AFRL-DE-TR-2003-1139 AFRL-DE-TR-2003-1139

MODES OF A DOUBLE-BAFFLED, CYLINDRICAL,COAXIAL WAVEGUIDE

Clifton C. Courtney and Donald E. Voss

Voss Scientific418 Washington St SEAlbuquerque, NM 87108

August 2003

Interim Report

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION IS UNLIMITED.

AIR FORCE RESEARCH LABORATORYDirected Energy Directorate3550 Aberdeen Ave SEAIR FORCE MATERIEL COMMANDKIRTLAND AIR FORCE BASE, NM 87117-5776

STINFO COPY

AFRL-DE-TR-2003-1139

Using Government drawings, specifications, or other data included in this document for anypurpose other than Government procurement does not in any way obligate the U.S. Government.The fact that the Government formulated or supplied the drawings, specifications, or other data,does not license the holder or any other person or corporation; or convey any rights or permissionto manufacture, use, or sell any patented invention that may relate to them.

This report has been reviewed by the Public Affairs Office and is releasable to the NationalTechnical Information Service (NTIS). At NTIS, it will be available to the general public,including foreign nationals.

If you change your address, wish to be removed from this mailing list, or your organization nolonger employs the addressee, please notify AFRL/DEHE, 3550 Aberdeen Ave SE, Kirtland AFB,NM 87117-5776.

Do not return copies of this report unless contractual obligations or notice on a specific documentrequires its return.

This report has been approved for publication.

//signed//ANDREW D. GREENWOODProject Manager

//signed// //signed//REBECCA N. SEEGER, Col, USAF L. BRUCE SIMPSON, SESChief, High Power Microwave Division Director, Directed Energy Directorate

i

REPORT DOCUMENTATION PAGEForm Approved

OMB No. 0704-0188Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining thedata needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducingthis burden to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currentlyvalid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.1. REPORT DATE (DD-MM-YYYY)18-08-2003

2. REPORT TYPEInterim

3. DATES COVERED (From - To) 09-07-2003 to 31-07-2003

4. TITLE AND SUBTITLEModes of a Double-Baffled, Cylindrical, Coaxial Waveguide

5a. CONTRACT NUMBERF29601-03-M-01015b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER65502F

6. AUTHOR(S)

Clifton C. Courtney and Donald E. Voss5d. PROJECT NUMBER30055e. TASK NUMBERDP5f. WORK UNIT NUMBERCE

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER

Voss Scientific418 Washington St SEAlbuquerque, NM 87108

9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)AFRL/DEHE3550 Aberdeen Ave SEKirtland AFB, NM 87117-5776 11. SPONSOR/MONITOR’S REPORT

NUMBER(S)

AFRL-DE-TR-2003-113912. DISTRIBUTION / AVAILABILITY STATEMENTApproved for public release; distribution is unlimited.

13. SUPPLEMENTARY NOTES

14. ABSTRACTThere is considerable interest in antenna and transmission line structures that are conformalto curved and cylindrical surfaces. The double-baffled, coaxial transmission line is definedby inner and outer radii, and an arc length. It is conformal to curved surfaces,particularly structures cylindrical in nature. In this note we derive the TE and TM, axiallypropagating modes of a double-baffled, coaxial transmission line. First, the characteristicequations that define the cutoff frequencies of each mode are derived, then the electricfields are explicitly expressed. Finally, an example double-baffled, coaxial transmissionline geometry is defined for which the lowest TE and TM mode cutoff frequencies are computedand graphs of the normalized field components are presented.

