Relations Between Attenuation and Phase in Feedback Amplifier Design

By H. W . BO D E

In t r o d u c t i o n

T

H E engineer w ho e m b a rk s upon th e design of a feedback am plifier m u st be a c re a tu re of m ixed em otions. On th e one h a n d , he can rejoice in th e im p ro v em e n ts in th e c h a ra c teristic s of th e s tru c tu re w hich feed b ack prom ises to secure h im.1 O n th e o th e r h a n d , he know s t h a t unless he can finally a d ju s t th e p h ase an d a tte n u a tio n c h a ra cteristics a ro u n d th e feed b ac k loop so th e am plifier will n o t sp o n tan eo u sly b u rs t in to u n co n tro llab le singing, none of th ese a d v a n ta g e s can a c tu a lly be realized. T h e em o tio n a l situ a tio n is m uch like t h a t of an im p ecu n io u s y o u n g m an w ho h a s im p etu o u sly in v ited th e la d y of his h e a rt to see a p lay , u n m in d fu l, for th e m o m en t, of th e lim ita tio n s of th e $2.65 in his p o ck ets. T h e ra p tu ro u s com m ents of th e girl on th e wrny to th e th e a te r w ould be v e ry p le a sa n t if th e y were n o t shadow ed b y his p riv a te specu latio n a b o u t th e c o st of th e tick ets.

In m a n y designs, p a rtic u la rly th o se req u irin g on ly m o d e rate a m o u n ts of feedback, th e b o g y of in s ta b ility tu rn s o u t n o t to be serious a fte r all.

In o th ers, how ever, th e s itu a tio n is like th a t of th e y o u n g m an wrho has ju s t a rriv e d a t th e box office an d finds th a t his w orst fears are realized. B u t th e y o ung m an a t least know s w here he stan d s. T h e en g in eer’s experience is m ore ta n ta liz in g . In ty p ica l designs th e loop ch ara c teristic is alw ays sa tisfa c to ry — ex ce p t for one little p o in t. W hen th e engineer changes th e c ircu it to c o rre ct th a t p o in t, how ever, diffi

culties a p p e a r som ew here else, a n d so on ad infinitum . T h e solution is alw ays ju s t a ro u n d th e corner.

A lth o u g h th e engineer ab so rb ed in chasing th is rainbow m ay n o t realize it, such an experience is alm o st as stro n g an in dication of th e existence of som e fu n d a m e n ta l physical lim itatio n as th e census which th e y o ung m an ta k e s of his p ockets. I t rem inds one of th e experience of th e in v e n to r of a p e rp e tu a l m otion m achine. T h e p e rp etu al m o

tion m achine, likewise, alw ays w orks— except for one little factor.

E v id e n tly , th is s o rt of fru stra tio n an d lost m otion is in ev itab le in 1 A general acquaintance with feedback circuits and the uses of feedback is as

sumed in th is paper. As a broad reference, see H. S. Black, “ Stabilized Feedback Am plifiers,” B. S. T. J January, 1934.

421

fe ed b ac k am plifier design as long as th e p ro b lem is a tta c k e d b lin d ly .

F E E D B A C K A M P L I F I E R D E S I G N 423 phase discovered by a variety of authors. T hey are derived typically from a Fourier analysis of the transient response of assumed structures and are frequently am bigu

relatio n betw een th e a tte n u a tio n a n d th e p h ase sh ift of a physical

