Phase distortion in operational amplifiers

J-34

Annales de l'Association internationale pour le Calcul analogique

N" 3 - - ]uillet 1962

PHASE DISTORTION IN OPERATIONAL AMPLIFIERS by L.K. IVADHIVA. ~

ABSTRACT The stray capacities in the input and feedback paths of an operational amplifier, its finite bandwidth and d.c. gain effectively narrow the bandwidth of an amplifier causing amplitude and phase errors in the computer solutions. A given degree of accuracy consequently becomes more difficult to obtain with an amplifier when used in a highspeed repetitive computer than when used in a slower machine. In a high-speed computer, phase distortion is more serious than the amplitude distortion. An attempt has been made in this paper to analyze an operational amplifier with its associated input-feedback networks and a practical method has been suggested for reducing the undesired phase shift.

Introduction. M~acnee [1] has shown how the presence of undesired time constants associated with operational amplifier networks effectively narrow the bandwidth of an operational amplifier and consequently introduce amplitude and phase errors in the computer solutions of the differential equations. The errors get larger, and hence a given degree of accuracy becomes more difficult to obtain with an amplifier when used in a high-speed repetitive computer than when used in a slower machine. Bell and Rideout [2] in their analysis of an operationaI amplifier have assumed an amplifier with a finite bandwidth and the effect of finite bandwidth has been represented by a single equivalent time constant. The authors, however, have neglected the presence of stray capacities that are invariably associated with input and feedback networks of the amplifier. The authors have pointed out and emphasised the need for guarding against phase errors which are more seriom than amplitude errors in the repetitive type of computers. _An attempt has been made in this paper to analyse a physically realizable operational amplKier along with its input and feedback networks. The effect of finite bandwidth of an amplifier is assumed to be represented by a single equivalent time constant and the stray capacities in the input and feedback networks are taken into account.

Analysis of an Operational Amplifier. Simplified block diagramatic representation of an operational amplifier is shown in figure 1 (a), where ZI = series input impedance Zl = amplifier input impedance Z2 = feedback impedance Z q = output impedance K = G. Ra = d.c. gain of the operational amplifier Y~ = G./(1 -k sT.) T. = amplifier time-constant. * De/ence Research & Development Delhi, India.

Organisation,

New

T ]

l (a) Block diagram of an operational amplifier.

E,

(

i

Eo

(b) Equivalent circuit of an operational amplifier. Fig, 1. - - Block diagram of an operational amplifier and its equivalent circuit.

An equivalent circuit of figure 1 (a) is shown in figttre 1 (b). T h e nodal equations by inspection may be written as : ElY1 = (Yt -b Y~ q- Y2) E~ - - Y oE o

(1)

0 = --(Y2 --Y.)

(2)

E~q- (Y_~ q- Y,0 E,)

Solving equations (1) and (2) for E, Eo

Y1 (Y, - - Y._,)

E~

Y2 (Y~ + Y , - ] - Ya -}- Y.) -}- Ys (Y~ -]- Y~)

(3)

Equation (3) is a general expression for the transfer function of an operational amplifier and its associated

135

L.K. Wadhwa : Phase distortion in operational amplifiers

input feedback networks. If the amplifier had infinite gain very wide bandwidth, infinite input impedance and very low output impedance, then equation (3) will reduce to the well known form: Eo

Yz

El

Yo

D = R, Ra (T~ + T.) 4- R, R a (T~ + T 0 4- R1Rl T. + T,_, {Ri Ra + R~ Ra 4- R~ Ri + K R~ Ri}

4- t~ {R, T~ 4- R, Ti 4- T~ (R~ 4- t,)} + Ra Ta T a (R~ T, 4- R~ T,) s'-'

(14)

The phase distortion from equation (10) is seen to be In a high-speed repetitive type of computer phase distortion is more serious than amplitude distortion as it causes large phase shifts with fewer amplifiers resulting in instabilities in the computer loop. It would be of interest to derive from equation (3) an expression for the phase distortion for the case of a scaler unit. In a scaler Z, and Z., are usually resistive. But due ~o the presence of stray capacities at the input and in the feedback path of the amplifier the reactive components are also present. In a well designed operational amplifier intended for precision analogue computers the output stage is usually a cathode follower with a very low output impedance and hence it would be reasonable to assume the output impedance Z a small and purely resistive (Za = Ra).