15. SUBJECT TERMSElectromagnetics; waveguide modes

16. SECURITY CLASSIFICATION OF: 17. LIMITATIONOF ABSTRACT

18. NUMBEROF PAGES

19a. NAME OF RESPONSIBLE PERSONAndrew Greenwood

a. REPORTUnclassified

b. ABSTRACTUnclassified

c. THIS PAGEUnclassified Unlimited 28

19b. TELEPHONE NUMBER (include areacode)505-846-6642

Standard Form 298 (Rev. 8-98)Prescribed by ANSI Std. 239.18

ii

iii

Contents1. Introduction .........................................................................................................................................12. Geometry .............................................................................................................................................13. Wave Equation ...................................................................................................................................14. Boundary Conditions .........................................................................................................................35. Solution of the Separated Wave Equation .......................................................................................36. TEz and TMz Field Components .......................................................................................................3

6.1 TMz Field Components .............................................................................................................36.2 TEz Field Components ...............................................................................................................4

7. Solution of the Separated Wave Equation Subject to the Boundary Conditions of theGeneralized Geometry ................................................................................................................................4

7.1 TMz Field Components .............................................................................................................57.2 TEz Field Component ................................................................................................................7

8. Example ...............................................................................................................................................99. Conclusion ........................................................................................................................................14

iv

List of Figures

Figure 1. The geometry of the double-baffled cylindrical coaxial waveguide................................2Figure 2. Plots of the Characteristic Equation for the double-baffled, coaxial waveguide

transmission line (n = 1)...................................................................................................................11Figure 3. Normalized distributions of the electric field components of the TE11 mode of the

double-baffled coaxial waveguide..................................................................................................12Figure 4. Normalized distributions of the magnetic field components of the TE11 mode of the

double-baffled coaxial waveguide..................................................................................................13Figure 5. Vector plot of the current density of the TE11 mode........................................................13Figure 6. Normalized distributions of the electric field components of the TM11 mode of the

double-baffled coaxial waveguide..................................................................................................14Figure 7. Normalized distributions of electric field components of TM 11 mode of double-

baffled coaxial waveguide...............................................................................................................15Figure 8. Normalized distributions of electric field components of TE11 mode of double-baffled

coaxial waveguide. ...........................................................................................................................16Figure 9. Normalized distributions of the electric field components of the TE21 mode of the

double-baffled coaxial waveguide..................................................................................................17

v

List of Tables

Table 1. Cutoff frequencies of the TE modes of a double-baffled, coaxial waveguide. ...............9Table 2. Cutoff frequencies of the TM modes of a double-baffled, coaxial waveguide...............9

vi

1

1. Introduction

The double-baffled, coaxial transmission line is defined by inner and outer radii, and an arc length,

and can be conformal to curved surfaces and cylindrical structures. This note describes the

propagating modes of a coaxial waveguide transmission line with two baffles, with propagation

assumed in the z-direction. First, the characteristic equations that define the cut off frequencies of

each mode are derived, then the electric fields are explicitly expressed. Finally, an example

geometry is defined for which the lowest TE and TM mode cutoff frequencies are computed and

graphs of the normalized field components are presented.

2. Geometry

The geometry of the double-baffled, coaxial waveguide transmission line is shown in Figure 1.

Note that the arc between the baffles has an angular extension of 0ϕ ϕ= .

3. Wave Equation

The natural coordinate system for the coaxial waveguide transmission line with two baffles is the

cylindrical coordinate system. The scalar Helmholtz wave equation in cylindrical coordinates is2 2

22 2 2

1 ( , , ) 1 ( , , ) ( , , )( , , ) 0

z z zk z

zψ ρ ϕ ψ ρ ϕ ψ ρ ϕ

ρ ψ ρ ϕρ ρ ρ ρ ϕ

∂ ∂ ∂ ∂+ + + = ∂ ∂ ∂ ∂

(1)

Using standard separation of variable techniques the wave equation can be written as

( )2 2( )( ) 0

d dRk n R

d d ρρ

ρ ρ ρ ρρ ρ

+ − = (2a)

22

2 ( ) ( ) 0d

nd

ϕ ϕϕ

Φ + Φ = (2b)

22

2( ) ( ) 0z

dZ z k Z z

dz+ = (2c)

where: ( , , ) ( ) ( ) ( )z R Z zψ ρ ϕ ρ ϕ= Φ , and 2 2 2zk k kρ + = .

