New concepts in traveling wave tubes based on multiple transmission lines

Mohamed Othman1, Filippo Capolino1, Alex Figotin2

August 1, 2014

1Department of Electrical Engineering and Computer Science 2Department of Mathematics

1

Outline

Multiple Transmission Line (MTL) concept and extension of

Pierce model

• Uniform MTL – e-beam dispersion relation and modes

Possible routes for gain enhancement in uniform finite

traveling wave tubes (TWT)

• Pierce parameters

• Gain

Periodic MTL TWT

• High-pass-type circuit

• Dispersion relation with wide band interaction

• Frozen mode regime in finite MTL TWT

2

The goal is to determine whether an MTL can enhance the gain of TWTs

Part I

Part II

Multiple transmission line (MTL) is a generic concept that can model a

variety of coupled EM guiding structure

The purpose of utilizing MTL is to enhance wave/matter interaction, and

introduce peculiar dispersion characteristics that cannot be attained

using single transmission line circuits

MTL involves rigorous mathematical formulation, since it deals with

matrices that are not necessarily diagonalizable

Multiple transmission line concept

We address two cases

Uniform

MTL TWT

Periodic

MTL TWT

Coupled cavity TWT

Examples

Coupled Helices

3

Pierce theory: review

Pierce theory: simple, analytical theory still in widespread use today

Ideal representation of electron beam as a fluid.

Complex slow wave structure idealized as a simple one-

dimensional transmission line (TL) described by two parameters:

inductance (L) and capacitance (C).

4 J. R. Pierce, Traveling Wave Tubes. Princeton, NJ: Van Nostrand, 1950.

C

L

,z z sV j LI I j CV i

0 0 0 0 0 0,b b bI I I u u v

2 0 0

0 00

z z z b zu v E k I j Eu

Equation of motion & continuity equation

Linearization

Transmission line equation

230

2 2 20

1 2 0( ) ( )

c

c

kC

k k

4 solutions for k: 2 complex, 1 forward,1 backward

Synchronism

0k

Dispersion: k is the modal propagating constant

J. R. Pierce, Traveling Wave Tubes. Princeton, NJ: Van Nostrand, 1950. 5

Bunching, space charge waves

Pierce theory : analysis

9.54 47.3G CN

/N H

H

Gain

j t jkze e

Varying as j t jkze e

Waves with

Extension of Pierce theory to MTL

6

Need for more detailed formulation to extend Pierce theory to MTL.

Complex slow wave structures and real waveguides are better

represented by Multiple transmission Lines (MTL) since most SWS or

real waveguides naturally have multiple modes.

Coupled helix waveguide traveling wave tube. This is an example of

two coupled slow-wave structures interacting with a single electron beam.

MTL formulation

1

( ) ( ) ( )N

t n n

n

V z

E r e ρ

1

( ) ( ) ( )N

t n n

n

I z

H r h ρ

Consider a waveguide system with a uniform cross-section

that is able to support fields in the form

{Transverse

fields

We allow coupling between the modes, and we define a state vector

1 2 1 2( )T

N b N bz V V V V I I I IΨ

The state vector defines the evolution of the EM waves/space charge

waves along the z-direction

Felsen and Marcuvitz. Radiation and scattering of waves. Vol. 31. John Wiley & Sons, 1994.

Cross section z

E-beam

Defining equivalent space charge voltage and currents 0 0 ,b b bI u u 0 bb

u vV

7

MTL development

ˆ ˆx y ρ x y

Modal functions

Notations

• MTL described by inductance and capacitance matrices L and C

• Assume that L and C are symmetric and positive-definite

• V and I vectors on MTL: and

• Strength of each shunt current generator can be scaled

• Assume Ez of nth TL is related only to its voltage:

• Describe how each TL affects beam dynamics:

• We assume s = a is the cold structure modal wavenumber

• This needs to be revisited in future (possibility of exotic dispersions)

• Assume solutions varying as

• k is complex propagation constant and ω is radian frequency

1 2[ , ,.., ]TNs s ss

1 2[ , ,.., ]TNV V VV 1 2[ , ,.., ]TNI I II

,z n n z nE a V

1 2[ , ,.., ]T

Na a aa

j t jkze e

c

8

Analysis of MTL TWT

Basic formulation involving evolution equations for the system

Extraction of modal characteristics

dispersion relation k – ω where k is the modal wavenumber

Evaluation of coupling parameters

Coupling impedance, Pierce parameters

Calculation of gain for finite size TWT

z

z s

j

j

V LI

I V iC e-beam is seen as a current generator Provides power to the TL

Objectives

9

Tamma, Capolino. "Extension of the Pierce Model to Multiple Transmission Lines Interacting With an Electron Beam." IEEE Trans. Plasma Science, vol. 42, no. 4, pp. 899-910, (2014)

