302 PROCEEDINGS O F THE IEEE March

in the design of the experiment and in the above signal estimate, but the fall-off in sig- nal may be aggravated by diffraction pat- terns and interference between reflectors in the array which could not be observed in the laboratory.

H. H. F'LOTKIN T. S. JOHNSON

P. SPADIN J. Mom

Goddard Space Flight Ctr. Nat'l Aeronautics Space Adm.

Greenbelt, Md.

Oscillator Noise In the recent paper by Grivet and Bla-

quiere [l 1, there is a discussion of oscillator circuits in IIA leading to the conclusion that shot noise may be made negligible compared with thermal noise of the main resonant cir- cuit, but the proof given is based on an un- realizable circuit.

While it must be true that an optimum arrangement can be found to minimize the shot-noise contribution in an oscillator, this condition of operation is not necessarily the best. With a quartz crystal, both the Q value and the resonant frequency are dependent upon the power dissipation. The design of a clock oscillator must, therefore, be a com- promise between short-term stability (high Q and high power) and low-aging rate (low power). Because of the practical limit to the power level set by the properties of the quartz, the electrodes and other details in- cluding thermal properties of the mounting, it is desirable to arrange that the equivalent noise temperature of the circuit is the same as the quartz temperature. This does not seem to be a simple proposition.

In a simple linear approximation of an oscillator, the equivalent temperature can be derived very easily. It can be shown that tube properties play an important part even when there is no loss in the grid circuit or complex-noise sources and saturation prob- lems. The simplified equivalent tuned cir- cuit is shown in Fig. 1, where

R, at temperature T, is the circuit natural- loss resistance Rt at temperature T t is the loss introduced by the tube Qu, QL are the unloaded and loaded Q values. Put Tt/T, =t (This is approximately equal to 3 p for a triode.)

We define a short-circuit power Psc of the maintaining device, as operated, equal to four times the available power.

Hence power in the circuit

reported III this pawr was sponsored by the Air ManuFript received March 3, 1964. The research

Force Electronic Systems Div., Air Force Systems Command, under Contract .4F19-(628)2390. The MITRE Corp. has obtained approval for the release of the information contained in this report.

R ,Tc

Rt 2%

Fig. 1.

And the equivalent noise temperature

I t is desired to maximize

PQL' Psc x*(l - z) -=- T, T, Qu2t(l - x ) + z

where x = QL/Q.. This has a maximum a t

x = 4t - 1 - (at + 1)"$

4t - 4

and, a t x = 3 for t = 1. Using this result we get the following table:

Line Width 1 Te, Factor for

I - - l-

The optimum value of QL/Q" is not critical. If the circuit dissipation does not limit the design, the factor PQL*/T. may be made to approximate the value PscQu2/Tt.

Thus it appears to be of major impor- tance to choose a tube (or other amplifying device) having a maximum value of Psc/Tt ("dynamic range") as would be chosen in- tuitively.

When the power dissipation of the high-Q element is a limiting factor, it appears to be still desirable to use a high dynamic-range amplifier and to adjust

to the maximum permissible value. In this way PQL'/T, will approach the value PQ,*/T,. This, I assume, is the point that Grivet and Blaquiere were making. The practical way of using a quartz crystal is to use circuit elements as impedance trans- formers without seriously degrading the frequency sensitivity in the region of reso- nance. Many practical factors will affect the design, and it seems unwise to draw any sweeping conclusion about the realizability of a swamping of shot noise.

I t would seem most desirable to use an extremely large crystal and to cool it to a low temperature, but there is the need to have a suitable value for crystal resistance in order to avoid frequency drift caused by changes in the amplifier characteristics as well as by shot noise.

The discussion of Robinson's oscillator [2] seems to be irrelevant to a discussion of ultra-stable oscillation. The only thing in common with a quartz oscillator is that, in

the investigation of paramagnetic resonance, it is necessary to avoid saturation of the specimen. In one instrument, the double- tuned circuit is arranged to give impedance- frequency characteristics determined by the high-Q element above all; in the other, the double-tuned circuit displays the effect of the weak coupling to a high-Q element as the frequency is controlled primarily by a low-Q circuit.

The use of a 'separate limiter" in Robin- son's circuit seems a misleading concept. Is there really any fundamental distinction between a single stage of nonlinear amplifi- cation and a cascading of amplifier stages with nonlinearity? What I interpret as the essential feature of Robinson's circuit is a modification of the amplification factor and dynamic range compared with a normal marginal oscillator, and consequently, the ability to achieve a better linewidth a t the maximum power in the unsaturated speci- men. Apart from this advantage I find it difficult to accept that a nonlinear resistance element will be free from LF excess-noise effects.

N. HOLTDING The Mitre Corp.

Bedford, Mass.

REFERENCES [l] Grivet. P.. and A. Blaquiere. Nonlinear effects of

noise in electronic clocks, PROC. IEEE. vol 51.

121 Robinson, F. N. H. Nuclear resonance absorption Nov 1963. pp 1606-1614.

nrcuit. 1. Sci. Insfr., vol36, Dec 1959. PP 481-487.

A Delayor Schematic for Polynomial Functions

A delayor is taken to mean, in this corre- spondence, to be a device which, for an in- put x(t )u(t ) , produces an output x ( t - a ) u(t - a), a being the delay and u(t ) unit step function starting at zero time. The ideal delayor, as is well known, is a lossless trans- mission line terminated in its characteristic resistance. A distortionless line, terminated in its characteristic resistance, also can be a delayor though the output appears attenu- ated. The main limitation in these two cases as well as their modifications is that the maximum delay obtainable is in the order of microseconds.l.2 Also, any lumped net- work approximant for the realization of transfer function e* is bound to create a certain amount of distortion.8 In this con- text the schematic presented herein for a delayor may be of some interest.