F E E D B A C K A M P L I F I E R D E S I G N 425 a large n u m b e r of re la tio n s betw een th e a tte n u a tio n an d p hase c h a r

a c te ristic s of a physical n etw o rk . One of th e sim plest is X ⁰⁰

B d u = %(A„ - A 0), (1)

w here u rep resen ts log f / f o , f o being an a rb itr a ry reference frequency, B is th e p h ase sh ift in rad ian s, a n d A 0 a n d A x are th e a tte n u a tio n s in nepers a t zero an d infinite frequency, resp ectiv ely . T h e theorem sta te s, in effect, t h a t th e to ta l are a u n d e r th e p h ase c h a ra c teristic p lo tte d on a lo g arith m ic freq u en cy scale dep en d s only u p o n th e differ

ence betw een th e a tte n u a tio n s a t zero a n d infinite frequency, an d n o t upon th e course of th e a tte n u a tio n betw een these lim its. N o r does it depend upon th e physical configuration of th e n e tw o rk unless a n o n m inim um p hase s tru c tu re is chosen, in w hich case th e a re a is necessarily increased. T h e e q u a lity of p hase areas for a tte n u a tio n c h a ra c teristic s of different ty p e s is illu stra te d b y th e sketch es of Fig. 1.

T h e significance of th e phase a re a re latio n for feedback am plifier design can be u n d e rsto o d b y supposing t h a t th e p ra c tic a l transm ission range of th e am plifier e x ten d s from zero to som e given finite frequency.

T h e q u a n tity A 0 — A x can th e n be identified w ith th e change in gain aro u n d th e feed b ack loop req u ired to secure a cut-off. A ssociated w ith it m u st be a c e rta in definite p hase area. If we suppose th a t th e m axi

m um p hase sh ift a t a n y freq u en cy is lim ited to som e ra th e r low value th e to ta l are a m u st be sp read o u t over a p ro p o rtio n a te ly b ro ad in terv al on th e freq u en cy scale. T h is m u st correspond ro u g h ly to th e cut-off region, a lth o u g h th e p o ssib ility t h a t som e of th e are a m ay be found abo v e or below th e cut-off ran g e p re v e n ts us from d eterm in in g th e necessary in te rv a l w ith precision.

A m ore d etailed s ta te m e n t of th e relatio n sh ip betw een phase shift a n d change in a tte n u a tio n can be o b ta in ed b y tu rn in g to a second

th e o re m . I t rea d s as follow s:

5 (/c ) = ^ I w^ lo g co th T ^ M' (2) w here B ( f c) re p re se n ts th e p h ase sh ift a t a n y a r b itra r ily chosen fre  q u e n c y f c a n d u = l o g / / / c. T h is e q u a tio n , like (1), h olds only for th e m in im u m p h ase sh ift case.

A lth o u g h e q u a tio n (2) is so m ew h at m ore c o m p licated th a n its predecessor, it lends itself to an e q u a lly sim ple p h y sical in te rp re ta tio n . I t is clear, to begin w ith , t h a t th e e q u a tio n im plies b ro a d ly t h a t th e

0.1 0.2 0.3 0 .4 0 5 0.6 0.6 1.0 2 3 4 5 6 6 10

F ig. 2— W eighting function in loss-phase formula.

p h ase sh ift a t a n y fre q u en cy is p ro p o rtio n a l to th e d e riv a tiv e of th e a tte n u a tio n on a lo g a rith m ic fre q u e n c y scale. F o r exam ple, if d A / d u is d o u b led B will also be d o u b led . T h e p h a se sh ift a t a n y p a rtic u la r freq u en cy , how ever, does n o t d e p e n d u p o n th e d e riv a tiv e of a t t e n u a tio n a t t h a t freq u e n cy alone, b u t u p o n th e d e riv a tiv e a t all freq u en cies, since it involves a su m m in g u p , or in te g ra tio n , of c o n trib u tio n s from th e com plete freq u en cy sp ec tru m . F in a lly , we n o tice t h a t th e c o n tr i

b u tio n s to th e to ta l p h ase sh ift from th e v a rio u s p o rtio n s of th e fre q u en cy sp e c tru m do n o t a d d u p e q u a lly , b u t r a th e r in acc o rd a n ce w ith th e function log co th | u | /2. T h is q u a n tity , th erefo re, a c ts as a w e ig h t

ing fu n ctio n . I t is p lo tte d in Fig. 2. A s we m ig h t ex p e c t p h y s ic a lly

