A Nonl inear Low-Frequency Instability Phenomenon in Audio Amplifiers

T H E R O N U S H E R JR. ASSOCIATE MEMBER AIEE

UN D E R certain conditions, audio amplifiers with feedback m a y exhibit

instabili ty a t low frequencies, which is no t predictable by conventional linear anal­ysis. Th i s instabili ty caused b y non­linear effects can occur when the final stage is in class Β operation and is pre­ceded b y a variable-gain stage, such as a pentode or te t rode.

Mathemat ica l analysis of the problem reveals t h a t a nonminimum phase zero appears in t he transfer function of t he driver and power stage of t he amplifier. This zero, which is no t revealed by linear analysis, depends on signal ampli tude, and tends to disappear as signal ampli tude becomes very small. T h e presence of t h e nonminimum phase zero causes an addit ional 90-degree phase lead a t low frequencies in the forward pa th of the amplifier. This addit ional phase lead, together wi th t h e phase lead from t h e capacitor-resistor coupling networks be­tween o ther stages, can therefore cause instabil i ty t o occur a t high-signal ampli­tudes when feedback is applied to t he amplifier.

T h e remaining sections of the paper contain t h e theoretical analysis of the problem, experimental results which suppor t the analysis, and a technique for reducing the nonlinear effect.

Analysis

simplicity, the variable-gain portion is treated by considering a single tetrode. The circuit diagram is shown in Fig. 1. The various signal component voltages are defined.

2. The tetrode stage in Fig. 1 is coupled to the power stage by a capacitor-resistor network. 3. The final stage is biased to plate current cutoff.

4. The characteristics of the triode are linear in the conducting regions.

5. The triode power tubes are coupled to a resistive load by an ideal transformer. The winding resistance and leakage in­ductance are neglected, and the magnetizing inductance is considered infinite.

6. Distortion effects In the tetrode are neglected.

T h e small-signal t ransfer function of t h e configuration in Fig. 1 will first be determined so t h a t t h e effects of t h e non-linearities m a y easily be seen b y compar­ing t h e results of t h e linear analysis wi th those obtained when t h e nonlinearities are included.

When the excitation, vu consists of a single-frequency sinusoidal voltage, t h e low-frequency gain of t h e t e t rode is given by equat ion 1, where gmo is t h e t r ans -conductance a t t he operat ing point .

V"=-gmoRi*-GT ( 1 ) ν i

T h e coupling ne twork yields t he transfer

function found in equat ion 2, where

VIA RgCs+1 s+ù)g

T h e gain of t h e final s tage is given in equa­t ion 3, where Ν is t h e ra t io of t h e t u r n s in one half t he p r imary of t h e t ransformer t o those in t h e secondary. T h e t r iode pa­rameters are μ and rp.

» o _ _ l ^ A - G o ( 3 )

T h e over-all transfer function derived b y linear analysis is t h e product of t h e results in equat ions 1, 2, and 3.

£ ? = G o G r _ ? _ ( 4 )

Vi S-f-CÛg

T h e asymptot ic frequency response dia­gram for t h e transfer function in equat ion 4 is found in Fig. 2.

T h e nonlinear effects of t h e variable-gain te t rode stage and t h e class Β power stage appear when the inpu t signal to t h e te t rode contains two sinusoidal compo­nents . One component has a frequency in the region of interest , near ωρ, and t h e other component is assumed to have a frequency much higher t h a n ωβ. T h e inpu t signal is expressed in the following equat ion.

VÎ = AL sin COLÎ+AH sin ( 5 )

AH>AL

A linear analysis would indicate t h a t t he response, v0> t o t h e i npu t signal in equa­tion 5 can be determined b y considering each component separately. However, the following analysis reveals t h a t t h e response of t he amplifier a t t h e frequency

T h e general assumptions pertaining to t he analysis of t he problem are listed as follows.

1. Since it has been indicated in the fore­going that the nonlinear effects occur in the variable gain portion of the amplifier and the class Β power stage, onfy these two portions will be investigated. For

Paper 57-1029, recommended by the AIEE Elec­tronics Committee and approved by the AIEE Technical Operations Department for presentation at the AIEE Fall General Meeting, Chicago, 111., October 7—11, 1957. Manuscript submitted March 7, 1957; made available for printing August 22, 1957. T H B R O N U S H E R , JR. , is with Yale University, New Haven, Conn. The author extends his thanks to the M. B. Manu­facturing Company for the use of their facilities, to Gait B. Booth for helpful suggestions, and to Gabriel Hajian for assistance in the experimental work.

