THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN QUARTERLY PROGRESS REPORT NO. 8 FOR BASIC RESEARCH IN MICROWAVE DEVICES AND QUANTUM ELECTRONICS This report covers the period February 1, 1965 to May 1, 1965 Electron Physics Laboratory Department of Electrical Engineering By: M. E. El-Shandwily~, APp?oveb B. Ho, J. E. Rowe CYgli C. Yeh "Pro ect Engineer A_ppro,ve,d by;. Rowe, Director Electron Physics Laboratory Project 05772 DEPARTMENT OF THE NAVY BUREAU OF SHIPS WASHINGTON 25, D. C. PROJECT~~~ SEIA NO R0001 AK99 CONTRACT NO. NObsr-8927 -

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ABSTRACT Large-signal trajectory computations for a d-c quadrupole amplifier with a twisted quadrafilar pump field structure are presented. Two modes of operation, the cyclotron-to-cyclotron mode and the cyclotron-to-synchronous mode are studied. The results are compared in terms of the paths of electrons for different pump field strength, source of r-f power supply and efficiency. It is believed that the cyclotronto-synchronous mode of operation is more efficient in many respects. The derivation of the equation for the V-I characteristic of tunnel diode is presented. It incorporates the effects of temperature semiconductor material, doping concentration and the bias voltage on the negative resistance characteristic of the diode. This equation will be used to study cross-modulation effects in the tunnel diode under multiple-input-signal operation. A small-signal nonlinear analysis of a crossed-field amplifier with n input frequencies is derived. It is found that besides the usual cross-modulation product of frequencies 2f - f as in the case TWA a newr p or the 0-type TWA, a new product in -the form of:-f + f - f becomes r pi t

TABLE OF CONTENTS Page ABSTRACT iii LIST OF ILLUSTRATIONS vi PERSONNEL vii 1. GENERAL INTRODUCTION 1 2. STUDY OF FREQUENCY MULTIPLICATION IN ANGULAR PROPAGATING CIRCUIT 2 ANALYSIS OF AMPLITUDE- AND PHASE-MODULATED TRAVELINGWAVE AMPLIFIERS 2 4. STUDY OF A D-C PUMPED QUADRUPOLE AMPLIFIER 4 Introduction 4.2 Cyclotron-Cyclotron Wave Interaction 4.2.1 Low Pump Field 4.2.2 High Pump Field 4 4k. 3 Cyclotron-Synchronous Wave Interaction 8 4.3.1 Nonanmplifying Case —Low Pump Field 8 4..2 Amplifying Case —High Pump Field 12 4.4 Energy Relations 12 4.5 Conclusion 16 4.6 Future Work 19 5. INVESTIGATION OF THE CROSS-MODULATION PRODUCTS IN A WIDEBAND TUNNhEL DIODE AMPLIFIER 19 5.1 The Quantum Mechanical Tunneling Effect 19 5.2 The Tunnel Diode 24 5.5 The V-I Characteristic 26 5.4 An A-pproximate Solution of the V-I Characteristic 55 5.5 Future Wo-)rk, -

Page 6. NOLIEAR ANALYSIS OF THE CROSSED-FIELD AMPLIFIER WITH MULTI-SIGNAL INPUT 537 ~6.1~ Introduction 37 6.2 Theoretical Analysis 57 6.5 Future Work 58 7. GENERAL CONCLUSIONS 58

LIST OF ILLUSTRATIONS Figure Page 4.1 Cyclotron-Cyclotron Wave Interaction for Low Pump Field Strength. 5 4.2 Electron Trajectory of Different Entrance Angle for Cyclotron-Cyclotron Wave Interaction and Low Pump Field. 6 4.5 Cyclotron-Cyclotron Wave Interaction for High Pump Field Strength. 7 4.4 Cyclotron-Synchronous Wave Interaction for Very Low Pump Field Strength. (V = 10 Volts) 9 P 4.5 Cyclotron-Synchronous Wave Interaction for Low Pump Field Strength. (Vp = 30 Volts) 10 4.6 Cyclotron-Synchronous Wave Interaction for Low Pump Field Strength. (V = 42 Volts) 11 p 4.7 Cyclotron-Synchronous Wave Interaction for High Pump Field Strength. (V = 53 Volts) 1 ngular Variation for Different Pump Field 4.8 Angular Variation for Different Pump Field Strengths,-. 14 4.9 Energy Variation of Cyclotron-Cy.clotron Wave Interaction. 17 4.10 Energy Variation of Cyclotron-Synchronous Wave Interaction. 18 5.1 quantum Mechanical Tunneling. 21 5.2 TbLe V-I Characteristic of a Typical Tunnel Diode. 25 5.5 Energy Level Diagram of a Tunnel Diode. 27 5.4 Energy Level Diagram., Density of States and the Definition of Terms Used in Analyzing the V-I Characteristics of a Tunnel Diode. 28 5.5 The Density of States Function for a Degenerate N-Type Semiconductor Showing the Effect of Broadening of the Impu.rity Level E. 532

PERSONNEL Time Worked in Scientific and Engineering Personnel Man Months* J. Rowe Professors of Electrical Engineering.05 C. Yeh 1.11 D. Solomon Associate Research Engineer.51 H. Detweiler Assistant Research Engineers 97 W. Rensel.11 M. El-Shandwily Research Associate 2.9 ~A. Cha ~Research Assistants 1.66 nA. Heath.55 13. Ho 1.27 R. Ying 1.59 Service Personnel 8.40 *Time Worked is based on 172 hours per month.

QUARTERLY PROGRESS REPORT NO. 8 FOR BASIC RESEARCH IN MICROWAVE DEVICES AND QUANTUM ELECTRONIC 1. General Introduction (C. Yeh) The broad purpose of this project is to investigate new ideas in the area of microwave devices and quantum electronics. The program is envisioned as a general and flexible one under which a wide variety of topics may be studied. At present, the following areas of investigation are in progress~ A. Study of frequency multiplication in an angular propagating structure. Fabrication of a low-frequency multiplier tube which mutiplies a 600 mc input signal to a 2400 mc output signal with adjutable fee dback control is in progress and extensive testing will follow. B. Analysis of amplitude- and. phase-modulated traveling-wave amplifiers. This phase of the investigation has been concluded anruL.,a final summary on the experimental findings will be given. C. Study of a d-c pumped quadrupole amplifier. Electron traj'ectories in a d-c pumped quadrupole structure are computed by solving the equations of motion for different modes of interaction. Frnom the computer results on trajectories energy relations will be derived and studied. D. Investigation of the cross-mo-du'lation products in a wideband tunnel diode amplifier. Theoretical expressions for the voltage-current