Bo

= tan -~ (~---)-

D~ tan-~ (~---)

and for small angles, equation (15) may be rewritten as Be0

q' -- (-Z-)

Do

-- (--d-)

(16)

Since K is very large, Rt, R.,, RI are large compared to Ra and T, is large compared to T1 , T=, TI and for values of m in the range of interest (f < 10 Kc/see). Equation (10) after necessary simplification reduces to: ~l

R.,

(1 + i o T 0

(iv)

Further if, 1

1 Yl -

Zl

-

-

1

R,

(4)

(1 4- s Ti) Rl

0)

+ s %0 Ro

(6)

(1

Y2 -- Z o -1 Y.

-

Y,

-

K

=

1 + j o {% + K ( 1 + R , + ~ ) }

(1 + s Tj)

Yx -- Zt --

1 -

Z~

(7)

G-~

K -

(1 + s T y )

(8)

R,(1 + s T y )

G.R.

(9)

Then substituting equation (4) through (9) in equation (3) and simplifying ¢ E,,

El

-

R,

(A

+ B s)

(C + D s)

(1o)

where

A =(KRo_--Ra)--{RaTaT. 4- RaT~(Ta 4- T.)} s'a (11) B = {T~ (K R2 -- Ra) -- Ra (T. 4- T=)) - - R a Ti To T. s -~

(15)

(12)

C = [R~ Ra + R, Ra + R1 Ri 4- K R~ Rt + R2 (Rt + R~)]

and consequently the phase distortion ,as given by equation (16) to: ~ o, [ T t -

~-

E))I

{T., 4- T, (i + ~, R,2 + R2

(18)

It is evident from equation (18) that the phase distortion is small for values of o in the range of interest if T~, T._,, T, very small, i.e. stray capacities small and wide bandwidth amplifier and K and Rl - - ~:he gain and input resistance of the amplifier - - very large. A practical method for reducing the phase distortion is to make T. R2 R2

T~ = T,, + -k-- (1 + ~

+ -~--)

(19)

by addition of a suitable capacitor in tither the input or the feedback network so as to satisfy equation (19). This method of phase correction has been successfully employed [3] in the design of WP Analogue Computer at the University of Wisconsin. The operational amplifier in the WP Computer consists of a plug-in: Philbrick K_~ x chopper-stabilized with a K= p amplifier with input and feedback resistors R1 and R2 each a i00 kilo-ohms. The phase distortion for an uncompensated scaler was measured' to be about 0.5° at frequency as low as 2 Kc/sec while for a compensated scaler the phase distortion was tmmeasurable (< 0.1'°) for frequencies ,as high as 40 Kc/sec.

+ IR.~T. (R, % + lq %) + T., {1~, 1%,(% + %)

Conclusions.

+ R,Ra (T. + T,) -4- R, R, T~}

The stray capacities that are invariably present in the input and feedback paths of an operational amplifier make greater contribution to phase errors than the finite bandwidth of the amplifier, A method for reducing these phase errors is to make the time con-

4-

R_. T,, (R, T, 4- R~ T,)] s'2

* Details are shown in the Appendix,

(15)

136

Annales de l'Association internationale pour le Calcul analogique

stants of the input and feedback networks approximately equal.

"

i

(i + s T=)

Z2

R=

Acknowledgement. Ys

The valuable suggestions and encouragement of Prof. V.C. Rideout of the University of Wisconsin, U.S.A., is gratefully acknowledged.

--

N ° 3 - - /uillet 1962

1

1

Z,

Ra

(29)

G~

y, -

K

(1 + s T y ) BIBLIOGRAPHY

Block diagrammatic representation of an operational amplifier and its assodated 'input feedback networks is shown in figure 1 (a) and its equivalent circuit is shown in fig. 1 (b), By inspection the nodal equations may be written as :-

EtYl ----- (Y, + Y~ + Y=) E ~ - Y~.Eo

[ (1+sTy) +

R,

(l+sTt)

R1

Then from equations (23), (26), (28) and (30)

(14-sTy)

K

Rl

(14-sT1) --RaT2Tj'--Rs(T=4-T.)s4-KR=--R,. O: - -

-

[

-

RI • '.