F E E D B A C K A M P L I F I E R D E S I G N 427 it is m uch larg er n e a r th e p o in t u = 0 th a n it is in o th e r regions. W e can, therefore, conclude th a t while th e d e riv a tiv e of a tte n u a tio n a t all frequencies e n te rs in to th e p h ase sh ift a t a n y p a rtic u la r frequency / = f c th e d e riv a tiv e in th e neighborhood of f c is rela tiv e ly m uch m ore

im p o rta n t th a n th e d e riv a tiv e in rem o te p a r ts of th e sp ectru m . As an illu stra tio n of (2), let it be supposed t h a t A = ku, w hich co r

responds to an a tte n u a tio n h av in g a c o n s ta n t slope of 6 k db p e r octave.

T h e associated p h ase sh ift is easily e v a lu a te d . I t tu rn s o u t, as we m ight expect, to be c o n s ta n t, a n d is eq u al n u m erically to &7t/2 radians.

T h is is illu stra te d b y Fig. 3. As a second exam ple, we m ay consider

Fig. 3— Phase characteristic corresponding to a constant slope attenuation.

a d isco n tin u o u s a tte n u a tio n ch a ra c teristic such as th a t shown in Fig. 4.

T h e asso ciated p h ase ch a ra c teristic , also shown in Fig. 4, is p ro p o r

tional to th e w eighting function of Fig. 2.

T h e final exam ple is show n b y Fig. 5. I t consists of an a tte n u a tio n c h a ra c teristic w hich is c o n s ta n t below a specified frequency f b a n d has a c o n s ta n t slope of 6 k d b p e r o ctav e above /&. T h e associated phase ch ara c teristic is sy m m etrical a b o u t th e tra n sitio n p o in t betw een th e tw o ranges. A t sufficiently high frequencies, th e p hase sh ift a p proaches th e lim itin g &7r/2 ra d ia n s w hich w ould be realized if the c o n s ta n t slope w ere m a in ta in e d over th e com plete sp ectru m . A t low frequencies th e p h ase sh ift is su b sta n tia lly p ro p o rtio n al to frequency an d is given b y th e eq u atio n

„ 2k f

S o lu tio n s dev elo p ed in th is w a y can be a d d e d to g e th e r, since i t is a p p a re n t from th e g en eral re la tio n u p o n w hich th e y are b a se d t h a t th e p h ase c h a ra c te ris tic c o rre sp o n d in g to th e sum of tw o a tte n u a tio n

20 K

<m 3 u 10k

F ig. 4— Phase characteristic corresponding to a discontin uity in atten uation.

Fig. 5— Phase characteristic corresponding to an atten uation which is constan t below a prescribed frequency and has a constan t slope above it.

c h a ra c te ristic s will be eq u a l to th e su m of th e p h a se c h a ra c te ristic s co rresp o n d in g to th e tw o a tte n u a tio n c h a ra c te ristic s se p a ra te ly . W e can th erefo re com bine e le m e n ta ry so lu tio n s to secure m ore co m p lic a te d

F E E D B A C K A M P L I F I E R D E S I G N 429 ch a ra c te ristic s. A n exam ple is furnished b y Fig. 6, w hich is b u ilt up from th re e so lu tio n s of th e ty p e show n b y F ig. 5. B y proceeding sufficiently far in th is w ay, an ap p ro x im a te c o m p u ta tio n of th e phase c h a ra c te ristic a sso ciated w ith alm o st a n y a tte n u a tio n c h a ra c teristic can be m ade, w ith o u t th e la b o r of a c tu a lly p erfo rm in g th e in teg ra tio n in (2).

F ig. 6— Diagram to illustrate addition of elem entary attenuation and phase char

acteristics to produce more elaborate solutions of the loss-phase formula.