-2the effects of temperature, doping concentration and the bias voltage to the characteristics. The portion of the characteristic which shows the negativ-e resistance will boe used to comnpute the cross-modulation ec7fects when multiple signal inputs are applied. E. Nonlinear analysis of the crossed-field amplifier with multifrequency input signals. A study of the nonlinear beam-circuit interaction in the M-type amplifier will be conducted for small multifrequency input signals. The output spectrum will contain frequencies not present at the input. Equations for the output spectrum will be derived and computed. 2. Study of Frequency Multiplication in Angular Propagating Circuit (C. Yeh and B. Ho) The work on fabrication of the experimental low-frequency multiplier tube is progressing. A quadrupole helix with reasonable uniformity has been successfully wound by inserting the wires in the groves cut on the supporting sapphire rods. Cuccia couplers of proper dimensions have been mounted on both ends of the cavity and inductances have been added to tune them to proper frequencies. A method of coupling an r-f signal into the coupler has been worked out. It consists of a coupling loop located outside of the envelope to affect the coupling inductively. 3. Analysis of Amplitude- and Phase-Modulated Traveling-Wave Amplifiers (M. E. El-Shandwily and J. E. Rowe) A summary technical report on the nonlinear operation of the traveling-wave amplifier with multi-signal input has been written. The computer results for the large-signal analysis which have not been

reported in the previous reports are included. It also contains all the available information concerning this problem. It is expected that the report will be issued during the next period. Study of a D-c Pumped Quadrupole Amplifier (C. Yeh and Ho) 4.1 Introduction. The equations of motion for the cyclotroncyclotron wave interaction and cyclotron-synchronous wave interaction of a d-c pumped quadrupole amplifier have been developed in Interim Scientific Reports No. 5 and No. 6 respectively. During this period, the computer solution of these equations for arious initial conditions has been obtained. The results are presented in the form of electron trajectories and energies. Some of these data will be plotted and discussed in this report. Th oputations are based upon the following pump structure constants: Cyclotron frequency 1200 mc Helix -pitch 8.4 m Helix radius 1 mm Magnetic fie'ld 450 gauss IBeam voltage 500 volts 4.2 Cyclotron-Cyclotron Wave Interaction. Under this condition, the electr on tr.ajectories and energy relations for two different pumpfed are studied. 4.2. LoT upFed For)he artiulapup4 field

-4can be considered as a very low pump field. The reason for choosing this value will be made clear later. The results of the solution can best be interpreted by plotting the relative motion of the electron with respect to the pump field. Figures 4.1 and 4.2 show the trajectories of electrons for different pump field strengths in relation to a stationary pump field. The trajectories for low pump field strengths are all of similar shape. Their radius of rotation increases with time after they enter the pump field structure, reaches a maximum approximately along a 45-degree line and then decreases. It will be clear in the next section that the 45degree line, where the positive helix wire is located, is the line which separates the amplifying and nonamplifying regions. In Fig. 4.1 only the trajectories of the most favorably phased electrons are plotted. However, in order to show the phase-focusing effect for the unfavorable electrons after their entrance., a plot of trajectories for different entrance phases under the same pumping voltage is given in Fig. 4.2. It can be seen that electrons entering at all angles will slip into -the trajectories of the most favorably phased ones within a few cyclotron periods after e'ntrance. Eventually, they group into two spokes. 4.2.2 High Pm Field-. When the pump voltage is raised, the shape of the trajectory changes. The electron tends to move toward the po' -itive helix wire. Consequently., the radius of rotation increases monotonically with time. Figure 4.5 shows some of these trajectories.

-53300 0 300 300QO 600 0.48 o.2s 2400 1200 p0 2100 1800 1500 FIG. 4.1 CYCLOTRON-CYCLOTRON WAVE INTERACTION FOR LOW PUMP FIELD

-63300 0 300 3000 600 p~~~~~~~ 27000o.2 2100 1800 15Q0 FIG. 4.2 ELECTRON TRAJECTIORY OF DIFFERENT ENTRANCE ANGLE FOR CYCLOTRON-CYCLOTRON~~~~~~~~ WAEITRCINAN4O UPFED

-73300 0 300 20V ~ 9Q 1o2 p 19v 200' o800 6500 /~~~~~04 P \ 2700 D9B X

-8straight line approximately 45 degress from the reference line. This is the location of the positive helix wire. An increase in radius of — rotation signifies that the rotational energy of the electron has been increased. If useful energy can be lderived from this electron, amplification or gain results. It can be seen from Fig. 4.3 that amplifying electrons stay above the 45-degree -ine whr-le'ie nonamplifying electrons reach a maximum at or around the 45-degree line and then decrease in radius of rotation (see Figs. 4.1 and 4.2). 4. Cyclotron-Synchronous Wave Interaction. The pump field structure used for this study is the same as in the case ofcyclotroncyclotron wave interaction. However, the beam voltage is reduced to one fourth as much to satisfy the requirement for this type of wave interaction, i.e.., Pq = 12 (e Interim Scientific Reports No. 5 and No. 6). Two cases have been studied; the nonamplifying case and the amplifying case. 4.3.1 Nonamplifying Case,~-Low Pump Field. When the pump voltage is smaller than the critical value, i.e., V. K Vo(PCa)'2/4., the pump field has very little effect upon the trajectory of the beam electrons. The beam keeps rotating at the cycl'otron frequency while the pump field rotates at half of the cyclotron frequency. Figure 4.4 shows the trajectories for a very low pump field. A stationary pump fie.ld is assumed in this plot. Under this pump field strength, the radius of rotation decreases as the beam moves along', and thus no gain can be

-93300 Q 300 /~~~~~~ 0.16 202 / \ ~~~0.04 2700 210 100050 FIG. 4K4 CYCLOTRON-SYNCHRONOUS WAVE INTERACTION FOR VERY LOW

-103300 0 300 p J~~~~~ 0.12

-113300 0 300 p -T~~~~~~~~~~~54T >X~~~~~~~. T 3000:F 3/ T X6v

-123.2 Amplifying Case —High Pump Field. When the pump voltage is raised above the critical value, the pump field gains control over the motion of the beam. After entering into the pump field structure, the electron beam starts to retard in phase and lock into step with the pup ield. The radius of rotation increases as the electrons gain energy from the pump field. Gain is therefore obtained in this case4 Figure k7 shows two of those trajectories. Equipotential lines of the pump field structure are also drawn in the figure to show how the electron gains energy during its course of motionc The angular displacement of the beam electronsforboth cases are plotted in. 4.8$ The slope of the curve shows that the beam rotates at a rate equal to the cyclotron frequency for the nonamplifying case, while in the amplifying case the beam rotates at the cyclotron frequency at the beginning, but it starts to slow down and finally rotates at half of the cyclotron frequency, which is the same rate as the pump field. 4i.4 Energy Relations. It is generally believed that the transverse energy of the rotating beam of a d-c quadrupole amplifier is converted from the axial energy of the beam. In other words, the r-f power is amplified at the expense of the axial beam power. As the beam passes along the pump field structure, the beam velocity would decrease. This energy conversion -process gives rise to the problem of synchronization between the beam waves and the'pump field. The problem becomes more serious es-pecially if the device is operated at a large-signal level. As has been shown in Interim Scientific Report No. 6, o phase slip of more than 45 degrees, the beam will fall into the~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ fo