F ig . 1 . Dr iver a n d p o w e r stages of class

Β ampl i f ie r

6 9 8 Usher, Jr.—Nonlinear Low-Frequency Instability in Audio Amplifiers JANUARY 1 9 5 8

G 0 G T

, l / 2 A ( C O M P O N E N T S )

Fig . 2 . A s y m p t o t i c f requency response, s ingle - f requency input

w L is dependent upon A H , t he ampl i tude of the high-frequency component.

T h e voltage in equat ion 5 m a y be though t of as a high-frequency input to the te t rode, the operating point of which is being varied a t t h e low-frequency ra te . Since the t ransconductance of mos t te t rodes is nearly linearly dependent upon the operating point, t he gain of t h e te t rode stage to the high-frequency signal component varies a t the low-frequency ra te . T h e plate voltage of the te t rode stage contains no t only a low-frequency component , b u t also an ampli tude-modula ted high-frequency component. T h e voltage waveform a t the plate of tube TIA is shown in Fig. 3. As shown in Fig. 3, t h e gain of t h e te t rode stage is highest when the plate voltage is low and vice-versa. The expressions for the signal frequency components of t he te t rode plate voltages are given in equations 6 and 6(A).

Via = · gmoRiAL sin <aLt+

dAL

A l sin ω / . it^RiAn sin ωΗΡ^ (6)

(6A)

T h e upper coupling network in Fig. 1 a t t enua tes t he ampli tude and shifts t he phase of t he low-frequency component of VIA, b u t t he modulated high-frequency

I L T 2 B (COMPONENTS)

(A)

t f k t If N. 11 § 1 w I

I B ) ' ( D )

F ig . 4 . G r i d vo l tage waveforms, p o w e r stage

component is vir tual ly unchanged if ωΗ

» ω α . T h e signal frequency com­ponents for t he grid voltages of tubes T2A

and TIB a re shown in Fig. 4. Fig. 4(A) depicts t he components of v2A. T h e phase shift of t he low-frequency component has been assumed to be 90 degrees. Fig. 4(B) shows the sum of t h e components of v2A. N o t e t h a t t h e upper envelope of VIA is shifted in phase relat ive to t h e lower envelope. Th i s is a resul t of v1A being modified b y t h e coupling network. Figs. 4(C) and (D) show the grid voltage relationships for t ube T2B.

Since t he power tr iodes are bo th biased to plate cur ren t cutoff, t ube T2A conducts only when v2A is positive, and tube T2B

conducts when v2B is positive. Due to t h e phase inversion from grid to plate, t he lower envelope of v3A is t h e negat ive of an amplified reproduction of t he upper en­velope of v2A. Similarly, t he lower en­velope of vzB is t h e negat ive of an ampli­fied reproduct ion of t he upper envelope of v2B. T h e voltage relat ionship for t he transformer is

equal in magni tude . T h e o u t p u t voltage, v0, therefore consists of a constant-ampl i tude high-frequency component and a low-frequency component .

A more critical examinat ion of Fig. 5 reveals t h a t t h e envelopes of v0 are no t in phase with t h e low-frequency component amplified from t h e grid of T2B. T h e demodulat ion effect of t h e class Β operated power stage has produced an addit ional low-frequency voltage com­ponent a t the plates of T2A and T2B, and also a t t h e transformer secondary. This addit ional low-frequency component is proport ional to t h e amplified envelope of t he modula ted high-frequency com­ponent a t t he plate of t he te t rode stage.

T h e equat ion for t he upper envelope of t h e high-frequency component of viA is obta ined from equat ions 6 and 6(A) and is found following.

VZA = —IZB = NVO

F ig . 3 . T e t r o d e p la te vo l tage waveform

a t all t imes. F rom equat ion 7, t he lower envelope

of t h e o u t p u t voltage, v0, has the same shape as t he negat ive of t he upper envelope of v2A, and t h e upper envelope of v0 bas t h e same shape as t he upper envelope of v2B in Fig. 4. T h e result for v0 is shown in Fig. 5. I t is seen from an inspection of Figs. 5, 4(B) , and 4(D) t h a t bo th envelopes of v0 are in phase, and are F ig . 5 . O u t p u t vo l tage waveforms

JANUARY 1 9 5 8 Ushery Jr.—Nonlinear Low-Frequency Instability in Audio Amplifiers 6 9 9