2

(a)

y

x

z

? 1

? 2

f

f 0

Baffle

Baffle

Direction of propagation

(b)

y

x

? 1

? 2

f

f 0

Baffle

Baffle

(c)

z

x

? 1

? 2

? 2 cos f 0 ? 1 cos f 0

Figure 1. The geometry of the double-baffled cylindrical coaxial waveguide: (a) 3-D

perspective drawing; (b) plane view o the xy-plane; and (c) plane view of the

xz-plane.

3

4. Boundary Conditions

The boundary conditions for the coaxial waveguide transmission line with two baffles are:

0Eρ = for 0ϕ = , and 0ϕ ϕ= (3a)

0Eϕ = for 1ρ ρ= , and 2ρ ρ= (3b)

0zE = for 1ρ ρ= , and 2ρ ρ= , and 0ϕ = , and 0ϕ ϕ= . (3c)

5. Solution of the Separated Wave Equation

The ( )ϕΦ and ( )Z z equations are harmonic equations with harmonic functions as solutions; these

will be denoted ( )h nϕ and ( )zh k z .

The equation in ( )R ρ is a Bessel equation, and has Bessel function solutions:

( )nJ kρ ρ = the Bessel function of the first kind of order n

( )nN kρ ρ = the Bessel function of the second kind of order n

(1) ( )nH kρ ρ = the Hankel function of the first kind of order n

(2) ( )nH kρ ρ = the Hankel function of the second kind of order n

Let the function ( )nB kρ ρ represent the linearly independent combination of two of the above.

Then, the general solution to the scalar Helmholtz wave equation is:

, , ( ) ( ) ( )zk n k n zB k h n h k z

ρ ρψ ρ ϕ= (4)

6. TEz and TMz Field Components

The electric and magnetic field components can be written in terms of fields that are TEz and TMz.

6.1 TMz Field Components

The TMz field components are found by letting zψ=A u , where =A the magnetic vector

potential, and z =u unit vector in the z-direction. Then

1( )jω

ωµε= − + ∇ ∇ ⋅E A A , (5a)

and 1µ

= ∇×H A . (5b)

4

When expanded in cylindrical coordinates these equations become:21

Ej zρ

ψωµε ρ

∂=

∂ ∂(6a)

1 1H ρ

ψµ ρ ϕ

∂=

∂(6d)

21 1E

j zϕψ

ωµε ρ ϕ∂

=∂ ∂

(6b)1

Hϕψ

µ ρ∂

= −∂

(6e)

22

2

1zE k

j zψ

ωµε ∂

= + ∂ (6c)

0zH = (6f)

6.2 TEz Field Components

The TEz field components are found by letting zψ=F u , where =F the electric vector potential,

and z =u unit vector in the z-direction. Then

1ε

= − ∇ ×E F , (7a)

1( )j

jω

ωµε= − + ∇ ∇ ⋅H F F . (7b)

When expanded in cylindrical coordinates these TEz field equations become:

1 1Eρ

ψε ρ ϕ

∂= −

∂(8a)

21H

j zρψ

ωµε ρ∂

=∂ ∂

(8d)

1Eϕ

ψε ρ

∂=

∂(8b)

21 1H

j zϕψ

ωµε ρ ϕ∂

=∂ ∂

(8e)

0zE = (8c) 22

2

1zH k

j zψ

ωµε ∂

= + ∂ (8f)

7. Solution of the Separated Wave Equation Subject to the

Boundary Conditions of the Generalized Geometry

Propagating waves in the z-direction in the double-baffled coaxial waveguide give rise to

( ) zj k zzh k z e−= (9)

The harmonic function ( )h nϕ can be written as

( ) sin( ) cos( )n nh n a n b nϕ ϕ ϕ= + (10)

5

Note that n is not necessarily an integer. The scalar wave function is then

, , ( ) ( ) z

z

j k zk n k nB k h n e

ρ ρψ ρ ϕ −=(11)

subject to the boundary conditions. The solutions for the TEz and TMz modes in the guide are as

follows.

