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Applied ElectronicsA First Course in Electronics,

Electron Tubes, and Associated Circuitry.By the Members of the Staff of the

Department of Electrical EngineeringMassachusetts Institute of Technology.

12. USE OF AN IDEAL OUTPUT TkANSFORMER FOR IMPEDANCEMATCHING

Advantages of adjustment of the load resistance to suit the tube aredescribed in the preceding articles. In practice, however, an arbitrarychoice of the load resistance to realize these advantages is usually notfeasible because, for example, the load may be a device already available,or one whose design involves inherent limitations of resistance. Hence, inamplifiers, output transformers are generally used between the tube andthe load. The characteristics of such transformers are described in thevolume on magnetic circuits and transformers. From the considerationsof Art. 11, it is apparent that a value of load resistance equal to the plateresistance is not desirable when maximum power with a prescribedamount of harmonic generation is wanted, and that a transformer ratioto cause the actual load resistance to have an apparent value in the platecircuit of about twice the plate resistance of the tube is-needed when thetube is a triode and the quiescent plate voltage is specified.

The circuit diagram for an amplifier with an output transformer ofturns ratio a and a resistance load, is shown in Fig. 28a. If the trans-former is assumed to be ideal, the path of operation on the plate charac-teristics is as shown in Fig. 28b. Because the ideal transformer has nolosses, the windings have no resistance, and the quiescent operating pointQ has an abscissa EbOthat equals Ebb-In an actual transformer, thequiescent point lies somewhat to the left of this abscissa by the amount ofthe direct voltage drop in the primary winding; that is, on a line throughpoint (Ebb,0) with a slope of - (l/Rdc), where Rdcis the direct-currentresistance of the winding in series with the plate battery (see Art. 4,ChI IX, for a somewhat similar condition in the resistance-capacitance-coupled amplifier). The path of the operating point on Fig. 28b is alonga load line having a slope - (1/42RL), since the apparent impedance asviewed from the tube into the transformer is 0,2RL Thus the resistancesused for the determination of the direct-current and alternating-currentconditions on the plate characteristics are different.

If the path of operation remains in the linear region of the plate charac-

+

Arl. /2] OUTPUT TRANSFORMERS FOR..lMPIIDANCB MATCHING 431

teristics, the equivalent circuit for alternating components is that shownin Fig..28c, which may be further simplffiedto that of Fig..2&1becauseof the impedance transformation property 'ofthe transformer. For ma:d-

+ +~jl*I'I'I'~~~-4---"""""'_ [1111111(1+

Ea: B""

(a)

0:1(c)

R"

(d)Flc.. 28. Linear ClaasAt triode operation with an ideal output transformer ..

mum power output 'Witha prescribed amount of harmonic generationand a prescribed quiescent plate voltage, the considerations of Art. 11showthat the turns ratio of the transformer should be

/I = ~~"z. [189]RL

The output transformer serves a threefold purpose; namely, (a) itmakes possible the realization of the conditions for maximum power

/-32 CLASS A SINGLE-STAGE AJfPUFIERS lei. VIII

output with almost any tube and load, since its turns ratio can be selected;(b) it eliminates J2.,RL,the direct-current component of power in theload, because 1'bis confined to the low-resistance primary winding of thetransformer and does not exist in RL; and (e) it serves as an electricalisolator between the tube and the load circuit, since with it the potentialof anyone point in the load circuit may be given any desired value with-out regard to potentials of points in the tube circuit.

13. PAR.AI.LEL OPERATION; CLASS AlOne method of obtaining a power output greater than that obtainable

-.-~. 2i.

t --::-I"D It+ '. NI ~ R,.)1111166

(a)

(b)FIc. 29. Circuit diagram and equiva1erttdrcuit for two identical triodes

toDDected in parallel

from a single tube is the use of two tubes connected in parallel. Anothermethod, which has some advantages over the parallel connection, isdiscussed in Art. 14.. When two identical tubes are connected in parallel, the total plate-

~,.t. 13] PARALLEL OPERATION; CLASS At 433

circuit current for particular values of the interelectrode voltages isdouble that of either tube. The circuit diagram is that shown in Fig. 29a.The equivalent circuit for varying components of current and voltagewhen the operation is restricted to the linear region of the plate charac-teristics is that shown in Fig. 29b. The equivalent tube that gives thesame performance as the two tubes in parallel is one with a plate resist-ance equal to one-half the plate resistance of either tube and a platecurrent double that of either tube. The ideal output transformer reflectsthe load resistance RL into the plate circuit of the tubes as the value(N1/N 2)2RL

The operating points for the two tubes move simultaneously along thesame path in the same direction on the plate characteristics. The analysisof Art. 11 then applies to the equivalent tube exactly as it does to eitherof the actual tubes. For a sinusoidal grid-signal voltage eQ) the harmonicgeneration caused by nonlinearity in the tubes comprises both odd andeven harmonics of the grid-signal-voltage frequency in the total plate-circuit current and the plate voltage.

