o w e r s the e f f e c t i v e v a l u e of ^ to R 4 APPENDIX 3o DERIVATION OP THE EACT PARAMETERS FOR AN EMITTER- FOLLOWER I t has been found d e s i r a b l e t o work i n terms of "mixed h-parameters", i*e<» n £ e * n r b » n 0t> s i n c e "these are u s u a l l y s p e c i f i e d and and h Q b are q u i t e p r e d i c t a b l e . The i m p o r t a n t mid-band parameters of an a m p l i f i e r are i t s v o l t a g e g a i n , A y» i t s c u r r e n t g a i n , A — , i t s i n p u t impedance, , and i t s output impedance, Z » out F i g u r e A3-1. E m i t t e r - F o l l o w e r I n terms of common c o l l e c t o r h-parameters v o l t a g e g a i n = A. vc h. 1 + h Z T , I C v oc L\ h r c ~ "ZT ( _ ) L h, f c •*..A3-1 c u r r e n t g a i n = A- = s i c f c 1 + h Z T oc L ...A3-2 61 output impedance = Z Q C = — »».A3-4 h„ h h _ f c r c I C g where Z T = l o a d impedance, and Z = g e n e r a t o r impedance. g These w i l l be c o n v e r t e d t o mixed h-parameters. ^h„, (1 - h J - h ,h., h f e = _ : thi o b _ i b ...A3-5 (1 + h f b ) ( l - h r b ) + h o b h . b 1 - h 1 + h f e = rb (1 + h f b ) ( l - h r b ) + h o b h . b ««*A3-6 ^ + W*1 ~ h r b } + h o b h i b = ~ ~ ~ ~ — A 3 " 7 1 + h f e , h ., h .., 1 + h f b = — i - 2 k ^ b _ i A 3 _ 8 1 1 +' h. 1 - h ' i e rb N - . . = — ...A3-9 (1 + h f b ) ( l - h r b ) + h o b h . b S u b s t i t u t i n g A3-7 i n t o A3-9 y i e l d s 1 + h f h i c = h . b — e- M . A 3 - 1 0 ^ ^"rb 62 1 + hp, h „ = «..A3-11 r c ^ + V ( l - h r h ) + h o b h i b S u b s t i t u t e A3-7 and A3—8 i n t o A3-11 to o b t a i n , h ,h., 1 + h„ b ^ : 1 _ ob i b ̂  , f e ^ " 1 + h f e 1 ~ h r b 1 ~ h r b i 1 + h-pa — — h o b > W ,*.A3-12 1 - h , 1 - h , rb rb h f c = - ( 1 + h f e ) .*.A3-13 h Q C = ^ = ...A3-14 (1 + h f b ) ( l - h r b ) . + h Q b h i b S u b s t i t u t e A3-7 i n t o A3-14 t o o b t a i n 1 + h h f e oc 1 - h ob •••A3-15 rb S u b s t i t u t i n g A3—10* A3-?129 A3-13, and A3-15 i n t o A3-1 g i v e s A 1-h 1 - ( l + h f e ) oh i b rb 1 - h rb h., l+h'„ i b f e Z L X - h r b 1+h 1 + f e 1-h h o b Z L rb 1 + h f e • * *A3-16 63 whi c h s i m p l i f i e s t o 1 - h r b V = T T T — • • • A 3 - 1 7 S u b s t i t u t e A3—13 and A3-15 i n t o A3-2 to get 1 + h f e A. = - — ...A3-18 l+h„ 1 + T T ^ h o b Z L ^ r b S u b s t i t u t i n g A3-10, A3-13 and A3-15 i n t o A3-3 g i v e s ( l + h f ) 1+h ( 1 + h f e ) 1 _ h r b Z. = h., — + — :— ,..A3-19 I C xb -, , n , 1, 1 - 1 : 1 T. -i . 