Prof. David R. Jackson Dept. of ECE Fall 2013 Notes 15 ECE 6340 Intermediate EM Waves 1.

1

Prof. David R. JacksonDept. of ECE

Fall 2013

Notes 15

ECE 6340 Intermediate EM Waves

2

0 0

0 0

( , , ) ( , ) ( , )

( , , ) ( , ) ( , )

z j z z

z j z z

E x y z E x y e E x y e e

H x y z H x y e H x y e e

At z = 0 :*

0 0

1ˆ(0) ( )

2f

S

P E H z dS

At z = Dz :* 2

0 0

1ˆ( ) ( )

2z

f

S

P z E H e z dS

Attenuation Formula

Waveguiding system (WG or TL): zS

Waveguiding system

3

Attenuation Formula (cont.)

Hence

If

2

2

( ) (0)

Re ( ) Re (0)

zf f

zf f

P z P e

P z P e

2( ) (0) zf fz e P P

1z

( ) (0) (1 2 )

(0) 2 (0)

f f

f f

z z

z

P P

P P

so

4


From conservation of energy:

(0) ( )

2 (0)

f f

f

z

z

P P

P

(0) ( ) ( / 2)lf f dz z z P P P

( )ld zP

where

= power dissipated per length at point z

( ) (0) 2 (0)f f fz z P P P

so0z

S

z z

5


Hence

( / 2)

2 (0)

ld

f

z z

z

P

P

(0)

2 (0)

ld

f

P

P

As z 0:

Note: The point z = 0 is arbitrary.

6


General formula:

0

0

( )

2 ( )

ld

f

z

z

P

P

z0

0( )f zP

This is a perturbational formula for the conductor attenuation.

The power flow and power dissipation are usually

calculated assuming the fields are those of the mode with

PEC conductors.

7

Attenuation on Transmission Line

A BC C C

2

ld

f

P

P

z

AC

szJ

LBC

z

Attenuation due to Conductor Loss

The current of the TEM mode flows in

the z direction.

c

8

Attenuation on Line (cont.)

2

2

2

1 1

2

1 1( )

2

1

2

ld s sz

S

s sz

C

s sz

C

R J dSz

z R J dlz

R J dl

P

C= CA+ CB

Power dissipation due to conductor loss:

2

0

1

2f Z IP

Power flowing on line:

zS

A

B

CA

CB

I

(Z0 is assumed to be approximately real.)

9

Hence

2

20

1

2A B

sc sz

C C

RJ dl

Z I

Attenuation on Line (cont.)

10

R on Transmission Line

Ignore G for the R calculation ( = c):

R Dz LDz

CDz GDz

Dz

2

ld

c

f

P

P

2

2

0

1

21

2

ld

f

R I

Z I

P

P

I

11

R on Transmission Line (cont.)

so

02c

R

Z

Hence

0(2 )cR Z

Substituting for ac ,

2

2

1( )s sz

C

R R J l dlI

12

Total Attenuation on Line

Method #1

c d

d TEM

TEMz dk j k k jk

so d k

c k Hence,

If we ignore conductor loss, we have a TEM mode.

13

Total Attenuation on Line (cont.)

Method #2

The two methods give approximately the same results.

1/ 2

Re

Re ( )( )R j L G j C

0(2 )cR Zwhere

c

c

G C

14

Example: Coax

)2

)2

sz

sz

IA J

aI

B Jb

2 2

20

1

2A B

sc sz sz

C C

RJ dl J dl

Z I

Coax

I

I z

a b

r

A

B

15

Example (cont.)

Hence

0

1

2 ln

rs

c

bR a

bba

2 22 2

20 0 0

0

1

2 2 2

1 1

2 2 2

sc

s

R I Ia d b d

Z a bI

R

Z a b

00 ln

2 r

bZ

a

Also,

Hence

(nepers/m)

16

Example (cont.)

0

00

(2 )

1 1(2 )

2 2 2

1 1

2

1 1 1

2

c

s

s

R Z

RZ

Z a b

R

a b

a b

Calculate R:

17

Example (cont.)

1 1

2 2R

a b

This agrees with the formula obtained from the “DC equivalent model.”