~

=

(20) 1 +--+ Ra

(33)

b = lq[T~T,+T,(%+%)I c = Ra(T,+Ta)--Tx(KR~--R,)

d = KR2--R, Now from equations (24) and (26) through (31) 1

K

] + ~ Ra (l + s T,)

(1 + s T,)

[

~1

+

R, Rl Ra (1 + s T.)

Y~ (Y~--Y2) - -

Y= (Y~ 4-Y i4- Y. + Y~) 4- Ya (Y~ -5Y~)

(21)

Ya (Ya -- Y=)

E o

~,

]

which on reduction to common denominator, rearrangement and simplification becomes :-

-E~

ot

alsa + bls 2 + cls + d

R~ R2 Ra R~ (1 + s T~)

(34)

(22)

where al = Ra T2 T~ (R1 Tl + Rt T1)

(23)

b~ -- Rs Y~ (R, TL + R, T~) + T2 {R, Ra (Y, + Y~)

and fl = Y= (Y~ + Yl 4- Ya + Ya) 4- Y. (Y~ + Yi) (24) then

(1 + s T,)

[.(RxT~ 4- R, T~) s 4- (RI 4- R 0 R, R,

=

~ =

(32)

R, R~ R, (1 4- s T~)

a = RaTLTuT.

Solving equations (1) and (2) for E

if

--as,--bs2--cs+d

R2

E0

-1

R, Ra (1 + s Ta

(1 + s T=)

0 = --(Y2--Y~) Eg 4- (Y.,+Y,)E o

let

(i +sTy)

R, R. {T, T~ s-"4- (T, q- T,) s 4- i) 4- R~ R~ {Tt T~ s=4- (Ti q- T,) s q- I} 4- R 1 R l (i 4- s T,) + K R, Rl

+N1

E,

(31)

where

APPF~DIX

(1+sT2)

(30)

Ra(1 + s T y )

K = G~ Ra

[1] MACNEE, A.B. < Proe. IRE, Vol. 40, No. 3, pp. 303-308, March t952. [2] BELL, H. Jr. & RIDEOUT, V.C. (( Precision in HighSpeed Electronic Differential Analysers>>, Cyclone Symposium II on ~(Simulation and Computing Techniques~> Pt II, pp. 61-77, April 28, May 1952. [3] WADHWA, L.K. (( A Detailed Study of Certain Linear and Non-linear High-speed Analog Computer Components >~ M.S. Thesis, Dept. of Elec. University of Wisconsin, Aug. 1956.

/~-

<28)

(25)

+ R~ l~. (Y, + %) + l~ ~, %) + R2 T~ (R~ T, + R~ Tx)

3

Y~-

i z~-

(i + s %) R,

(26)

Yi =

1 (1 + s Tl) Zl --'-Rl

(27)

+ T~ {RL Ra + R, Rs + R, R, + KR~ R,} d, = R~ R. 4- R~ Ra 4- R, Rl 4- R~ (R, 4- R,) 4- K R~ Rl

(35)

137

L.K. Wadhwa : Phase distortion in operational amplifiers

Substituting equations (32) and (34) in (25) Eo E,

-

R,

( a + B s) ( c + D s)

and therefore from equations (37) and (38):B ~ TiKR=

(36)

dl >> bl d-'

where

and therefore from equations (35) and (37):-

A = (d - - b s =) t3 =

(-- ~ --

a s2)

C ~ K R1 Rl

(37)

and therefore from equations (35) and (37):-

D = (cl + a,s a) Now substituting s = j ~o in equation (36)

D = T~KIR i +R1R, (38)

Since K is very large, R~, R=, and R 1 very much greater than Ra, T. greater than T~, TR and Ti, and for values of m in the region of practical interest (f K 10 Kc/sec) d~> boo~ and therefore from equations (37) and (33):A ~ K R~_

T,,+R,,(R,+RI)

T,,

(42)

On substituting equation (39) through (42) in (38) and simplifying:R,_,

(~ + j ~, T,)

(43)

R~

R2

~+i~(T~+ (~+~+~)} and the phase distortion, since the angles are small, is therefore

(39) q~ ~ m I T s - -

C )) a£o 2

(41)

¢~ )) a t o;-'

C = (d, + b,s'-')

[F._~_] _ (a + j ~ B) s =j~o - - - P,l ( c + j ~, D)

(40)

T~ R,. R° {T,, -k ~ (1 -b ~ - -{- g:-)}] sal

(44)