The maximum power output with a prescribed amount of harmonicgeneration and a prescribed quiescent plate voltage for the two tubes istwice that for a single tube, and the load resistance that corresponds tothis output is one that is about twice the plate resistance of the equivalenttube. Thus, for maximum power output with a prescribed amount ofharmonic generation and a prescribed quiescent plate voltage,

[190]

and the transformer ratio is

[191.]

14.. PUSH-PULL OPERATION; CLASS A lA method of connecting two tubes that is often preferable to the parallel

connection discussed in the previous article when increased power out-put is desired is shown in Fig. 30. The tubes are connected so that the~late current in one tube decreases when that in the other tube increases.This type of connection is commonly called the push-pull connection.The push-pull connection has numerous advantages over the parallelconnection, one of the most important being the elimination of even-harmonic generation. As a result, the maximum p~wer output with aprescribed amount of harmonic generation is greater than that from twotubes in parallel, and the push-pull circuit is extensively used not onlyfor Class A operation but also for Class B and Class C operation.

434 CLASS A SINGLE-STAGE AMPUFIERS [Ch. VIII

It is assumed in the following analysis of the push-pull amplifier that:First, the operation is restricted to the negative-grid region of the tubecharacteristics and consequently the grid current is considered to benegligible; second, the transformers are ideal; third, the load is a pure

I +e,+ Ie. II

Eue'11

...

+

t't

+

1+

. +-t~

+

ell p

N1

11(11111 Ebb N'l

'(.e'p

.f +1l"

and

and

Also

FIG. 30. Push-pull connection of two triodes.

resistance; and, fourth, the tubes have identical characteristics. Primedsymbols are used to distinguish the quantities in one tube from those inthe other, but the arbitrary assignments of direction and polarity aremade symmetrically with respect to the common-cathode point on thediagram. The effects of nonlinearity on the alternating-current operationare neglected at first; later they are taken into account.

14a. Determinationoj QuiescentOperatingPoint. - The determinationof the quiescent operating point Q on the plate characteristics is made inthe same manner as for a single tube in Art. 12, Fig. 28b. Thus when

el = 0, [192]e, = e~ = 0, [193]

e, = e: = Ecc [194J

t p = e~ = 0 [195]

eb = e = EbO = Ebb. [196JThe quiescent plate currents 106 and l~o are therefore equal in the twotubes and correspond to the point on the plate characteristics at the volt-age co-ordinates given by Eqs, 194 and 196. The total quiescent platecurrent through the plate-power supply or battery is thus twice thecurrent for one tube, and, as far as the quiescent operating conditionsare concerned, the tubes operate in parallel.

Arl. ill PUSH-PULL OPERATION; CLASS Al 435

The windingdirections in the output transformer with its center-tappedprimary winding are shown in Fig..32. The dots used to indicate polarityhave the significance that a late of change of flux in the core that makesone of the dot-marked coil ends instantaneously positive with respect tothe corresponding unmarked end also makes the other two dot-markedcoil ends instantaneously positive with respect to their respective un-marked ends. For the zero-grid-signal condition

[197]the magnetomotive force in the upper winding of N 1 turns tends to sendflux in a counterclockwise direction in the core, and that in the lowerwinding of N 1 turns tends to send flux in the clockwisedirection. The net

p-.

.. t p

---:""If

+ II No. R,e l Nt

.t~p

--pi

-i p

FIG. 31. Equivalent circuit for identical tubes with the varying components of current andvoltage restricted to the linear region of the tube characteristics.

magnetization of the core resulting from the quiescent components of theplate currents is therefore zero. This cancellation of magnetization is oneof the principal advantages of the push-pull connection over the single-tube or parallel connection. For a given power output and amount ofharmonic generation caused by nonlinearity in the transformer, thetransformer in a push-pull amplifier may be lighter in weight and lessexpensive than the one in a parallel-tube amplifierJ because it is notnecessary to provide a large core with an air gap to prevent the magneticsaturation and resulting waveform distortion caused by the averagecomponent of plate current. The effect of a direct current superposed onthe alternating current in an iron-cored coil is discussed in the volume OPmagnetic circuits and transformers.

