Field and Wave Electromagnetic

(1)

Field and Wave Electromagnetic

Chapter10

Waveguides and Cavity Resonators

Introduction (1)

- TEM waves are not the only mode of guided waves

- The three types of transmission lines (parallel-plate, two-wire, and coaxial) are not the only possible wave-guiding structur

* Waveguide

1( )

2

2( ^s) 2 ^c

c

C L

R G R

L C

R f

w w

α α

π μ σ

≅ + ∝

= = ∝

∴

⇒

e.

- Attenuation constant for loss line

Attenuation of TEM waves tends to increase monotonically with frequency prohibitively high in the microwave range.

0

(2)

Electromagnetic Theory 2 3

Introduction (2)

0 ( 0, 0)

z z

E H

H E

= =

= ≠

- TEM waves :

- TM waves : transverse magnetic waves - TE waves : transverse electric(TE) waves - single conductor wave guide

: rectangular and cylindrical wave guide.

- dielectric-slab waveguide : surface waves

General Wave Behaviors along Uniform Guiding Structures (1)

* General wave behaviors along uniform g uiding structures - straight guiding structures with a uniform cross section.

- Assume that the waves propagate in the +z direction with a propagation constant

- For harmonic time dependence on z and t for all field components :

γ α= +jβ

( ) ( )

z j t j t z z j t z

e⁻^γ e^ω =e ^{ω γ}⁻ =e⁻^α e ^{ω β}⁻

(3)

General Wave Behaviors along Uniform Guiding Structures (2)

0 ( )

0

( , , ; ) [ ( , ) ] ( , ) :

,

j t z

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

t jw z

ω γ

γ

= ℜ −

∂ → ∂ → − ,

∂ ∂

- For a cosine reference

where two-dimensional vector phasor - in using a phasor representation - In the charge-free dielectr

2

2 2

0 0,

( _xy _z) ( _xy E k E

H k H k

E E

ω με

∇ + =

∇ + = =

∇ = ∇ + ∇ = ∇

2

ic region inside, Helmholtz's equations should be satisfied

where

- In Cartesian coordinates, rectangular wave guide

² ²₂)E ²_xyE E

z

γ

+ ∂ = ∇ +

∂

2

General Wave Behaviors along Uniform Guiding Structures (3)

2 2

2 2 2 2

( ) 0

( ) ( )( ) 0

( ) 0, ( ) 0

xy

xy x y z x y z

xy x x xy y y

E k E

H k H

E k E

xE yE zE k xE yE zE

E k E E k E

γ γ

γ

γ γ

∴ ∇ + + =

∇ + + =

→ ∇ + + + + + + =

∇ + + = ∇ + + =

2 2

2

2 2

cf)

i.e

² ²

2 2

( ) 0

xy z z

r xy

E k E

φ

γ

∇ + + =

∇ ∇

2

The solution of above equations depends on the cross-sectional geometry and the boundary conditions

cf) instead of for waveguides with a circular cross section

(4)

General Wave Behaviors along Uniform Guiding Structures (4)

0 0

0 0 0 0

0

z z

y x y x

x

E j H H j E

E H

E j H H j E

y y

E

ωμ ωε

γ ωμ γ ωε

γ

∇× = − ∇× =

∂ + = − ∂ + =

∂ ∂

− −∂

- Interrelationships among the six components in Cartesian coordinates

1 4

2 ⁰ ⁰ ⁰ ⁰ ⁰

0 0 0 0

0 0

z z

y x y

y x y x

z z

z y

x z

y x

x y z

E H

j H H j E

x x

E E H H

j H j E

x y x y

E E

x y z y z

E E

j H

x y z z x

E E

E E E

x

ωμ γ ωε

ωμ ωε

ωμ

= − − −∂ =

∂ ∂

∂ ∂ ∂ ∂

− = − − =

∂ ∂ ∂ ∂

∂ −∂

∂ ∂

∂ ∂ ∂ = − −

∂ ∂ ∂ ∂ ∂

∂ ∂

∂ − 5

3 6

cf)

x

y

z

j H j H j H y

z e

γ

ωμ ωμ ωμ

γ ⁻

⎛ ⎞

⎜ ⎟

⎜ ⎟ ⎛− ⎞

⎜ ⎟ = −⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜⎝− ⎟⎠

⎜ ⎟

⎜ ∂ ⎟

⎝ ⎠

∂ → −

where ∂ and is surpressed

General Wave Behaviors along Uniform Guiding Structures (5)

