âStormer's'' Numbers in Generating Phase. Coherent Electronic Systems. Abstract-In microwave systems requiring the various frequency components to be ...
PROCEEDINGS OF THE IEEE, AUGUST 1969
1460 defects, impurities, etc. However, the relative dependence and functional relationships for the different variables should remain valid, so that the basic form of the optimizationshould be a good guideline. Although availinformation able is limited, performance estimates are in reasonable accord with published data. JOHNB. HOPKINS NASA Electronics Research Ctr. Cambridge, Mass. 02139
TABLE I t=1
.~ 1
f
=2
2
1 1
’
I
176
1
I
t =3
3 t=4
I
I
REFERENCFS [I ] S . T. E n g “Low-noise properties of microwave backward diodes.” IRE Trans. Microwace Theory and Techniques, vol. MTT-9. pp. 41%425. September 1961. 121 C. A. B u m . Jr... “Backward diode for low-level millimeter-wave detection,” IEEE Tram. MicrowaDe Theory and Techniques, vol. MTT-11, pp. 357-362, September 1963. [3] J.B. Hopkins, “Characteristin of tunneling p-n junctions,” NASA Tech. Note D 4 3 , March 1968. (41L. Esaki, “New phenomenon in narrow germanium p-n junctions.” Phys. Rec., vol. 109, pp. 603404. 1958. [SI E. D. Kane, “Theory of tunneling,” 1.&I. Phys., vol.32, pp. 8S94, January 1961. [a] A. M.Cowley and H. 0. Sorensen, “Quantitative comparison of solid-state microwave detectors,” IEEE Trans. Microwate Theory and Techniques, vol. MTT-14, PP. 58-2. December 1966. [7] J. Karlovsky, “The curvature coefficient of germanium tUMd and backward diodes.” Solid Srare Electronics, vol. IO, pp. 1109-1 113, November 1967.
1
3 4 9
In microwave applications there is clearly a use for electronic systems which generate a relatively high carrier frequency, and a local oscillator frequency differing by a relatively low IF or modulating frequency, with all three locked in phase. The development of the step recovery diode has made convenient the efficient multiplying of frequency by relatively large factors in a multiplying train to produce relatively high output frequencies. If two large numbers are both factorizable into convenient factors and differ by 1, they are ideal for generating phase coherent electronic systems. Typical pairs of such numbers are
225 = 32 x 52J 9800 = 23
X
5’
9801 = 34 X 11’
X
7’ p4 - 2p2 + 1,
1
p2(p2 - 2)
with simple algebraic generation, theproblem in the last case being to find a p2 - 2 which is factorizable. The author was directed by his colleague Prof. E. Barnes, a puremathematician, to a most interesting paper by D. H. Lehmer.’ Such number pairs were of great interest in calculating logarithms of some large adjacent integers to great accuracy in the form of logarithms of relatively small prime numbers. On page 69 of the reference, Lehmer presents a table in which integers N , greater than 1, are given such that the largest prime factor of N ( N - 1)is the tth member of the series of prime numbers 2, 3, 5.7, 11, etc. A portion of this tableis reproduced here in Table I. Lehmer’s original table continues to t = 13, but only the figures shown in Table I are given here, as sufficient number pairs exist for t = 5 and below to cover most practical needs. If n and n + 1 are two such numbers, a series of m relatively large numbers differing by 1 can be generated as follows : Manuscript received May 16, 1969; revised June 9, 1969. ’ D. H. Lehmer, “On a problem of Starmer,” Illinois 1.Math., vol. 8, p. 57, March 1964
’
1 5 50 7 540 99 I 11 8 64 12 ! 6 ‘ 1 0 126 15 121 22 33 16225 21 243 45 25 240128 36 4375 385 55 81 49 1 56
100
3025 9801
1 ~
~
~
I
441
Integers N , greater than 1, such that the largest prime factor of N ( N - 1) is the tth prime number, for 1 2 t 5 5.
mn, (m - 1)n
where
+ n + 1, ( m - 2)n + 2(n + l), . . ’ , m(n + 1)
rarely exceeds or in practical electronic applications,
These numbers seem to have direct applications to Doppler navigation equipment and their types of CW radar. E. 0. WILLOUGHBY Dept. of Elec. Engrg. University of Adelaide Adelaide, So. Australia 5001
“Stormer’s’’ Numbers in Generating Phase Coherent Electronic Systems Abstract-In microwave systems requiring the various frequency components t o be locked in phase, such as the carrier and the local oscillator, an ideal basis for the design of the frequency multiplying trains is afforded by selecting t w o large numbers, differing by1. each of which is made up of convenient factors.
r=5
A Driven Subharmonic Oscillator Abstract-A new type of drivensubharmonic oxillator is describedwhich dependsfor its actionontheincorporation of a nonlinearelementwith a hysteresis type of current-voltogeCharacteristic.Twocircuitsthatgiveclose approximationstothecharacteristic of thenonlinearelement are presented.
Thebasic circuit of the subharmonic oscillator, as illustrated in Fig. l(a), consists essentially of a sinusoidal drive u d t ) in series with a nonlinearelement and a tuned circuitacross which the sinusoidal subharmonic output voltage appears. The direct bias supply VBis shunted by a bypass capacitor and uAr) is supplied from a generator of low internal impedance. The circuitdevelops a sinusoidal output, phaselocked to the input, of frequency fi/n, where n is an integer and fi is the input frequency. In addition, frequencies can be produced of the formfifix(m/n).where m and n are,integers and m