1 rb r b l+b„ 1 + — h , Z, 1-h v o b L rb w hich s i m p l i f i e s t o l+h„ + Z T f e l b L ...A3-20 i c 1-h , , rb 1+h,, 1 + — h , Z T 1-h 0 rb S u b s t i t u t i n g A3-10, A3-12, A3-13 and A3-15 i n t o A3-4 g i v e s oc ...A3-21 1+h l ^ h rb 1+h (1 - 1+h f e h o b + 1-h f e 1-h rb h i b h o b > rb 1+h f e 1-h rb h., + Z i b i 64 S i m p l i f y i n g , 1-h h., + -, ,, r b Z l b 1 + n f e « 2 — • • • • » o o A 3 ~* 2 2 OC • 1 + h , Z ob g I n summary; 1-h rb 1+h vc ..A3-17 f e 1 + l b ' i c 1+h 1 + f e 1-h rb h vZ T ob L ...A3-18 1+h Z. i c f e h., + ZT l b L 1-h ...A3-20 rb 1+h 1 + f e 1-h rb ob L h., + y-T-^ Z i b l + h f e g oc ..A3-22 1+h , Z ob g APPENDIX 4. DERIVATION OF THE EXACT PARAMETERS FOR A COMMON EMITTER AMPLIFIER •AAV—» Zg <• out in - • o F i g , A 4 - l 0 Common—Emitter A m p l i f i e r P u r s u i n g the l i n e of r e a s o n i n g used i n Appendix 3 we s t a r t from the e q u a t i o n s v o l t a g e g a i n , A y E h. 1+h Z T l e oe L h re f e a o A. 4 c u r r e n t g a i n , Aj-g = f e 1+h. Z T oe L o. A4 h„ h Z T 1 e re L i n p u t impedance, Z j E = t u g - Y+h—Z— oe L output impedance, Z OE h„ h fe re oe h. +Z l e g h 9 o A.4- 66 where Z-̂  = l o a d impedance and Z^ = g e n e r a t o r impedance 1+h h h. = h f e i e i c i b 1-h rb from Appendix 3 ..»A4-5 h = 1-h = -z—^-re r c 1-h rb l + h f 1 - h ^ o b ^ i b ^ r b rb. u s i n g h ^ c from Appendix 3 -A4-6 1+h fe h = h = n i . ^ - u oe oc 1-h , ob rb h , from Appendix 3, .oA4-7 h ^ e i s used as i s e E q u a t i o n s A4-5, A4-r6, and A4-7 are s u b s t i t u t e d i n t o A4-1, g i v i n g L-hrb 1+h f e h vh..-h 1—h ^ pb i b rb 1 + ± e h 7 1+h. h., 1-h , ob L f e i b rb 1-h , Z T rb L ,A4-8 S i m p l i f y i n g we have h f e 1-h rb *VE i b 1+h f e 1 + ( h f e h rb h ob 1+h f e i b 1-h )Z T rb .A4-9 67 E q u a t i o n 7 i s s u b s t i t u t e d i n t o e q u a t i o n 2 g i v i n g A = tS. ...A4-10 ^ r b E q u a t i o n s 5, 6, and 7 are s u b s t i t u t e d i n t o e q u a t i o n 3 l + h f e - . i ^ f e ' l ^ T faobhib-"rb'ZL ^ I E ~ i b 1-h , " 1-h , rb r b -. , 1 + h f e 1-h , ob L rb S i m p l i f y i n g 11!. y i e l d s ..*A4-11 h~ h , h , 1 + 4+h- h . K + 1-h , J Z L ; t± h . » — ...*A4-12 'IE - l - h r b \ b 1 + h 1 + I Z h T h o b Z L rb E q u a t i o n s 5* 6* and 7 are s u b s t i t u t e d i n t o e q u a t i o n 4, to g i v e , Z 0 E " !+ bfe h f e ^ . h h - h _ ! £ h _ - I f - i - h , W i b " h r b l - h r b ob l - h r b r b _ 1 + h f e -, u i e h.,+ Z l - h r b i b g «<> *A4-13 68 S i m p l i f i e d 13 y i e l d s 1-h , Z r b OE ~ h ^ h„ h o b + <hob + -fj 1+h- h., 1 + _ i b + 1-h , Z rb g .