prof. david r. jackson dept. of ece notes 6 ece 5317-6351 microwave engineering fall 2011 waveguides...

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Prof. David R. JacksonDept. of ECE

Notes 6

ECE 5317-6351 Microwave Engineering

Fall 2011

Waveguides Part 3:Attenuation

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For most practical waveguides and transmission lines the loss associated with dielectric loss and conductor loss is relatively small.

To account for these losses we will assume

zk j

attenuation constant Phase constant for lossless wave guide

c d

Attenuation constant due to conductor loss

Attenuation constant due to dielectric loss

Attenuation on Waveguiding Structures

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Attenuation due to Dielectric Loss: d

Lossy dielectric complex permittivity

1

(1 tan )

c

c c

cc

c

c

j

j

j

j

2 2 2 (1 tan )c ck j

tan c

c

0

0

rc

rc

complex wavenumber k

ck Note:

k k jk

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Thus

2 2z d ck j k k

Attenuation due to Dielectric Loss (cont.)

ck

Remember: The value kc is always real, regardless of whether the waveguide filling material is lossy or not.

Note: The radical sign denotes the principal square root:

Arg z

This is an exact formula for attenuation due to dielectric loss. It works for both waveguides and TEM transmission lines (kc = 0).

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2 (1 tan )

(1 tan )

c

c

k j

j

tan 1

1 1 / 2 1z z z for

Small dielectric loss in medium:

Approximate Dielectric Attenuation

Use

1 tan / 2ck j

tan / 2

ck

k k

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2 2

2 2

2 2 2

(1 tan )

tan

z d c

c c

c c c

k j k k

j k

k j

2 2 2tan ( )c c ck

1 11 / 1

2 2

z za z a z a a a z a

a a

for

Small dielectric loss:

Approximate Dielectric Attenuation (cont.)

Use

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2

22

2

2

2

2

2

tan

an

2

t

c

c

cc

z c c c

c

d

k k j

j

k jk

2 tan

2c

d

Attenuation due to Dielectric Loss (cont.)

2 2c ck 2 2

ck k

2 tan

2d

k

We assume here that we are above cutoff.

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d

k

k

zk j k k jk

For TEM mode

Attenuation due to Dielectric Loss (cont.)

tan

2d

k

ck

k jk

We can simply put kc = 0 in the previous formulas. Or, we can start with the following:

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Assuming a small amount of conductor loss:

We can assume fields of the lossy guide are approximately the same as those for lossless guide, except with a small amount of attenuation.

We can use a perturbation method to determine c .

Attenuation due to Conductor Loss

Note: Dielectric loss does not change the shape of the fields at all, since the boundary conditions remain the same (PEC). Conductor loss does disturb the fields slightly.

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This is a very important concept for calculating loss at a metal surface.

Surface Resistance

z

0 0, , , x

Plane wave in a good conductor

Note: In this figure, z is the direction normal to the metal surface, not the axis of the waveguide. Also, the electric field is assumed to be in the x direction for simplicity.

C

S

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Surface Resistance (cont.)

Assume

1/2 1/21

(1 )22

jk j j j

Note that

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k k jk j

2k k

The we have

Hence

1

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Surface Resistance (cont.)

Denote

Then we have

2k k

1

"pdk

“skin depth” = “depth of penetration”

2

1' "

pd

k k

At 3 GHz, the skin depth for copper is about 1.2 microns.

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Surface Resistance (cont.)

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Example: copper7

0 4 10 [H/m]

75.8 10 [S/m]

Frequency

1 [Hz] 6.6 [cm]

10 [Hz] 2.1 [cm]

100 [Hz] 6.6 [mm]

1 [kHz] 2.1 [mm]

10 [kHz] 0.66 [mm]

100 [kHz] 21 [mm]

1 [MHz] 66 [m]

10 [MHz] 20.1 [m]

100 [MHz] 6.6 [m]

1 [GHz] 2.1 [m]

10 [GHz] 0.66 [m]

100 [GHz] 0.21 [m]

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Surface Resistance (cont.)

dP = time-average power dissipated / m2 on S

* *0

1 1ˆRe( ) Re( )

2 2d x y zE H z E H P

z

x

S

fields evaluated on this plane

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Surface Resistance (cont.)

where

x yE H

1

2

(1 )2

(1 )

c

s

s

jj j

j

j

j R

Z

Inside conductor:

2sR

“Surface resistance ()”

“Surface impedance ()” 1s sZ j R

ˆt tE n H Note: To be more general:

n̂ outward normal

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Surface Resistance (cont.)

(1 )

1

2

s

s s

s

Z

Z j R

R

Summary for a Good Conductor

ˆt s tE Z n H

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ˆt s tE Z n H

ˆ

t

t

E

H

n

tangential electric field at surface

tangential magnetic field at surface

outward unit normal to conductor surface

n̂conductor

ˆeffs tJ n H

efft s sE Z J

“Effective surface current”

The surface impedance gives us the ratio of the tangential electric field at the surface to the effective surface current flowing on the object.