E q u a tio n s (1) an d (2) are th e m o st sa tisfa c to ry expressions to use in stu d y in g th e rela tio n betw een loss an d p hase in a b ro ad physical sense. T h e m echanics of c o n stru c tin g d etailed loop cut-off c h a ra c te r

istics, how ever, are sim plified b y th e inclusion of one o th er, som ew hat m ore com plicated, form ula. I t ap p e a rs as

/ "

A d f

Vo2

~ P ( P

- fc2) + X

B d f

\ F I ! ( P - P 2) _ 7T B ( f c)

2fcSfo2 - fc2 ’ 7T A ( P ) 2fcSfc2 - / o 2’

fc < f o

fc > fOi (4)

w here / 0 is som e a rb itra rily chosen frequency an d th e o th e r sym bols h av e th e ir p rev io u s significance.

T h e m eaning of (4) can be u n d e rsto o d if it is recalled th a t (2) im plies th a t th e m inim um p h ase sh ift a t a n y frequency can be co m p u ted if th e

a tte n u a tio n is p re sc rib ed a t all frequencies. In th e sam e w ay (4) show s how th e c o m p lete a tte n u a tio n a n d p h ase c h a ra c te ris tic s can be d e te rm in e d if we begin b y p re scrib in g th e a tte n u a tio n b e lo w/ 0 an d th e p h a se sh ift a b o v e fo- Since f 0 can be chosen a r b itr a rily larg e or sm all th is is e v id e n tly a m ore general fo rm u la th a n e ith e r (1) or (2), w hile it can itself be generalized, b y th e in tro d u c tio n of a d d itio n a l irra tio n a l facto rs, to p ro v id e for m ore e la b o ra te p a tte r n s of b a n d s in w hich A an d B are specified a lte rn a te ly .

As a n exam ple of th is fo rm u la, le t i t be assu m ed t h a t A = K for / < / o a n d t h a t B = kir/2 for / > f 0. T h ese a re show n b y th e solid lines in Fig. 7. S u b s titu tio n in (4) gives th e A an d B c h a ra c te ristic s in th e re st of th e sp e c tru m as

B = k sin 1 j , Jo

A = K + k log

/ < / o

V ^ 2 _ 1 + / o ] ’

/ > / 0 ‘ ⁽⁵⁾ T h ese are in d ic a te d b y b ro k en lines in Fig. 7. In th is p a rtic u la rly

Fig. 7— Construction of com plete characteristics from an atten uation character

istic specified below a certain frequency and a phase characteristic above it. T he solid lines represent the specified atten uation and phase characteristics, and the broken lines their com puted extensions to th e rest of the spectrum.

sim ple case all four fra g m e n ts can be com bined in to th e single a n a ly tic fo rm u la

A + i B = K + k log ( «

F E E D B A C K A M P L I F I E R D E S I G N 431 T h is expression will be used as th e fu n d a m e n ta l fo rm u la for th e loop cut-off c h a ra c te ristic in th e n ex t section.

Ov e r a l l Fe e d b a c k Lo o p Ch a r a c t e r i s t i c s

T h e su rv e y ju s t concluded shows w h a t co m b in atio n s of a tte n u a tio n an d p hase c h a ra c teristic s are p h y sica lly possible. W e h av e n e x t to d eterm in e w hich of th e ava ila b le co m b in atio n s is to be regarded as rep resen tin g th e transm ission a ro u n d th e overall feedback loop. T h e choice will n a tu ra lly dep en d som ew h at upon e x actly w h at we assum e t h a t th e am plifier o u g h t to do, b u t w ith a n y given set of assu m p tio n s it is possible, a t lea st in th e o ry , to d eterm in e w h a t co m b in atio n is m ost a p p ro p ria te .

/

>X< 'N

/ \

/

^{- 1 , 0} f=co f= o

V /

I s

1 ( /

'm ill

li

" ^S^S_\

\

^M\

-— - ____ - —

Fig. 8— N yq uist stability diagrams for various amplifiers. Curve I represents

“ ab so lu te” stab ility, Curve II instability, and Curve III “ conditional” stability.