•.A4-14 AI so, 1 _ ob Z n„ ~ g 0 E 1-h , OE rb + * r h o b h f e 'fe 1-h rb l + h f e  h i b h , h ,h„ rb \, i / ob f e ) + 2 \ '1-h rb 1+h f e h 1-h •f e rb rb 1+h f e h »>*A4-15 i b 1+h 1 + f e 1-h rb ^ i b Z g APPENDIX 5. DERIVATION OP THE EXACT PARAMETERS FOR A COMMON- EMITTER AMPLIFIER WITH EXTERNAL EMITTER-IMPEDANCE. -AAA—o— A II o F i g u r e A5—1* Common-Emitter A m p l i f i e r w i t h E x t e r n a l Emitter-Impedance The. a m p l i f i e r of F i g u r e A5-1 d i f f e r s from a common—emitte a m p l i f i e r o n l y i n h a v i n g an e x t e r n a l emitter-impedance* There- f o r e i f a s e t of h—parameters (the h'-parameters) i s found f o r a d e v i c e c o n s i s t i n g of a t r a n s i s t o r w i t h a r e s i s t o r i n the e m i t t e r l e a d , the h*—parameters may be used w i t h the r e s u l t s of Appendix 4 t o p r e d i c t the performance of the a m p l i f i e r . h'-parameters e e "ib 'e o - —A A A — o — A A A — > F i g u r e A5—2; S m a l l — s i g n a l H y b r i d E q u i v a l e n t C i r c u i t of Common- Base T r a n s i s t o r w i t h E x t e r n a l Emitter-Impedance 70 Prom F i g u r e A5—2 i t i s c l e a r t h a t h* = h., + Z„ »..A5-1 l b l b E h* = h , •••A5-2 rb rb h ' f b = h f b /.*.AM h , o b = h o b — A 5 - 4 I t o n l y remains to determine h ' ^ e t o complete the s e t of mixed h-parameters* I f i n F i g u r e A5-1 i s reduced t o zero an e m i t t e r — f o l l o w e r r e s u l t s , w i t h l o a d Z-g* The c u r r e n t g a i n of t h i s e m i t t e r - f o l l o w e r i s j u s t h*-. = — (l+h' £ e ) o From A3—18 i n Appendix 3 h» = - ^ ••••A5-5 1 + h f e 1 + I ^ T T h o b Z E rb T h e r e f o r e , f e 1 + h f e 1 + h f e 1-h , ob E rb ••»A5-6 1+h f e h f e ™ 1-h , h o b Z E rb 1+h- 1-h , ob E rb 71 I n E q u a t i o n s A4—9, A4-10, A4-12 and A4-14, h ^ e appears i n the forms h f g , h f e / ( l + h f e ) and ( l + h f e ) / ( l - h r b ) . Hence, h* „ h„ h , Z-n, f e f e ob E AC- 7 1+h* - l+h„ " 1-h , *».A>W f e f e rb and, 1+h 1 f e rb 1+h 1-h rb f e (I+h f e > h o b Z E ...A5-8 A m p l i f i e r Performance I f E q u a t i o n s ( l ) t h r o u g h (4) and (6) through (8) are s u b s t i t u t e d i n t o E q u a t i o n A4—9 of Appendix 4, the r e s u l t may be ex p r e s s e d as shown i n E q u a t i o n ( 9 ) . h„ h ,Z„ 1-h , AT r / f e ob E \ rb Q VE = ~ G ( I T h — - T=h^ > 1 + 7 - ***A5-9 where G = Z L / Z E t and h., h„ h , h . , h , i b , n, „ , n 1 f e r b , i b ob \ g =; -g- + Gh z + G ( j ^ — - + -T-hT ) E f e rb I t i s noteworthy t h a t g may be m i n i m i z e d f o r g i v e n v a l u e s of G and h—parameters by a p r o p e r c h o i c e of Zg t h u s , 72 d Z E Z E o b S e t t i n g Bg/dZg = 0 and d e s i g n a t i n g the v a l u e of Zg which r e s u l t s Z-gQ y i e l d s h i b Z E 0 = / O h ^ - ***A5-11 and h., Z-r,n Z-n h„ h , h.,h , xb / EG , E \ . r i f e rb , l b o b x , 0 Z E 0 Z E Z E 0 i + h f e r b I f E q u a t i o n s ( l ) t h r o u g h (4) and (6) t h r o u g h (8) are s u b s t i t u t e d i n t o E q u a t i o n