-13-'53300 0 300 p~~~~~~~~~~~~~~~~ 300" 210 106050 0.8- I/2 Tc) FIG. 4I.7 CYCLOThON-SYNCHIRONOUS WAVE INTERACTION FOR HIGH PUMP

-1424 20 SLOPE EQUAL TO CYCLOTRON FREQUENCY SLOPE EQUAL TO TWISTED PUMP FIELD FREQUENCY c' 16 z 30 VOLTS z 12 / r/ -J I, SLOPE EQUAL TO D 8 CYCLOTRON FREQUENCY z 4 / 0 6 12 18 24 30 36 TIME, T

-15inherent difficulty, the overall efficiency of this type of device is comparatively low at a theoretical maximum of 33 percent even with depressed collector operation'. From the energy relation study of the computer solution it can be shown that under some beam-wave interactions, the axial velocity remains almost constant. This indicates that the transverse energy of the roating beam must come from the pump field. This is contrary to the common belief. If this is true, then the off-synchronization problem would no longer exist. Consequently, a high efficiency transverse wave amplifier is thus possible. The anomalous gain mechanism reported previously belongs to this type of interaction. Let us now evaluate the energy balance relations. In Interim Scientific Reort No. 6, the transverse energy in terms of the radial displacement p, angular displacement O., was given as Etrans Erad Etan where E =eV p2 rad 0 and Etan =eVp (OP) The axial energy is given by 2 axial eV

puter solution for the energies in the cylotron-cyclotron wave interaction is given in Fig. 4.9. It can be seen that as the radius of rotation increases the axial energy decreases rapidly. This conforms with the general belief that energy is derived from axial energy. However, when the device is operated at a very large signal level p approaches unity and the total energy exceeds the initial beam energy, which means that even in cyclotron-cyclotron wave interaction an energy ptransfer from ump field to the beam would also occur. Figure 4.10 shows the energy variation for the cyclotronsynchronous wave interaction. Notice that the axial energy is essentially constant during the entire amplifying process, while the tangential and radial energy grow as indicated. This result definitely indicates that the transverse energy is coming entirely from the pump field. Another point of interest is that the magnitude of the"' radial comp'onent is' much smaller than the tangential component. This indicates that it would give a high efficiency operation since the amplified output is t.aken. from a Cuccia coupler which depends on the tangential component on-ly. Compaigtiswt ig. 4.9, it is obvious that the cylotronsynchronous mode of operation is more efficient. 4.5 Con'clusion. From the computer study of the transverse wave amplifier., the exact trajectory of the electron is known. From this the gain of the interaction can also be determined. This condition as well as the amount of interaction obtained from this computer study is entI-ire-ly in agreement with the cou'pled-mode analysis given in Interim

cd~-lw w 0 -z w IY 0- Z~Z w w~~~~~~~~~~ II. II,i~~~~~~~~~f NN C\j CD~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~z

-i8-.4 w~~~~~ z < ~ ~ ~ ~ 0) -J~~~~~~~~~~~~~~~~~~~~~~~~~C o ~~~~~~~~~~~~0 0 I- P 0 _ (.~~~~~~~~~~~~~~~~( Z~~~~~ 6~~~~~~~~~~~~ 0 If) 0 C\j CD cli~~~~~~~~~~~~~~~~4

-19modes of interaction. It is contrary to the general belief tht the dc pump could possibly feed energy to the orbiting electron. With the conventional concept of axial-transverse energy conversion, the ficiecy is limited to a low value; however, with this pump-ransverse enrgy coversion, a hig,h efficiency device of thllis type is enirely possible. -tLuure Work. It seems quite feasible that a high igh eficiency d-c quadrupole amplifier can b'e designed. Te elements Ived would be simple and t'he construction would be simle. I is he intention of the authors to try this approach. vesigation of the Cross-Modulation Products in a Wideband Tunnel Dode Amplifier (C. Yelh) 5.1 The Quantum Mechanical Tunneling Effect. The time-invariant SchIrodinger wave equation can be expressed in the form.2 2m where 4r is the probability density function such that VV or in other words th-at' the amiplitude of squared, whien properly normalized, indicates the -probability of finding a particle in the volume.

-20Whereas in classical mechanics a negative kinetic energy is not permitted, it is allowable in quantum theory. In fact the solutio the condition that E - U < 0 is fundamental to the theory of tunneling. Assume that one has a potential barrier U0 in a one-dimensional space, and that the particle energies are less than U as shown in p ~~~~~~~~~~0 Fig. 5.1. The solutions to the wave equations in the three regions are, respectively, for x < 0, A exp (j - B exp ( - j (5.2) for 0 K x K x p2x~~~~ 2 C e x,p - + D exp(5) and for x >x 1 F exp where p- 2mE~ p2'12m(U - E)" The boundary conditions are that the functions *V,l2 and and their derivatives 6t1* x 2/x and 6*,/6x must be continuous at x = 0 and x = 0 repectveXy Ths odiin nal n2o ov orfu u

-21U = U POTENTIAL BARRIER n E s uJ L u=O Ix x=O X=Xo I I I I I, POSSIBLE SQIUTIONS I ~OF THE SCHRODINGER FIG. 5.1 QUANTUM MECHANICAL TUNNELING.

-22) = A+B, 2() = C + D () = C exp (2 ) + D exp (P2 ad = F exp (j liXo) 3 0 p *1 Pi 1i j A - j B P2 P2 C - Do, - exp ) xpa) 6x 0 6x 0 Then A +B C C+D C exp (PX0 +Dexp( )= F exp(.~) p 2 p2 j~A j~B C=

-23If A, B, C and D are solved in terms of F, one obtains - p r A ~F 2j + j exp (iP + P ) 4 l-j ex+j- exp P2 P2 2 - p p x B 1 -+ 2x )0j 1L 0 j - ( j exp (JPl +P) M — 2e +,jp j 2 = (P2 P1 F2 D F2 l j expJ(jp + P2 — If one defines a transmission coefficient T as the ratio of the power of the transmitted to incident waves,, then F 2 _ T =. (5.6) A 2 22 cosh p1 + 1 sinh2 1 2

-24In the case ofa tunnel diode, the barrier thickness is thin. It is possible to have T - 1 in a limiting case of x = 0. The transmitting wave is drawn in the lower part of Fig. 5.1 just to indicate that there is a definite probability of transmission even as E - U 0. To calculate the tunneling of carriers from the valence band through the orbidden region due to the applied field as in the case of a tunnel diode, Zener1 attempted to use Block's solution in a periodic field U(x) perturbed by a uniform field 8. The resulting tunneling probability is is~~~~~~~/ L a expL (2m*),/2 Eg/2 where ~ is the applied field, a is the lattice constant, m is the effective mnass of the carrier and E9 is the ga*p energy. The tunneling probability indicates thaLG electrons are actually diffusing into regions where the kinetic energy is computed to be negative. 5.2 The Tunnel Diode. A tunnel diode is a p- junction diode in which the p- and n-type regions have exceptionally large concentrations of acceptor and donor impurities. A tunnel diode exhibits two unusual regions in the V-I characteristic as shown in Fig. 5.2. At a very small forward bias,, a region of negative resistance is observed. In t he reverse bias region, the usual saturation region of the junction diode is absent. These characteristics will be explained in terms of the tunneling theory discussed in the above section.