(The DC equivalent model assumes that the current is uniform around the boundary, so it is a less general method.)

b

a

DC equivalent model of coax

18

Skin Inductance

R Dz

C Dz G Dz

DL Dz L0 Dz

This extra (internal) inductance consumes imaginary (reactive) power.

The “external inductance” L0 accounts for magnetic energy only in the external region (between the conductors). This is what we

get by assuming PEC conductors.

An extra inductance per unit length L is added to the TL model in order to account for the internal inductance of the conductors.

0L L L

19

Skin Inductance (cont.)

R Dz

C Dz G Dz

DL Dz L0 Dz

21

2A B

I s sz

C C

P X J dl

Imaginary (reactive) power per meter consumed by the extra inductance:

21

2IP L I

Skin-effect formula:

I

Circuit model:Equate

20


Hence:

2

2

2

2

1 1 1

2 2

1 1

2

1

2

A B

A B

s sz

C C

s sz

C C

L X J dlI

R J dlI

R

21


Hence

RL

1 1

2 2L R

X R

or

22

Summary of High-Frequency Formulas for Coax

1

2HFaR

a

1

2HFbR

b

1

2HF HFa aX L

a

1

2HF HFb bX L

b

Assumption: << a

HF HF HFa bR R R

HF HF HFa bL L L

23

Low Frequency (DC) Coax Model

At low frequency (DC) we have:

DC DC DCa bR R R

2

1DCaR

a

1

2DCbR

bt

a

b

c

t = c - b

0

8DCaL

DC DC DCa bL L L

42 2

02 2 22 2

ln3

2 4DCb

cc

b cbL

c bc b

Derivation omitted

24

Tesche Model

This empirical model combines the low-frequency (DC) and the high-frequency (HF) skin-effect results together into one result by using an approximate circuit model to get R() and L().

F. M. Tesche, “A Simple model for the line parameters of a lossy coaxial cable filled with a nondispersive dielectric,” IEEE Trans. EMC, vol. 49, no. 1, pp. 12-17, Feb. 2007.

Note: The method was applied in the above reference for a coaxial cable, but it should work for any type of transmission line.

(Please see the Appendix for a discussion of the Tesche model.)

25

Twin Lead

a

x

y

h

a

x

y

h

DC equivalent model

Twin Lead

Assume uniform current density on each

conductor (h >> a).

1 1

2 2R

a a

26

Twin Lead

a

x

y

h

Twin Lead

1 1 1

2 2R

a a a

sRR

a

or

(A more accurate formula will come later.)

27

Wheeler Incremental Inductance Rule

0

0

1s

LR R

n

x

y

A B

n̂

L0 is the external inductance (calculated assuming PEC conductors) and n is an increase in the dimension of the conductors (expanding the surface into the active field region).

2

2

1( )s sz

C

R R J l dlI

Wheeler showed that R could be expressed in a way that is easy to calculate (provided we have a formula for L0):

H. Wheeler, "Formulas for the skin-effect," Proc. IRE, vol. 30, pp. 412-424, 1942.

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The boundaries are incremented a small amount n into the field region

Wheeler Incremental Inductance Rule (cont.)

PEC conductors

x

y

A B

n̂Sext

n

0

0

1s

LR R

n

L0 = external inductance (assuming perfect conductors).

29

Derivation of Wheeler Incremental Inductance rule


2

2

1( )s sz

C

R R J l dlI

200 2

extS

L H dSI

2

0

2

0

1

41

4ext

H

H

S

W L I

W H dS

2 20 0 02 2 ( )sz

C C

LH dl J l dl

n I I

Hence 20

0 2C

L n H dlI

We then have

PEC conductors

x

y

A B

n̂Sext

n

30


2

2

1( )s sz

C

R R J l dlI

2 20 0 02 2

0

1 1( ) ( )sz sz

C C

L LJ l dl J l dl

n nI I

PEC conductors

x

y

A B

n̂Sext

n

From the last slide,

0

0

1s

LR R

n

Hence

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Example 1: Coax

a

b

c

00 ln

2

bL

a

1 1

0 0 0 0 02

0

1 11

2 2

1 1

2

L L L b bb

n a b a a a a

a b

1 1

2 2sR Ra b

0

0

1s

LR R

n

32

Example 2: Twin Lead

100 cosh

2

hL

a

a

x

y

h


01

1

cosh2

Cha

100 cosh

2

hZ

a

From image theory:

00 ln ,

hZ a h

a

33

Example 2: Twin Lead (cont.)

