1 74 IEEE TRANSACTIONS ON ELECTRON DEVICES, MARCH 1967

The subscript 1, 2, or 3 affixed to E will denote the first, second, or third drift-tube discontinuities. A prime associated with E indicates the growing wave amplitude immediately preceding the disconti- nuity, while the absence of a prime indicates the amplitude of the growing wave immediately following the discontinuity, i.e., a t the beginning of the slow-wave structure region. For example, after the growing wave has traversed the 4 xQ slow-wave region, calculations show that at the first discontinuity

This corresponds to a power gain of -0.815 dB. This is what one normally expects, a reduction in the growing wave at the exit of a discontinuity as compared with its value immediately preceding the discontinuity. This result would appear to belie the original assertion that one would obtain an increase in gain. However, two conditions must be met before gain can be realized via discontinuities. First, it is necessary to have modulation on the beam; second, the growing wave should not be dominant. In order t o introduce modulation on the unmodulated beam, the slow-wave structure was made relatively long (4 kg). Hence, a t the discontinuity, the growing wave is domi- nant and conditions for gain at the discontinuity are not satisfied. However, now there is sufficient modulation on the beam, and at the exit of this discontinuity all three waves are again present. After the growing wave has traveled 1 h, t o the next discontinuity, the results are

or a power gain of 4.2 dB. At the third discontinuity we obtain

or a power gain of 3.9 dB. A similar calculation was made again but the initial slow-wave

structure was only 1 hQ long. These and the previous results are summarized in Table I. One observes in the table that if the initial slow-wave structure is only 1 x, long we do have some gain across the first discontinuity. I t is small because the modulation is low, but there is gain because the growing wave is not yet dominant.

TABLE I COMPARISON OF SQUARE OF ABSOLUTE VALUE OF GROWING

WAVE ON BOTH SIDES OF A DISCONTINUITY

I (a) I ih)

S I 0.829 1 -0.815 1 1.317 I t S . 2

2.637 2.015 4-3 . O ~~

3 2.450 2.335 - I +3.7

(a) Initial slow-wave structure is a 4 X, long. The remainder of the slow-wave structures are 1 X long.

number of discontinuities traversed.)

* -

(b) Initial slo&wave structure is 1 X, long also. (Subscript r represents the

These results indicate that it is possible to enhance crossed-field gain via discontinuities. The cutoff waves excited at the discon- tinuity give rise to a phase shift and a change in the amplitudes of the propagating waves without violating the conservation of energy. I t is this change in the complex amplitudes of the propagating waves which provides the novel gain. This scheme does not appear to be restricted to M-type devices but should be true for 0-type as well. L. S. Nergaard3 formulated a simplified model for an 0-type

3 L. S. Nergaard, RCA Laboratories, Princeton, N. J., private communication.

device and introduced a discontinuity. His results appear to support our conclusion. In previous work with 0-type devices, severs were placed where the growing wave was dominant. I t is stili necessary t o have all three waves present. If the initial sever were followed by another it should be possible to obtain gain. Of course, after the addition of the required number of severs, the final slow-wave region must be sufficiently long so that at the output the growing wave is dominant.

B. HERSHENOV RCA Laboratories

Princeton, N. J.

Nonreciprocal Tunnel Diode Distributed Amplifier This correspondence suggests a method for the construction of a

distributed tunnel diode amplifier. The configuration is a strip-line geometry with a narrow tunnel diode along the center line of a metallic strip of appreciable width. [l] Nonreciprocal properties are achieved with the placement of ferrite material on both sides of the diode. The ferrite material is magnetized uniformly in the + x direc- tion. A resistive shunt placed at both outer edges of the strip line forms the final termination. This configuration is shown in Fig. 1. At high frequencies the circulating magnetic fields will be weak a t the outer edges. Thus the resistive shunt will not cause a serious amount of attenuation, At low frequencies the resistive shunt can cause substantial attenuation. Dc stability conditions are used to determine the parameters of the ferrite and the resistive shunt.

METALLIC STRIP -,

-a $0 JUNCTION REGION

Fig. 1. Distributed amplifier configuration. in which 1 =resistive and 2 =ferrite.

For a TE mode, the solutions to the field equations in the ferrite for y<O and z the direction of propagation are given by

The solutions for y>O are obtained from the above by substituting - K for K . In the above

"/e - (n2 + KI ' ) , K1' Wz€(l.r' - K e ) / p .

The quantities p and K are defined by the permeability tensor for the anisotropic ferrite material [2 1.

The aim is to determine y and p when the negative conductance of the diode active region, the diode spreading conductance, and the admittance looking outward from the diode are included. The method of attack is to determine the admittance per unit length looking outward from the diode junction, giving the propagation constant y and the transverse propagation constant p as a function of frequency. If a voltage is t o exist, this admittance must be the negative of the admittance per unit length of the diode junction

by the Joint Services Electronics Program under Grant AF-AFOSR-766-66. Manuscript received August 26, 1966. The work reported here was supported

CORRESPONDENCE 175

region. This is based on the assumption that the current leaving the diode enters the strip line.