Evolution of space charge waves along the z-direction

• Assume beam induces current in every TL (shunt current generator)

• Use linearized beam equations like in Pierce theory

10

Uniform MTL TWT

,z z sj j V LI I CV i

Uniform MTL TWT: Modal properties

Tamma, Capolino. " IEEE Trans. Plasma Science, vol. 42, no. 4, pp. 899-910, (2014)

. . . .

bY

,s Ni

,1si

,2si

bjkI+

-

+

-

+

-

Y

1V

2V

NV

11: a

1: Na

21: a

MTL

Transverse resonance condition

Dispersion relation

0b Y Y V

2 22 2 0 0

2

0

det 0Tkk

k

1 LC Lsa

det 0b Y Y

Need to find complex propagation constant k of the resonant modes

Establish transverse resonance condition in terms of beam impedance

Yb and MTL impedance Y

Transverse resonance

condition from circuit point of

view for s = a

0 ,1 ,2 ,max , ,...,c c c N

Condition for growing waves

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Uniform MTL: Growing waves

• Natural propagation constants of MTL:

• Growing wave solutions always exists if electron propagation

constant satisfies

Illustrative example: 2-TL system coupled to beam with

,1 ,2 ,, ,...,c c c N

0 ,1 ,2 ,max , ,...,c c c N

β0 [rad/m]

Re(

k) [

m-1

]

Im(k

) [m

-1]

β0 [rad/m]

0 ,1c 0 ,2c

Real part of k Imaginary

part of k

,2 ,1c c

12

1 21 2 1 2

0 0 0 0

, ,(4 / ) (4 / )

T

p p p p pC C C CV I V I

C

,1 ,2 0c c

Small signal gain [dB]

0 1 1 2 2dB p pG C C L

Gain can be enhanced using MTL!

Represent the coupling between TLs

and the e-beam

are the coupling

impedances to the

lines

1 2and

(0) 0bI

(0) 0bV No pre-modulation

Boundary conditions

No reflection from boundaries

Gain can be represented in terms of

Pierce parameters

2

dB

( )10log

(0)

V LG

V

13

We assume 0k

Gain for 2 TL: Pierce model

As in Pierce model

0 1 2, , are constants

Coupling parameters

1pC

Cp2

,1 0/c

,2 0/c

0 0 0Beam parameters : 6 KV, I 3 A, 0.15V u c

11 22 11 22TL parameters : 80 pF, 7μHC C L L

Condition for synchronism Trade off between Gain and Synchronism

Better gain synchronism

14

• The dispersion relation is written as a polynomial

0 ,1 ,2( , , , , ) 0c cF k

k has only one pair of complex roots

Theorem: For uniform MTL described by L and C, there exists only one

unique growing wave mode*

Uniform MTL TWT: remarks

*Figotin and Reyes. "Multi-transmission-line-beam interactive system." Journal of Mathematical Physics, vol. 54 no. 11, pp. 111901 (2013)

Is it possible to have more than one growing mode ?

15

Summary for uniform MTL

16

Uniform MTL TWT concept:

Developing the Pierce theory for more than one transmission line

Finding the modes – dispersion characteristics:

Calculating the gain for finite size structures: possible gain

enhancement is observed

Assessing the coupling parameters, especially impedance (matrix!):

provide intuition on how to properly design the TWT

Following steps:

Developing concrete boundary conditions for finite MTL TWT

Designing real structures that support 2 modes such as coupled helixes

or coupled ladder circuits

Next topic Periodic circuit

Periodic TWT circuit In contrast with helix TWTs , CCTWTs operate at higher-average power

levels and higher frequencies but with smaller bandwidths than helix TWTs.

The interaction structure in a CCTWT is composed of a series of cavities

that are connected via slots and a beam tunnel

The voltage across the gaps modulates the electron beam velocity

Future designs will allow interaction with more than one mode

Curnow, IEEE T-MTT (1965) Gittins, Power travelling-wave tubes. New York (1965).

