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Slow-Wave Transmission Line Model and Cold Test Experiment of Curved Ring-Bar Structures

Muhammed Zuboraj and John L. Volakis

Outline

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  Objective

  Review of Curved Ring-Bar Structures

  Coupled Transmission Line Model of Curved Ring-Bar

  Experiment and Measurement of ω- diagram (Cold Test)

  Conclusion

Ø  Design Slow Wave Structure for high power TWTs

Ø  Fabricate and characterize the SWS (Cold Test)

Ø  Design TWT with SWS designed

Goal

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Features: 1.  A Ring-loop type structure with loops are

modified to ellipses instead of circles 2.  High phase velocity 3.  Strong TM01 mode 4.  Generated harmonics are much lower

strength implying higher efficiency

Curved Ring-Bar (Features)

E-field Profile(V/m)

H-field Profile(V/m)

H-field cancellation

E-field enhancement

Strong TM01 mode

  Elliptic bars provide inductive coupling (LM)   Presence of L,C elements responsible for

wave slowdown!!   L,C parameters are controlled by geometry!!   Coupling can reverse the mode profile

Coupled Transmission Lines

TL-1

TL-2

Inductively coupled bars

Waveguide wall

Waveguide wall

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  A curved Ring-Bar is employed instead of grooves/corrugation

Electron beam

Curved Ring-Bar (Performance Paramters)

* M. Zuboraj, N. K. Nahar, K. Sertel, J. L. Volakis, “High power microwave slow wave structure for relativistic beams:, URSI-2014, Boulder ,CO, Jan ,2014

Dimensions: a = 4.5mm, b = 60mm p = 22mm, w = δ = 2mm h1= 6.5mm, h2 = 11.65mm

p

w

2h1 h2

δ 2a

ü  0.80c<ʋp<0.7c (1-2.5GHz) ü  Average K0=45Ω ü  Strong E-field at center ü  Suitable for bunching

  K0(Ω) is higher compared to Ring-Loop SWS (≈30Ω)

  Phase velocity is higher >0.7c i.e. suitable for high power

Slow  waves  when  CRB  is  inside  the  waveguide  

gλ

High Power TWT

E-field

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Phase Velocity & Interaction Impedance Profile

Geometry

Coupled Transmission Line Model of Curved Ring Bar

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( )z ML L Cβ ω= ± +

Coupled TL Solutions

Inside waveguide environment

TL-1

Straight Bars instead of ellipses

Coupled TL can behave as artificial dielectric!!!

Transmission Line Equations

TL-2

2 2 2 4 2 2( ) 0z MLC L Cω β ω− − =

1( )z

z McL L C

ωυ

β= = <

+

Coupled Transmission Line Model of Curved Ring Bar (Contd.)

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TL-1

TL-2

  Coupled lines form Passband and Stopband   Circuit model analysis matches very well with Eigen mode solution

Pierce’s Coupled Mode Theory1,2

Linear Coupling Employed

1.  J. R Pierce, “Coupling of Modes of Propagation, Journal of Applied Physics, 1954. 2.  Dean. A. Watkins, Topics in Electromagnetic Theory, John Wiley & Sons Inc. New York.

Coupled Transmission Line Model of Curved Ring Bar (Contd.)

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TL-1

TL-2

  Coupled lines form Passband and Stopband   Circuit model analysis matches very well with

Eigen mode solution

Coupling

L,C Parameters of the Circuit Model

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Where, t = thickness of the wire = 2mm

Curved Ring-Bar Case:

1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

βp (rad)

f = ω

/2π(G

Hz)

Full wave simulation Dual TL model

Curved Ring-Bar ω- diagram Phase velocity as function of axial ratio

  Circuit model satisfactorily predicts the coupled line slow wave phenomena   Control of wave slowdown is mathematically predicted by parameter, m

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E-field

H-field

E-field

H-field

Inside WG

TE mode becomes TM

  Insertion of rings introduces coupling and reverses the modes!!   TM mode has parallel field to the electron beam therefore much improved coupling

Cold Test Experiment (Fabrication)

