Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Volume 16 issue 2 1965 [doi 10.1007_bf015876 - 302 Kurze Mitteilungen Brief Reports

Zeitschrift für Angewandte Mathematik und Physik (ZAMP) Volume 16 issue 2 1965 [doi 10.1007_bf015876

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Unformatted text preview: 302 Kurze Mitteilungen - Brief Reports - Communications braves ZAMP Differential Amplifier with Transistors B y STANKO TURK and LEo BUI)IN, E l e c t r o t e c h n i c a l F a c u l t y , U n i v e r s i t y of Zagreb and I n s t i t u t e ' R u d e r Bo~kovi6', Zagreb, Yugoslavia Introduction Differential amplifiers h a v e a h i g h gain for i n p u t signals low gain for i n p u t signals of t h e s a m e phase. As it is well a m p l i f y a small s y m m e t r i c a l signal in t h e presence of h i g h The differential m o d e gain Ad of t h e circuit shown in Figure gain of one transistor ill the c o m m o n e m i t t e r configuration. of t h e c o m m o n e m i t t e r resistance in Figure 1 and it is Azl -- v ~ -- v~ v i i -- vi~ -- of the opposite phase and a known, this enables one to a s y m m e t r i c a l disturbances. 1 is t h e same as the voltage The gain AA is i n d e p e n d e n t h~l (h2~ + 1 / R L ) ( R + hzl) -- hl~ hzz " (1) +Vcc +Vcc RL Rc Vol Vo2 4 th o '.21~ -V~E "VEE Figure 1 The differential amplifier with transistors. Figure 2 The equivalent circuit. The c o m m o n m o d e discrimination factor is defined as t7 = (vol~ \ Vi 1 / (Vii = -- Viz ) (Vol t AA - B " (2) vii I (vii = + v i z ) To calculate B in (2), one m a y consider, using t h e bisection theorem, t h e reduced circuit shown in F i g u r e 2, where the e m i t t e r resistance is equal to 2 R~. F r o m this circuit one readily obtains RL (2 RE h~ -- h~l) B 2RE (l+h~x--h]~+h22 where A = hxz hz~ -- hl~ h~l. RL+ Rh~2 ) +A (RL+ 2RE) + R(h~RL+ 1) + h i 1 ' (3) Vol. 16, 1965 Kurze Mitteilungen - Brief Reports - Communications br~ves 303 T h e d e n o m i n a t o r B will b e e q u a l t o zero w h e n RE h21 2 h~2 -- REo~. (4) H e n c e , for R ~ = REoo t h e d i s c r i m i n a t i o n f a c t o r F b e c o m e s i n f i n i t e a n d will b e n e g a t i v e :for t h e v a l u e s RE > REoo. T h e p l o t of F v e r s u s RE is s h o w n i n F i g u r e 3. [ \ 2...2~ h 2h2z Ri S h2-~RL[1* Ah*1~1-hlz")'h22(Rt+R) Figure 3 T h e d i s c r i m i n a t i o n f a c t o r F as a f u n c t i o n of R E. D i f f e r e n t i a l a m p l i f i e r s w i t h t u b e s h a v e t h e s a m e c h a r a c t e r i s t i c b e h a v i o u r , if t h e e x i s t e n c e of t h e grid c u r r e n t - a l t h o u g h t h i s c u r r e n t is s m a l l - is n o t neglected. Since t u b e s h a v e a v e r y h i g h c u r r e n t a m p l i f i c a t i o n factor, t h e v a l u e of REoo will b e of t h e o r d e r o f m a g n i t u d e of 109 o h m s . T h e circuits w i t h t r a n s i s t o r s h a v e REoo a p p r o x i m a t e l y 10 ~ o h m s . Also, b e c a u s e t h e ' e q u i v a l e n t h21' for t u b e s is e x t r e m e l y large, t h e a d e q u a t e f o r m u l a s a r e m u c h simpler. To