seven-league oscillator

PROCEEDINGS OF THE I.R.E.

the radio transmitter, sends out a carrier from themobile unit, and brings in the mobile service operator.After the called party is on the line and the propercharges are deposited in the coin box, the operator com-pletes the connection. At the conclusion of the call,when the customer hangs up, the radio transmitter isturned off and the equipment restored to normal.

CONCLUSION

Experiments so far have indicated that radio trans-mission between moving trains and Bell System mobilefacilities along rail routes is generally satisfactory forpublic telephone service. A number of installations re-quiring an attendant on the train for their operationhave been made. The public reaction to these installa-

tions has been favorable, but the railroads have foundthe cost of an attendant burdensome. Experiments us-ing coin-box operation to eliminate the attendant areunder way, and this method appears promising.

It should be noted, however, that a train telephoneservice of the type described, using either urban or high-way channels, requires base-station equipment suitablylocated along the route of the railroad and a radio chan-nel capable of handling the additional traffic. At thepresent time neither of these requirements is met formost of the railroad mileage in the United States. It ap-pears, therefore, that provision of train passenger serv-ice on any extended scale will be dependent upon theavailability of frequencies and the addition of the nec-essary base-station equipment.

Seven-League Oscillator.*F. B. ANDERSONt, SENIOR MEMBER, IRE

Summary-A bridge-type RC oscillator is described which iscontinuously adjustable over a frequency range of 20 cps to 3 mc inone sweep of a two-gang linear potentiometer control. Tracking re-quirements of the two-gang control are not severe. The output isavailable in four phases, and the frequency is an approximatelylogarithmic function of the linear potentiometer setting. Practicallimits of the frequency range are tentatively 0.01 cps and 10 mc.Accuracy of setting of the order of one per cent is attainable withordinary components. Frequency stability is of the order of 2 per centper db of tube gain variation.

HXERETOFORE a continuous sweep of a wide fre-quency range with a single oscillator control hasbeen difficult to realize. Frequency bands of 3 to

1 ratio are obtainable with fixed-inductance variable-capacitance tuned oscillators for a 9 to 1 variation ofcapacitance. Still wider ranges are available withvariable inductor tuning, and with variable resistance-capacitance tuning in configurations such as Wienbridges,' bridged-T circuits2 and double-T circuits.3Continuous ranges of perhaps 1,000 to 1 may be ob-tained with variation of both resistance and capacity.Still other methods of achieving wide frequency rangeshave been presented in different types of phase-shiftoscillators.45The heterodyne type of oscillator alone has been

capable of frequency variation from the order of 1 cpsor less to 1 mc or more in one continuous sweep. Because

* Decimal classification: R355.914.3 XR355.911.5. Original man-uscript received by the Institute, March 16, 1950; revised manuscriptreceived, October 26, 1950.

t Bell Telephone Laboratories, Inc., New York, N. Y.I W. R. Hewlett, United States Patent No. 2,268,872, January 6,

1942.2 P. G. Sulzer, "Wide-range RC oscillator," Electronics, vol. 23,

p. 88; S&ptember, 1950.3 H. H. Scott, "A new type of selective circuit and some applica-

tions," PROC. I.R.E., vol. 26, pp. 226-235; February, 1938.4 G. Willoner and F. Tihelka, "A phase-shift oscillator with wide-

range tuning," PROC. I.R.E., vol. 36, pp. 1096; September, 1948.6 M. E. Ames, "Wide-range deviable oscillator," Electronics, vol.

22, p. 96; May, 1949.

the output frequency at the low end of the range de-pends on the small difference of two large quantities,the stability is poor at low frequencies. Low frequencyoutputs also require elaborate precautions to preventlocking into step of the beating oscillators.A considerable simplification of control and extreme

widening of the frequency-band ratio is possible with atwo-bridge circuit, in which three arms of each bridgemay be resistive and the fourth arms are RC combina-tions capable of covering about four octaves for eachtwo pairs of elements included, one pair in each bridge.Such bridges are readily designed to provide a nearlyconstant transmission at the desired frequency settingsover wide ranges of adjustment. The frequency range islimited mainly by the tube gain available.A simple version of an oscillator based on such

bridges is shown in Fig. 1. The RC arms are shown asmade up of two sections in series, but may be extended

Fig. 1-Basic wide-range variable oscillator.

to eight or ten sections each to provide frequency rangesof a billion or more extreme ratio.

1951 881


Tuning is accomplished by adjustment of the two-gang potentiometer. Two outputs are shown, which aresubstantially in quadrature.