14b. Operationin a Linea' Region.- For a smaIl grid-signal voltagewhen the path of the operating point is restrictedto thelinearregionof thetube characteristics,the operation for varying components of current and

436 CLASS A SINGLE-STAGE AMPLIFIERS [en.VIIIvoltage may be obtained through use of the equivalent circuit for thetube. The total plate current and voltage are then obtainable throughsuperposition of the quiescent and varying components. The equivalentcircuit for varying components is that shown in Fig. 31.

If the input transformer is wound in a manner similar to the outputtransformer of Fig. 32, with equal numbers of turns on the two halvesof the center-tapped secondary winding, then

e~ = -eo' [198]Because of the linearity and symmetry of the circuit, it follows that

i~ = -i p ' [199]Thus the total current through the plate-power supply or battery is

i b + ib= loo + i p + 1bO - ip [200]= 2l so = constant [201]

and contains no varying component. For this reason, it is not impor-tant that the plate-power supply have low internal impedance whenoperation is restricted to the linear region of the curves. Also, if aself-bias resistor is used (see .Arts. 4 and 11, Ch. IX), no by-pass capacitor

is required to prevent fluctuationsin plate current from affecting thegrid voltage during strictly linear_.

1.1 operation.The ideal output transformer and

load introduce an apparent resistanceequal to 4 (N 1/N2)2RL between points

FIG. 32. Winding directions in the out- P and p' in the equivalent circuit ofput transformer. Fig. 31 because of the impedance-

transformation property of the trans-former. The equivalent circuit including this apparent resistance isthus that of Fig. 33a, where the center connection shown dotted isunnecessary, because no varying component of current exists in it. Fromthis circuit, the apparent load resistance for each tube Is seen to be2(N1/N2)2R L t or half the plate-to-plateresistanceR p p , where

(Nl)2Rl'l' = 4 N2

RL ~[202]If one tube were removed from its socket in the circuit of Fig. 30, thesecond tube would then have an apparent load resistance of (N1/N 2 )2RL Re-insertion of the first tube would therefore double the effective loadresistance of the second tube. This reaction of one circuit on the otherwould not occur if the output transformer were eliminated and a center-tapped load resistor were used. Thus it may be concluded that the effect

Arl. 11] PUSH-PUll OPERATION, CLASS Al 437

is associated with the autotransformer effect or coupling between thetwo plate circuits by the output transformer.

This analysis indicates that for small grid-signal voltages the loadline for the path of operation of one tube on the plate characteristics inthe linear region passes through the quiescent operating point, as shownin Fig. 28, but has a slope of

1

01(c)

(b)

(3)

-i,

- 2ip

-~-----------~--~

However, since the analysis here is restricted to operation in the linearregion of the tube characteristics,it must not be inferred that themaximum power output as well asthe corresponding optimum loadresistance and harmonic genera-tion can be obtained from thisload line by the methods of Arts.8 and 11. The reaction of the sec-ond tube through the transformeraffects the path of operation in thefirst, and it is shown subsequentlythat over an extended region thepath of operation is nol a straightline on the plate characteristics ofeither tube, even though RL isa pure resistance and the outputtransformer is ideal.

The circuit of Fig. 33a reducesto that of Fig. 33b ..This diagram.shows that as far as varying com-ponents are concemed the twotubes are in series but, as was pre-viously stated, they are in parallelas far as quiescent components areconcerned.

However, the foregoing is not (d)the only possible point of view" FlO. 33. Equivalent circuits for the linealThe circuit of Fig. 33c is equiva- Class AJ push-pull amplifier.lent to Figs ..33a and 33b for powerconsiderations alone; it is not equivalent for the voltage and current atthe load resistor. Also, Fig ..33c is the equivalent circuit including the ap ..parent resistance offered by the transformer and load in Fig. 33d. Figure

/.38 CLASS A SINGLE-STAGE AMPliFIERS [eh. VIII

33c is iden tical with Fig. 29b; thus it may be stated that the Rush-pull COD-nection is equivalent, as far as varW-g components and power considera-tions are concerned, to two parallel-connected tubes op.erating into one-halfthe center-tapp~d winding of the outp.ut transformer. Either of the fore-gQing alternative concepts may' be useful, but they are app'licable only_to operation over the linear re~on of the tube's characteristic curves.