In accordance with the convention used in this paper the diagram is rotated through 180° from its normal position so that the critical point occurs at — 1 ,0 rather than + 1, 0.

T h e situ a tio n is co n v en ien tly in v estig ated b y m eans of th e N y q u ist s ta b ility d iag ram 5 illu s tra te d b y Fig. 8. T h e d iag ram gives th e p a th

6 Bell System Technical Journal, July, 1932. See also Peterson, Kreer, and Ware, Bell System Technical Journal, October, 1934. The N yquist diagrams in the present paper are rotated through 180° from the positions in which they are usually drawn, turning the diagrams in reality into plots of — m/3. In a normal amplifier there is one net phase reversal due to the tubes in addition to an y phase shifts chargeable directly to the passive networks in the circuit. T he rotation of the diagram allows this phase reversal to be ignored, so that the phase shifts actually shown are the same as those which are directly of design interest.

tra c e d b y th e v e c to r re p re se n tin g th e tran sm issio n a ro u n d th e feed b ack

F E E D B A C K A M P L I F I E R D E S I G N 433 useful ran g e is decreased as th e assum ed m arg in s a re increased, so t h a t it is g en erally d esirab le to choose as sm all m arg in s as is safe.

T h e essential fe a tu re in th is s itu a tio n is th e re q u ire m e n t t h a t th e d im in u tio n of th e loop gain in th e cu to ff region should n o t be acco m panied b y a p h ase sh ift exceeding som e p rescrib ed a m o u n t. In view of th e close connection betw een p h ase sh ift an d th e slope of th e a tte n u atio n c h a ra c teristic evid en ced b y (2) th is e v id e n tly d e m a n d s t h a t th e am plifier should c u t off, on th e w hole, a t a well defined r a te w hich is n o t to o fast. As a first ap p ro x im a tio n , in fact, we can choose th e cu to ff

loop.

c h a ra c teristic as an e x a c tly c o n s ta n t slope from th e edge of th e useful b an d o u tw a rd . Such a c h a ra c te ristic h as a lre a d y been illu stra te d b y Fig. 5 a n d is show n, r e p lo tte d,6 b y th e broken lines in Fig. 10. If we choose th e p a ra m e te r c o rresp o n d in g to k in Fig. 5 as 2 th e cutoff ra te is 12 d b p e r o c ta v e a n d th e p h ase sh ift is s u b s ta n tia lly 180° a t high frequencies. T h is choice th u s leads to zero p h ase m argin. B y choos

ing a so m ew h at sm aller k on th e o th e r h a n d , we can pro v id e a definite 6 T o prevent confusion it should be noticed that the general attenuation-phase diagrams are plotted in term s of relative loss while loop cutoff characteristics, here and a t later points, are p lotted in term s of relative gain.

m arg in a g a in st singing, a t th e c o st of a less ra p id cutoff. F o r exam ple, if we choose k = 1.5 th e lim itin g p h ase sh ift in th e ///3 loop becom es 135°, w hich p ro v id es a m argin of 45° a g a in st in s ta b ility , w hile th e ra te of cutoff is red u ced to 9 d b p e r o c tav e. T h e v a lu e k = 1.67, w hich c o r

resp o n d s to a cu to ff ra te of 10 d b p e r o c ta v e a n d a p h a se m arg in of 30°, h a s been chosen for illu s tra tiv e p u rp o ses in p re p a rin g Fig. 10. T h e loss m argin d ep en d s u p o n co n sid e ra tio n s w hich will a p p e a r a t a la te r p o in t.