Fig. 2 shows a schematic of an oscillator which coversa tuning range of 20 cps to 3 mc. This is a simplifiedpreliminary model. The RC networks shown in thebridges have six sections. The bridges shown are alike;it is not essential nor even desirable that both bridgesbe alike, as will be shown later.The thermistor shunted across the second bridge in

the plate circuit of tube VI serves to limit the oscilla-tion amplitude. The thermistor and RC network in theoutput circuit of tube V4 provide additional regulationof the output to a power amplifier.A resistance potentiometer across the plate voltage

supply provides a positive bias to the tube grids in orderto permit large resistances in the cathode returns to im-prove frequency stability.

Fig. 3 shows the dial calibration of the oscillator ofFig. 2. The measured points oscillate slightly about asemilogarithmic asymptote over the low frequencyrange. The deviation from the straight line asymptoteat high frequency is caused by parasitic capacitance, butit serves to spread out this part of the range somewhat.

PRINCIPLE OF OSCILLATOR

The oscillator is based on the dissymmetrical bridgeof Fig. 4. Three arms of the bridge are resistive. Thefourth arm is composed of an RC network which pro-vides over the whole frequency range a nearly constantreactive component of transmission to the output of thebridge. The resistive component of transmission throughthe RC network varies with frequency, and is balanced

itz0

a)

11 - -- - - -

90C

8C - -

70

50 -f _ _-

2C- -- --20 gX X

C-

- -

10 31OO-0 01 .-a 3 MC- 10'10 30 R2 103 104 105

FREQUENCY IN CYCLES PER SECOND

Fig. 3 Dial calibration.

at the desired frequency of oscillation with equal resis-tive component from adjustable arms on the other sideof the bridge. This leaves only the reactive com-ponent at the frequency of oscillation. Ninety-degreephase shift between bridge input and output results.

Fig. 4-A type of bridge useful in an oscillator.

Fig. 2-Seven-league oscillator, 20 cps-3 mc.

A ugust882

10'r 3 MC 1077

Anderson: Seven-League Oscillator

The output of the bridge is split nearly equally be-tween the grid and cathode of one tube, and the cathodeand grid of the other tube, because of the large cathodecoupling impedance employed. The outputs of the twotubes are nearly equal, and of opposite phase. One out-put of the required phase is connected to a secondbridge, which is adjusted along with the first by a gangcontrol, to provide a phase reversal and an additionalninety-degree phase shift. One of the outputs from thissecond bridge is fed back into the first bridge to com-plete the oscillator loop. The two quadrature phaseshifts and the phase reversal provide the 3600 or 0°phase shift required at the frequency of oscillation. Thetheory outlined is idealized, and can be approximatedwell enough if attention is given the networks andparasitics comprising a practical oscillator loop.

BRIDGE DESIGN

The design of the adjustable bridge is based on circlediagrams of transmission through RC networks.

Fig. 5 shows a resistance R, tapped at aR and abR.The portion (1-a)R is bridged with a capacitance C.The transmission ratio is

a + jVl ~fc- b where f,= . (1)e + f 27rCR(l - a)a

(i-a)R c

a-b)R =

-L

f,cV

a - I

ab I

/ I

0

Fig. 5-Transmission to a resistance through an RC network.

Equation (1) is of bilinear form, and gives rise to a

semicircular plot in the complex plane as in Fig. 5. Thesemicircles have been shifted along the imaginary axisto emphasize the scale factor b.

If the resistance aR is tapped at abR and the portiona(1-b)R is bridged with a capacitance C1, as in Fig. 6,we have for the transmission ratio

b+j-1

_ fl, where f - (2)

v 4 * f 27rCjaR(1-b)b1 +j±3

- .,

If the critical frequencies f, and fc, are chosen so thatfC1»f,c the transmission ratio for the combination ofFig. 7 tends to approach the semicircle between ab andb in the region near and below fc,, and the semicircle be-

tween b and 1 in the region near and above f',. Betweenf, and f,1 the transmission ratio swings gradually fromone semicircle to the other.

a (l-b) R C.

abR v

__- -1 A

a fc,a v

ab ,, b

/

0 1


'fc fc Ia

ab

(1-a)R c ,, b

e a(i-b)R c, e =e

V

abR vZ ,f,

of=o1f=@


If fel is chosen such that f', = 16fC, more or less, thetransmission envelope will be a smoothly flowing en-velope of the two semicircles of Fig. 7.

If the factors a and b are chosen so as to provide semi-circles of equal diameters, the reactance component ofthe over-all transmission will be nearly constant be-tween f, and f,1.