-25I NEGATIVE RESISTANCE I~~/ \ -NORMAL DIODE CURRENT REVERSE BIAS O. 3 Mv FORWARD BIAS REVERSE _ CURRENT FIG. 5.2 THE V-I CHARACTERISTIC OF A TYPICAL TUIEL DIODE.

-26The energy-level diagram of a tunnel diode is shown in Fig. 5.3. For a heavily doped semiconductor, the Fermi level for a p-type material is below the valence band energy and that for an n-type material is above the conduction band energy. When these materials are brought together into intimate contact, and when a small forward bias is applied, the electrons in the neighborhood of the Fermi-level in the conduction band of the n-type material are Lacing the iholes in the neighborhood of the Fermi-level in the valence band of the p-type material directly across a forbidden gap. The tunneling probability is a maximum for the electrons in the n-type material to fall into the holes of the p-type material through the gap. This tunneling produces a negative resistance over a range of low forward bias. For a larger forward bias, tunneling ceases because there are no longer any conductor band states on the n-side at the same energy level as the valence band states on the p-side, and the current becomes the normal forward current observed in ordinary diodes. The behavior for reverse bias can also be predicted. In view of the energy level diagram in Fig. 5.3, tunneling also occurs when the Fermi-level on the p-side is higher than on the n-side. In fact, the higher the reverse voltage, the wider the range of levels that participate in tunneling and the larger the reverse current. Thus saturation under reverse bias will be absent. 5.3 The V-I Characteristic. Figure 5.4a shows an enlarged energy level diagram of a tunnel diode under the condition of a small forward bias similar to Fig. 5.3b. The forward bias voltage is V. Efp and Efn are the Fermi-level energies of the p- and n-type semiconductors

27CL~~~O 0 ~~ >U ccl ~ 0 UJ t~J cL~~~~~~~~~C X ff |}w LU u 0 LLI'" o 0-~~~~~~O Cf. 0 ~ o~ W c.3 ~0 WJ UJ 9 o g W~~~~~~~~~~~~~~~~~~~~~~~~ c c c,,) >> z CL UJ ui~~~~~~~~~~~c

-28Ecp Cp N (E Ncn(E Cp Ecp Eg Eg/2 E E E'Ax EA_P E vp VP EpE V Lb~~~~~T eVE eV E P E Efp Ep.4 Ecn ~~~~cn Cn-. Cn E J ~~~~~ED ~~Eg/2 E d Eg~~~~~~~~~~~~~~~~~ Nv (E) V v n Vn a.) ENERGY LEVEL DIAGRAM b.) DENSITY OF STATES c.) RELATIVE OF A TUNNEL DIODE WITH IN A TUNNEL DIODE POSITIONS OF FORWARD BIAS WITH FORWARD BIAS THE ENERGY LEVELS FIG. 5)-4 ENERGY LEVEL DIAGRAM, DENSITY OF STATES AN~D TEE DEFINITION

-29respectively. If one defines the forward current as the tunneling current caused by electrons tunneling from the conduction band to the valence band, and the reverse current as the field emission current (or Zener current) caused by electrons passing from the valence band to the conduction band, then the total current at any time is the algebraic sum of these current, i.e., J = Jf + J~ f r The tunneling current density is proportional to the product of the density of occupied states on one side and the density of unoccupied states on the other side multiplied by the tunneling probability. Let Ncn(E) and NV (E) be the number of quantum states and fcn(E) and f (E) are the Fermi-Dirac probability functions of occupancy in the conduction and valence band of the n-type and p-type material respectively. Then Nc (E)f c (E) is the number of occupied quantum states of the energy E in the conduction band of the n-type material and N [1 - fV (E)] is v P the number of vacant quantum states of the samecenergy E in the valence band of the p-type material. Let ~,_> be the probability of tunneling from c to v, then the forward tunneling current density Jf may be written as E Jf.= N ~(E) Np (E) ~(E) f (E) Ll fiB)]E dE.(5.8) c,n Notic that the Intgaini from B to A B_ Tis sbeas

Similarly, the tunneling current density in the reverse direction J can be expressed as E cn Ncn(E) Nvp(E) vc(E) f (E) 1 - f n(E) dE cn V V —C V p cn Thus the total tunneling current density becomes E n N(E) Nvp vf (E)c L1 - f (E)1 dE NVP(E)c-v cn f VP E cn - Ncn VPE)V-> (E) NVPp v-*c - f'cn(E)j dE lo5.0 E cn Assuming equal probability of tunneling from c to v as from v to c., the current density is written as EVrP j ~N (E) N VP(E)Qfcn (l VfP (E) - VP (E )( - fcn(E.) dE E cn - Ncn(E) NVP (E) fcn (E) - fVP(E)j dE. (5.11) E cn The integr-al of Eq. 5.11 is extpremely diff'icult to evaluate due to the following reasons:

-31impurities. Doping levels of more than 1019 atoms/cm3 e necessary to obtain significant tunneling. The heavy doping degenerates the semiconductor in which the Fermi-level will be located within the bands. In the n-type semiconductor, the Fermi-level is found in the conductor band, and in the p-type semiconductor it is found in the valence band. Tae high doping densities also cause the donor and acceptor impurity levels to be broadened into bands themselves and merge with the main bands. Figure 5.5 shows the denrsity of states function for a degenerate n-type semiconductor. It can also be seen that band gap E g narrows considerably to E' due to heavy doping. A similar sketch for g ~the density of states function of a degenerate p-type semiconductor tan also be drawn. The product of these two density of states functions is a highly irregular function of E and depends very much upon the doping concentrations.2. The tunneling probability depends very much upon the potential barrier width and height and thus is a function of the junction field and the energy gap. It is further complicated by the fact that, althougli tunneling conditions require that energy and momentum be conserved,, t-he conservation of momentum does not require the momentum of the electrons to be the same before and after the tunneling. This results in the two possible types of tunneling, the direct and indirect tunneling. In the direct tunneling,, all particles participating in the tunneling must have the same energy on both sides of the barrier and they must tunnel from the vicinity of the minimum of the conduction band energy-momentum space