A lth o u g h c h a ra c te ristic s of th e ty p e show n b y Fig. 5 are re a so n a b ly sa tisfa c to ry as am plifier cutoffs th e y e v id e n tly p ro v id e a g re a te r p h ase

Fig. 10— Ideal loop cutoff characteristics. Drawn for a 30° phase margin.

m argin a g a in st in s ta b ility in th e region ju s t b ey o n d th e useful b a n d th a n th e y do a t high frequencies. In v irtu e of th e p h a se a re a law th is m u s t be inefficient if, as is supp o sed here, th e o p tim u m c h a ra c te ris tic is one w hich w ould p ro v id e a c o n s ta n t m arg in th ro u g h o u t th e cutoff in te rv a l. T h e re la tio n b etw een th e p h ase a n d th e slope of th e a t t e n u a tio n suggests t h a t a c o n s ta n t p h a se m argin can be o b ta in e d b y in c re a s

ing th e slope of th e cu to ff c h a ra c te ristic n e a r th e edge of th e b a n d , leav in g its slope a t m ore rem o te frequencies u n c h a n g e d , as show n by th e solid lines in Fig. 10. T h e e x a c t expression fo r th e req u ire d cu rv e can be found from (6), w here th e p ro b le m of d e te rm in in g such a c h a r

a c te ristic a p p e a re d as an exam p le of th e use of th e g eneral fo rm u la (4).

F E E D B A C K A M P L I F I E R D E S I G N 435

c h a ra c te ris tic to be sim u la te d w ith sufficient precision. F o r exam ple, we are o b v io u sly in p h y sical difficulties if th e cu to ff c h a ra c te ris tic specifies a n e t gain a ro u n d th e loop a t a freq u en cy so high t h a t th e tu b e s th em selv es w o rk in g in to th e ir ow n p a ra s itic c a p a c ita n c e s d o n o t give a gain.

T h is lim ita tio n is s tu d ie d m o st easily if th e effects of th e p a ra sitic e lem en ts are lu m p ed to g e th e r b y re p re se n tin g th e m in te rm s of th e a s y m p to tic c h a ra c te ris tic of th e loop as a w hole a t ex tre m e ly high frequencies. A n ex am p le is show n b y Fig. 11. T h e s tr u c tu re is a

Fig. 11— E lem ents which determ ine the asym p totic loop transm ission characteristic in a typ ical amplifier.

s h u n t feed b ack am plifier. T h e ¡3 c irc u it is re p re se n te d b y th e T co m  p osed of n etw o rk s As, N e a n d A?. T h e in p u t a n d o u tp u t c irc u its are re p re se n te d b y A j a n d Ah a n d th e in te rsta g e im p ed an ces b y Ah a n d A3. T h e C ’s are p a ra s itic c a p a c ita n c e s w ith th e ex cep tio n of C5 a n d C§, w hich m a y be reg a rd ed as design elem en ts a d d e d d e lib e ra te ly to A5 an d A6 to o b ta in an efficient high freq u e n cy tran sm issio n p a th fro m o u tp u t to in p u t. A t sufficiently high frequencies th e loop tra n sm issio n will d e p e n d o n ly u p o n th ese v a rio u s ca p a c itan c e s, w ith o u t re g a rd to th e A ’s. T h u s, if th e tra n sc o n d u c ta n c e s of th e tu b e s are re p re se n te d by G\, Gi, a n d G3 th e a s y m p to tic g ain s of th e first tw o tu b e s are Gi/coCi an d G2/wC3. T h e re s t of th e loop includes th e th ird tu b e a n d th e p o te n tio m e te r form ed b y th e c a p a c itan c e s C1, C 4, C5 a n d C6. Its a s y m p to tic tran sm issio n can be w ritte n as G3/uC, w here

C = Cl + C 4 + - ^ (C5 + Ce).

5 '-'6

E ach of th ese te rm s d im inishes a t a ra te of 6 d b p e r o c tav e . T h e co m p le te a s y m p to te is GiG2G3/w3CC2C 3. I t a p p e a rs as a s tra ig h t line w ith

W dokumencie The Bell System Technical Journal : devoted to the Scientific and Engineering aspects of Electrical Communication, Vol. 19, No. 3 (Stron 83-106)