Several RC combinations can be connected in series toextend the frequency range as in Fig. 8. Design relationsfor the RC network will be discussed later.

BRIDGEX

Fig. 8-Output voltage from an adjustable bridge.

If a variable frequency voltage e is connected to theRC network in Fig. 8, a voltage to be defined as Ne willbe developed across the shunt arm of network. Thereactive component of Ne will be nearly constant over awide band of frequencies.

If a variable control, such as a resistance potentiom-eter, is connected to the same generator of voltage e,a voltage to be defined as Pe will be developed betweenthe slider and the generator return. The control can beadjusted to obtain a voltage Pe equal to the in-phasecomponent of the voltage Ne at any desired frequency.

1951 883


This adjustment will be different for every frequency.At the frequency of in-phase component balance theoutput voltage (Ne-Pe) will be only the reactive com-ponent of Ne as in Fig. 9. This voltage is in quadraturewith e, and nearly constant over a wide range of po-tentiometer settings at the corresponding frequencies ofbalance.

_ v

e Ne-PePef=o

Fig. 9-Output voltage from an adjustable bridge.

The output (Ne-Pe) will lead the applied voltage eby an angle decreasing from 1800 at zero frequency to00 at infinite frequency. Over the range of potentiometersettings between the extreme reactive peaks of the RCnetwork, the output (Ne-Pe) will be maximum forzero frequency, reduce to a minimum in the region ofthe 900 balance point, and increase toward anothermaximum at infinite frequency. At settings well beyondthe extreme reactive peaks of the RC network, as forfi orf2, the bridge output will exhibit a single peak withno dip to a minimum as in Fig. 10.

(a) (b)Fig. 10-Output voltage from an adjustable bridge. (a) Typical

medium potentiometer setting. (b) Extreme potentiometer set-tings.

The effect of parasitic capacitance will first be con-sidered for the potentiometer. The transmission through

the potentiometer side of the bridge is determined bythe tap k at low frequencies. At high frequencies theparasitic capacitance C in Fig. 11 bridged on the sliderand its connected equipment changes the transmissionas indicated in (3).

1 -jf-k(t - k)T! fce2

1 + fi)k2( I- k

Equation (3) also gives rise to semicircle transmissionplots as in Figs. 11 and 12. The curves for several fre-quency ratiosf/f, are plotted in Fig. 12 by constructingcircles for k(1-k)=0.25, 0.2105, 0.177, 0.1485, and so

+cj2K2G-K)2 K[I-i-LK(O-K)]V _ K [1 -j K (1 - K)] K -jEK(1 )

e [1+W2K2(i-K) O 1+(S-) K2 (1-K)

= 2TTf fc= 1 OR Rp=27TRpC 22Tfc_CREAL AXIS, K

Fig. 12-Vle voltage ratio diagram for resistance potentiometerwith capacitance between slider and ground.

forth (1.5-db intervals) and plotting intercepts withlines through the origin having slopes of these values.These lines represent v/e transmission ratios havingphase angles

= arc tan - k(l -k)

Table I in the appendix shows corresponding values ofk and k(I -k), over a range of four octaves, which areuseful in plotting the curves of Fig. 12.The effect of parasitic capacitance on the resistance

arm connected to the RC network is considered next.In the frequency range where this capacitance is effec-tive, the reactances of all the RC network capacitances

SEMICIRCLE FOR Cp=O

(3)

V i Kc- K - K)-

(fc

Ko tK 1

@ ~=arc tan ±-fK (1-K)

....(3)

-tane = f K(I- K)

KIi +tane

OR 2 + i -[K(I-K)]

Fig. 11 Transmission through a capacitancebridged resistance potentiometer,

cn2 (cp+cn)(t-n)(ab...mn)R

___n f7- Cn >nn-,f= /-->Cn+Cp

C, j1k jn p I

4=CD \,--

Fig. 13-Effect of parasitic capacitance onshunt arm of AS network.