0

0 0

0

0 0

z

y x

z

x y

E E j H

y

H H j E

x

γ ωμ

γ ωε

∂ + = −

∂

− −∂ =

∂

- Transverse component can be expressed in terms of longitudinal components.

Ex) Combining 1 and 5 1 :

5 : eliminati ⁰

0

0 2 0

0

2 0 0

0 0

2 2 0

0 0

0 2 2

2

)

1 ( )

y

z

y x

z

x y

z z

x

z z

x

E

j E j E H

y

H H j E

x

E H

k H j

y x

E H

H j h k

h y x

ωε γ ωε ω με

γ γ γ ωε

γ ωε γ

ωε γ

∂ + =

∂

− − ∂ =

∂

∂ ∂

+ = −

∂ ∂

∴ = − − + =

∂ ∂

ng from 1,5 1' :

5' :

(

where +γ²

(5)

General Wave Behaviors along Uniform Guiding Structures (6)

0 0

0

2

0 0

0

2

0 0

0

2

0 0

0 2 2 2

2

1 ( )

z z

x

z z

y

z z

x

z z

y

x

E H

H j

h y x

E

n

H j H

h x y

H E

E j

h y x

H E

E j h k

h

o

y

e

x

t

ωε γ

ωμ γ

ωμ γ γ

∂ ∂

= − − +

∂ ∂

= − + +

∂ ∂

= − +

∂ ∂

= − − + = +

∂ ∂

∇ - i.e

where

First, solve

² ²

2 2

0 0 , , ,

y

xy

x y x y

E h E H h H H H E E

+ =

∇ + =

and for longitudinal components then find using above equation

TEM Waves (1)

2

2 2

2

0, 0

0 0

0

z z

x y x y

TEM TEM

E H

E E H H h

k jk j γ

γ ω με

= =

∴ = = = = =

+ =

= =

- for TEM waves

unless

- TEM waves exist only where

or : propagation constant of a uniform plan

* TEM wave

( )

1

p TEM

k μ ω

= = με

e wave on a lossless transmission line.

- Phase velocity

(6)

TEM Waves (2)

0

x TEM

y

TEM

y TEM

x

n

E j

Z H j

Z Z E te

H o

γ ωμ

γ ωε

μ ηε

μ ε

ΤΕΜ ΤΕΜ

= = =

= =

= − = −

- Wave impedance from 2,4

is the same as the intrinsic impedance of the dielectric medium

1

TEM

H z E

∴ = Z ×

TEM Waves (3)

B

B H

Why?

1. lines always close upon themselves

2. For TEM waves to exist, and lines would form closed loops in a t

* Single-conductor waveguides cannot support TEM waves.

c d

H dl I I I I

∫

ⁱ = +

ransverse plane.

3. By the Ampere's circuital law.

transverse plane : conductor current : displacement current

c 0 I =

4. Without an inner conductor

(7)

TEM Waves (4)

0 0 0

z

d s

z

E

I D ds

t E

= →

= ∂ =

∂

=

∫

ⁱ

∵

5. For TEM wave, no longitudinal displacement current cf) in the z direction

6. Therefore there can be no closed loops of magnetic field lines in any transverse plane

7. Assuming perfect conductors, a coaxial transmission line having an inner conductor can support TEM waves

8. When the conductors have losses, no longer TEM waves

TM / TE Waves (1)

solving solving

For TE waves For TM waves

2 0 2 0

0 0

2

0 0

2

0 0

2

0 0

2

0 .