-52I/ 2 Ncn(E)a (E) Ncn(~ Efn Ecn — ED I~~ Eg Eg Evj VP ( FIG. 5.5 THE DENSITY OF STATES FUNCTION FOR A DEGENERATE N-TYPE SEMICONIDUCTOIR SHOWING TEE EFFECT OF BROADENING OF THE IMPUIRITY LEVEL ED

-7 -7 not occur at the same value of momentum space as the valence-band maximum, a condition which prevented the direct tunneling because of the violation of the law of- conservation of momentum, indirect tunneling can exist. In this case, the difference in momentum is supplied by some scattering agents, such as phonons or impurities. It is thus very difficult to estimate the transition probability. Fortunately, the indirecttunneling probability is generally considerably lower than the direct tunneling probability when direct tunneling is possible. j3 The Fermi-Dirac distribution function is temperature dependent. At absolute zero temperature, the Fermi-Dirac distribution function is a pure step function. At higher temperature the available states above the Fermi-level can be occupied with a nonzero probability. All these factors introduce complications into the actual evalu&, ation of Eq. 5.11. In fact, a general solution of Eq. 5.11 has not yet been found. Up to date, the only solution available is that for absolute zero temperature and in an assumed constant electric field configuration. 5.4 An Approximate Solution of the V-I Characteristic. To facilitate the solution of the integral of Eq. 5.11, it is desirable to investigate each.term of the integral carefully. The density of states terms, Nn (E) and N (E), will be discussed first. Figure 5.4b is a sketch of the density of states graph of a tunnel diode with a forward bias. On the left-hand side of this figure, Fig. 5.4a, the energy level diagram of the tunnel diode under the above stated operating condition is reproduced for clarity. A slight change in notation is in effect in order to simplify the subscripts. For example Ecp, the conductor level on p-side, is abbreviated as Cp, E as Vp, etc. On v-pp

-34the right-hand side of the graph, shown as Fig. 5.4c, notations and reference of the energy levels are indicated. In Fig. 5.4b, N (E) and N (E) are drawn as solid curves. cn VP These are the modified density of states curves in which the effect of the broadening of the donor or acceptor band are taken into account. If there were no broadening effect of the impurity levels, these curves would follow the dotted extensions of the curves. In fact, the broadening effect of the impurity level is the result of heavy doping. Although the detailed calculation of the broadening effect is complicated, one could make a rough estimate by assuming that Ncn(E) would start from an energy level E' somewhere in the band gap and increase gradually g toward the original Ncn(E) at Vp. Similarly, N (E) is also shown. Let us assume that for equal impurity concentration on both sides, Eg g is approximately halfway between the band gap and that Ncn(E) = A1(E')4/3* and N (E) - A (E' - E')4/3, (5.12) Vp 2 x where the relative positions of E' and E' with respect to other levels x in shown in Fig. 5.4c, and E' =E + E /2, g E' = E +Bg, (5.13) x x g The choice of 4/3 power is arbitrary. It is determined from the best available experimental data on broadening by heavy doping.

E being the energy level with respect to C (or zero). Notice that without the broadening effect, N (E) = A E1/2 and N (E) = A(E$ -E)1/2 The Fermi-factors are defined as cn(E) (E'-E)/kT (514a) 1 + e f~~VP( =[E'-(E'-E')]/kT 1+e x p and 2kT/ f (E) - f (E) Scn (E) - fp(E)=eV eV + 2(E' - E) sinh e + cosh 2kT n 2'kT 4.1 2kT where eV = E' -E' = E - E n p n p Define E' E +-E + E - eV (.16) x g n p and (2 E'/E) (.17) x and assuming for each irmpurity concentration, E' = E' = eV /2, (5.18)

then eV + 2(E' - El) n e 1 e (V - V) x 2kT 2 kT m Equation 5.15 can be modified as s inh (eV fcn(E)- fVp(E) eV 1 e (V - V) x cosh ) T + cosh kT m V) x Assume also for the time being that the transition probability between n and p are equal and 100 percent, then Eq. 5.11 can be expressed in terms of x as -= e( m-V 5/3 _O (1-x)4/ dx A-2 C2 1 Kcosh v + cosh " (V-V)x T1)2 h(Vm V)]5/3 0_1 (l-x2) / dx L 1 oshk/eV + e2 -l cosh +coshn (Vm-V) (5.20) where B is a constant. Equation e.20 is useful in predicting the effect of several factors on the tunneling current density. The doping concentration will appear in the choice of V, the temperature effect in T and the bias effect in V. 5.5 Future Work. Equation 5.20 will be evaluated under various prescribed operating conditions, particularly when:.the forward bias is

-37varied. For a particular sample of semiconductor material, and at a particular operating temperature, the variation of V, the forward or reverse bias voltage should result in a V-I characteristic which can be checked experimentally~ When the tunnel..diode is used as an amplifier, the negative resistance portion of the V-I characteristic is of importance. To compute the cross-modulation products due to the nonlinear portion of the characteristic, Eq. 5.20 or its modification will be used for multiple-signal operation. It is planned that series development of the terms involved in this equation will be used for this analysis. The effect of tunneling probability ~ can also be included if an adequate approximation can be found. 6. Nonlinear Analysis of the Crossed-Field Amplifier with Multi-Signal Input (M. E. El-Shandwily, J. E. Rowe) 6.1 Introduction. During the previous period a study of the nonlinear operation of the injected-beam forward-wave crossed-field. amplifier with multi-signal input was initiated. A small-signal nonlinear analysis has been completed during this period. A brief discussion of the analysis is given in the following section. 6.2 Theoretical Analysis. It is assumed that at the tube input the voltage applied consists of N-signals which can be written as N NR jV t Vin = Re Vn e (61) n=1 where the frequencies cu,.O.,N are within the bandwidth of the tube. Due to the nonlinearity of the beam-circuit interaction the output

-38spectrum will contain frequency components which are not present in the input. It is the purpose of this analysis to derive expressions for the different frequency components in the output. The starting point is to neglect the nonlinearity and consider the circuit voltage as a superposition of the effect of the individual input signals according to the linear theoryo This circuit voltage is then assumed to act on the electrons whose motion.can be calculated by successive approximations. It will therefore be possible to express the longitudinal and transverse coordinates of the electrons as a power series of the input voltages. From these relations one can calculate the arrival time of the electrons at some longitudinal distance in terms of their departure time at the input plane and therefore the r-f charge in the beam. The induced voltage in the circuit is due to two effects, the first is the r-f charge whose effect is similar to the O-type case, the second is the r-f displacement which in the presence of a longittudinal field that -varies with the transverse position will produce a circuit voltage even if the density of the beam does not vary along the direction of propagation. The induced voltage on the circuit is given by -j r K r r2 PE' (6.2) where r, r are the propagation constants of the circuit in the absence and presence of the beam respectively, K is the circuit impedance and PE is the effective charge density of the beam at the frequency w. The effective charge density, PE, can be written as