7.1 TMz Field Components

The TMz electric field components in terms of the wave function are

( )( )( ) ( ) zz j k z

n

jk kE B k h n e

jρ

ρ ρ ρ ϕωµε

−−′= (12a)

( )( ) 1( ) ( ) zj k zz

njk n

E B k h n ejϕ ρρ ϕωµε ρ

−− ′= (12b)

( )2 21( ) ( ) zj k z

z z nE k k B k h n ej ρ ρ ϕωµε

−= − (12c)

Since

0zE = for 1ρ ρ= , and 2ρ ρ= , and 0ϕ = , and 0ϕ ϕ= ;

then

0 00, 0,( ) | ( sin( ) cos( )) | 0n nh n a n b nϕ ϕ ϕ ϕϕ ϕ ϕ= == + =

is satisfied if

1na = , 0nb = , 0

mn

πϕ

= , and 1,2,3,m = K . (13)

Note that

( ) ( )( )n n

dB k B k

d kρ ρρ

ρ ρρ

′ = .

The general Bessel function, ( )nB kρ ρ , also satisfies the boundary conditions if

1 2,( ) | 0nB kρ ρ ρ ρρ = =

Let

( ) ( ) ( )n n n n nB k a J k b N kρ ρ ρρ ρ ρ= + .

Then

1 2,( ) ( ) | 0n n n na J k b N kρ ρ ρ ρ ρρ ρ =+ =

1 1( ) ( ) 0n n n na J k b N kρ ρρ ρ+ =

2 2( ) ( ) 0n n n na J k b N kρ ρρ ρ+ =

6

Solving the first equation for na gives

1

1

( )

( )n

n nn

N ka b

J kρ

ρ

ρ

ρ= (14)

Substitution into the second equation yields:

12 2 2 2

1

( )( ) ( ) ( ) ( ) 0

( )n

n n n n n n n nn

N ka J k b N k b J k a N k

J kρ

ρ ρ ρ ρρ

ρρ ρ ρ ρ

ρ+ = − + = ,

and rearranging terms gives

12 2

1

( )( ) ( ) 0

( )n

n n nn

N kb N k J k

J kρ

ρ ρρ

ρρ ρ

ρ

− =

For specific values of n , 1ρ and 2ρ , the values of kρ that solve

2 1

2 1

( ) ( )

( ) ( )n n

n n

N k N k

J k J kρ ρ

ρ ρ

ρ ρ

ρ ρ= (15)

are the sought after mode numbers that are true for any non-zero value of nb . Hence,

1nb = and 1

1

( )

( )n

nn

N ka

J kρ

ρ

ρ

ρ= − .

Finally, the scalar wave function for the TMz modes is:

1, ,

1

( )( ) ( ) sin( )

( )z

z

n j k zk n k n n

n

N kN k J k n e

J kρ

ρρ ρ

ρ

ρψ ρ ρ ϕ

ρ−

= −

for

0

mn

πϕ

= , 1,2,3,m = K , and 2 2 2zk k kρ + = .

The convention for the zeros of the Characteristic Equation is 1 2 3, , ,p p p p= K , where the 1p is

the first zero solution, 2p is the second solution (with increasing numerical value, and so forth).

The TMz field components are then found explicitly as:

1

1

( )sin( ) ( ) ( )

( )zz n j k z

n nn

k k N kE n N k J k e

J kρ ρ

ρ ρ ρρ

ρϕ ρ ρ

ωµε ρ−

−′ ′= −

(16a)

1

1

( )1( ) ( ) cos( )

( )zn j k zz

n nn

N kk nE N k J k n e

J kρ

ϕ ρ ρρ

ρρ ρ ϕ

ωµε ρ ρ−

= − −

(16b)

2 21

1

( )( ) ( ) sin( )

( )zn j k zz

z n nn

N kk kE N k J k n e

j J kρ

ρ ρρ

ρρ ρ ϕ

ωµε ρ−

−= −

(16c)