A4^12 of Appendix 4, the r e s u l t may be e x p r e s s e d as shown i n ( 1 3 ) . Z ' l E = ( l + h f e ) Z E • i ...A5-13 i - h r b + ( i + G ) ( i + h f e ) h o b z E 1±ZL ( l - h r b ) + (l+G) h d + h f e ) z E • o b where g and G are as p r e v i o u s l y d e f i n e d . S i m i l a r l y A' T i s 73 g i v e n by E q u a t i o n (14) a s , 11 ( ^ r b ) h f e - ( l + h f e ) h Q B Z E I E - l - h r b + U . + G M l + h f e ) h o b Z E - •••A5-14 and Z'QJ, by E q u a t i o n ( l 5 ) > 1-h rb OE h + (h + ^ ( l " h r b ) h f e - < I + h f e > h o b Z E W ( h o b + Ty h T - T z g i - h r b + ( i + h f e ) ( h o b z E + fe- i b _ E> ...A5-15 APPENDIX 6. DERIVATION OF THE PARAMETERS OF THE COMPOUND STAGE USED IN THE DIFFERENTIAL AMPLIFIER I f two t r a n s i s t o r s are connected as shown i n F i g u r e A6—1, t h e y can be r e g a r d e d as one d e v i c e , A s e t of h—parameters (the h M — p a r a m e t e r s ) may be d e r i v e d i n terms of the h-parameters of the i n d i v i d u a l t r a n s i s t o r s t o d e s c r i b e the o p e r a t i o n of the compound p a i r * F i g u r e A6-1* Two-^Transistor Compound i n the Common—Emitter C o n f i g u r a t i o n The commons-emitter h"—parameters are d e f i n e d by E q u a t i o n s ( l ) and ( 2 ) * v " = h". i " + h" v" .**A6-1 b l e b re c h" i " + h" v" ..*A6-2 f e b oe c Ql i s connected i n the common-emitter c o n f i g u r a t i o n and can be d e s c r i b e d by E q u a t i o n s (3) and ( 4 ) , v r t, = h. , i " + h .v . b i e l b r e l c l •••A6-3 i , = h„ ,i'\ + h ,v , •*«A6-4 c l f e l b o e l c l Q2 i s connected i n the common-collector mode and i s t h e r e f o r e d e s c r i b e d by E q u a t i o n s (5) and ( 6 ) . • v , = h. ^ i , ^ + h -v" = -h. _ i , + h ~v" c l i c 2 b2 r c 2 c i c 2 c l r c 2 c .••A6-5 i " = h„ 0 i , 0 + h _v" = -h„ 0 i , + h 0 v " c f c 2 b2 oc2 c f c 2 c l r c 2 c S o l v i n g E q u a t i o n (3) f o r v ^ g i v e s V L = I T T v " b - h ^ 7 i M b •••A6-7 r e l r e l whic h when s u b s t i t u t e d i n t o E q u a t i o n (4) g i v e s the r e s u l t i - (h - h i e l h ° e l ) i " + h ° e l v» c l ~ ^ n f e l h , ; 1 b + h , v b r e l r e l ...A6-8 E q u a t i o n s (7) and (8) are combined w i t h E q u a t i o n (5) and the r e s u l t i s s o l v e d f o r v", . b /, b f e l b " r e l ' ' 1 i c 2 \ . ,. , b r e l b ' r c 2 i e l 1+h -, h. 0 ' b 1+h -, h. ~ o e l i c 2 o e l i c 2 . A6-9 76 E q u a t i o n (9) i s s u b s t i t u t e d i n t o E q u a t i o n (8) t o y i e l d , h f e l h o e l h r c 2 ^ 1 = 1+h A. , l"b + 1+h , V " c — a 6 " 1 0 o e l i c 2 o e l i c 2 E q u a t i o n (10) i s s u b s t i t u t e d i n t o E q u a t i o n (6) t o g i v e , i« _ _ h f c 2 h f e l .