Y4-te -K)RppKRP:

884 A ugust

Anderson: Seven-League Oscillator

but one, designated C,n in Fig. 13, are negligible. The as-sociated resistances may be considered short-circuitedfor practical purposes. This leaves the resistancesab ** m(1-n)R and (ab ... mn)R, bridged withcapacitance Cn and the parasitic capacitance C,.The capacitance C,, across the resistance (ab * m)

(1 -n)R will form with Cp a potentiometer of some lossat infinite frequency, so that transmission will never be-come unity. The transmission for these two RC sectionswill be represented by (4), which is of the bilinear form,indicating a semicircular locus.

v 1 + jc&,,R(ab ... mn)-=n -*(4)

e 1 + jC'.(Cn + Cp)R(l- n)(ab m**n)

If the infinite frequency transmission is less than the dctransmission, the semicircle swings below the real axis.We can now consider the RC arm and the capacitance-

shunted resistance arm connected to it. In the extensionof the process used for Fig. 7 we aim to divide a re-sistance R into the desired number of sections (n+1) asin Fig. 14. The first section lies between the top end of

aR, abR, and so forth, are shown in Fig. 14. The inter-cept p is set by trial, and may be readjusted if necessaryafter the completion of a trial design. The ratio of thetwo intercepts of the lowest frequency RC semicircle isab ... mn/b ... mn or a, of the next higher frequencysection is b, and so on. Equation (5) gives a convenientmethod of evaluating the factors a, ab, and the like,which is to divide all the semicircle intercepts into theproduct (ab . .. mn), which may be set as desired

ab * * mn- 1

ab . . . mn

ab** mnb = a

b ..mn

ab.. imn= ab * * * m

n

ab * * * mnL 1 = ab*

a, b, . . mi, n<1

imn

(5)

nn I mn n24fca 24(n-2)fca 24(n-1)fc

'fcb =fc =fcm

Fig. 14-Relations between transmission andelements of an RC network.

the resistance, at value R, and the tap, at value aR. Thesecond section lies between the taps at values aR andabR. The last tap lies between the tap at value(ab ... mn)R and the bottom end of the resistance, atvalue zero.The first step is to lay down the semicircles required

for the desired RC network transmission. These semi-circles may be equal, but, as shown later on, should betapered a bit.The relations between the real axis intercepts of the

component semicircles, n, mn, and so forth, and the taps

The critical frequencies are chosen, by proportioningof capacitances and resistances, to increase in geometricratio, as fc, 24fc, 28fc, 212fc, * e 24mfc. The capacitancevalues are determined by equating reactance at thedesired fc to the resistance facing the capacitance, as-suming all lower frequency RC sections to be shorted toa zero impedance generator, all higher frequency RC sec-tions to be pure resistances.The value of the lowest tap on the resistance R, at

value (ab . . . mn)R, should be in general less than thereactance of the shunting capacitance Cp at the highestfrequency of oscillation. It should be kept large enoughto sustain the gain required to overcome the bridge lossat the highest frequency. The total resistance R is notcritical, and is determined by the values (ab ... mn)Rand (ab mn) selected. It should be, in general, large,and may be set to fit with favorable values of capaci-tances as well as can be done.

Because the RC system shunts the paralleled po-tentiometer, as in Fig. 4, the impedance ZPL presentedto the tube driving the bridge falls as frequency in-creases. The transmission between the grid circuit ofthe driving tube and the output of a bridge with a con-stant reactive transmission will likewise fall. This fallingoff can be compensated by suitable shaping of the RCarm transmission as follows.The plate load impedance ZPL is approximately

RRp(ab..* mn)-

rZPL =

RRp + (ab - * -mn)-

r

where r is the transmission ratio of the RC system.The voltage gain of the tube driving the parallel com-

1951 885

(6)


bination will be gmZPL, where gm is the transconductanceof the tube. The transmission from the grid of the tubeto the output of the RC network will be NgmZPL, andshould be unity or slightly larger. This requires that

N.-gmZPL

This gives rise to a straight-line minimum boundary ofthe form

r 1 1

N = +- =

(ab * - * mn)Rgm Rpgm gmZPL

This is shown in Fig. 15.

_ NEcn

R+RpgmR Rp

1 -

gn]Rp0 0.25 0.50 0.75

TRANSMISSION RATIO, r

Fig. 15-Method of approximating required transmissioncharacteristic of RC network.

This straight line is the minimum requirement for N.The semicircles representing the RC network are fittedunder this line as a guide as in Fig. 16. Some marginshould be allowed for network deviations and gm varia-tion. The effective gm is reduced by about one half in thecircuit, as will be shown later.

z

Lowen a

5lnzz

gm Rp'I0 0.25 0.50

TRANSMISSION RATIO, r0.75 1.0

Fig. 16-Method of approximating required transmissioncharacteristic of RC network.

The low-frequency end of the guide line is bent up-ward as required to allow for plate-grid coupling net-work loss. The high-frequency end is bent upward toallow for parasitic capacitance losses.

