Prof. David R. Jackson Dept. of ECE Notes 9 ECE 5317-6351 Microwave Engineering Fall 2011 Waveguides...

1

Prof. David R. JacksonDept. of ECE

Notes 9

ECE 5317-6351 Microwave Engineering

Fall 2011

Waveguides Part 6:Coaxial Cable

2

To find the TEM mode fieldsWe need to solve

200 ; ( )

( ) 0t a V

b

0

0

( ) l

( ) l

n

n

C

C

D

or

Zero volt potential reference location (0 = b).

0

@

ln

a

aV C

b

0

ln

VC

ab

0

z

y

x

b

a

, ,

Coaxial Line: TEM Mode

3

0( ) lnln

V bba

Thus,

, , , ˆ zzjk zt

jk zE x y z x y ee

0ˆ, ,ln

zjk zVE x y z e

ba

1ˆH z E

Coaxial Line: TEM Mode (cont.)

Hence

z

y

x

b

a

, ,

0ˆ

ln

zjk zVH e

ba

: z ck k k jk TEM

c j

c

4

0

2 2

0 0

20

0

0

0

ˆˆ ˆ ˆ( )

ln

ˆ( )

( )

;

ln

2( )

ln

z

z

z

z

B B b

A A a

bjk z

a

sz

j

jk z

jk z

k z

V z E dr E d d zdz E d

Ve d

ba

V z V

I z J d H d a b

Ve d

e

VI z e

ba

ba

0

( )

( )

V zZ

I z


z

y

x

b

a

, ,

0 ln2

bZ

a

Hence

Note: This does not account for conductor loss.

5


z

y

x

b

a

, ,

d c

: d k TEM

0

(0)

2l

c

P

P

*0

0

*

2

0

1ˆRe

2

1Re

2

1Re

2

S z

P E H z dS

VI

Z I

Attenuation:

Dielectric attenuation:

Conductor attenuation:

2

0 0

1

2losslessP Z I (We remove all loss from the

dielectric in Z0lossless.)

6


z

y

x

b

a

, ,

1 2

2

0

2 22 2

0 0

2 22 2

0 0

2 22 22 2

0 0

2 2

2

(0)2

2 2

2 2 2 2

1 1

2 2 2 2

1 1

2 2 2 2

1

4

sl s

C C z

sa sbsz sz

sa sb

sa sb

sa sb

sa sb

RP J d

R RJ ad J bd

R RI Iad bd

a b

R RI ad I bd

a b

R RI I

a b

R RI

a b


2sR

7


z

y

x

b

a

, ,

2 1(0)

4sa sb

l

R RP I

a b


2sR

2

0 0

1

2losslessP Z I

0

(0)

2l

c

P

P

2

2

0

141

22

sa sb

clossless

R RI

a b

Z I

Hence we have

0

1 1

4sa sb

c lossless

R R

Z a b

or

8


z

y

x

b

a

, ,

2sR

Let’s redo the calculation of conductor attenuation using the Wheeler incremental inductance formula.

0

02

losslesss

c lossless lossless

R dZ

Z d

Wheeler’s formula:

In this formula, dl (for a given conductor) is the distance by which the conducting boundary is receded away from the field region.

The formula is applied for each conductor and the conductor attenuation from each of the two conductors is then added. lossless

9


z

y

x

b

a

, ,

2sR

0

0

0

2

ln2

losslesss

c lossless lossless

losslesslossless

R dZ

Z d

bZ

a

0

0

0

0

2

2

losslessa sac lossless lossless

losslessb sbc lossless lossless

R dZ

Z da

R dZ

Z db

d da

d db

0

0

1

2 2

1

2 2

losslessa sac lossless lossless

losslessb sbc lossless lossless

R

Z a

R

Z b

0

1

2 2

losslesssa sb

c lossless lossless

R R

Z a b

0

1 1

4sa sb

c lossless

R R

Z a b

Hence

so

or

10


z

y

x

b

a

, ,

where

We can also calculate the fundamental per-unit-length parameters of the coaxial line.

tanG C

0losslessL Z

0/ losslessC Z

02 losslesscR Z

0 ln2

losslesslossless b

Za

From previous calculations:

(Derived as a homework problem)

(Formulas from Notes 1)

11

Coaxial Line: Higher-Order Modes

z

y

x

b

a

, ,

We look at the higher-order modes of a coaxial line.

The lowest mode is the TE11 mode.