Staggered slots Interaction gaps

Beam tunnel

17

j t jkze e

z

Periodic MTL – e-beam interaction

A. Figotin and I. Vitebsky, Frozen light in photonic crystals with degenerate band edge,

Phys. Rev. E, 2006

C. Locker, K. Sertel and J. L. Volakis, “Emulation of propagation in layered anisotropic

media with equivalent coupled microstrip lines,” IEEE Microw. Wireless Compon. Lett.,

Dec. 2006.

A B C

d

AdBd Cd

AC

. . . . . .

ABX BCXCAX

zBeam

j t jkze e All a.c. quantities: and k is the Block wavenumber

2 TL

z

amplification k j ,i jX model interfaces between different

waveguide cross-sections

18

MTL high pass type CC-TWT circuit

High pass type periodic circuit z

11L11cC

11C

22L11cC 22C

21L21C

One segment of MTL

DBE

RBE

Beam line

Cold structure dispersion + electron beam

(0) (0)jkdeT Ψ ΨDispersion relation is found

Transfer matrix of one unit cell

The dispersion may develop a regular band edge (RBE) Or a degenerate band edge (DBE)

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k : Bloch wave-

number 4k

2k

Unit cell composed of THREE MTL segments with different coupling

Dispersion of coupled system

Wide band interaction

Interaction with two modes at two different frequencies creates a wide band

region where amplification is possible

More than one growing wave

Unlike the uniform MTL, here it is possible to have two

growing waves simultaneously!

Two possible growing waves!

Wid

e b

and

amp

lification

20

Blue: exponential growing mode Green: exponential growing mode Black: exponential decaying mode Red: real k mode

Only forward modes With positive real k

Interaction near a degenerate band edge

Growing waves can be observed in the vicinity of DBE

We are investigating the backward wave excitation at that condition.

Growing waves

21

Interaction near the edge of transmission band

Time-Domain analysis!

Green: uncoupled system

DBE

Frozen wave amplifying regime

22

Field distribution at DBE resonance

The unique transmission properties of Fabry-Perot resonators with DBE can be utilized*

DBE

A. Figotin, and I. Vitebskiy, "Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers.” Phys. Rev. E 72(3) (2005)

Gigantic gain (very narrow frequency)!

sV

N = 32 unit cells

z

u

Electron super bunching as theymove along z-axis with velocity u

d

Frozen DBE mode magnitude profile

p(z)

TWT with super amplification via the DBE mode. A, B, and C are three different waveguide sections with distinct transverse anisotropy.

Injected electron beam

Frozen mode axial electric field phase profile

+- -

A B C A B C

RBE mode magnitude profile

outV

Will be implemented using waveguides with rotated elliptical cross sections

very low energy electron beam

(1)V (2)V (3)V ( 1)V N

Conclusions

Complex slow wave structures and real waveguides are better

represented by MTLs

MTL brings a promise in enhancing the gain of TWT, through optimizing

the coupling parameters

Periodic MTL offers phenomenological paradigm change since all modes

can be amplified

Future Research Directions Investigate wave amplification in finite uniform MTL TWT

Investigate finite periodic TWTs, excitations, and termination

Develop the theory of interaction near a regular band edge (RBE) and a

degenerate band edge (DBE)

Analyze TWT MTL in time domain including nonlinear effects (software

development) 23

Future Research Directions (continued)

0

0

z

b bt

b b tz t z t

b bt z t z z

b bt

t

uu

A D B V L R IA B V DV L R I

C G V E F I H IE H F I C G V

1 2 3

1 2 3 1

, , ,......, , , [ ,0]

, , ,......, , , [ , ...... ,0]

T T

N b b

T T

N b b 2 N

V V V V V

I I I I I s s , s

V a a

I s

0

0

0

0

, , , ,1 0 0 0

, , , ,1 0 0

T

b b b

b b

u

uu

I 0 0 0 L 0 R 0 I 0A B L R D a

0 0 0 0 0

C 0I 0 0 0 G 0 0 s

E F G C H00 0 0 0

Considering the following definitions

Time dependent equations can be written as

where

Finite Difference Time Domain (FDTD) analysis of MTL interaction with e-beam

* Collaboration with Mehdi Veysi, UC Irvine

Objective: investigate evolution of MTL TWT and study oscillations

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Thank you

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