Prof. David R. Jackson Dept. of ECE Notes 11 ECE 5317-6351 Microwave Engineering Fall 2011 Waveguides Part 8: Dispersion and Wave Velocities 1

Prof. David R. JacksonDept. of ECE

Notes 11

ECE 5317-6351 Microwave Engineering

Fall 2011

Waveguides Part 8:Dispersion and Wave Velocities

1

Dispersion => Signal distortion due to “non-constant” z phase velocity

=> Phase relationships in original signal spectrum are changed as the signal propagates down the guide.

In waveguides, distortion is due to:

Frequency-dependent phase velocity (frequency dispersion) Frequency-dependent attenuation => distorted amplitude

relationships

Propagation of multiple modes that have different phase velocities (modal dispersion)

Dispersion

2

1 2

1 1

1 21 2

1 21

2 2

21 1 2 2

cos cos

( ) ( )co

cos cos

cos ( ) c

s cos

os ( )

in a b

out a

a

a b

b

b

v A t B t

v A t L

A t B t

LB

B

LA t t

t L

0Zoutv+

-0 , ( )Z inv

+

-

L

Dispersion (cont.)

3

Consider two different frequencies applied at the input:

Matched load

2

1

a

a

b

b

jbin

j Lb

jain

j Lao

o

ut

ut

V Ae

V A

V Be

B

e

V e

1 21 2

1

1 1 2

1 2 2

2

( ) ( )cos co

cos ( ) cos ( )

s

out a

a b

bv A t L

L LA t

t

B

B

t

L

11( )p

Lt

v pv

0Zoutv+

-0 , ( )Z inv

+

-

L

Dispersion (cont.)

22( )p

Lt

v

4

Matched load

Recall:

No dispersion (dispersionless) ( )pv f

1 2( ) ( )p pv v

1 2t t

Dispersion ( )pv f

1 2( ) ( )p pv v

1 2t t

Phase relationship at end of the line is different than that at the beginning.

Dispersion (cont.)

5

( )iV o ( )V ( )Z

Consider the following system:

( ) j zZ A e

o ( ) ( ) ( )iV V Z

Signal Propagation

Amplitude Phase

2 f

The system will represent, for us, a waveguiding system.

6

Waveguiding system:

( )iS t o ( )S t( )Z

( ) ( )

1( ) ( )

2

j ti i

j ti i

S S t e dt

S t S e d

Fourier transform pair

*( ) ( )i iS S

Input signal

*

*

*

*

*

( )

1 1( ) ( )

2 2

1 1( ) ( )

2 2

1 1( ) ( )

2 2

( ) ( )

i i

j t j ti i

j t j ti i

j t j ti i

i i

S t S t

S e d S e d

S e d S e d

S e d S e d

S S

Proof:

Output signal

Property of real-valued signal:

Signal Propagation (cont.)

7

We can then show

0

1 1( ) ( ) ( ) Re ( )

2j t j t

i i i iS t S e d S t S e d

(See the derivation on the next slide.)


8

The form on the right is convenient, since it only involves positive values of .(In this case, has the nice interpretation of being radian frequency: = 2 f . )

0

0

0

0

0 0

*

0 0

1( ) ( )

2

1 1( ) ( )

2 2

1 11 ( ) ( )

2 2

1 1( ) ( )

2 2

1 1( ) ( )

2 2

1

2

j ti i

j t j ti i

j t j ti i

j t j ti i

j t j ti i

S t S e d

S e d S e d

S e d S e d

S e d S e d

S e d S e d

*

0

0

( ) ( )

1Re ( )

j t j ti i

j ti

S e S e d

S e d


9

o

0

0

1( ) Re (

1Re

)

)

(

(

)

j z

i

t

j t

jiA e

S t Z

S

e d

e

S

d

0

1( )( ) Re j

it

iS t e dS

Using the transfer function, we have

Interpreted as a phasor


10

Hence, we have

(for a waveguiding structure)

o

0

1( ) Re ( ) ( ) j t

iS t Z S e d

Summary


11

( )iS t o ( )S t( )Z

A) Dispersionless System with Constant Attenuation

( ) j zZ A e

0

o

0

0

0

1( ) Re )

1Re )

(

( p

j z j

zj t

i

i

v

t

A S

S t A e S e d

e d

0A A constant

0pv

Constant phase velocity (not a function of frequency)

o 00

( ) ip

zS t A S t

v

The output is a delayed and scaled version of input.

The output has no distortion.

Dispersionless System

12

Now consider a narrow-band input signal of the form

Narrow band

0m

mm

( )E

B) Low-Loss System with Dispersion and Narrow-Band Signal

00( ) ( )cos( ) Re ( ) j t

iS t E t t E t e

(Physically, the envelope is slowing varying compared with the carrier.)