1

1

( )1( ) ( ) cos( )

( )zn j k z

n nn

N knH N k J k n e

J kρ

ρ ρ ρρ

ρρ ρ ϕ

µ ρ ρ−

= −

(16d)

7

1

1

( )( ) ( ) sin( )

( )zn j k z

n nn

k N kH N k J k n e

J kρ ρ

ϕ ρ ρρ

ρρ ρ ϕ

µ ρ−

′ ′= − −

(16e)

0zH = (16f)

7.2 TEz Field Component

The TEz electric field components in terms of the wave function are:

1 1( ) ( ) zj k z

nd

E B k h n edρ ρ ρ ϕ

ε ρ ϕ−= − (17a)

1( ) ( ) zj k z

nd

E B k h n edϕ ρ ρ ϕ

ε ρ−= (17b)

0zE = (17c)

Since

0Eρ = for 0ϕ = , and 0ϕ ϕ=

0Eϕ = for 1ρ ρ= , and 2ρ ρ=

then

0 0

0

0, 0,

0,

( ) | ( sin( ) cos( ))|

( cos( ) sin( )) | 0

n n

n n

d dh n a n b n

d dn a n b n

ϕ ϕ ϕ ϕ

ϕ ϕ

ϕ ϕ ϕϕ ϕ

ϕ ϕ

= =

=

= +

= − =

is satisfied if

0na = , 1nb = , 0

mn

πϕ

= , and 1,2,3,m = K . (18)

The general Bessel function, ( )nB kρ ρ , also satisfies the boundary conditions if

1 2,( ) | 0nd

B kd ρ ρ ρ ρρ

ρ = =

Let

( ) ( ) ( )n n n n nB k a J k b N kρ ρ ρρ ρ ρ= + ,

then

{ } 1 2,( ) ( ) | 0n n n nd

a J k b N kd ρ ρ ρ ρ ρρ ρ

ρ =+ =

1 1( ) ( ) 0n n n na J k b N kρ ρρ ρ′ ′+ =

2 2( ) ( ) 0n n n na J k b N kρ ρρ ρ′ ′+ = .

8

Solving the first equation for na gives

1

1

( )

( )n

n nn

N ka b

J kρ

ρ

ρ

ρ

′= −

′. (19)

Substitution into the second equation yields:

12 2 2 2

1

( )( ) ( ) ( ) ( ) 0

( )n

n n n n n n n nn

N ka J k b N k b J k b N k

J kρ

ρ ρ ρ ρρ

ρρ ρ ρ ρ

ρ

′′ ′ ′ ′+ = − + =

′.

Rearranging terms gives

12 2

1

( )( ) ( ) 0

( )n

n n nn

N kb N k J k

J kρ

ρ ρρ

ρρ ρ

ρ

′′ ′− = ′

For specific values of n , 1ρ and 2ρ , the values of kρ that solve

2 1

2 1

( ) ( )

( ) ( )n n

n n

N k N k

J k J kρ ρ

ρ ρ

ρ ρ

ρ ρ

′ ′=

′ ′(20)

are the sought after mode numbers that are true for any non-zero value of nb . Hence,

1nb = and 1

1

( )

( )n

nn

N ka

J kρ

ρ

ρ

ρ

′= −

′.

Finally, the scalar wave function for the TEz modes is:

1, ,

1

( )( ) ( ) cos( )

( )z

z

n j k zk n k n n

n

N kN k J k n e

J kρ

ρρ ρ

ρ

ρψ ρ ρ ϕ

ρ−

′= − ′

for 0

mn

πϕ

= , 1,2,3,m = K and 2 2 2zk k kρ + = .