h a v e a h i g h e m i t t e r r e s i s t a n c e RE c o m b i n e d w i t h a low v o l t a g e drop, a t r a n s i s t o r is o f t e n u s e d as a ' c u r r e n t source' as s h o w n i n F i g u r e 4. T h e d y n a m i c r e s i s t a n c e of t h i s c i r c u i t is :Yd 1 (1 hl~ h2z/h2~ [ + (h~l -- his ) / h ~ -- REE (5) w h e r e RB = ( R 1 II R2)I f t h e r e s i s t a n c e s RB, REE a n d hll a r e n e a r l y of t h e s a m e m a g n i t u d e t h e a p p r o x i m a t e v a l u e of t h e d y n a m i c r e s i s t a n c e is r d ,m - 3 h~l h22 (6) 304 Kurze Mitteilungen - Brief Reports - Communications br~ves ZAMP REE -vE Figure 4 The transistor current source. T h i s v a l u e is less t h a n REoo. If REE is v e r y large, t h e d y n a m i c r e s i s t a n c e will b e a p p r o x i m a t e l y h,a/h22, w h i c h is g r e a t e r t h a n REoo. A f u r t h e r r e m a r k seems t o b e necessary. I n m a n y d i f f e r e n t i a l a m p l i f i e r s a f e e d b a c k of t h e c o m m o n m o d e signal t o t h e b a s e of t h e c u r r e n t source t r a n s i s t o r in F i g u r e 4 is used in o r d e r t o i m p r o v e t h e d i s c r i m i n a t i o n f a c t o r F. Since for RE > REoo t h e sign of F reverses, t h e circuit m a y b e c o m e i n s t a b l e . Experimental Results M e a s u r e m e n t s of t h e c o m m o n m o d e d i s c r i m i n a t i o n f a c t o r are m a d e o n a m o d e l w i t h t w o 2 N 7 0 8 t r a n s i s t o r s . Since t h e c h a r a c t e r i s t i c s of t h e s e t r a n s i s t o r s are n o t equal, t h e d i s c r i m i n a t i o n factors were d e t e r m i n e d as (vol) F1 = \ vii ] (vil= Vi I / ( V i l = (%2_] -- vi2) Yi2 ) , a n d as F2= \ vi~ / ( v i 2 = - vil) \ Vi 2 ! (Vi2 = Y i l ) B o t h v a l u e s of t h e d i s c r i m i n a t i o n f a c t o r s differ. T h e i r p l o t is s h o w n in F i g u r e 5 a n d n e e d s s o m e e x p l a n a t i o n . I n a s y m m e t r i c a l d i f f e r e n t i a l a m p l i f i e r e a c h t r a n s i s t o r sees t h e r e s i s t a n c e 2 RE ill its e m i t t e r , if v i i ~ vi~. If t h e t r a n s i s t o r s are n o t equal, e a c h t r a n s i s t o r for v i l ~ v i z sees t h e p a r a l l e l c o m b i n a t i o n of RE a n d t h e e m i t t e r i n p u t r e s i s t a n c e of t h e o t h e r t r a n s i s t o r as t h e a c t u a l e m i t t e r resistance. T h i s e m i t t e r i n p u t r e s i s t a n c e is (B/A + B'/A') R ~ = -- R E B / A + R E ( C / A -- C ' / A ' ) ' (7) where A = l+h22RL+h2x, A'~I+h~RL+h~I B = h~x + R + R L (h~z R + Ah) , B" ~ h'H + R ' + R L (hgz R ' + A'n) , C :A;~--has+h~R, C" ~ A ~ - - h ' 1 2 + h ~ R . P a r a m e t e r s "h' b e l o n g t o t h e t r a n s i s t o r for w h i c h t h e d i s c r i m i n a t i o n f a c t o r is b e i n g d e t e r m i n e d , w h i l e p a r a m e t e r s ' h " b e l o n g t o t h e o t h e r t r a n s i s t o r . T h e m a x i m a of t h e d i s c r i m i n a t i o n f a c t o r s were t h e r e f o r e o b t a i n e d for t w o d i f f e r e n t values of RE. I f t h e t r a n s i s