:

: : :

xy z z

z

z x

z y

z x

z y

E h E

fo r E

E H j

h y

E H j

h x

E E

h x

E E

h y

ω ε ω ε γ γ

∇ + =

= ∂

∂

= − ∂

∂

= − ∂

∂

= − ∂

∂

①

②

③

④ w ith proper bo u n dary co n ditio n s

z

0

H

=

E_z

= 0

2 2

0 0

2

0 0

2

0 0

2

0 0

2

0 .

: : : :

xy z z

z

z x

z y

z x

z y

H h H

fo r H

H H

h x

H H

h y

H E j

h y

H E j

h x

γ γ

ω μ ω μ

∇ + =

= − ∂

∂

= − ∂

∂

= − ∂

∂

= ∂

∂

⑤

⑥

⑦

⑧ w ith pro per bo u n dary co n ditio n s

(8)

TM / TE Waves (2)

0 0 0 0

2

0 0 0 0

0 0

ˆ ˆ

( )

, , ,

)

,

T T M x y T z

x y x y

x y T M

y x

T M

E xE yE E

h

E E H H

E E

Z H H j

cf Z j

j

γ

γ ωε ωμ

γ γ

ω με

= + = − ∇

= = − =

≠

∵

② ③ ① ④

are given can be

determ ined from the w ave im pedance for the TM m ode from , and ,

for TM is not equal to w hic

1 (ˆ )

T E M

T M

H z E

Z

γ

∴ = ×

h is

0 0 0 0

2

0 0

ˆ ˆ

( )

, ,

)

(ˆ ) ˆ

( )

T TE x y T z

x y

x y TE

y x

T E TM

T E T E

H xH yH H

h E E

H H

E E j

Z H H

cf Z Z

j

E Z z H

Z H z

γ

ωμ γ γ ωε

= + = − ∇

= = − =

≠ =

∴ = − ×

= ×

⑥ ⑦ ⑤ ⑧

sim ilar way, can be obtained from

from , and ,

TM / TE Waves (3)

2 0 2 0

xyEz h Ez 0

h h

h

∇ + =

⇒

solution of

for a given boundary condition are possible only for discrete values of

infinity of 's

but solutions are not possible for all values of

eigenvalues or characteri

2 2 2

2 2

h k

h γ γ

ω με

= +

= −

stic values

2 0 2 0

xyHz h Hz 0

∇ + =

eigen va lu es

(9)

TM / TE Waves (4)

2 2

2

0,

2 :

1 ( )

c

for h f h

f

h f

f γ

ω με

π με

γ

=

= −

cutoff frequency

cf) The value of for a particular m ode in a w aveguide depends on the eigenvalue of this m ode

2 0 2 0

xyHz h Hz 0

∇ + =

eigen va lu es

TM / TE Waves (5)

2

2 2

2

1 ( )

( ) 1

1 ( )

1

c

c c

c

h f

f

f f f

f

h j jk h

k jk f

f

k γ

ω με γ

γ β

β β

= −

> >

⇒ >

= = −

= −

⇒

= −

(a) or

in this range, and is imaginary

propagation mode with a phase constant (f^c)² (rad/m)

f

(10)

TM / TE Waves (6)

2

2 2

2 2 2 2 2

2 2 2

2 2 1

1 ( )

1 1

(1 ( ) )

1 1 1 1

( )

1 1 1

g

c

c c

c

g

c

g c

u

k f

f f

f

u f f

f f f

u f

π λ π

λ λ λ

β με

λ

λ λ

λ λ λ λ

λ λ λ

= = > = = =

−

=

= −

= − = −

∴ + =

- Guided wavelength

where

let cutoff wavelength, then

TM / TE Waves (7)

1 ( )2

g p

c

u u u u

f f

ω λ

β λ

= = = >

− - Phase velocity

cf) 1. Phase velocity of guided wave is faster than that of unbounded medium.