Again, the convention for the zeros of the TE Characteristic Equation is 1 2 3, , ,p p p p= K , where

the 1p is the first solution, 2p is the second solution (with increasing numerical value, and so

forth. The TEz field components are then found explicitly as:

1

1

( )1( ) ( ) sin( )

( )zn j k z

n nn

N knE N k J k n e

J kρ

ρ ρ ρρ

ρρ ρ ϕ

ε ρ ρ−

′= − ′

(21a)

1

1

( )( ) ( ) cos( )

( )zn j k z

n nn

k N kE N k J k n e

J kρ ρ

ϕ ρ ρρ

ρρ ρ ϕ

ε ρ−

′′ ′= − ′

(21b)

0zE = (21c)

1

1

( )( ) ( ) cos( )

( )zz n j k z

n nn

k k N kH N k J k n e

J kρ ρ

ρ ρ ρρ

ρρ ρ ϕ

ωµε ρ−

′−′ ′= − ′

(21d)

1

1

( )( ) ( ) sin( )

( )zn j k zz

n nn

N kk nH N k J k n e

J kρ

ϕ ρ ρρ

ρρ ρ ϕ

ωµε ρ ρ−

′= − ′

(21e)

9

21

1

( )( ) ( ) cos( )

( )zn j k z

z n nn

k N kH N k J k n e

j J kρ ρ

ρ ρρ

ρρ ρ ϕ

ωµε ρ−

′= − ′

(21f)

Note that the Characteristic Equations for the TE and TM mode are similar, but not exact, to the

forms given in [Ref. 2]. The reason for the differences is at this time unknown.

8. Example

Determine the first few cutoff frequencies of the TE and TM modes of a Double-baffled,

Cylindrical, Coaxial Waveguide defined by the parameters : 1 5 0.127in mρ = = ,

2 6 0.1524in mρ = = and 0 2 /3ϕ π= .

The cutoff frequencies for the first few the TE and TM modes have been computed and are

presented in Table 1 and Table 2 below.

Table 1. Cutoff frequencies (GHz) of the TE modes of a double-baffled, coaxial

waveguide ( 1 5 0.127in mρ = = , 2 6 0.1524in mρ = = and 0 2 /3ϕ π= ).

M / p 1 2 3

1 0.513006 5.93146 11.8179

2 1.02589 5.99859 11.8516

3 1.53851 6.10889 11.9074

Table 2. Cutoff frequencies (GHz) of the TM modes of a double-baffled, coaxial

waveguide ( 1 5 0.127in mρ = = , 2 6 0.1524in mρ = = and 0 2 /3ϕ π= ).

M / p 1 2 3

1 5.92129 11.8129 17.7111

2 5.98762 11.8464 17.7335

3 6.09656 11.9021 17.7707

The characteristic equations for the TE11 and TM11 cases are plotted in Figure 2a and c. The

characteristic equation for the TE1p mode is plotted in Figure 2b showing the first 3 roots that

characterize the first 3 modes. Note that the radial gap between the conductors is 1-inch, about ½

of the freespace wavelength of the 1st TM cutoff frequency. And the median arc length between

10

the inner and outer radii is 11.52 inches, about ½ of the freespace wavelength of the 1st TE cutoff

frequency.

(a)

0 10 20 30 40 50 k?

-4

-2

0

2

4

Cha

ract

eris

tic E

quat

ion

TE Modes

(b)

0 50 100 150 200 250 300 kρ

-4

-2

0

2

4

Ch

arac

teri

stic

Eq

uat

ion

TE Modes

(c)

100 110 120 130 140 150 kρ

-0.075 -0.05

-0.025

0

0.025 0.05

0.075 0.1

Ch

arac

teri

stic

Eq

uat

ion

TM Modes

11

Figure 2. Plots of the Characteristic Equation for the double-baffled, coaxial waveguide

transmission line (n = 1): (a) TE11 mode; (b) TE1p , for 1,2,3p = modes; and

(c) TM11 mode.

Plot the electric field components over a cross section of the guide at a frequency that is 1.2 times

the cutoff frequency of the lowest mode. For 1m = ,

0

1.52 / 3

mn

π πϕ π

= = = .

For the TM11 mode, 5.921cf = GHz. For 1m = ,

0

1.52 / 3

mn

π πϕ π

= = = .