POTENTIOMETER-RC NETWORK

BRIDGE TRANSMISSION

We are now ready to combine the transmission of theRC network and the potentiometer to derive the trans-mission through the bridge as in Fig. 17. The voltageoutput from the bridge (Ne-Pe) is determined, for a

particular setting of the potentiometer, by the vectordifference between the RC network output Ne and thepotentiometer output Pe, over the 0-oo frequencyrange. All settings of the potentiometer from k=O tok = 1 must be considered.The transmission characteristics of Figs. 12 and 14

(7)fl f f3 f4FREQUENCY

Fig. 17-Bridge output voltage (Ne-Pe) variation with frequenlcy.

are combined in Fig. 17 to evaluate the transmissionthrough the bridge. The curves of Fig. 17 show themagnitude and phase of this transmission.The transmission from the bridge through the tubes

is next considered. Fig. 18 shows the voltage relations atthe bridge output and the connected tube grid-cathode

1.0

LN = 9gmNVN

Ne

-z-

II ip = 9mPgVp

ePZ (9NVN+9PVP)

9mN = GRID-PLATE TRANS-CONDUCTANCE OFTUBE (N)

9N= GRID -PLATE + GRID-SCREEN TRANSCON-DUCTANCE OF TUBE (N)

SUBSCRIPT P REPLACES NFOR TUBE (P)

Fig. 18-Input voltage to tubes.

systems. Equations (8a-8d) express the transmissionbetween bridge output and the tube grid-cathode sys-

tems.

GmNv=grid-plate transconductance of tube (N).N= grid-plate +grid-screen transconductance of tube

(N).

Subscript p replaces N for tube (P).

VN N(1 + Zgp) - pzgp- =* (8a)e 1 + Z(gN +gP)

Vp P(1 + ZgN) - NZ0N_ = * (8b)

e I +Z(gN + gP)

For Zgv and Zgp>> 1, and gN = gp, (8a) and (8b) reduce to

VN N-P

e 2

vp P-N-- ore 2

(8c)

(8d)VN

e

CATHODE COUPLING IMPEDANCE FACTOR

As shown in (8a) and (8b), the multiplying factorsoperating on the N and P vectors depend on the cathode

impedance Z. These factors are readily analyzed bymeans of the approximation formulas given in (9a) and

(9b). For Zap> 1, and gN=gP=gm,

COMPENSATION /COMPENSATION FOR PARASITIC IFOR PLATE-GRID CAPACITANCE /COUPLING LOSS \ E

, .- arc tan, ,- arc tan (a b ... mn) Rgm

--~--~ b~ ~(b1_ nr gT 9mZPL(ab..mn)Rg9m.. pg9m

SEMICIRCLES OFMULTISECTION RCNETWORK APPROX-

REQUIRED IMATING REQUIREDCHARACTERISTIC N CHARACTERISTIC N-

I~~~~~~~'i

"' \\\d~~~~~~~~~~~~~~~~~~

886 A ugust

A nderson: Seven-League Oscillator

11 (I + 1 ) (9a)

1+ Z(gN± gp) 2\2Zgm/,

P __ 1-8 ~~~~(9b)1 + Z(gN + gP) 2\ 2Zgm/

Normally gN and gp will be almost equal and (9a) and(9b) will show the effects of the Zgm factor for valuesgreater than unity. Fig. 19 shows how this factor affectsthe multipliers of the N and P vectors of (8a) and (8b).

1 +2Zgm2ZgM

(a) (b)Fig. 19-Cathode impedance effects. (a) Z capacitive.

(b) Z inductive.

For Z having a negative phase angle, an important high-frequency case, the factor Zm likewise has a negativephase angle. The quantity

2 2Zgm)'

which multiplies the N vector in (8a), then has a posi-tive phase angle, and rotates the multiplied N vector ina positive direction.

Likewise the quantity

2+( 2Zgm)'which multiplies the P vector, has a negative phaseangle, and rotates the multiplied P vector in a negativedirection. The N vector is also lengthened slightly bythe multiplying factor, the P vector shortened. Thedifference between the multiplied vectors, as expressedby (8a), is a measure of the transmission between thebridge output and a tube grid-cathode system, such asof V1 of Fig. 2. This difference N-P, as operated on bythe multipliers, is increased by the capacitive impedanceZ. These relations hold since N multiplied will ordinarilylead P multiplied by an angle between 0° and 1800, andnot between 1800 and 3600. However, in (8b), the rolesof the P and N vectors are interchanged. The effect of acapacitance Zgm factor is to rotate the multiplied Pvector in a positive direction, the multiplied N vectorin a negative direction. The difference between themultiplied vectors, as expressed by (8b), is a measureof the transmission between the bridge output and atube grid-cathode system, such as of V3 of Fig. 2. Thisdifference N-P, as operated on by the multipliers, isdecreased by the capacitive impedance Z. To compen-sate for this decrease the RC network driving tube V4may be designed with a larger reactive transmissioncomponent at the highest frequencies.