-39PE (Po + P) c(y), (6.3) where p is the d-c charge density, p is the r-f charge density and cp(y) is the coupling function. The coupling function will be expanded in a Taylor series, about the unperturbed position of the beam ) c(+ d~ 2 dfl (y 0)2 therefore the effective r-f charge density can be written as PE = PP(Yo) + Po(Y-Yo) d2Y + P(Y-Yo) y + 2 Po(Y-Yo)2 dy y ydy o 1 d. +29' P(Y-Yo)2 _diY ~ 3! Po(YY) d+ +- * (6. 5) 0 dy2 dyY() 3 Yo5 The first term, pe(yo), gives the effective charge density due to the longitudinal r-f charge at the unperturbed beam position. The terms p (y-yo)(dm/dy) y, (1/2) po (y-yo) (d2p/dy ) ly and (1/5) Po(y-yo)3(d3p/dy3)ly give the effective charge due to the r-f transverse displacement of the d-c charge of the beam. The remaining two terms, namely p(y-yo)(d/dy) y~ and (1/2) p(y-yo)2(dp2p/dy2),y give the effective charge due to r-f displacement of the r-f charge density. It should be noted that since pE is a real quantity then

-40when p and y-y are expressed in a complex form the real part has to be taken first and then the multiplication is performed. As has been mentioned previously, the equations of motion and the successive approximations described above are used to obtain expressions for the r-f charge density, p, and the r-f transverse displacement, (y-yo). There will be an expression for each frequency component of the charge density. The r-f transverse displacement (y-yo) which is obtained from the equations of motion will have components at all possible frequency combinations. Since in this analysis the perturbation is.. carried only to the third order, that is, the y and z displacements are obtained in terms of the input voltages, their squares and cubes, then only the components with the following frequencies will be found both in p and (y-y ) f f ~ f, 2f - f and f + f - f p where r, p, or s = 1, 2,,.,N. To obtain the effective charge density at a particular frequency care must be taken to include all the contributions of the different terms in Eq. 6~5. For example the r-f charge density at f ~ f will contribute an effective charge density at f ~ f ~ f r p r p s when multiplied by the appropriate part of (y-y ) in the third term of Eqo 6.5o Physically, this is because an r-f charge density at f + f r p will have an r-f transverse displacement at f and therefore it produces s a circuit voltage at the combined frequency f + fp + fs r p s To simplify the analysis the following assumptions are made: lo The beam is assumed to be very thin so that the difference between the fields on both sides of the beam is neglected. 20 Space-charge forces are neglected. 35 Only the increasing wave of each input signal is considered. 40 The adiabatic equations of motion are used.

In addition to the above assumptions the usual assumptions which are commonly used in the analysis of such devices are also utilized, namely nonrelativistic mechanics, no thermal velocity distribution and no reflections. According to these assumptions the equations of motion of any electron that entered the interaction space at time t are written as: dz 1_ I N V d~n d2c~n (Y-Yo) 1 d3p (y- nz - ReX VILdIY + 2l+ lYJ- e _ dT- c n Ldycy0 dy2Yo 2 dy3 y0 y n=n e nJ no (6.6) and dy __cR N (y) dn d2_pn) Re 0 + (y-y) + n (y-y)2 dT L nn n dy dy y n=1 d3 ~ f do (6.7) 3! d3P (Y-Y )3j en e n no (6) where q is the electron charge-to-mass ratio, c is the cyclotron radian frequency, V' is the amplitude of the increasing wave of the nth signal n at the input plane z = 0, Pn jen (1 + jDn n) is the progagation constant of the increasing wave of the nth signal, ~en = n/Uo D = is the interaction parameter, T - t-t is the transit time, 0

-42t is the entrance time and o Re denotes the real part. The first approximation to solve Eqs. 6.6 and 6.7 is to substitute y = y and z = u T on the right-hand side of Eqs. 6.6 and 6.7 and o o0 integrate with respect to T. The result is Z = uoT _1 \ n () e n n cos ( t (1) + nnT) 0 2P v nD n n o n n n co n n=l (6.8) and N o + 2 V 7 n X(1) e nYn cos (o t - + nT) - o 2P V L D n n o n n n-1 (6.9). (1) where X(1) e n = - D u, n en n o 6 = +jGD, n n + jan' c c%/UO' Qn = cosh yenyo/sinh en d and n s= sinh eno/sinh end. In writing the above equations it has been assumed that CPn(y) sinh eny/sinh end and the anode-sole spacing is d. The second approximation is obtained by substituting from Eqs. 6.8 and 6.9 into the right-hand side of Eqs. 6.6 and 6.7 and neglecting the second-order

-435-r z terms of (y-yo). Also both Eqs. 6.6 and 6.7 will be written as N -r u iy 7 YT 1 e n [1 n v Vi X( e i cos (.tt () + YT) L 2PVL D. i i i ii j 1-1 and then the integration is performed with respect to T. The process is repeated to obtain the third-order approximation. The result of these steps is an expression for z and y in terms of the input voltages, their squares and their cubes and the departure and transit times. The continuity equation of the charge density is used to obtain expressions for the r-f current (or charge density) at the various frequencies as it has been done in the O-type amplifier. After going through extremely complicated algebra the following components of the output voltage are derived:

-44V'f 1 + D b -jD d r r r rr r r VV. = -- (r +jb + d ) e inr inr r r r r V' 3 + inr 4r )( r) (1 + Drbr jDrdr) + 27r) + I v V. r r r r eDrdr r o oinr j(r + 27r) - b + jdr) ( r e rrrr r r _rH r r D ~ Pini i +J rJ i=r ~ ~r ~, 5 + 2 -- r -b + jdr r D 7i i r r r r} r r1 jQ + P + S + T,(6.10)

-45where ()(i) -je VrIx( X 1' er X(1) (1) () r (a)~~~(I ~r rr rr L (Kr1r r G 16 X Xr X r er r r rr G = ~c - S2 r ~ ~ ~~r r rr 2~~~~2 *r1o ((1 s ) (1) (3)) Cs(e(r' err + sin (er - 9 rr + x(l)x(2) e(2 (rr ~2 5) ~ rr rr r r rI+, 12) 22~~ r r rr( +(q(* ~ ~ 4~) rr) (6) 2 + ~ X(3) X(6) X(r err -j (3) r rr mrrr j Cos (Br rr ~r 2 2 j(6) rr r (1 - j) rr~ ~ O cos (3) (1) -(1 e(s)) rrj Q sn r rr - r mr rre r r r 6~( r sin (e (m) -e(s>))~ + X(1) X(z) X(6) -jSTrr(-j - X(j) X(2) (5) -j(e + 9(2) _( r5) 2 + 1 (i) x e r mrr rr +( 2j 2 r r rrr _~~~~~~~~~~~~~~ 2 -J(2 + i X(1) X(1) X(5) e rrrr 2 +' r r rrr