Sketch of field lines for TE11 mode

x

y

12

Coaxial Line: Higher-Order Modes (cont.)

z

y

x

b

a

, , TEz:

We look at the higher-order modes of a coaxial line.

2 20 0, , 0z c zH k H

2 2 2z ck k k

The solution in cylindrical coordinates is:

0

( ) sin( ),

( ) cos( )n c

zn c

J k nH

Y k n

0, , , zjk zz zH z H e

Note: The value n must be an integer to have unique fields.

13

Plot of Bessel Functions

0 1 2 3 4 5 6 7 8 9 100.6

0.4

0.2

0

0.2

0.4

0.6

0.8

11

0.403

J0 x( )

J1 x( )

Jn 2 x( )

100 x

x

Jn (x)

n = 0

n = 1

n = 2

(0)nJ is finite

2( ) ~ cos

2 4n

nJ x x

x

14

Plot of Bessel Functions (cont.)

0 1 2 3 4 5 6 7 8 9 107

6

5

4

3

2

1

0

10.521

6.206

Y0 x( )

Y1 x( )

Yn 2 x( )

100 x

x

Yn (x)

n = 0

n = 1

n = 2

(0)nY is infinite

2( ) ~ sin

2 4n

nY x x

x

15

z

y

x

b

a

, ,

We choose (somewhat arbitrarily) the cosine function for the angle variation.

0 , cos( ) ( ) ( )z n c n cH n AJ k BY k

0, , , zjk zz zH z H e

Wave traveling in +z direction:

The cosine choice corresponds to having the transverse electric field E being an even function of, which is the field that would be excited by a probe located at = 0.


16

z

y

x

b

a

, ,

Boundary Conditions:

, 0E a

2z

c

j HE

k

,

0a b

zH

( ) ( ) 0

( ) ( ) 0

c n c n c

c n c n c

k AJ k a BY k a

k AJ k b BY k b

Hence

, 0E b

Note: The prime denotes derivative with respect to the argument.


17

z

y

x

b

a

, ,

( ) ( ) 0

( ) ( ) 0n c n c

n c n c

AJ k a BY k a

AJ k b BY k b

Hence

In order for this homogenous system of equations for the unknowns A and B to have a non-trivial solution, we require the determinant to be zero.

( ) ( )

Det 0( ) ( )

n c n cc

n c n c

J k a Y k ak

J k b Y k b

( ) ( ) ( ) ( ) 0n c n c n c n cJ k a Y k b J k b Y k a


18

z

y

x

b

a

, ,

Denote

( ) ( ) ( ) ( ) 0n c n c n c n cJ k a Y k b J k b Y k a

cx k a

( ; , / ) ( ) / / ( ) 0n n n nF x n b a J x Y x b a J x b a Y x

For a given choice of n and a given value of b/a, we can solve the above equation for x to find the zeros.

The we have


19

xn1 xn2

xn3 x

A graph of the determinant reveals the zeros of the determinant.

npx xc npk a x

; , /F x n b a

thnpx p zero


cx k a

Note: These values are not the same as those of the circular waveguide, although the same notation for the zeros is being used.

20

Fig. 3.16 from the Pozar book.

n = 1

Approximate solution:

2

1 /ck ab a


Exact solution

21

Coaxial Line: Lossless CaseWavenumber:

2 2z ck k k

2

ccf f

c c

k k

f k

22c c

c

r

k c k af

a

1 1

1 /c

r

cf

b aa

82.99792458 10 [m/s]c

TE11 mode:

22

Coaxial Line: Lossless Case (cont.)1 1

1 /c

r

cf

b aa

1 /

d

r

c

f

a b a

At the cutoff frequency, the wavelength (in the dielectric) is

d a b

2/ 2da b

so

or

Compare with the cutoff frequency condition of the TE10 mode of RWG:

2da

a

b

2b

23

Example

4

4

0.035 inches 8.89 10 [m]

0.116 inches 29.46 10 [m]

2.2r

a

b

Page 129 of the Pozar book:

RG 142 coax:

/ 3.31b a 1 1

1 /c

r

cf

b aa

16.8 [GHz]cf

Prof. David R. Jackson Dept. of ECE Notes 9 ECE 5317-6351 Microwave Engineering Fall 2011 Waveguides...

Documents

Transcript of Prof. David R. Jackson Dept. of ECE Notes 9 ECE 5317-6351 Microwave Engineering Fall 2011 Waveguides...