t o r s were identical, b o t h e m i t t e r i n p u t r e s i s t a n c e s w o u l d b e e q u a l to -- 2 RE. T h e p a r a l l e l c o n n e c t i o n of t h i s e m i t t e r i n p u t r e s i s t a n c e a n d t h e a c t u a l e m i t t e r r e s i s t a n c e R E g i v e s 2 RE. T h i s is t h e r e a s o n w h y t h e q u a n t i t y 2 RE a p p e a r s in t h e d e n o m i n a t o r of (2). Vol. 16, 1965 Kurze Mitteilungen - Brief Reports - Communications br~ves 305 % t~ u % 2 g u iS "'~101 2 3 /, 56789102 2 Emiter resistonce 3 " S 6 78910~ RE [k~] 2.103 Figure 5 The experimental results. Conclusions Generally considered, t h e p r o p e r t i e s of t h e differential amplifier w i t h t r a n s i s t o r s are v e r y close to t h o s e of t h e differential amplifier w i t h v a c u u m tubes. The m o d i f i c a t i o n in i n p u t a n d o u t p u t i m p e d a n c e s a n d a g r e a t d e p e n d e n c e on t h e t e m p e r a t u r e has b e e n e x p e c t e d . I n fact, t h e g r e a t e s t difference is in t h e b e h a v i o u r of t h e d i s c r i m i n a t i o n factor. I n a real a m p l i f i e r t h e p a r a m e t e r s of t h e t r a n s i s t o r s are n o t identical, a n d as a conseq u e n c e a talse s y m m e t r i c a l c o m p o n e n t a p p e a r s a t t h e o u t p u t in s p i t e of equal i n p u t s . T h e choice of g r e a t e r r e s i s t a n c e RE will n o t a l w a y s lead to b e t t e r results because t h e region w h e r e t w o d i s c r i m i n a t i o n f a c t o r s h a v e t h e o p p o s i t e sign can be reached. I n a way, b e s t results c a n be e x p e c t e d w i t h R~ for w h i c h one of t h e d i s c r i m i n a t i o n factors b e c o m e s intinite. U n f o r t u n a t e l y , t h i s p o i n t should be a v o i d e d b e c a u s e t h e t r a n s i s t o r p a r a m e t e r s a n d t h e r e f o r e b o t h t h e p o s i t i o n of t h e m a x i m u m d i s c r i m i n a t i o n f a c t o r a n d t h e sign of t h e false s y m m e t r i c a l c o m p o n e n t in t h e o u t p u t are t e m p e r a t u r e d e p e n d e n t . I n f e e d b a c k s y s t e m s t h i s m a y i n t r o d u c e instabilities. Acknowledgement T h e a u t h o r s are i n d e b t e d t o Prof. E. BALOINGER for m a n y suggestions c o n c e r n i n g this paper. ZAMP I6/20 306 Kurze Mitteilungen - Brief Reports - Communications br~ves ZAMP REFERENCES E. J . ANGELO, JR., Electronic Circuits (McGraw Hill B o o k Co. Inc., N e w Y o r k 1958). R. I). MIDDLEBROOK, Di[/erential A m p l i / i e r s ( J o h n W i l e y a n d Sons, Inc., N e w Y o r k 1963). D. W . SLAUGHTER, The E m i t t e r - C o u p l e d Di[[erential A m p l i ] i e r , I R E T r a n s . o n Circuit T h e o r y CT-3, 51, M a r c h (1956). J . H . SIMPSON a n d R. S. RICHARDS, P h y s i c a l P r i n c i p l e s a n d A p p l i c a t i o n s on J u n c t i o n Transistors (Oxford U n i v . Press, L o n d o n 1962). Zusammen/assung I n D i f f e r e n t i a l v e r s t ~ i r k e r n m i t T r a n s i s t o r e n erh~ilt m a n die b e s t e U n t e r d r i i c k u n g