The remainder of the oscillator loop design offersproblems similar to those met in negative feedbackamplifiers.! In the case of the oscillator, one frequency ofoscillation is required. The network design problemover the 0-oo frequency range, except for this one fre-quency of oscillation, consists of avoiding any tendencyto oscillate at another frequency. This means that ifthe oscillator loop transmission passes through a 00phase shift at any frequency except the desired fre-quency of oscillation, it must do so with a magnitude ofless than unity, or 0 db. Exceptions may be made forconditional stability as shown by Nyquist,7 althoughspecial measures may be necessary to establish oscilla-tion in the desired mode.The shaping of the transmission loop frequency char-

acteristic is readily accomplished with proper choice ofcomponents. No further discussion is necessary.

SUMMARY OF LooP TRANSMISSION

Figs. 20 through 22 show transmission magnitude andphase for the feedback loop of the oscillator of Fig. 2 atextreme and middle frequency settings. Prominentlydemonstrated are the extremely broad bands, stretchingwell beyond the actual frequency adjustment range, inwhich negative feedback exists. The feedback magni-tudes range up to some 50 db.

360 ~~~~~~~~~~20CPS SETTING

270 _PHASE

_40w 1B(

--..GAIN~90z /'

w - -.0

-270

-3600.01 0.1 I 10 100 10 100 10 100

I. LUelT.-LLccKFILOCYCLESFREQUENCY

tuco

u3

z

z

0

Fig. 20-Loop transmission.

SMOOTHING OF TRANSMISSION RIPPLESThe ripples in the RC network transmission can be

compensated somewhat by staggering the critical fre-quencies fc of the two RC networks. The configurationof the RC arm also may be modified to advantage, andthe frequency ratios of 16 to 1 for succes- ive sections maybe changed.

TRANSIT-TIME EFFECTSAt frequencies in the megacycles, tube transit times

add to the loop phase shift, and cannot be neglected.

I H. W. Bode, "Network Analysis and Feedback Amplifier De-sign," D. Van Nostrand Co., New York, N. Y., 1945.

7 H. Nyquist, 'Regeneration theory," Bell Sys. Tech. Jour., vol. 11,p. 126; January, 1932.

8871951

CYCLES MEGACYCLES


For a 6AK5 tube this factor may add a phase shift ofthe order of 0.20 per mc per stage. The physical size ofthe loop also contributes, but is a smaller factor in anormal design.

360 X - 10-KC SETTING

270 - _

(11 P~~~~HASE180 _ _ ._cr

ZN 20

< _9C -- - - __d -20

go.. GAIN.

-180 _<

-270

-3601 .,0.01 O.t 1 10 100 I 10 100 -1 10

CYCLES KILOCYCLES MlEGACYCLESFREQUENCY

Fig. 21 Loop tranlsmission.

360_9C 3MC SETTING 60

a-q270-

uC270 ' GAIN .EGAYCLE

cx 180 7 ~~PHAS a%E2

Z 20ci.21Lo tasisin

< ^ ,

90 7 XA9C-g - -4---_-2

0.01 0.1 1.0 10CYCLES

100 1 10 100KILOCYCLES

FREQUENCY

1 10MEGACYCLES

Fig. 22-Loop transmission.

FREQUENCY RANGEExtreme frequency range depends on the gain avail-

able to compensate for the bridge losses. The top fre-quency gain is limited by the parasitic capacitance ofthe amplifier system. If the top frequency is reduced byone half, the capacity reactance limiting the gain isdoubled, and the realizable gain is doubled too. Thebridge loss then can be doubled by halving the semi-circle diameters. Twice as many semicircles can be ac-

commodated and the extreme highest-to-lowest fre-quency ratio is roughly squared. The lowest frequencycan be reduced decades below 1 cps to regions in whichinsulation resistance will be a limiting factor.

FREQUENCY CALIBRATION OF DIALThe frequency calibration of the dial is nearly a

logarithmic function of linear scale divisions with linearcontrol potentiometers. This relation arises from thenearly linear variation of the resistive component of theRC network impedance with the logarithm of frequency.The linear asymptote-through which the real com-

ponent of the RC arm transmission oscillates wouldgive rise to a logarithmic frequency adjustment dial

scale, but for several factors. Most prominent are thatthe RC arm envelope asymptote has a slope and thatthe plate grid coupling networks and parasitic capaci-tance introduce phase and magnitude changes as theextreme frequencies are approached.