Consider the relation

p = - dk dP '

Manuscript received September IS, 1964.

1965 Correspondence 303

where z ( t ) is a polynomial of degree n given by, say, x ( t ) = (a0 + ad + adZ + - + ants)&). (2)

Introducing (2) in (I), and after suitable manipulation, it can be shown that

y ( t ) = x(t - a)u(t). (3)

Thus the differential operation indicated on the right-hand side of (1) shifts the function r(t) to the right along the time axis, pro-

Fig. 1-Delayor schematic based on (1).

Fig. 2-Response of schematic of Fig. 1. but without switch on output side, to ramp input.

vided z(t ) is, or can be approximated by, a polynomial, but, as can be seen from (3), y ( t ) is not zero in the interval 0 <t <a, a condition required in a pure delayor. This situation can, however, be corrected by hav- ing a switch on the output side as shown in Fig. 1 which is the schematic for a delayor on the basis of (1). In ( 2 ) when n=O, the input is a step function, and the schematic of Fig. 1 reduces to a mere switch on the out- put side.

The major weakness of the scheme is that it involves differentiators which are an un- favorable feature from the point of view of noise. However, i t may be possible to mini- mize the noise effects by careful adjustments on the differentiators used. Again, the de- lay a and the maximum slope of the func- tion, with reference to ( l ) , should be such that it does not “overloadn the differenti- ators.

In any case the scheme was tried for z(t ) =t and a = 1 sec on an analog computer (EAI, PAGE, TR-48). Fig. 2 is a direct re- cording of the input to, and output of, the computer setup. The reason for the fact that the output is not zero in the interval 0 <t < 1 is that a switch on the output side as shown in Fig. 1 was not used. It is seen from the re- cording that the result is good, and on the basis of this fact it is conceivable that the schematic presented may find an applica- tion a t least in the generation of functions, the graphs of which are approximated with linear segments.

Also, in (1) a does not interfere with the

differentiation of ~ ( t ) . Therefore (3) is still valid even if it is time variant. This means that by making a time variant the sche- matic of Fig. 1, but without the switch in the output side, will give a continuously shifted output.

ACKKOWLEDGMENT The author wishes to express his thanks

to N. Jafri of the Department of Electrical Engineering, University of Ottawa for get- ting the experimental result.

S. G. S. SHIVA Dept. of Elec. Engrg.

Univ. of Ottawa Ottawa, Ontario, Canada

Prentice Hall. Inc., Englewood Cliffs. N. J., pp. 101- f S. Moskowitzand J. Racker, “Pulse Techniques.”

117; 1954. 2 R. Blitzer. ‘Basic Pulse Circuits,” McGraa-Hill

Book Company, Inc., New York. N. Y., pp. 311-312; 1964.

8 B. Liu, “A time domain approximation Tethod and its application to lumped delay lines, IRE TRANS. ON CIRCUIT THEORY. vol. CT-9, pp. 256261; September, 1962.

S B a n d GaAs Gunn Effect Oscillators

We have obtained from GaAs Gunn ef- fect oscillators [1]-[5] a t room temperature 2.5 watts of peak power a t about 3 Gc/s with a power efficiency of 7 per cent. The devices were fabricated from n-type GaAs which had a room temperature resistivity about 0.75 ohm-cm and mobility of 5500 cmV-1s-1. “Ohmic” contacts were made to (100)- oriented chips of chemically-polished GaAs by alloying one surface to a disc of Sn-clad Mo and alloying a Sn sphere into the other surface. The alloying was done in a hydro- gen atmosphere a t 360’C. The devices have been assembled in a small varactor package as shown in Fig. 1. All devices investigated were less than 40-microns thick. The low field resistances of the devices have shown variations, and only those samples whose resistances were consistent with the bulk resistivity have made good oscillators. These devices have shown consistent results as to frequency of oscillation, threshold voltage, and efficiency, while devices having a resistance other than the expected resist- ance have shown current instabilities but no coherence and very small amounts of microwave power.

The devices were initially checked in a resistive circuit where it was possible to maintain a nearly constant voltage across the device and observe the current with a sampling scope. Figure 2(a) shows the cur- rent instability and the corresponding volt- age waveform of a typical oscillator when pulsed with a 30-volt 20-11s pulse. The oscil- lation frequency for this device as observed in the circuit was about 2.5 Gc/s; any har- monic content in the waveform would be suppressed by the sampling scope. The threshold voltage was a t 12 volts. The aver-

Manuscript received January 28. 1965.

OFHC

Sn SPHERE-

,,. kLUMiNA

1.- Sn CLAD Ma

t---OFHC COPPER

Fig, 1 Artist’s representation of a GaAs Gunn effect oscillator mounted In a varactor package.

Fig. 2. (a) Current and voltage waveform of a GaAs

A/div. Time scale is 0.2 ns/div. (b) Frequency Gunn effect oscillator. V = 2 0 V/div. I =0.21

spectrum when pulsed wlth a O S - N S 25-voltpulse.

TABLE I

Peak Peak Micro- Per cent wave Power Efficiency

20 volt 1 16 volt (0.5-as pulsz) 0 . 5 watts 3 Percent

25 volt 1 .1 5 .5 Percent 1 . 6 6 . 4 Percent

35 volt 2 . 5 6 . 7 Percent 7 Percent

6 a As ---, ,,‘-EPOXY

Fig. 3. Artist’s representation of a CW oscillator.

age current above threshold was 1 A. This oscillator was about 35-microns thick, had a top contact diameter of 270 microns, and a measured low field resistance of 5.2 ohms.

The devices were then inserted in a tun- able coaxial cavity and the output power