Narrow-Band Signal

13

E t

t

iS t

0 0

0

0 0

( ) cos

1 1

2 21

( ) ( )2

i

j t j t

S F E t t

F E t e F E t e

E E

mm

( )E

00

iS

00( ) ( )cos( ) Re ( ) j t

iS t E t t E t e

Narrow-Band Signal (cont.)

14

( )0

o

0

0

( )0

0

1 1Re ( )

2

1Re ( )

2

1( ) Re ( )

j t z

j z j t

t

i

j z

A E e d

A E e d

S t A S e e d

Hence, we have


15

Since the signal is narrow band, using a Taylor series expansion about 0 results in:

00

0

00

0 0 00

0 0 0

0 ( )( ) ( ) ( ) ...

( ) ( ) ...A

neglect

d

d

dAA A

dA

Low loss assumption


16

Thus,

0 0 0

0 0 0 0 0

0 0 0

0

(

( )0 0

0

( ) ( )00

0

(

)o

)00

0

0

1Re ( )

2

Re (

1( ) Re ( )

2

)2

Re ( )2

s s

j z j zj t

j z j t j z j t

j t z j z j ts

j t

s

z

s

A E e e e d

Ae e E e e

S t A

d

Ae E e

E e d

e d

0 0 0( )0 Re ( )2

s sj t z j z j ts s

Ae E e e d

The spectrum of E is concentrated near = 0.


17

0 0 0

0 0

( ) ( )0o

( )0

00 0 0

0

0

( )

cos (

( ) Re ( )2

)

Re

sj t z j t zs s

j t z E t z

AS t e E e d

A e

A t z E t z

Define

phase velocity @ 0

Define

00

1g

dv

d

group velocity @ 0

0

0pv


18

Envelope travelswith group velocity

Carrier phase travels with phase velocity

p gv v

No dispersion1

1

1

1

p

g

cv

c

dc

d

v c

Proof :

constant

o 0 0( ) cosg p

z zS t A E t t

v v


19

z

o ,S t z vg

vp

/ gE t z v

o 0 0( ) cosg p

z zS t A E t t

v v


20

Recall2

2

a

Phase velocity:

Group velocity:1

g

d dv

d d

pv

22

pv

a

221

gva

21p g dv v c

Example: TE10 Mode of Rectangular Waveguide

After simple calculation:

Observation:

21

Example (cont.)

y

z a

b

, x

o

o

PEC

1

10c

pv slope

Lossless Case c

cf f

(“Light line”)

gv slope

22

( )iS t o ( )S t( )Z

Filter ResponseInput signal

What we have done also applies to a filter, but here we use the transfer function phase directly, and do not introduce a phase constant.

Output signal

23

o

0

1( ) Re ( )j j t

iS t A e S e d

From the previous results, we have

( ) jZ A e

( )iS t o ( )S t( )Z

Filter Response (cont.)Input signal

( ) jZ A e

Assume we have our modulated input signal:

Output signal

0o 0 0 0

0

( ) ( ) cosS t A E t t

0( ) ( )cos( )iS t E t t

0 0

0 0

d

d

whereThe output is:

0 0A A

24

0o 0 0 0

0

( ) cos ( )S t A t z E t z

Let z -

( )iS t o ( )S t( )Z

Filter Response (cont.)Input signal Output signal

0o 0 0 0

0

( ) ( ) cosS t A E t t

Phase delay:

0

0p

Group delay:

0g

d

d

This motivates the following definitions: If the phase is a linear function of frequency, then

p g constant

In this case we have no signal distortion.

25

o 0 0( ) ( ) cosg pS t A E t t

( )iS t o ( )S t( )Z

Linear-Phase Filter ResponseInput signal Output signal

o

0

1( ) Re ( )j j t

iS t A e S e d

( ) jZ A e

o 0

0

1( ) Re ( )j j t

iS t A e S e d

0A A Linear phase filter:

Hence

26

Linear-Phase Filter Response (cont.)

o 0

0

0

0

0

1( ) Re ( )

1Re ( )

j j ti

j ti

i

S t A e S e d

A S e d

A S t

A linear-phase filter does not distort the signal.

We then have

o 0( ) iS t A S t

It may be desirable to have a filter maintain a linear phase, at least over the bandwidth of the filter. This will tend to minimize signal distortion.

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Prof. David R. Jackson Dept. of ECE Notes 11 ECE 5317-6351 Microwave Engineering Fall 2011 Waveguides Part 8: Dispersion and Wave Velocities 1