14c. Operation over a Range Extending beyond the Linear Region.-When the path of operation extends beyond the linear region of the platecharacteristics., harmonics are generated in the plate current that are notpresent in the grid-signal voltage. For example, if the grid-signal voltageis sinusoidal and expressible as

eg = ViE, cos wt, [203]the plate current under conditions of no harmonic generation is

it;= I bO + ViI l' cos wt; [204Jbut with harmonic generation it is expressible as the Fourier series,

i b = I bo + [po + Vi(I 1'1 COSwt+ Ip2cos2wt + Ip 3 cos 3wt + .. '), [205]as is demonstrated in Art. 8. Because of the symmetry in the push-pullcircuit, the plate current in the second tube is similar to Eq, 205, but'with wt replaced by 6)t+ 180. Thus

i~ = I bO + 11'0+ V2[I pi cos (wI + 180) + 11'2cos (2wl + 360)+ I p3 cos (3wt + 540) + ..] [206]

= I bo + 1p o + Vi[ -11'1 cos wt + 11'2 cos 2wt - I p3 cos 3wt+ ...J. [207J

One feature assumed in an ideal transformer is that the exciting currentis negligible, which is equivalent to the statement that the magnetomo-tive force required to magnetize the core is zero. Consequently, the sumof the magnetomotive forces caused by currents in the windings is zeroin a given direction around the core; and in a two-winding ideal trans-former the current ratio is the inverse of the turns ratio. In the idealpush-pull output transformer, the sum of the magnetomotive forces in agiven direction around the core caused by currents in the three windingsis zero, just as in the two-winding transformer. Thus, for the transformerin Fig. 32,

[208Jor

[209]

Art. Jj} PUSH-PULL OPERATION,' CLASS Al 439

A rigorous analysis of the push-pull amplifier requires a consideration 1.of the finite magnetizing impedance and also of the leakage reactancesamong the three windings of the output transformer, but their effectsare neglected here.

Substitution of Eqs. 205 and 207 in Eq. 209 gives the current in theload as

i 2 = 2 ~~ V'2(I p 1 cos wt+ I p 3 cos 3wt+ -). [21OJThus the effect of the symmetrical arrangement is to cause a cancellationof the average components, the second harmonics, and all other evenharmonics generated within the tube. However, if the grid-signalvoltageis nonsinusoidal, all frequencies present in it, including even harmonics,are amplified as usual. The absence from the output signal of componentsresulting from even-harmonic generation is one of the advantages of thepush-pull connection over the parallel connection of tubes.

The current through the plate-power supply or battery is the sum ofthe currents through the two tubes. From Eqs. 205 and 207, this isCurrent through plate-powersupply = i b + i~

= 21b{)+ 211'0+

140 CLASS A SINGLE-STAGE AMPLIFIERS [Cn.VIII

Compositetube

21'l"" : wt

I , I V " I('(I 1/ '\. I

---1------,/1 --------" c: / I',? ,/ I

,-_....... IItII ,- ......I ;' "I /1//

c

o

i 2 contains only odd harmonlcs.P as was deduced analytically in Eq. 210.A graphical analysis for the path of operation on the plate charac-

teristics over an extended range is not readily made for an individualtube, because of the couplingbetween the plate circuits ofthe two tubes through the out-put transformer. However, asis shown subsequently, the op-eration of the circuit can berepresented graphically by con-struction of the plate charac-teristics for a composite tube,the composite tube being de-fined as one which, operatinginto one-half the output trans-former primary winding with

wt the other half open-circuited,gives the same current andpower in the load as the twotubes in push-pull. The path ofoperation on these composite

wl characteristicsis a straight line,and the methods of finding thepower output and harmonicgeneration given in Art. 8 are

FtG. 34. Waveforms in a push-pull amplifierwhen the operation extends into the nonlinear applicable. It is assumed againregion of the tube characteristics. in this analysis that the output

transformer is ideal, thus hav-ing no resistance, leakage reactance, exciting current, or losses, and thatthe load is a pure resistance. The circuit diagram is again that of Fig.30, and Eq. 209 applies to theplate circuit.

Equation 209 shows that theoutpu t current i 2 is the same asthe output current that wouldexist if an equivalent current

FIG. 35. Composite tube and circuit equivalentexisted in one-half the trans- to that of Fig. 30.former primary winding. Thusthe operation of the circuit in Fig. 30 is the same as that in Fig. 35, where

15 P. Franklin, Differential Equations [or Electrical Engineers (New York: John Wiley &Sons, 1933) I 65.

,4,:.141 PUSH-PULL OPERATION,' CLASS At 141

the composite tube has a plate current id and the relationships amongid' eb,and et:are yet to be found.