100 cosh

2

hL

a

10 0 0 0 022 2

1 1 2cosh

2 21 1

2 2

hL L h h an a a a a ah h

a a

2

1 2

12

s

ha

R Ra h

a

0

0

1s

LR R

n

a

x

y

h


Note: By incrementing a, we increment both conductors

simultaneously.

34

Example 2: Twin Lead (cont.)

2

1 2

12

s

ha

R Ra h

a

a

x

y

h


100 cosh

2

hZ

a

100 cosh

2

hL

a

01

1

cosh2

Cha

tanG C

Summary

35

Attenuation in Waveguide

2

ld

c

f

P

P

2

2

1 1

2

1

2

c

lsd s

S

ss

C

R J dSz

R J dl

P

z

C

cSS

z

A waveguide mode is traveling in the

positive z direction.

We consider here conductor loss for a waveguide mode.

36

Attenuation in Waveguide (cont.)

or21

ˆ2

ld s

C

R n H dl P

0 0ˆ( ) orWG WG TE TMt tE Z z H Z Z Z

Hence

*0

1ˆ ˆRe ( )

2WG

f t t

S

Z z H H z dS

P

Power flow:*1

ˆRe ( )2f t t

S

E H z dS P

Next, use

37

Assume Z0WG

= real ( f > fc and no dielectric loss)

Hence

* * *

2

ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )t t t t t t

t

z H H z z H H H z H z

H

2

0

1Re

2WG

f t

S

Z H dS P

2

0

1

2WG

f t

S

Z H dS P


Vector identity: B C A B A C C A B

38

Then we have

2

20

ˆ

2s C

c WG

t

S

n H dlR

Z H dS


C

S

n̂

x

y

39

Total Attenuation:


c d

1/ 22 2z ck j k k

1/ 22 2Imd ck k

ck

Calculate d (assume PEC wall):

where

so

40

Attenuation in dB

( ) (0) z j zV z V e e

10 10

( )dB 20log 20log ( )

(0)zV z

eV

10

lnlog

ln10

xx Use

z = 0 z

zS

Waveguiding system(WG or TL)

41

soln( )

dB 20ln10

( )20

ln10

ze

z

20Attenuation [dB/m]

ln10

Hence

Attenuation in dB (cont.)

42

Attenuation 8.6859 [dB/m]

or

Attenuation in dB (cont.)

43

Appendix: Tesche Model

CZa Zb G

z

L0

02

ln

rcCba

tan c

c

G

C

0

0 ln2

bL

a

0a bZ Z Z j L

Y G j C

The series elements Za and Zb account for the finite conductivity, and give us R and L for each conductor at any frequency.

44

Appendix: Tesche Model (cont.)

Inner conductor of coax

Outer conductor of coax

a a aZ R j L

b b bZ R j L

The impedance of this circuit is denoted as

The impedance of this circuit is denoted as

aZ

bZ

DCaR

HFaR HF

aL

DCaL

DCbR

HFbR HF

bL

DCbL

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At low frequency the HF resistance gets small and the HF inductance gets large.

DCaR

HFaR HF

aL

DCaL

DCaR

HFaR HF

aL

DCaL


46


At high frequency the DC inductance gets very large compared to the HF inductance, and the DC resistance is small compared with the HF resistance.

HFaR HF

aL

DCaR

HFaR HF

aL

DCaL


47

The formulas are summarized as follows:

2

1DCaR

a

1

2DCbR

bt

0

8DCaL

42 2

02 2 22 2

ln3

2 4DCb

cc

b cbL

c bc b

1

2HF HFa aR L

a

1

2HF HFb bR L

b


Prof. David R. Jackson Dept. of ECE Fall 2013 Notes 15 ECE 6340 Intermediate EM Waves 1.

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Transcript of Prof. David R. Jackson Dept. of ECE Fall 2013 Notes 15 ECE 6340 Intermediate EM Waves 1.