-46f~r I;1) (3) -(6) je 2(6 PC 16 -r rr rrr jJ (Or rr r r + J) sin (X(l) - e +.~ X(1) X() -e)rr + -J( r rr r r r rrr 2 r ~ x(2) s) 3(&rr r + e(r2) r8- 2 r r r rrr 2 rr rr rr X } r r + r j xjx2 e r {e (1) (1) 2' ( 1 jir X(l) X(l) x(5) e J2r rrr 2 rr r r rrr 2r r~~~~~~~~~~~~~ *r (1) 3) eos ( 41) (3) )-sn( r ~, S2 e~~~~~~~~~~~~~~~~~~ r r rr rr rr 1r (1~~~~~) X(l)i~ X(I) X(-i) e re r r r r jo(2)2 -- h-....~~r ())(2)J2( ) rr 3()~ + x x x(2) r r r rr e 2 r,(1) X(1) X(2) e -jerr 3 3~

-47-.. X(1) X2(1) X() -e 16 i r r rr cos ((1) - (3) ) - sin ((1)_ e(3))} r rr r rr + ( )x(1) x(1) X() e rr (1 + j) r r rr,- (X(1) X(1) X(l) e r M = ( ) X X X() e r 52 c r r r Q =16 [ 1 B,r- r r ( )x((1) x2 ) X(1) e jr L crr + " ~ ~- er f(AF>2 +, r ii. {31. ), 3 J2~ - r ) I. e i i i r, Q J X()X3 j0rcr) lr r r~ z 2 c r r 1 PXr (1(3) (J(e() - e 3) 41 Xri e r

-48(Q Contd.) i }{S (1)(1( -je(2) I- r -— i i r #) X(') x' (2 ri 2 R R I f 2 ri c r r r 1 r (e (1) e-l (2)) j(e 8~ir 5 r (1 ~ (6) -je(6 + 1) (1j) X(2) e +'ri i r ir i ii rii 2~~~~~~ ~- 4r ~f (t - Cos (e() - (3) ( S2.~~~~~~i 11 r r r {1 Si Sr R/i (i)2 r r r ~~~~~~iii S2. Ilr, I", ~rr(3,_ 1 ~i (6) -jO(6) ~r 2i ~r ~ i ( () -jo irii s)r 1) (3)1 r i r i ) j ri i 2: I )X~1' xK e 1 J ni i *r X e iri 2, R i ri cc...... r c r r in~~ f a.i(' je(X) e - 2 r r ~~~2ir i r mi (s) r rr r r r _ (1(1) _() 3 (s) e r lr )Ji 1 (6) rX 8, S1, jrir +~c - - -+ j(i -X e X e r i 22 I 1r nir r r r 0(i) e 6 f2 B 1~~(1) X () (l r ir 1L 1 r iri r - Xr r r r r ~r ~ ~'~(1) e(l (6) -Jr.'~ ~~~~~r r r B~i i, n rii r - ~

-49(Q Contd.) (1) 2 ((2) (5) + 1 i x(') x(2) x(5) -J(e + eir) iir) +, r rir iire C r r r r r r j(e(i) + e(2) e(5) +1) (2) X (X) e i r iir cr +i. - x.2 x2. 2 *r. ) + ai r + i r r r r r r r r j( rii {e(i e( 5) 1i'r'3' i *i ( x1) x() r iir. X) 1) Xii( ( e ( P = 1- Q r_Qi =i.i ri j + — co (r ) r r r rr r r r r r (3) +l Br (1)(3) (6) ri iri

-50(P Contd.) B j(- ~e(l) + - X X +1 i x') x(3) x(6) e r ir - ri 2 r ir irl r 22 2 r r r r r r r r +Dr 5 r ~ i XY. e((1)X{ Q Qr Q ) iQ ri 2 r Q2 IQ c r r r + 1 Fr ( (( ) X(X) X X r 2iri 2 PC r1 r l1 Jf Xi. 2 i( 2 r r r r r r r (2)~-~'~ ( ) + 22)) 5) 2 1 ri 1 i r _+ 2 2 )r zTi r r r r r r r r () j(() + e(2) (5) + Xi (1) ) X.) rr iir 2 PC r ir iir r r r r r r ~ -j(Oe'. + + 2 B, r I mmr C B r 1 llr

-51(P Contd.) 2 Q 2 a 1 / r r r r r i l (3) r (1) (3) r (11) (3) + j~ ~~PX X e O1Cs (19. e sin - r 1 11I +~1 r X(') X ) X (3) ri *r,e1 (2) + J ~ ~ — ~- r 1 1 2 B 1 ri 1 r c r 1 r I B~~ ~~~~~~~i -. -j. s in (8. -. x(z) () 31) - ig r -i3 - 1r X( X e)X3 r11 ~2 Si r I ir C r r r r (2) r D i (1) 1) (2) ri (3) 2 X X1 ri c r r r r (1.) (1) ( r 2)i i is ii i-r 2 2 Z r 1. ir'?T- a c r r r r, Pr(~) Pr) -j(e() - ) r ~) + (*) r - ri B (l) %!l x! i *~jr 2i X r r r rIir --- ~B -r R- re iir ~ ~+~ (2) 1 Pr Pr *i (2)l)x!) -Joi -j 1 2 5 i I 1 ri P r (1) ( 1) (le) _(2) + rj2 1 +rr 1ir - -- + ~ ~ ~ ~ ~~ ~,X X. X. e 2 Q r i ir c r r r r

-52(P Contd.) +i O Or Or _i ___X(1) Xl Xl e r pr 2 r c Pi r r r i i e ~~~~~~~~~~r -~ ~ —-- --- X X. r1. 1 A +r r 1 r e3 a.~ r 1 rr ~(3) rir 1 8,B, J X!1 X.1 X 3) e +i.(-e( ) - 0 + R ) 0 i (1) X(') 3) r ir 2 r\ 1 ir 1 r r i r r r *. *r je() ('(1)J -r 5 5) 2 "r X-+ )-3 ) e r -2 72 r 1 r r c~~r r r Or (e >5' >5)x2)e() si+( 3())f ~~~~~~ 1 1 ri 1 r e r r r ire.... $i(1)(1)X() J-(1)- (2(1) B r X Xi xi Xri e j~~~~~f f r r r 1t *r 1 i r > 5i) (1) (3) ((1) + 1 e3 X = 1 —) -X(1 X1) X..s 3-jrO(i 1r r i r zl r r r r r (1) ((1) (2) ri r + X. X. X. eS X X X.X e + e'I r r ) r ~~~~(1) r 1 l -j~- (r!)X1) X(S -J() 0(1 -i 0()r Ri $li r i ir2 fi fi + f r r r r (z)()(2-J( 2O(1) + el (+e2) ( + -x x.l x! X2 e r r r r r ~i ir Xti X!)X1)ee r ~i