In functional notation, Eq. 213 becomesid(ecJ eb) = ib(ec, eh) - i'(e~, e'), [214]

where the two terms on the right-hand side of the equation represent thecharacteristics of the individual tubes. These can be combined inaccordance with the circuit restrictions as follows: Since the input trans-former is ideal,

thus[215]

[216JAlso, since the ideal output transformer acts as an autotransformerbetween the t\VOplate circuits, it makes

~[217JWith two separate load resistors substituted for the transformer and load,Eq ..217 would not be correct, and the entire analysis that jol1011JSis there-fore true only when the transformer is used. Equation 199, which was ob-tained in the linear analysis, does not apply to the nonlinear operationof the tube.

Since the resistance of the transformer is negligible,E bO = E~ = Ebb; [218]

thuse~ = Ebb+ e~ = Ebb - ep [219J

andeb = Ebb+ ep , [220]

whencee' = 2E bb - eo. [221]

Since the tubes are assumed to be identical, the function i b is the sameform as the function i~, but they are functions of different variables; thusthe prime may be dropped if the variables are indicated, and substitutionof Eqs. 216 and 221 in Eq. 214 gives

id(E cc +.eo, eb) = ib(E cc + eo,eb) - ib(Et:c- eg , 2Ebo - eb). [222]In this way J once the power-supply voltages Ebband Ect:are selected, theindependent variables are reduced to two, namely, ell and eo, and thecharacteristics may be graphically constructed as shown in Fig. 36, wherethe curves for zero grid-signal voltage e(Jare drawn. The plate-currentcharacteristic curve for the composite tube is obtained through rotatingthe plate characteristic for an individual tube through 180 degrees aboutthe origin, then displacing the curve along the axis of abscissas until its

142 CLASS A SINGLE-STAGE AMPLIFIERS [eh. VII]

new origin falls at the point at which eb is equal to 2Ebb on the originalscale, and, finally, subtracting the magnitudes of the ordinates of the twocurves. Thus the length of the ordinate Xy equals .xzminus xv in Fig. 36.Several features of the composite characteristics are at once apparent;namely, (a) the quiescent plate current in the composite tube is zero;

i

Compositecharacteristic for ell-O;,__ -i,,

Art. II] PUSH-PULL OPERATION; CLASS Al 443

The ordinates of this latter curve are then added algebraically to those 'ofthe curve for the particular positive value of grid-signal voltage, denotedby ib(Ecc + ea) eb), giving the left-hand dotted line with positive slope.Again, on this line, the length of the ordinate xvis subtracted fromthat of the ordinate Xi to give the length xy, and the resulting curveis the characteristic of the composite tube for the particular positivevalue of ego The construction of the right-hand dotted line, which isthe characteristic of the composite tube for the particular negative

IIIiI-~~r..r--~~-- 2Ebh-e.-- ......

-i b(Ecc- eg . 2E/J6- e. >

7/

II

I Z Compositecharacteristic for eg=O;i

CLASS A SINGLE-STAGE AMPLIFIERS [eh. VIII

is -SO volts, and the heavy dotted lines with positive slopes are the twoindividual tube characteristics for zero grid signal voltage. When theordinates of these curves are added algebraically, the result is the heavysolid line with positive slope, which is the composite characteristic for

Path ofoperation for

composite tube;1

slope> --(~)'ZRL~V'l

70

50

60

80 Ec:::

90 :-.'-

__ -Path ofoperation for

one tube

Type 45 tubesEbbte: 240 voltsEre = - W volts

ec .. 0 IJI

JIII

I/I

I

1'--+ ..................,.,:;.+-t1~lP4-~~~........~~~~--+--h...,.--+-~ 0 -ell in volts

10

.10090

8Q

70

CIS 60Ec 50,-

.......40

30IblJ20

10

0

1...--------------..:..-------'-100--- Compositecharacteristics----.Cha:raeteristi~ of individualtubes

FIG. 38. Composite characteristics for two Type 45 tubes in a push-pull circuit at specificvalues of grid-bias voltage and quiescent plate voltage.

zero grid-signal voltages. The light solid lines with positive slopes are thecomposite characteristics for lO-volt increments of grid voltage con-structed by the method shown in Fig. 37. The solid and dotted lines withnegative slope are discussed subsequently.

Not only the characteristic corresponding to the zero grid-signal-volt-

Ari.N] PUSH-PULL OPERATION,' CLASS Al 145

age condition but all the composite characteristics over a. wide rangeof grid voltage are essentially straight lines. The grid-bias voltage Etc,chosen in Fig. 38 as -50 volts, is one for Class A operation. In later dis-cussions of Class ABand ClassB push-pull operation, it is shown that thecomposite characteristics are not always straight for those operatingconditions. Note that the characteristics of the composite tube as definedhere are dependent upon the values of Ece and Ebb and thus depend onquantities external to the tube. The composite tube differs from an ordi-nary tube in this respect.