„ , h o e l h r c 2 h f c 2 x „ c - 1+h n h . 0 1 b + v o c 2 1+h v h . . ; v c o e l i c 2 o e l xc2 .•.A6-11 I f E q u a t i o n s (9) and (11) are now compared w i t h E q u a t i o n s ( l ) and ( 2 ) , i t i s c l e a r t h a t , h f e l h r e l h i c 2 h". = h. , - , , f f ...A6-12 l e l e i 1 +h , h. _ o e l i c 2 h " r e - l ^ h 2 ' "-A6-13 o e l i c 2 h " f e = l + h f 6 ' h f C ' ..»A6-14 o e l i c 2 h ,h nh„ ^ I I o e l r c 2 f c2 k , , K n ' Q = h - r—T- r . . .A6-15 oe oc2 1+h , h. o e l i c 2 These e q u a t i o n s are now c o n v e r t e d t o g i v e the h"^-parameters i n terms of the mixed h-parameters by s u b s t i t u t i n g (see Appendices 3 and 4)s 1 + h f e h. = h. = h., ., , X t f ...A6-16 l e i c i b 1-h , rb 1 l + h f h = 1-h = -T~~T-— (^—T h ,h.,—h , ) re r c 1—h , v l — h , ob i b r b ' rb r b 77 1+h h = h oe oc f e 1-h rb lob where h f c = - ( l + h f e ) These s u b s t i t u t i o n s y i e l d the r e s u l t s h l e = B h i b l 1+h 1 ° h r b 2 A , , h o b l h i b l h f e l + h i b 2 (I=E + f e 2 r b l 1+h f e l ,•.A6-17 r e v 1-h h o b l h i b l r b l r b l 1+h f e l h o b 2 h i b 2 . 1+h fe 2 1-h rb2 *.A6-18 h f e h f e l ( l + h f e 2 } 1+h 1 + f e l 1+h f e 2 1-h r b l 1-h rb2 h o b l h i b 2 ...A6-19 h " = B ( h . , + t-T^ oe x o b i l+h„ -| i e l hob2> ...A6-20 1+h f e l 1+h f e 2 1-h B = r b l 1-h rb2 1+h 1 + f e l 1+h f e 2 1-h r b l 1-h rb2 h o b l h i b 2 .A6—21 I t i s u s u a l l y the case t h a t h ^ g ^ 1 and h b << 1* Making the a p p r o x i m a t i o n s l + h f e = h ^ g and 1—'h- ̂  = 1 i n E q u a t i o n s (16) t h r o u g h (21) one o b t a i n s . 78 and h h. rJ i e i b h f e •..A6-22 h ob h f e h h h. t\j oe i e rb 'fe re h". = h % i e f e L i b l l f e 2 + h i b 2 ( h o b l h i b l + h r b l } J ••••A6-23 h h » r e * h»,„ ( h ^ h r b l f e v l l o b l i X i b l ~ h f e i / V h f e 2 " o b 2 i b 2 ' h» = h" (h , , oe f e o b i n + ^ 2 ) f e l h" % h f e l h f e 2 f e l + h f e I h f e 2 h o b l h . b 2 E q u a t i o n s (23) are s u b s t i t u t e d i n t o E q u a t i o n s (22) h " i b = hft + h i b 2 ( h o b l h i b l + h r b l > «A6-24 i e 2 h " r b = h * b l + h i b l h o b 2 + h " f e h i b l h i b 2 ^ o b l ^ ob2 ob o b i h f e l h f e l h f e 2 f e l + h f e l h f e 2 h o b l h i b 2 REFERENCES Johnson, S . D * a n d S i n g e r , J.R., Rev. S c i . I n s t * 29* 1026, (1958). G a r v i n , R,L.», Rev. S c i . I n s t . 29, 223, (1958) The E n g i n e e r i n g S t a f f of Texas I n s t r u m e n t s L t d . , T r a n s i s t o r C i r c u i t D e s i g n , New Yo r k , T o r o n t o , London, M c G r a w - H i l l Book Company I n c . , 1963, pages 96-98.

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