*Courtesy: Manufacturing Process Laboratory, The Ohio State University

Rings Welding/ soldering Fabrication

Steps

Copper sheet

Water-Jet cutter* Precision > 1.52mm Sample

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Soldering

Objective: To verify slow wave phenomena

Cold Test Experiment (Cavity Design for Cold Test Experiment)

  Goal of this experiment is to observe 7 resonances in presence of curved-ring bar

  TM01 mode overlaps with TE11 (Shaded Region)

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20 40 60 80 100 120 140 160 1800

1

2

3

4

5Eigenmode solution

βp in degrees

Freq

uenc

y (ω

/2π) i

n G

Hz

TM01TE11TE11degenerate

Resonances to be measured /

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⎣⎢⎢⎢⎢⎢⎡  𝑓𝑓0  𝑓𝑓1  𝑓𝑓2  ⋮  ⋮  𝑓𝑓𝑚𝑚  ⎦⎥⎥⎥⎥⎥⎤

=

⎣⎢⎢⎢⎢⎢⎡

 𝑐𝑐𝑐𝑐𝑐𝑐0  𝑐𝑐𝑐𝑐𝑐𝑐0  𝑐𝑐𝑐𝑐𝑐𝑐0  ⋮  ⋮  

𝑐𝑐𝑐𝑐𝑐𝑐0  

 𝑐𝑐𝑐𝑐𝑐𝑐0  

 𝑐𝑐𝑐𝑐𝑐𝑐(𝜋𝜋/𝑚𝑚)      𝑐𝑐𝑐𝑐𝑐𝑐(2𝜋𝜋/𝑚𝑚)

⋮  ⋮  

     𝑐𝑐𝑐𝑐𝑐𝑐(𝑚𝑚𝜋𝜋/𝑚𝑚)  

 𝑐𝑐𝑐𝑐𝑐𝑐0    

 𝑐𝑐𝑐𝑐𝑐𝑐(2𝜋𝜋/𝑚𝑚)  𝑐𝑐𝑐𝑐𝑐𝑐(4𝜋𝜋/𝑚𝑚)

⋮  ⋮  

 𝑐𝑐𝑐𝑐𝑐𝑐(2𝑚𝑚𝜋𝜋/𝑚𝑚)

         𝑐𝑐𝑐𝑐𝑐𝑐0    …  …      ⋮        ⋮      …

 𝑐𝑐𝑐𝑐𝑐𝑐0  

     𝑐𝑐𝑐𝑐𝑐𝑐(𝑚𝑚𝜋𝜋/𝑚𝑚)    𝑐𝑐𝑐𝑐𝑐𝑐(2𝜋𝜋/𝑚𝑚)

⋮  ⋮  

   𝑐𝑐𝑐𝑐𝑐𝑐(𝑚𝑚𝜋𝜋) ⎦⎥⎥⎥⎥⎥⎤

 

⎣⎢⎢⎢⎢⎢⎡  𝑎𝑎0  𝑎𝑎1  𝑎𝑎2  ⋮  ⋮  𝑎𝑎𝑚𝑚  ⎦

⎥⎥⎥⎥⎥⎤

(mode representation of field within waveguide)

  TE11 mode will not be excited   Enhances TM01 mode

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Cold Test Measured Data

-6-4-200

0.5

1

1.5

2

2.5

3

|S11| in dBω

/2π

in G

Hz

0 50 100 1500

0.5

1

1.5

2

2.5

3

(βp) in degrees

ω/2π

in G

Hz

NumericalExperiment

  Measured νp/c at midband : 0.744c

  TE mode is pretty much suppressed

  TM is dominant as expected

  Velocity curve is nonlinear at the edges due to cavity

TE11

TM01

TM01

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Summary and Future Work

Summary: Ø  Curved Ring-Bar inside of cylindrical waveguide for much improved efficiency Ø  Achieved TM01 mode dominance Ø  Developed dual TL model to understand and optimize parameters Ø  Validated the slow wave concept with experiments

Questions ??

Future Work: Ø Measure Interaction Impedance Ø Hot test simulations Ø Determine Power and TWT performance