The composite plate characteristics from Fig. 38 are reproduced inFig. 39. As far as the current, voltage, and power in the load resistor are

o

2Ebh

FIG. 39. Plate characteristics of the composite tube with load line superposed.

concerned, the operation of the push-pull circuit is equivalent to that ofFig. 35, where the characteristics of the composite tube are given byFig. 39. The quiescent operating point is on the abscissa axis at ebequals

Ebb.where id equals zero, and a load line having a slope - (Nl)2- RLN2

gives the path of operation on the characteristics, as is shown by thesolid line of negative slope in Figs. 38 and 39. The waveform distortionthat occurs because of harmonic generation in the tubes may be obtainedfrom this load line by the methods of Art. 8, if i b is replaced by id, andnegative values of id are recognized. Because of the symmetry of thecomposite characteristics mentioned previously, no steady or even-harmonic components appear in 1,4or in the load current.

446 CLASS A SINGLE-STAGE AMPUFIERS [Ci. VIII

Although the plate currents in the individual tubes may decrease tozero and remain there for an appreciable fraction of a half-cycle, suchbehavior is not apparent on the composite plate characteristics; since theoperation along the load line is entirely symmetrical about the quiescentoperating point for positive or negative values of grid-signal voltage.However, the paths of operation for the individual tubes can be foundreadily by a process which is the reverse of the one by which the compositecharacteristics are constructed. Thus, in Fig. 38, a vertical line throughthe intersection at A of the path of operation and a particular compositecharacteristic intersects at B and C the two individual tube character-istics from which the composite characteristic is constructed, therebydisclosing the individual tube plate currents for a particular value ofgrid-signal voltage. By this method, the paths of operation for the indi-vidual tubes shown by the dotted curves with negative slopes are con-structed. They are curved, even though the load is purely resistive andthe output transformer is ideal.

For the particular conditions illustrated in Fig. 38, the individual tubeplate currents do not fall to zero when the range of operation is limitedby the tv....o curves corresponding to zero grid voltage on the two tubes;thus the operation is Class Ai. However, if the grid-bias voltage is chosensomewhat larger, the individual tube currents may be zero for an appreci-able fraction of the cycle, and operation changes to Class ABI, which isdiscussed in more detail in Ch. X. Figure 40 shows a limiting example forClass Al operation, since the individual plate currents in it just reachzero when the grid voltage of the opposite tube reaches zero.

The considerations that govern the maximum power output with aprescribed amount of harmonic generation from a push-pull amplifier arequite different from those for a single-tube amplifier given in Art. 11.Since the even harmonics generated in the tubes are canceled in the outpu ttransformer, the total generation of harmonics in the amplifier is smallerin the push-pull amplifier than in the single-tube amplifier when the tubeshave the same operating voltages and deliver the same power outputindividually. Consequently it follows that, for the same total harmonicgeneration in the amplifier, the maximum power output from each tubeis larger in the push-pull amplifier than in the single-tube amplifier. Theincrease may be as much as 50 per cent. This increased power output ismade possible by the fact that changes both in the operating voltagesand in the load resistance effective in the plate circuits of the individualtubes may be made under the specified conditions. Since the even bar-monies are canceled, the path of operation may be extended farther intothe lower region of the tube characteristics in a push-pull amplifier thanis indicated in Fig. 26, Art. 11, for a single-tube amplifier when a pre-scribed amount of harmonic generation is not to be exceeded. Accordingly,

Art.N] PUSH-PULL OPERATIONj CLASS Al 147

for the same amount of harmonic generation, a larger magnitude of grid-bias voltage and a larger grid-signal voltage amplitude may be used in

lOO,------:-':"r--------------,90 Type45 tubes

Ew;'"250volts80 E '" 55 voltsee70

~ 6Q.5 50...

'-40

30

ItIJ 20

10

0

Path ofoperation for

composite tubewhen

50 N tRL (N 1) "" 1020ohms,60 ~or Rpp"= 4080ohms

70

80 E.S

90 "":....

'-------------,.;......L..:.:..-----..LI00

FIG. 40. Composite characteristics for two Type 45 tubes in a. push-pull circuit for thelimiting condition of Class At operation,"

the push-pull amplifier than in the single-tube amplifier, and the poweroutput from each tube is therefore larger.