2. Phase velocity depends on frequency so that single conductor waveguides are dispersive transmission systems

(11)

TM / TE Waves (8)

2

2 2

1 1 ( )

( 1 ( ) [2 1 ( ) ]

(2 ) 1 (

c g

g

g p

c c

c

u u f u u

d d f

u u u

f f

d k d f

f f

d

d d d f

f d f

df u f

λ

β ω λ

π με β

ω ω π

= = − = <

/

∴ =

− −

= =

= −

- Group velocity

cf)

²

2

)

1 1

1 ( ^c)

u f

f

=

−

TM / TE Waves (9)

2

2 2

2

1 ( )

1 ( ) 1 ( )

1 ( )

c

TM

c c

TE

c

jk f Z f

j j

f f

j j

Z f

jk f

γ

ωε ωε

μ η

ε

ωμ ωμ

γ

−

= =

= − = −

= =

− -

; purely resistive and less than the intrinsic impedance of the dielectric medium

-

2 2

1

1 (f^c) 1 (f^c)

f f

μ η

= ε =

− −

; purely resistive larger than the intrinsic impedance of the dielectric medium

(12)

TM / TE Waves (10)

2

( ) 1

1 ( )

z

c c

c

z z

f f f

f h f

f e ^γ e ^α γ α

− −

< <

= = −

∴ = ⇒

⇒

(b) or

: real number

wave diminishes rapidly with and is said to be evanescent

waveguide : high-pass filt

2

1 ( )

1 ( ) ,

c

TM c

c

TE

h f

f h f

Z j f f

j j f

Z j

γ

ωε ωε ωε

ωμ γ

−

= = = − − <

⇒ ⇒

= =

er

purely reactive no power flow associated with evanescent mode

1 ( )2 c

j f

h f

ωμ

−

: purely reactive. no power flow.

TM / TE Waves (11)

1 ( ^c)2

u ω β

ω β ω

ω

−

= −

- diagram

(13)

Parallel-Plate Waveguide (1)

( )

- ( _z 0)

j t z

x H

e ^{ω γ}

ε μ

−

= 1. Assuming and

2. Infinite in extent in the direction 3. TM waves

4.

- Parallel plate waveguide can support TM and TE waves as well as TEM waves

* TM waves between parallel plates

Parallel-Plate Waveguide (2)

0

2 0

2 0 2

2 2 2

0

( , ) ( ) -

( ) ( ) 0

( ) 0 0

( ) sin( )

z

z z

z

z n

n

E y z E y e x

d E y

h E y dy

h k

E y y y b

n y n

E y A h

b b

A

γ

π π

=

−

+ =

= +

= = =

∴ = =

(no variation along direction)

where B.C.

at and

from

where depends on the strength of excitation of

the particular TM wave

(14)

Parallel-Plate Waveguide (3)

0 0

2

0 0

2 0 0

2

2 2

( ) cos( )

( ) 0

( ) cos( )

( )

2

z

x n

z y

z x

z

y n

c

E

j j n y

H y A

h y h b

E H y j

h x

E y E

h x

E n y

E y A

h y h b

n b

f n b

ωε ωε π

ωε γ

γ γ π

γ π ω με

γ με

∴ = ∂ =

∂

= − ∂ =

∂

= − ∂ =

∂

= − ∂ = −

∂

= −

= 0 ∴ =

Cutoff frequency that makes

Parallel-Plate Waveguide (4)

1 1

2 2

0

1 2

2 2

0 0

c

z

f TM

b

f TM

b

TM f

E με με

=

∵

cf) for mode with n=1

for mode with n=2

cf) mode is the TEM mode with

- Dominant mode of the waveguide = the mode having the lowest cutoff frequency

- For parallel plate waveguides, the dominant mode is the TEM mode

(15)

Ex. 10-3(1)

ˆ ˆ ˆ (ˆ _x ˆ _y ˆ _z)

x y z

dl xdx ydy zdz kE k xE yE zE dx dy dz

E E E k

= + + = = + +

= = = ⇒

cf) Field line : the direction of the field in space i.e.

field line

Ex. 10-3(2)

1

( , ; 0) ( , ; 0)

( , ; 0) cos( ) sin 0

y

z

x

E y z t y z dy

dz E y x t

b y

H y z A z

b

y y b

ωε π β

π

∴ − = =

=

−

=

− = =

1

in the plane For TM mode at t=0,

At and

- There are surface currents because of a discontinuity in the tangential magnetic field.