Hence we have

Surface Resistance (cont.)

For the “effective” surface current density we imagine the actual volume current density to be collapsed into a planar surface current.

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Surface Resistance (cont.)

1

2sR

Example: copper7

0 4 10 [H/m]

75.8 10 [S/m]

Frequency Rs

1 [Hz] 2.6110-7 []

10 [Hz] 8.2510-7 []

100 [Hz] 2.6110-6 []

1 [kHz] 8.2510-6 []

10 [kHz] 2.6110-5 []

100 [kHz] 8.2510-5 []

1 [MHz] 2.6110-4 []

10 [MHz] 8.2510-4 []

100 [MHz] 0.00261 []6.6

1 [GHz] 0.00825 []

10 [GHz] 0.0261 []

100 [GHz] 0.0825 []

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Surface Resistance (cont.)

2

0

1

2d s tR HPIn general,

2 2*0 0 0

1 1 1Re( ) Re

2 2 2d x y z y s yE H H R H P

We then have

For a good conductor, 0ˆeffs tJ n H

Hence21

2eff

d s sR JPThis gives us the power dissipated per square meter of conductor surface, if we know the effective surface current density flowing on the surface.

PEC limit: effs sJ J eff

s sJ JPerturbation method : Assume that

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Perturbation Method for c

Power flow along the guide: 20( ) zP z P e

0 (0)P P

Power @ z = 0 is calculated from the lossless case.

Power loss (dissipated) per unit length: l

dPP

dz

20( ) 2 2 ( )z

lP z P e P z

0

( ) (0)

2 ( ) 2l lP z P

P z P

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There is a single conducting boundary.0

(0)

2l

c

P

P

*0

0

1ˆRe

2S z

P E H z dS

2

0

(0)2

sl s

C z

RP J d

Surface resistance of metal conductors:

2sR

ˆsJ n H Note :

On PEC conductor

Perturbation Method: Waveguide Mode

C

S

For these calculations, we neglect dielectric loss.

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0

(0)

2l

c

P

P

*0

0

2

0

1ˆRe

2

1

2

S z

lossless

P E H z dS

Z I

1 2

2

0

(0)2

sl s

C C z

RP J d

Surface resistance of metal conductors: 2sR

ˆsJ n H Note :

On PEC conductor

2C

S

1C

Perturbation Method: TEM Mode

There are two conducting boundaries.

For these calculations, we remove dielectric loss.

0 0losslessZ Z

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0

02

losslesss

c lossless lossless

R dZ

Z d

2C

S

1C

Wheeler Incremental Inductance Rule

The Wheeler incremental inductance rule gives an alternative method for calculating the conductor attenuation on a transmission line (TEM mode): It is useful when Z0 is already known.

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

The top plate of a PPW line is shown being receded.

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

d

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Example: TEM Mode Parallel-Plate Waveguide Previously, we showed

2

00 * 2

0 0

2

0 *

2

0

1ˆ ˆRe

2

1 1Re

2

1

2

w d

lossless

VP z z dydx

d

wV

d

wV

d

0

0

ˆ

ˆ

jkz

jkz

VE y e

dV

H x ed

0

ˆ

ˆ

tops

jkz

J y H

Vz e

d

0ˆ ˆbot jkzs

VJ y H z e

d

y

z w

d , ,

x

On the top plate:

On the bottom plate:

j

lossless

c

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2 2

, 0 0, 00 0

(0)2

w wtop bots

l s sy d z y z

RP J dx J dx

2

0

0

w

s

VR dx

d

2

02( )s lossless

VR w

d

2

0 2

200

( )(0)

2 12

2

s losslessl

c

lossless

wR V

dP

P wV

d

sc lossless

R

d

Example: TEM Mode PPW (cont.)

(equal contributions from both plates)

The final result is then

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sc lossless

R

d

Example: TEM Mode PPW (cont.)

Let’s try the same calculation using the Wheeler incremental inductance rule.

0

02

losslesss

c lossless lossless

R Z

Z

0

0lossless lossless

dZ

w

dZ

w

From previous calculations:top bot

c c c

0 0lossless losslessZ dZ

d

d

w

, ,

0 02 2 22

top losslesss s s sc lossless lossless lossless lossless

lossless

R R R RddZ d w wZ dww

0 02 2 22

bot losslesss s s sc lossless lossless lossless lossless

lossless

R R R RddZ d w wZ dww

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Results for TM/TE Modes (above cutoff): (derivation omitted)

TMn modes of parallel-plate2

, 0scn lossless

k Rn

d

TEn modes of parallel-plate22

, 0c scn lossless

k Rn

k d

c

nk

d

Example: TMz/TEz Modes of PPWy

z w

d , ,

x