For the TE11 mode, 513.00cf = MHz.

Then, compute the field distributions at 1.2 1.2 513.00 615.6cf f= × = × = MHz and

1.2 1.2 5.921 7.105cf f= × = × = GHz.

The wavelength at the operating frequency of the guide for the TE11 mode, 615.6f = MHz, in the

axial direction of the guide , gλ , is defined as

2 20.881036

7.13159gzkπ π

λ = = = meters

where 2 2zk k kρ= − . The wavelength at the operating frequency of the guide for the TM 11 mode,

7.105f = GHz, in the axial direction of the guide , gλ , is

2 20.076347

82.2978gzkπ π

λ = = = meters.

Normalized distributions of the electric field components of the TE11 mode of the double-baffled

coaxial waveguide for 1 5 0.127in mρ = = , 2 6 0.1524in mρ = = , 0 2 /3ϕ π= , and 615.6f = MHz

are shown in Figure 3. Normalized distributions of the magnetic field components are shown in

Figure 4. Since ˆs = ×J n H , we can plot the current density on the interior surfaces of the

waveguide. Shown in Figure 5 is a vector plot of the current density of the TE11 mode at

615.6f = MHz on the 2ρ ρ= , 00 2 /3ϕ ϕ π< < = surface.

12

Normalized distributions of the electric field components of the TM11 mode of the double-baffled

coaxial waveguide for 1 5 0.127in mρ = = , 2 6.0 0.1524in mρ = = , 0 2 /3ϕ π= , and 7.105f =

GHz are shown in Figure 6.

0 20 40 60 80 100 120 ϕ degrees

0

0.2

0.4

0.6

0.8

1 E

ρ

ρ=(ρ1+ρ2)/2

0 20 40 60 80 100 1200

0.002 0.004 0.006 0.008 0.01

0.012 ρ=(ρ1+ρ2)/2

ϕ degrees

E ϕ

(a) (b)

Figure 3. Normalized distributions of the electric field components of the TE11 mode of the double-

baffled coaxial waveguide ( 1 5 0.127in mρ = = , 2 6.0 0.1524in mρ = = , 0 2 /3ϕ π= , and

615.6f = MHz): (a) 1 2(( ) / 2 , )Eρ ρ ρ ϕ+ ; and (b) 1 2(( ) /2 , )Eϕ ρ ρ ϕ+ . 0zE = .

0 20 40 60 80 100 1200

0.002

0.004

0.006

0.008

ρ=(ρ1+ρ2)/2 Hρ

ϕ degrees

0 20 40 60 80 100 1200

0.1 0.2 0.3 0.4 0.5 0.6

ϕ degrees

ρ=(ρ1+ρ2)/2 Hϕ

(a) (b)

0 20 40 60 80 100 120 0

0.2

0.4

0.6

0.8

1

ϕ degrees

ρ=(ρ1+ρ2)/2

Hz

(c)

13

Figure 4. Normalized distributions of the magnetic field components of the TE11 mode of

the double-baffled coaxial waveguide ( 1 5 inρ = , 2 6.0inρ = , 0 2 /3ϕ π= , and

615.6f = MHz): (a) H ρ ; (b) Hϕ ; and (c) zH .

60 70 80 90 100 110 120

0

0.5

1

1.5

2

z/λ g

ϕ degrees

Figure 5. Vector plot of the current density of the TE11 mode at 615.6f = MHz on the

2ρ ρ= , 00 2 /3ϕ ϕ π< < = surface.

0 20 40 60 80 100 1200

0.0025 0.005

0.0075 0.01

0.0125 0.015

0.0175

ϕ degrees

Eρ

ρ=(ρ1+ρ2)/2

0 20 40 60 80 100 1200

0.01

0.02

0.03 0.04 0.05

ϕ degrees

Eϕ

ρ=(ρ1+ρ2)/2

(a) (b)

(c)

0 20 40 60 80 100 120 0

0.2

0.4

0.6

0.8

1

ϕ degrees

Ez

ρ=(ρ1+ρ2)/2

14

Figure 6. Normalized distributions of the electric field components of the TM11 mode of

the double-baffled coaxial waveguide ( 1 5 inρ = , 2 6inρ = , 0 2 /3ϕ π= , and

7.105f = GHz): (a) Eρ ; (b) Eϕ ; and (c) zE .