PHASE ANGLE BETWEEN OUTPUTS

The phase angle between stage outputs is nearly 90°.This arises from the 180° reversal in the two bridges,leaving 1800 of phase shift to be split between the twobridge units. The split is almost even, with some un-balance caused by the two different multiplying factorsoperating on the N and P vectors (see equations (8) and(9)). Deviations of the frequency-adjusting potentiom-eters from perfect alignment with each other alsocontribute to the dissymmetry.The phase angle between the outputs of the N and P

tubes of the same stage is roughly 1800, which may bemodified by the different multipliers of the N and Pvectors, and different plate load impedances.Thus the outputs of Vt and V2 in Fig. 2 are roughly

opposite in phase, as are those of V3 and V4. The out-put of V3 is displaced roughly 90° from that of Vl. Itfollows that the outputs of V2 and V4 are similarlyrelated. Thus a four-phase output, with phases dis-tributed at roughly 90° intervals, may be derived fromthe four plate circuits of the tubes.

Direct connection may be made to plates of tubes V2and V4 without seriously disturbing the oscillator. Con-nection to either of plates of Vl and V3 preferably ismade through a buffer amplifier to avoid reaction onthe oscillator from different connected loads.

ALIGNMENT OF FREQUENCY-ADJUSTINGPOTENTIOMETERS

The effect of misalignment of the frequency-adjustingpotentiometers results in a frequency rou4ghly halfwaybetween those corresponding to both potentiometers setin turn to align with each other.

AMPLITUDE STABILIZATION

If the oscillator output is limited by extreme tubeoverloading, the wave form is degraded. It is preferableto use some other sort of gain control, such as grid-biasautomatic voltage control, or a thermistor as shown inthe plate circuit of tube VI of Fig. 2. The resistance by-passing the condenser in series with the thermistor re-duces the voltage peak of the thermistor characteristicby providing a dc bias current. The resistance in serieswith the thermistor serves to adjust the slope of theregulation characteristic.The oscillator output is taken from the plate circuit

of V4. A second shunting thermistor circuit helps to re-

duce the output voltage variations. Further reduction ofthese variations over the frequency range is provided by

-1801

888 A4 ugust

,L--

LIt

IILc

I

t

iI

LI

i

A nderson: Seven-League Oscillator

an RC network, similar to those in the bridges, shiuntedacross the plate load of the tube V4. An article on funda-mentals of thermistors has appeared recently in thisjournal.8

HARMONIC OUTPUT

Harmonic output is proportional to the excursions ofthe tube currents. The tube system will act as an ampli-fier being driven by the fundamental frequency of theoscillator, and the harmonics incidental to the funda-mental output will be present. In general, the smallerthis output, the smaller the distortion.

FREQUENCY STABILITY

Frequency stability is dependent on many factors,the most prominent of which is the tube gain stability.The nonlinearity of the transmission loop also con-tributes. Factors such as stability of the frequency-control elements will not be discussed here.

Fig. 23 shows typical frequency shifts for inter-changes of Vl and V2 or V3 and V4 in cases where thetransconductance of VI is 12 per cent higher than thatof V2. The frequency shift is 5 per cent per db of tubegain change for the simple cathode impedance of aresistance and a small coil, bridged with parasiticcapacitance. Above 100 kc the parasitic capacitances inthe circuit begin to exact toll, and frequency stabilitybecomes poorer.

44z

Z O

VD

a.zUJw

uJ z

UluZ

Da0

c

0

_L

89-

LOW IMPEDANCE6 CATHODE NETWORK

HIGH IMPEDANCE2 CATHODE NETWORK7

01 I102 300 103 104 105

FREQUENCY tN CYCLES PER SECOND101

V<> g V2

4i-

SCREENGENERATOR lIMPEDANCES

< 2~~V

I63

0

Equations (lOa) and (lOb) give the changes in the com-ponents of the effective grid-cathode voltages VN andvp for changes in the tube gain factors gN and gp.

d (=d)N1-dPj.(e)

(10)

For gP=gN=-gm,

d F(3-dgp I- )dgNdNi=- -[(--- 1--]2L- 2Zgm} gp 2Zgm gN

p1i 1 \dgp (1 12 dgAN

2 2Zgm/, gp 2Zg m/,gNJ-

(lOa)

(lOb)