The change of voltages described in the preceding paragraph is onefactor that contributes to the increased value of maximum power outputin the push-pull amplifier. Another factor is the change that may be madein the effectiveload resistance for each tube. Since the composite charac-

This diagram is adapted from B.]. Thompson, 11 Graphical Determination of Performanceof Push-Pull Audio Amplifiers," I.R.E. Proc., 21 (1933), rig.8, p. 595, with permission.

#8 CLASS A SINGLE-STAGE AMPliFIERS [CA.VIII

teristics are symmetrical about the curve for zero grid-signal voltage, andare practically straight and parallel for Class At operation, the amountof harmonic generation is essentially independent of the plate-to-plateload resistance for a particular value of grid-signal-voltage amplitude.Consequently, the considerations of Mt. 11 and the result that the effec-tive load resistance for a tube must be approximately equal to twice theplate resistance of the tube for maximum power output with a prescribedamount of harmonic generation and a prescribed quiescent plate voltagedo not apply to the composite tube. Instead, the considerations of Art. 10apply, and the slope of the path of operation on the composite charac-teristics for maximum power output is equal to the negative of the slopeof the composite characteristics. The slope of the path of operation on thecomposite characteristics corresponds to (Nt/N 2)2R L , which is one-fourth the plate-to-plate resistance given by 4 (Nt/N 2)2RL Thus theplate-to-plate resistance should equal four times the plate resistance ofthe composite tube. The plate resistance of the composite tube is, how-ever, approximately one-half the plate resistance of the individual tubesat the quiescent operating point. The optimum plate-to-plate resistancefor the push-pull amplifier therefore is twice the plate resistance of theindividual tubes, and the optimum value of the load resistance effective inthe plate circuit of each tube hence is equal to the plate resistance of thetube. In accordance with Eq, 177, this condition results in an increase ofpower output from each tube over the value obtained when an effectiveload resistance equal to twice the plate resistance of the tube is used.

The discussion in this article applies to triodes used with an outputtransformer for delivering power to a resistance load. The use of thepush-pull connection as a balanced voltage amplifier without an outputtransformer is discussed in Ch. IX and Class AB, Class B, and Class Coperation is discussed in Ch. X.

15. SYMBOLS FOR VACUUM-TUBE CIRCUIT ANALYSISIn the preceding articles of this chapter a number of special SYmbols

are introduced and defined. A large number of SYmbolsare needed inanalysis of vacuum-tube circuits, because the operation is complicatedby the superposition of direct quantities and alternating quantities havingseveral harmonic components. Confusion is likely to result if a consistentset of SYmbolsis not defined and adhered to through all the analysis.The methods of circuit analysis given in the volume on electric circuitsare directly applicable to vacuum-tube circuits and may be used witharbitrarily assigned positive directions of currents and voltages, Theconsistent set of definitions adopted in this volume is merely one of theinnumerable possible sets. It is adopted because it is in substantial agree-

Art.15l SYMBOLS FOR VACUUM-TUBE CIRCUIT ANALYSIS #9

ment with the latest standards'" available; thus it is the set most likelyto be encountered by the reader in other publications in the future. Toeliminate one additional source of confusion, the definitions here areextended to include the assigned positive direction of the quantities. Ifsymbols defined for the directions opposite to those chosen here areneeded, other symbols can be used.

Table I summarizes the symbols for the electrical parameters of thetube and circuit and for some of the currents and voltages that do notenter into most of the problems. Table II gives the definitions and sym-bols of the current and voltage components that are fundamental to theoperation of triode circuits. In Table II, the first four rows contain sym-bols pertaining to the total quantities, and the fifth, sixth, and seventhrows, symbols which pertain to the varying components and are usefulin circuit analysis when harmonic generation is neglected. The last fourrows contain symbols useful in representing nonsinusoidal varying com-ponents as a Fourier series. Complex quantities are indicated by romantype.

TABLE ISYMBOLS FOR VACUU~I-TRIODE CIRCUITS

gp = plate conductance"1' = plate resistancegil = grid conductance'/1= grid resistancegrn = grid-plate transconductance (mutual conductance)

l ' fi . f debIJJ.= amp 1 cation actor = - -d .e~ I~ con~tanl

Cg p = grid-plate capacitanceCg k = grid-cathode capacitanceCpk = plate-cathode capacitanceEbb= plate-supply voltage rise from the cathode toward the plate

E cc or Ecd = control-grid supply voltage rise from the cathode toward the gridE~c2 = screen-grid supply voltage rise from the cathode toward the screen gridElf = filament or heater supply voltage (effective or direct value)E, = filament or heater terminal voltage (effective or direct value)It = filament or heater current (effective or direct value)I, = total electron-emission (saturation) current from the cathode

Tubes with more than one grid require additional symbols which aresupplied as follows'P:

"Generalized System for 1.l11tltigridTubes. The following scheme ofsymbols for multigrid tubes avoids the extension of letter subscripts andprovides a framework of symbols for tubes with any number of grids. Inthis system the grids are numbered according to position, the grid immedi-ately adjacent to the cathode or filament being No.1, the next grid No. 2t

III Stand4rds on Elutronics (New York: The Institute of Radio Engineers) 1938)t 11-14.