eines G l e i c h t a k t s i g n a l s , w e n n d e r g e m e i n s a m e E m i t t e r w i d e r s t a n d e t w a d e n W e f t h21/2 hz2 ann~ihert. E i n gr6sserer E m i t t e r w i d e r s t a n d h a t eine P h a s e n k e h r des G l e i c h t a k t s i g n a l s a m A u s g a n g z u r Folge, w a s in Verst~irkern m i t R i i c k f i i h r u n g e n (zur b e s s e r e n U n t e r d r i i c k u n g dieses Signals) Stabilit~itsschwierigkeiten z u r Folge h a b e n k a n n . (Received: December 12, 1964.) S l o w L a m i n a r F l o w in a C o n v e r g i n g or D i v e r g i n g Channel w i t h S u c t i o n at One Wall and B l o w i n g at the Other Wall B y I~OBERT M. T E R R I L L , D p t . of A p p l i e d M a t h e m a t i c s , U n i v e r s i t y of Liverpool, Liverpool, G r e a t B r i t a i n T h e s o l u t i o n for s t e a d y t w o - d i m e n s i o n a l , i n c o m p r e s s i b l e l a m i n a r flow in a c o n v e r g i n g or d i v e r g i n g c h a n n e l w i t h i m p e r m e a b l e wall is well k n o w n Ell 1). F u r t h e r if t h e R e y n o l d s n u m b e r is large a n d if t h e r e is s u c t i o n or b l o w i n g a t t h e walls w h o s e m a g n i t u d e is i n v e r s e l y p r o p o r t i o n a l t o t h e d i s t a n c e a l o n g t h e wall f r o m t h e origin of t h e c h a n n e l , a s o l u t i o n of t h e l a m i n a r b o u n d a r y l a y e r e q u a t i o n s c a n b e o b t a i n e d E21. I n t h i s n o t e a d i f f e r e n t i a l e q u a t i o n for t h e case of s u c t i o n a t one w a l l a n d e q u a l b l o w i n g a t t h e o t h e r wall is d e r i v e d f r o m t h e N a v i e r - S t o k e s ' e q u a t i o n s a n d t h e s o l u t i o n is o b t a i n e d for slow flow t h r o u g h t h e c h a n n e l . T h e e q u a t i o n s of m o t i o n for l a m i n a r flow in a c o n v e r g i n g or d i v e r g i n g c h a n n e l a r e DVr Vr b r Dr0 vr ~ r + V0 DVr U~ r r D0 v0 D~0 + r- 1 -- vr v0 - -DO + ~ 1 = Dp ( Q Dr + v _ V 2 V r e Dp ( r DO + v , V 2 v ~ + Ur 2 r2 r ~ D0 . ' DUO) 2 0vr v0 ) r2 r2 , DO (1) (2) where V2 _ D3 1 D Dr2 + - r O r + 1 D~ r 2 D02 a n d w h e r e r, 0 are p o l a r co-ordinates, v r a n d v o are t h e v e l o c i t y c o m p o n e n t s in t h e d i r e c t i o n s of r a n d 0 i n c r e a s i n g r e s p e c t i v e l y , p is t h e pressure, v t h e v i s c o s i t y a n d 0 t h e d e n s i t y . T h e e q u a t i o n of c o n t i n u i t y is D (r %) + Dvo Dr- T0-= 0. (3) L e t t h e b o u n d a r i e s of t h e c h a n n e l b e g i v e n b y 0 = 4- ~ a n d a s s u m e t h a t t h e velocities of b l o w i n g a t 0 = -- ~ a n d of s u c t i o n a t 0 = ~ are b o t h v 0 = V ro/r w h e r e r 0 is a t y p i c a l l e n g t h . T h e n , f r o m t h e e q u a t i o n of c o n t i n u i t y , v r = K F(O)/r w h e r e K is a c o n s t a n t w h i c h we will w r i t e e q u a l t o v R w h e r e R is t h e R e y n o l d ' s n u m b e r U ro/v of t h e flow. x) N u m b e r s in b r a c k e t s refer t o References, page 308. ...
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