- There are surface charges because of the presence of a normal electric field

(16)

Ex. 10-4 (1)

/ /

1

( , ) 1sin( ) ( )

2

j z j y b j y b

z

A

E y z A y e e e

b j

β π π

π − −

= = −

(a) A propagating TM wave = the superposition of two plane 1

waves bouncing back and forth obliquely between the two conducting plates

proof>

( / ) ( / )

1[ ]

2

j z

j z y b j z y b

e

A e e

j

β

β π β π

−

− − − +

= −

1 2

Ex. 10-4 (2)

z y

b

z y

β π

+ −

+ +

1 Term : A plane wave propagating obliquely in the and directions with phase constants and

2 Term : A plane wave propagating obliquely in the and directions w

0 0 0 0

0 0

ˆ

ˆ sin ˆ cos ˆ sin ˆ cos

ˆ sin ˆ cos

ˆ cos ˆ sin ˆ cos ˆ sin

x

i i i i i r r i r i

i i i i

i i i r i i

H xH

E yE zE E yE zE

yE zE

y z y z

θ θ θ θ

θ θ

β β₁ θ β₁ θ β β₁ θ β₁ θ

= −

= − = − −

= +

= + = − +

ith the same phase constants

(17)

Ex. 10-4 (3)

1 cos 1 cos 1sin

0

2 2 2 2

( , ) cos ( )

sin , cos

( ) ( )

cos 2

2 1, 1

2 2

0

i i i

j y j y j z

z i i

i i

i

E y z E e e e

b

b b

b b b f u

b b

β θ β θ β θ

θ

β θ β β θ π

π π

β β ω με

π λ

θ λ

β λ

λ με

θ

− −

1 1

1

= −

∴ = =

= − = −

= = ⇒ ≤

= = = ⇒

= ⇒

solution exists only for

at cutoff frequency

then waves bounce ba

2

- -

2 .

cos sin 1 ( )

c c

i i

c g p

y z

b f f

f u f

λ λ

θ θ

λ λ

⇒ < = >

= = = = = −

1

ck and forth in the direction and no propagation in the direction

TM mode propagates only when or

TE Waves between Parallel Plates (1)

2 0

2

0

0 0

2 0

0

0, 0

( ) ( ) 0

( , ) ( ) 0

( ) 0 0

( ) cos( )

z

z z

z x

z

z n

E x

d H y

h H y dy

H y z H y e H

E j

h y

dH y

y y b

dy H y B n y

b

γ

ωμ

π

−

= ∂ =

∂

∴ + =

=

− = − ∂ =

∂

= = =

∴ =

* TE waves

We note that B.C.

i.e at and

(18)

TE Waves between Parallel Plates (2)

0 0

2

0 0

2

0 0

2

0 0

2

2 2 2 2

( ) 0 ( 0)

( ) sin( )

( ) 0 ( 0)

( )

z Z

x

z

y n

z

x n

z Z

y

H H

H y

h x x

H n y

H y B

h y h b

H

j j n y

E y B

h y h b

H H

E y j

h x x

h k n

b γ

γ γ π

ωμ ωμ π

ωμ

γ π ω με

∂ ∂

∴ = − = =

∂ ∂

= − ∂ =

∂

= − ∂ =

∂

∂ ∂

= = =

∂ ∂

⇒ = − = − ⇒

⇒

∵

the same as that for TM waves The cutoff frequency is

0, _y 0 _x 0

n H E

⇒ = = =

the same For and

TE Waves between Parallel Plates (3)

0

01 10

) )

)

cf TM TEM

cf TM TM

=

i.e, TE mode doesn't exist0

or does not exist

or does exist

for the rectangular waveguide

0

( , ) 0sin( ) sin( )

z

m x m y E x y E

a b

π π

=

0

( , ) 0cos( ) cos( )

z

m x m y

H x y H

a b

π π

=

(19)

Energy-transport Velocity (1)