TABLE II. SYMBOLS FOR TRIODE CiRCUITS-

Narneand Directionof QuantiJy

CompOTUml Currentthroughthe Currentthroughthe Vollagedrop (JQ'O$$Vollagerisefrom Vollagerisefrom theloadin thecathodeto grid cathode to plou externalcircuit externalcircuit directionof pomiveImvardthegrid towardthe plate plate current

Instantaneous total value ec tb J( tb eLQuiescent value; steady

160value when varying com- Eco EbO leo ELOnonentof grid voltaze iszeroAverage value of the total Ee Eb It: lit ELouantitvInstantaneous maximum of Ecm E6ff1 lem Ibm ELmthe total QuantityInstantaneous value of the

eo e1' ..., 11' e. componentvEffective value of the vary- E, E1' If! 11' E.inR'comoonen tAmplitude of the varying E(1ffI Ep m 1(1m [pm s:componentAverage value of the vary- E,o Epo 100 11'0 E.oina component

,

Instantaneous value of the . .harmonic components Cob eQ2t , Cpt, ep 21 , J"h 1112, t 1'l 1 t p 2 1 e.1I es2, ...

Effective value of the har- E 1 E 2 ... E 1 E 2 ... 1011 lv'll' I ph 11'21... Est, E. 2, monic components til gl PIP IAmplitude of the harmonic Eg 1m ) Ef/2 ff1 1 E p 1m , Ep 2m l I glm, I ,,2m, I plm, I p2:m, Ed m , Ed m , components

en.VIII] PROBLEMS 451

etc. In designating the voltages or currents associated with a particulargrid, the symbols given on the preceding pages will be used with the gridnumber as a subscript .... Control-grid symbols are frequently usedwhere reference is not made to other grids. The number of the grid neednot be used in this case. It will be understood that, when no numberappears in the subscript, the reference is to the control grid."

It should be noted that one possible source of confusion lies in the factthat some of the symbols in the last three rows of Table II are also usedwith a different meaning for multigrid tubes. However, this does notlead to difficulty in any of the problems treated in this text.

PROBLE~IS

100,000ohms

FIG. 41. Triode circuit forProb.3.

400v

+

1. A triode having the plate characteristics of Fig. 7J Ch. IV, is used with a plate-supply voltage Ebb of 400 volts, a load resistance RL of 100,000 ohms, and a grid-biasvoltage E ee of - 3 volts. \Vhat is the quiescent plate current I bO?

2. A relay having a resistance of 1,000 ohms is to be operated by the plate cur-rent of a high-vacuum triode. If the available direct grid-signal voltage is 5 voltsand the relay closes at 30 rna and opens at 20 rna)which of the triodes wbose plate characteristicsappear in some" one manufacturer's literatureshould be satisfactory ? For each triode selected,specify the plate-supply and grid-bias voltages thatmust be used.

3. A triode has the plate characteristics givenin Fig. 7) Ch, IV, except that the grid-voltagescale is to be multiplied by ten - that is, the in-crement in grid voltage between adjacent curvesis 10 volts instead of 1 volt. The tube is connectedas shown in Fig. 41 with a 400-volt battery asa plate-power supply and a plate-load resistanceof 100,000 ohms. The resistor Ric is so adjusted that there is a voltage of 50 voltsbetween the grid and the cathode.

Find the quiescent plate current I bO, the quiescent plate voltage EM, and therequired value of Rs,

4. The plate current of a particular triode is satisfactorily given by the expression

i&= 17 X 10-6 (ec + e;)\.7amp.where " and eb are in volts.

(a) Determine the plate current ib corresponding to a grid voltage ec of -15volts and a plate voltage eb of 200 volts.

(b) Find the dynamic plate resistance ,~ and the mutual conductance gm of thetube at tbe operating point specified in (a).

(c) If the tube is usedas a Class Al voltage amplifier with a load resistance 0110,000 ohms and a grid-bias voltage Eco of -15 volts, what plate-supplyvoltage Ebb is required to produce a quiescent plate current equal to thatdetermined in (a)?

(d) Determine the voltage gain of the amplifier for the conditions in (c).