The transverse admittance can be determined by solving the boundary-value problem at y= -a. At the boundary it is assumed the resistive material forms the total terminating admittance. Hence

where YB is the total admittance of the resistive shunt. Substituting the field solutions into the above, it can be shown that

Suppose the diode width is small compared with the width of the ferrite. Then the admittance looking out from the diode may be approximated by

Y*I = - j H S

u=o ZE, jwZ(pz - K') 1 + B / A Suppose the strip fine is sufficiently wide and the frequency is suffi- ciently high that B/A+O. Then

Y, = . P4 f j K Y . $dl(/.&' - K')

The junction admittance may be characterized by YD= -GD+~wCD mhos per unit length, and the spreading conductance may be taken as G= ub/h mhos per unit length. The total admittance looking out- ward from the diode junction is

1 yo = 1/G + 1/2Ys

and the admittance equation is written as Y, = - YD. The admittance equation is solved for q and substituted into y2= - ( q ' + ~ ~ ~ ) . This gives a quadratic in y with the solution as

y = - WZKY w.\/1'Y'p' - PE

where Y is determined by YD and G. Near the gyromganetic resonant frequency WH, and using anticipated values for the parameters, the propagation constant becomes

Y = W l Y ( - K k M). From the assumptions that B/A+O and y= 5- jq , the circulating magnetic fields are circularly polarized. Then ( p + ~ ) corresponds to the +z direction and ( p - K ) to the -z direction. The propagation constants in the z direction are then given by

where the ( :riticaI frequency wc is defined as

wc = (GD/CD).\/~/RGD.

For w<we, RE(^+^) <0, predicting gain in the $2 direction, and Re(y-J >0, predicting attenuation in the -2 direction.

The maximum line length is calculated from the condition that the voltage across the diode at the receiving end is such as t o keep the diode operating in the negative slope region. Taking the line integral of the electric field and operating at the gyromagnetic resonant frequency, the maximum line length is given by

where VO is the input voltage and V, is the maximum permissible voltage at the load. The maximum voltage is determined from the diode volt-ampere characteristic.

I t is possible that standing waves may develop from reflections at the load. These standing waves will eventually saturate the de- vice. The loop gain expression

loop gain = r,rle-z(y+z+k)

shows that if Re(y+,+y-,) >0, the loop gain magnitude is less than unity provided the load is not totally reflecting. For operation a t the gyromagnetic resonant frequency there is total attenuation around the loop for any length of line.

Using the diode parameters 131

GD = 102 kmho/m h = 0.01 in (0.254 mm) CD = 40.6 pF/m b = 0.02 in (0.508 mm)

G = 200 kmhojm WH = 1/3 lo8 rad/s

and taking the maximum voltage gain V,/ VO = 10, the critical fre- quency is 35 X108 rad/s and the maximum line length is 0.04 in (1.02 mm). The approximation used in determining the strip ad- mittance Y, is valid for a ferrite width as small as 0.1 in (2.54 mm). Ferramic I ferrite material has a low-frequency relative dielectric constant of lo4 and a conductivity of 0.2 x mho/m [4]. For the dimensions given, the ferrite capacitance is 0.885 pF/m, and the ferrite conductance is 0.002 mho/m. The dc stability conditions [5] give the resistive shunt width of 0.01 in (0.254 mm) for a conduc- tivity of lo5 mho/m. This assumes that the load resistance is very large compared with the shunt material.

RANDOLPH C. MATSON CYRUS 0. HARBOURT

OREN B. KESLER Dept. of Elec. Engrg.

University of Texas Austin, Tex.

REFERENCES [l] M. E. Hines, "High-frequency negative-resistance circuit principles for Esaki

121 C . L. Hogan, "The microwave gyrator, Bell Sys. Tech. J., vol. 31, pp. 1-31. diode applications,= Bell Sys. Tech. J. , pol. 34, pp. 477-513, May 1960.

131 A. C . Scott, 'Distributed effects in large area Esaki diodes," I R E Tvons. on January 1952.

[Sl L. G. Van Uitert, "Dielectric properties of and conductivity in ferrites," Proc. Electron Devices, vol. ED-9. pp. 417-422. September 1962.

[51 W. H. Card, Parasitic oscillations and their suppression in tunnel-diode cir- I R E , vol. 44, gp. 1294-1303, October 1956.

cuits." Ph.D. dissertation, Dept. of Elec. Engrg., Syracuse University, Syracuse, N. Y., 1961; University Microfilms, Ann Arbor, Mich.