The differential factors operating on the componentsof vN/e as shown in the last two equations are almostequal, and become more so as Zgm-c*. The variationcoming from gp contributes to the inequality, and so tothe frequency shift. The contribution from gv tends tocancel. This would suggest special attention in the wayof regulation of supply voltages and perhaps a negativefeedback for the P tube. Similar measures are indicatedfor the N tube of the pair using the output of the P tubeto drive a bridge.The high dc resistance of the common cathode return

tends to keep the total current of the N and P tubes con-stant. If the current in one tube starts to fail, the cur-rent in the other tube increases to make up the differ-ence. Since transconductance is somewhat proportionalto space current, the common cathode connection tendsto accentuate frequency drift caused by tube aging.

FOR 9N - 9P 9m

Zgp_ .~~~~i[( ~____\p / dgI,I+Z9NZNT 2 (1 2Zg) 9ml (d 2Z9g) 9mji + Z9P+Z9N d, - Ps2 gm2Zgm) 9( Zgm

Fig. 23-Frequency stability with gain variation.

Increase of the cathode impedance by a more com-plex network improves the frequency stability. Thetube screen grid generator resistances tend to limit themaximum impedance and the improvement possible.The tube gain variation is the largest source of fre-

quency drift, and demonstrates the need for a largecathode feedback. The components of the the transmis-sion ratio vN/e, shown in Fig. 24, indicate how the phaseshift resulting in frequency change is derived fromchanges in the multiplied N and P vectors, N1 and P1.

8 J. H. Bollman and J. G. Kreer, Jr., "Application of thermistorsto control networks," PROC. I.R.E., Vol. 38, pp. 20-26; January,1950.

Fig. 24 Frequency stability with gain variation.

Fig. 25 shows separate cathode resistances for eachtube. The cathodes are coupled together through con-densers which reduce the performance to that of theparalleled cathode circuit impedances at sufficientlylarge frequencies. The large dc feedbacks acting on eachtube separately tend to hold the space currents constant.The transconductances tend to stay more nearly con-stant over much of the tube life, and the oscillator fre-quency is thus more stable.

8891951


The cathode coupling condensers enter into the low-frequency transmission around the oscillator loop, andmust be designed with the cathode return impedancesand grid-plate coupling systems to make a stable feed-back gain and phase characteristic. The cathode returnsmay be complex networks instead of plain resistances toprovide large impedances in the oscillator workingband.

df I22 j sinO2 1'Af = _ 2 -

dO f=fI - I12 1 -A2(12)

whereAf= frequency shift in cycles

dO/df= slope of phase-frequency characteristic offeedback loop in radians per cycle

12 = second harmonic current corresponding to I1,measured with no feedback

A2=feedback to second harmonic.

1 COS 021 - A2 - 1 - A2I

sin S21 - A21

Fig. 25-Direct-current stabilization of gain and frequency.

OSCILLATOR FREQUENCY SHIFT AS A FUNCTION OFOUTPUT AMPLITUDE

Frequency shift of an amplitude-regulated oscillatoras a function of output well below overload may beevaluated by means of a power series analysis. The fre-quency shift is caused by phase shift of the fundamentalfrequency resulting from quadrature components of thesecond-order difference product of fundamental modu-lating with second harmonic, and the third-order differ-ence product of fundamental modulating with thirdharmonic, and so forth. For small amplitudes of funda-mental, say 30 db below full output, only the second-order modulation is noticeable, and a two-term powerseries suffices

I = I1 + a21I2 (1 1)

whereI= total current in a tubeI,= fundamental frequency currenta2= a factor of proportionality.Evaluation of the actions in the feedback loop results

in this relation for frequency shift

dO/df is negative at fi; for 02 between 0° and - 1800,sin O2 is negative and Af is negative. Thus increasingamplitudes of A1 result in a downward shift of frequency,proportional to the square of the amplitude, as long as12 is proportional to the square of I1. This holds for smallamplitudes. For larger amplitudes the power series of(11) must include more terms.

APPENDIXTABLE I

1 /1 ~~tan01k= 2 +/4±+ +12tr4f 2

c

k(l-k)

0.25 0.500.21 0.300.177 0.230.149 0.18

0.125 0.150.105 0.120.088 0.100.074 0.08

0.063 0.070.053 0.060.044 0.050.037 0.04

0.031 0.030.026 0.030.022 0.020.019 0.02

-k(l-k)

k

0.500.700.770.82

0.850.880.900.92

0.930.940.950.96

0.970.970.980.98

C7AD

890 A ugust

seven-league oscillator

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