Contour and 3D projection graphs of the normalized distributions of the electric field components

of the TM11 modes of the double-baffled coaxial waveguide for 1 5 0.127in mρ = = ,

2 6.0 0.1524in mρ = = , 0 2 /3ϕ π= , and 7.105f = GHz) are shown in Figure 7. Likewise, plots

of the normalized distributions of the electric field components of the TE11 mode of the double-

baffled coaxial waveguide are shown in Figure 8 for 615.6f = MHz

Finally, the form of the next higher order mode is of interest. Referring to Table 1, the next

propagating mode is the TE21 mode, with a cutoff frequency of 1.02589cf = GHz. The non-zero

electric fields at (ρ1+ρ2)/2, as a function of ϕ are plotted in Figure 9.

9. Conclusion

This report derives expressions for waveguide modes and cutoff frequencies of the double-baffled,

coaxial transmission line. Example computations are also shown. The derived expressions are

useful for the design of antenna and transmission line structures that are conformal to curved and

cylindrical surfaces.

15

Figure 7. Normalized distributions of electric field components of TM11 mode of double-

baffled coaxial waveguide ( 1 5 0.127in mρ = = , 2 6.0 0.1524in mρ = = , 0 2 /3ϕ π= , and

7.105f = GHz): (a) Eρ ; (b) Eϕ ; and (c) zE .

16

Ez = 0 Ez = 0

Figure 8. Normalized distributions of electric field components of TE11 mode of double-

baffled coaxial waveguide ( 1 5 0.127in mρ = = , 2 6.0 0.1524in mρ = = ,

0 2 /3ϕ π= , and 0.6156f = GHz): (a) Eρ ; (b) Eϕ ; and (c) zE .

17

0 20 40 60 80 100 120 0

0.2

0.4

0.6

0.8

1

ϕ degrees

Eρ

ρ=(ρ1+ρ2)/2

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

ϕ degrees

Eϕ

ρ=(ρ1+ρ2)/2

(a) (b)

Figure 9. Normalized distributions of the electric field components of the TE21 mode of

the double-baffled coaxial waveguide ( 1 5 inρ = , 2 6.0inρ = , 0 2 /3ϕ π= , and

1.3f = MHz): (a) Eρ ; and (b) Eϕ . 0zE = .

18

References

1. Time Harmonic Electromagnetic Fields, R. Harrington, pg. 199, McGraw-Hill, NY, 1961.2. Encyclopedia of Physics , ed. S. Flugge, pg. 345, Springer-Verlag, Berlin, 1958.

19

DDIISSTTRRII BBUUTTII OONN LLIISSTT

DDTTIICC//OOCCPP88772255 JJoohhnn JJ.. KKiinnggmmaann RRdd ,, SSuu iittee 00994444FFtt BBeellvvoo iirr,, VVAA 2222006600--66221188 11 ccyy

AA FFRRLL// VVSSII LLKKiirrtt llaanndd AA FFBB,, NNMM 8877111177--55777766 22 ccyyss

AA FFRRLL// VVSSIIHHKKiirrtt llaanndd AA FFBB,, NNMM 8877111177--55777766 11 ccyy

AA FFRRLL//DD EEHHPP//DDrr.. TThhoommaass SSppeenncceerrKKiirrtt llaanndd AA FFBB,, NNMM 9977111177--55777766 11ccyy

OOffffiicc iiaall RReeccoorrdd CCooppyyAA FFRRLL//DD EEHH EE//DDrr.. AA nnddrreeww GGrreeeennwwoooodd 22 ccyyss

20