-533 6 r 1 (i+Y X(1) X(1) X(1) e r The output at the second harmonic frequency 2f is given by V2f Vt 2 D2 r 1 inr r 2r r r v. IV V. D inr 00 r r c 2r r 1+D b - jD d 2r 6 2r 2r 2r 2r r r X() e'j b - b e+ j r r p _ 4inr [ ~r l) X( r) (e21) r 2r + 2r 2 - 2r r + X)X(2) e-j(r() + o9(2) ) er rrW(2) r + rr ()) The output a 2f f is given by r r r pr r 2p mnr r j(1) (2) -J( r r V. 4 I VQ. V. D VD n n inp o0 inr np r 2r-p 2r-p 2r-p 2r-p 2r-]p 25 D 6 D r P - b + 2r-p P +D / 2r-P 2r-p D 2r-p 2 1 r 2 - 1 r r r ~~~j+ r-+ 2~ - 2- r Z,(.2

-54where 7 2Br (e) 2e(l 1~ - 2 x(') x(1) x(1) e pr U 16 LT4K r C __r -r 2r p ( -x 2 x(l) e r + r X(l x (3) e p rp 2, r j ~p rp r ~r r r c [1) (1 (3) ) ~i2 P X(1) e -e ( ( + + -p r SB xr(1) (3) e r rp -r -r r r c ( r 3rr (r) j (ep rr +-X e e (3)) 2 rrp t~~~%\~j2~ ~r r p 4r p r )(' +(r re *r e pr p X( ) ( e r 2rrpr r pr r rp r r r P e r x(l X")j( ()_e() - + E+ [- O+ xpp rQ -- X rrp ~i-~ ~- f, r X(6) e r r r r r r r(3) r'(1) (1) p~X X e rP( r rP r r- rr r r (0(1)~~~~~~~~~! ( ~)5 rX(1) X(3)-(ee r pr B, S1 ~j ~- + -j — e'~ 2 n - ~ — n 2 2 -2 r rpr'/ *r r r r r r r rr ~rr~~~~~~~~~~1 ((1'_p_ r --- ~p(1) X(1) 1) - (1),8- J~ccY~n ( r r r r r, B ~~~-Jfe(1) -() 1~~~~~ _gX1 ()e L prr) 2 5, r prr + J ~'- ~~~~~~x(2) -j rr 2 n + + J [~S ~ 2 rr - r~ r r r r r` r r +~r-X~1 e j > Br Rr r r r.

-55k X(6) e - rp6)2 j (3) W: 3-2 X e rL Xl X(3) e P rp 52 rrp rp rr~p r r + x x2 e p X(l) (3) e r pr r rr n r pr C r r r r p r p 52 p ~2 r r prrP r + - X X e B, n n n n p r 3 r r r r r % rj( p j(2() - (5) 1 P 1j) )(2) (5) X rr prr 32 BP r rr prr +~-cr J2 (rr ~-~+ -2pr r r r r r p p -j(20 (1) ) ( + x( (1) x(1) -(5) r prr +( _ + 32 p Rer r prr 2 r5 r r r -jo() j (1) 0(3)) ~ J X(l) r * j X(I) (3) e _ rp 32 r PC Qr 2 Qr p rp (1 (3)) + x(1) x(3) e r pr P Qr r nr r pr - + 2j r x( z)) (l) eJ r2(z e() nr r.p LBr \ R r /r pr rr r r + X(1 _ j() r ( B~~~~~~ ~_ r p P -- () s j(e~~ () (j r~ X( X(2 e P "~~~r Br Sr r

-56(W contd. ) 32\ r rr ri (e xl+ ( - Qpr X(l) x(3) e ( P r c r r r p + —. -k + ~r X() X(3) eJ r pr - DC - 2r r y *r _e r - r( + #-)X(l )X e P rp 6 r r rP r p __*p p r x x(2) e r )( c~ r r r r rpr r1 c r cB r 2 r r 52~ r~ r rpr cr r 1 B,~ Jlr r (e( 1r,(G( e 20 r ~p r r X(1) X(l) X(2) e - r() - () + 1 —- n, 5 2 p Q er r )' Br r~ jl-X(l) -j _ _ e9 r r cr = -D z = 2 D N n 6~

-57(2) (2) -jni 1 ni.e -D. n D -i n n n pnDn i (4) (3) nj ni 1 Xe ni Xn( e =D I iDi 6 +.. n nD n and( X(6) -J ni ii n + D D i nn nn

-586.3 Future Work. The equations derived above give the output voltage at the input frequencies fr up to the third-order approximation. The effect of all the input signals on one of them (cross-modulation) can be evaluated from these expressions. Expressions are also given for the output voltage at some of the generated frequencies (intermodulation components). The equations are being programmed for numerical computation on a digital coimputer. It should be noted that this analysis fails in the saturation region due to the limited number of successive approximations used. In order to describe the operation of the amplifier in the saturation region a different approach has to be used. A large-signal analysis using a Lagrangian formulation will be worked out in the next period. 7. General Conclusions (C. Yeh) Computer results on the large-signal trajectory calculations for a d-c quadrupole amplifier operating under both the cyclotron-tocyclotron mode and the cyclotron-to-synchronous mode have been obtained. Energy relations have also been computed. Results for these two mades of operation are compared for a certain quadrupole structure. It is found that under strong pump field operation, a cyclotron-tosynchronous mode gives high efficiency operation compared to the cyclotron-to-cyclotron mode. It is further believed that in this mode of operation, the r-f power is supplied by the pu p at the expense of its d-c power. Based upon the quantum mechanical tunneling effect in semiconductor materials, an equation expressing the V-I characteristics of a tunnel diode which include the effect of temperature, doping concentration

-59and bias voltage has been derived. The equation will be checked for accuracy for a typical semiconductor and, if satisfactory, will be used'o compute the cross-modulation effect in multiple-input-signal operation. General equations for the cross-modulation product of a crossedfield amplifier under multiple-input operation have been derived. The small-signal nonlinear version of this analysis is being programmed for computer calculations. Due to the additional dimension to be considered in the crossed-field operation, a new product of frequencies f + f - f becomes important. Because of the fact that less gain r p s per unit length of the crossed-field amplifier structure is involved, it is believed that the required accuracy to get correct results from the large-signal analysis for this type of structure is far less strict than that for the O-type TWA. The theoretical and experimental work on cross-modulation products in an O-type traveling-wave amplifier have been terminated and a summary of the complete work in the form of a technical report will be issued soon.

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