( I - K ) G 0 G T

0 ( I - K ) G 0 G T

kl + 1 / 1 ( L O G S C A L E S )

K G 0 6 T

0 /

1 j 1 - K 9

o ) g ω

( A )

" Π < ω ς ( L O G S C A L E )

( B )

F ig . 6 . A m p l i t u d e a n d phase response; 2 - f r e q u e n c y input

upper envelope of vlA = +~TRL<ALAH dAL

sin ω χ , / ( 8 )

Mult iplying equat ion 8 by — G0} the gain of t he final stage, yields t he low-frequency component of v0 contr ibuted by the demodulat ion effect.

dgm

vo" = — Go - 7 - 7 " RLALAH sin œLt dAL

T h e over-all gain for t h e low-frequency component produced b y demodulation is

— — —Go -tt-RLAH vi dAL

( 10 )

T h e complete expression for the transfer function of t h e low-frequency signal in t he presence of high-frequency signal is found by adding equat ion 4 and equation 10.

vo] Vi

^GoGr^—^-Goy-- RLAH ( 1 1 )

T h e result in equation 11 m a y be simplified considerably b y defining a non-

dimensional variable, K, in t he following fashion

~dALgmo 2 ( 1 2 )

T h e transfer function for t he low-fre­quency component becomes

(1-K)s-K<aa vo Vi

= GQGJ ( 1 3 )

T h e variable, K, is dependent upon t h e nonlinear gain characterist ics of t h e te t rode stage, a n d t h e ampl i tude of t h e high-frequency signal component . By using t he nondimensional variable, K, t h e result in equat ion 13 is easily extended to include t h e effect of variable gain occur­ring in any number of stages preceding t h e ou tpu t stage. I n general, Κ is defined

C r y ( 1 2 A )

where GT

R is t h e der ivat ive of gain with respect t o i npu t signal ampl i tude, and A η is the ampl i tude of t h e high-frequency signal a t t h e input .

T h e asymptot ic frequency response diagram is found in Fig. 6(A) and t h e corresponding phase diagram in Fig. 6(B). F rom Fig. 6(B), i t can be seen t h a t t h e nonminimum phase zero a t (K/l—K) ωα

in equat ion 13 causes an ext ra 90 degree phase lead a t low frequencies. As Κ increases, t h e level of t h e zero-slope asymptote a t high frequencies gets lower, and t h a t of t h e a sympto te a t low frequen­cies gets higher unt i l K= 1/2, when bo th asymptotes reach t h e same level.

If feedback is applied from ei ther t h e plates of t he power tr iodes, or t h e second­ary ôf t he o u t p u t transformer, a round the gain variable s tage and o ther preced­ing stages, t h e nonlinear effects m u s t either be t aken into account or effectively eliminated. T h e feedback amplifier would certainly be stable a t low-signal ampli tudes, if designed b y considering only the linear transfer function in equa­tion 4. However as t he signal ampl i tude

P R E A M P L I F I E R

P H A S E - S P L I T T E R

A N D

T E T R O D E

D R I V E R S

0 . 0 6 / i F

F i g . 7 . Se tup for e x p e r i m e n t a l tests

1 - K y ( L O G S C A L E )

( B )

Fig . 8 . A m p l i t u d e a n d phase response o i p o w e r stage

is raised, t he extra loop gain and phase lead a t low frequencies can eventual ly cause t he amplifier to become unstable and exhibit a t ype of motorboat ing action in addit ion to t h e reproduction of t he applied signal.

T h e next section gives a historical no te concerning t h e experimental work performed on a part icular amplifier and also t h e results of experimental tes ts using signals containing a high-frequency component and a low-frequency com­ponent .

Experimental Results

I n recent work wi th high-power audio amplifiers, t h e baffling t y p e of instabil i ty described in t he preceding section was noticed. Loop gain tes ts wi th single frequency signals revealed t h a t t h e amplifier should have a wide margin of stabil i ty when feedback was applied. However, t h e motorboat ing type of instabil i ty occurred when t h e signal ampl i tude was raised above a certain level, after feedback was applied.

Chronologically, t h e experimental work wi th a low-frequency voltage superposed on a high-frequency voltage was per­formed before t h e theory in t h e pre­ceding section was developed. T h e ex­perimental se tup is shown in Fig. 7.