⇒

* Energy-transport velocity

- Wave guide high pass filter

- Broadband signal 1. low frequency components may be below cutoff

2. high frequency components will travel widely different velocity

- Energy-transport velocity : veloc

( )

(m/s) ( )

z av en

av

z av s av

u P

W

P P ds

= ′

=

∫

ⁱ

ity at which energy propagates along a waveguide

: the time average power

Energy-transport Velocity (2)

2

[( ) ( ) ]

1 ( )

[

(

av e av m av

s

c en

W w w ds

u u f

f

w

′ = +

= −

∫

: the time average stored energy per unit length

H.W] prove that

*

) ( )

4

( ) ( )

4

e av

m av

e E E

w e H H

ε μ

= ℜ

= ℜ i

i

(20)

Energy-transport Velocity (3)

2

2 2 2

2

* 2

2

2 2 2

2 2

0

2 2

2

2 2 2 2 2

2 2

0

2 2

0 2

( ) [sin ( ) cos ( )]

4

( ) ( )

( ) [1 ]

8 8

( ) ( ) cos ( )

4

( ) ( )

8 8

( ) ˆ

cos (

e av n

b

e av n n

m av n

b

m av n n

b

z av av

b

n

n y n y

w A

b h b

E E j j

b b

w dy A k A

h h

w A n y

h b

b b

w dy A k A

h h

P P zdy

h A

ε π β π

β β β

ε β ε

μ ω ε π

μ ω ε ε

ωεβ

= +

⇒ − =

= + =

=

= =

=

= 2

∫

i i

i

cf)

2

) 2

4 ⁿ

n y b

dy A

b h

π ₌ωεβ

Energy-transport Velocity (4)

*

0 0* 0 0*

0 0*

2 2

2

2 2

) 1 ( )

2 1 ( ˆ ˆ )

2 ( ) 1 ( )

2 cos ( )

( ) 1 ( )

av

y x z x

av y x

n

c en

cf P e E H

e zE H yE H

P z e E H

A n y

h b

u u f

k k k f

ωεβ π

ωβ ω β

= ℜ ×

= ℜ − +

= − ℜ

= 2

= = = −

i

(21)

Attenuation in Parallel-plate Waveguides (1)

( )

d c

P zL

α α= +α

=

* Attenuation in parallel-plate waveguide - Losses are very small

-

For TEM mode

cf) For a lossy transmission line the time-average power loss per unit length

2

2 2 0 2 2

2 0 0 2

* 0 2

2 0 0

1[ ( ) ( ) ] ( )

2 2

( ) 1 [ ( ) ( )]

2 2

z

I z R V z G V R G Z e Z

P z e V z I z V R e Z

α

−

+ = +

= ℜ =

Dielectric losses

Ohmic losses

Attenuation in Parallel-plate Waveguides (2)

2 0 0

0 0 0

0

( ) ( ) 2 ( ) ( ) 1

( )

2 ( ) 2

2 2 2 (

L

d

P z P z P z

z

P z R G Z

P z R

GR Z R

G b

where R b

α α

σ μ σ

α η

ε σω

ωη

−∂ = =

∂

∴ = = +

∴ = = =

⎛ =⎜

⎜⎜ =

⎜⎝

∵ for low loss conductor)

independent of frequency

(22)

Attenuation in Parallel-plate Waveguides (3)

0

1 2

2 ( )

c

c c

d c

d

R f

R b f

b b

R R f

f f j

α π ε

σ η μ

ω ω ε

π μ

ω σ

α ε ε σ

ω

∴ = = ∝

= =

=

>

= + cf)

For TM mode

to find dielectric losses, at -

Attenuation in Parallel-plate Waveguides (4)

2 2 1/ 2

2 2 2 2 1 1/ 2

2 2 2 2 1

2 2

[ (1 ) ( ) ]

( ) {1 ( ) ] }

2 ( )

j n

j b

n n

j j

b b

n j n

j b b

n b

σ π

γ ω με

ωε

π π

ω με ωμσ ω με

π ωμσ π

ω με ω με

ωμσ ω με π

−