T h e double frequency inpu t was applied t o an early s tage of t h e amplifier. T h e grid voltage of t he upper E imac 3 X 3000 power tr iode was applied to

7 0 0 Usher, Jr.—Nonlinear Low-Frequency Instability in Audio Amplifiers JANUARY 1958

I

0.2 L 0.2

4 0 0 ~ OUTPUT

100 V f —< )~ο\Γ

0.5 I 2

FREQUENCY ( C P S )

3

2 7 Γ

F i g . 9 ( l e f t ) . E x ­per imenta l a m p ­l i t u d e responses for p o w e r stage, 2 - f r e q u e n c y i n ­

put

F ig . 1 0 ( r ight ) . E x p e r i m e n t a l phase responses for p o w e r stage, 2 - f r e q u e n c y i n ­

put

9 0 °

45°

4 0 0 O U T P U T

0 .2 0.5

FREQUENCY ( C P S )

a low-pass filter and the filter o u t p u t applied to t h e horizontal channel of an oscilloscope. T h e ou tpu t voltage from the amplifier was applied to an identical low-pass filter, and the filter ou tpu t applied to t he vertical channel of the oscilloscope. T h e identical low-pass filter sections allowed gain-phase measure­ment s for t h e final stage to be m a d e directly on t h e oscilloscope for the low-frequency portion of the signal only.

T h e theoretical result for the transfer function of the final stage is found by dividing equat ion 13 by — GT s/s+a)g.

t hen

A.

tB $2A - f t * 1 - * ' - * * ( 1 4 )

L S

Equat ion 14 yields the asymptot ic gain diagram in Fig. 8(A) and the phase diagram in Fig. 8(B). The experimental results for the frequency response of the low-frequency component for various levels of t he 400 cps (cycles per second) high-frequency component are shown in Figs. 9 and 10. F rom Fig. 10, the phase curves a t low frequencies are seen to rise above 90 degrees. I t is believed t h a t this is caused by the low-frequency effect of the internal resistance and magnetizing inductance of the ou tpu t transformer.

T h e variable Κ can be calculated from the experimental results by not ing t he frequency, ω / , for which there is 45-degree phase shift.

F rom Fig. 8,

K-- ω0 (16)

Κ \ - K C

(15)

T h e values of Κ de termined by equat ion 16 are p lo t ted as a function of t he 400 cps o u t p u t voltage in Fig. 11. T h e value of Κ is near ly linearly dependent upon t he 400 cps o u t p u t voltage, as it should be.

F u r t h e r suppor t of the demodulat ion theory is afforded by the fact t h a t chang­ing t he frequency, b u t no t t he ampli tude, of the high-frequency component did no t affect the gain-phase characteristics of t he low-frequency signal. Experi­mental ly, value of the high frequency was varied from 60 to 2,000 cps wi thout changing t he results.

Conclusions

I t has been shown t h a t t he combina­tion of variat ion in gain wi th operat ing point in early amplifier stages and a demodulat ion phenomenon in t h e class Β power stage can cause stabil i ty prob­lems when feedback is applied. One solution for combat ing t h e stabil i ty problem is to reduce t he value of Κ for a given ampl i tude of high-frequency signal. This essentially means t h a t t he variat ion in te t rode gain with operat ing point m u s t be reduced.

F rom the s tandpoin t of practical circuitry, a reduct ion in gain variat ion is accomplished b y inserting a common cathode resistor in the te t rode stage of

0 . 1 5 y ?

>

0 1 0 0 2 0 0 3 0 0 4 0 0 4 0 0 CPS OUTPUT VOLTAGE

( RMS VOLTS)

F ig . 1 1 . N o n d i m e n s i o n a l factor Κ as a funct ion of the h i g h - f r e q u e n c y ou tpu t vo l tage

Fig. 1. F rom Fig. 4, t he gain of TiA is high when t h a t of TXB is low and vice-versa. T h e cross-feed caused b y the common cathode resistor tends to make the gain per side more nearly uniform, and, fur thermore, does no t reduce the over-all gain for push-pull operation. T h e ca thode resistor can also be used to provide bias for b o t h te trodes. This technique m a y also be used on other variable gain stages in t h e amplifier.

References 1. T H E O R Y A N D APPLICATIONS OP ELECTRON T U B E S (book), Herbert J. Reich. McGraw-Hill Book Company, Inc., New York, Ν. Y., 1944 , pp. 2 2 3 - 3 4 8 .

2. S E R V O M E C H A N I S M S A N D R E G U L A T I N G SYSTEM D E S I G N (book), H. Chestnut, R. W. Mayer. John Wiley & Sons, Inc., New York, Ν. Y., 1 9 5 1 .

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