waveGuide

(1)

We consider the conditions under which EM waves can propagate when confined to the interior of some kind of “hollow” pipe – also known as a wave guide. In the real world wave guides consisting of e.g. rectangular, cylindrical, or arbitrarily cross-section shaped conducting and/or superconducting hollow metal pipes can be used to transport EM waves and EM energy in the radio and microwave region of the EM spectrum, whereas, e.g. glass or plastic optical fibers act as wave guides in the infrared, visible and even the UV portions of the EM spectrum.

We consider first the simplest type of a wave guide – a perfect conductor (ρ=1/σ=0) such that inside the walls of the perfect conductor: 𝐸 = 0 & 𝐵 = 0

The boundary conditions at the inner walls of a perfect conductor are:

𝐸 _𝐶 . 𝑑𝑙 = − 𝐵_𝑆 . 𝑑𝑎 = 0 (1) ⇒Tangential 𝐸 continuous: E = 0 𝐵_𝑆 . 𝑑𝑎 = 0 (2) ⇒Normal Bcontinuous: B⊥= 0

We are interested in/seek monochromatic/single-frequency plane traveling wave-type solutions - that propagate down the inside of the wave guide, e.g. in the +zˆ direction.

Using phasors & assuming waveguide filled with  lossless dielectric material and

 walls of perfect conductor, the wave inside should obey…

c

k2 2

where 

Then applying on the z-component…

Fields inside the waveguide

0 0

2 2

 



  

H k H

E k E

2 2 2 2 2 2 2 2

: obtain we where from

) ( ) ( ) ( ) , , (

: Variables of

Separation of

method by

Solving

0

k Z Z Y Y X X

z Z y Y x X z y x E

E k z

E y

E x

E

'' '' '' z

z z

   



 

       

0

2

2  

 E_z k E_z

z z

''

y y

y ''

x x

x ''

y x

'' '' ''

e c e c z Z Z

Z

y k c y k c Y(y) Y

k Y

x k c x k c X(x) X

k X

k k

k

k Z Z Y Y X X

 



  

 



 

 



 

 



    

   

6 5

2

4 3

2

2 1

2

2 2

2

) ( 0

sin cos

0

sin cos

0 : s expression in the

results which

2 2 2 2 2

y x k

k k

(2)

Substituting

Other components

From Faraday and Ampere Laws we can find the remaining four components:

So once we know Ezand Hz, we can find all the other fields.

Modes of propagation

From these equations we can conclude:  TEM (Ez=Hz=0) can’t propagate.

 TE (Ez=0) transverse electric

 In TE mode, the electric lines of flux are perpendicular to the axis of the waveguide

 TM (Hz=0) transverse magnetic, Ezexists

 In TM mode, the magnetic lines of flux are perpendicular to the axis of the waveguide.

HE hybrid modes in which all components exists )

( ) ( ) ( ) , ,

(x y z X x Y y Z z

E_z 

z z

y y

x x

e c e c z Z

y k c y k c Y(y)

x k c x k c X(x) where



 _ 



 

6 5

4 3

2 1

) (

sin cos





























z

y y

x x

z

z y y

x x

z

z z

y y

x x

z

e y k B y k B x k B x k B H

e y k A y k A x k A x k A E

z

e c e c y k c y k c x k c x k c E

 

 



 



 





 

 

sin cos

, field magnetic for the

Similarly

sin cos

: direction

-in traveling wave

at the looking only

If

sin cos

4 3

2 1

4 3

2 1

6 5

4 3

2 1

2 2 2 2 2

2 2

y x

z z

y

z z

x

z z

y

z z

x

k k k h

where

y H h x E h j H

x H h y E h j H

x H h j y E h E

y H h j x E h E

   

      

     

  

   

  

   



 

 

(3)

TM Mode

 Boundary conditions: From these, we conclude:

X(x) is in the form of sin kxx, where kx=mp/a, m=1,2,3,…

Y(y) is in the form of sin kyy, where ky=np/b, n=1,2,3,…

So the solution for Ez(x,y,z) is

Substituting

TM

mn

 Other components are

 The m and n represent the mode of propagation and indicates the number of variations of the field in the x and y directions

 Note that for the TM mode, if n or m is zero, all fields are zero

TM Cutoff

 The cutoff frequency occurs when



2 2 2 2

2

sin

sin k

b n a

m h

where e

y b n x

a m E

E_z _o j z   

             

        

 

       _









z

y y

x x

z A k x A k x A k y A k ye

E  ₁cos  ₂sin ₃cos  ₄sin  ,a x E

,b y E

z z

0 at 0

 









j z

y x

z A A k x k ye

E  ₂ ₄ sin sin 

0

sin sin



           

 

z

z j o

z

H

e y b n x

a m E

E   

z o

y z

y

z o

x z

x

z o

y z

y

z o

x z

x

e b

y n a

x m E

a m h j H

x E h j H

e b

y n a

x m E

b n h j H y

E h j H

e b

y n a

x m E

b n h E

y E h E

e b

y n a

x m E

a m h E

x E h E

  



 

  

 

  

 

  

 

  

  



            

        

  

            

       

 

            

        

  

            

        

  

sin cos

cos sin

sin cos

2 2





   

 2

2 2

2 

                

b n a

m k

k k_x _y

2 2

2

1 2

1 or

0 then

When

             

   

            

b n a

m f

j b

n a

m

c c

 

 

   

 

(4)

 Evanescent:

 Means no propagation, everything is attenuated Propagation:

 This is the case we are interested since is when the wave is allowed to travel through the guide.

Cutoff

 The cutoff frequency is the frequency below which attenuation occurs and above which propagation takes place. (High Pass)

 The phase constant becomes

Phase velocity and impedance

 The phase velocity is defined as

 And the intrinsic impedance of the mode is

Summary of TM modes

Wave in Dielectric Medium Inside the waveguide 0 and When

2 2

2 _  

           

     

 

b n a

m

0 and When

2 2

2 _  

           

     



 j

b n a

m

2 2

2 '

             

b n a

m u fc_mn

2 2

1

' _

      

              

f f b

n a

m  _ _c

  

f u

up   _  p

  

 2

'

2

1

' _

         

f f H

E H

E _c

x y y

x

TM 



2 2 2 2

1 ' /

' '

/ 1

' /

1 '

/ '

1 ' /

'

1 ' '

/ '

        

        

  

       



       

 

f f f

u

f f u

f u

f f f f u

c c p

c TM

c

 



  

 

  

  

 

  

 

(5)

TE Mode

 Substituting

 Note that n and m cannot be both zero because the fields will all be zero.

TE

mn

Cutoff

 The cutoff frequency is the same expression as for the TM mode

 But the lowest attainable frequencies are lowest because here n or m can be zero.

Dominant Mode

 The dominant mode is the mode with lowest cutoff frequency.  It’s always TE10

 The order of the next modes change depending on the dimensions of the guide.

Summary of TE modes

Wave in the dielectric medium Inside the waveguide

2 2

2

again where cos

cos 

             

          

 

b n a

m h

e y b n a

x m H

H_z _o   jz  

0

cos cos



           

 

z

z j o

z

E

e y b n x

a m H

H   

z o

y z

y

z o

x z

x

z o

y z

y

z o

x z

x

e b

y n a

x m H

b n h

j H y

H h H

e b

y n a

x m H

a m h

j H x

H h H

e b

y n a

x m H

a m h j E

x H h j E

e b

y n a

x m H

b n h j E y

H h j E

 

 

  

 

  

 

  

 

  

 

            

       

  

            

       

  

            

        

 



            

       

  

sin cos

cos sin

sin cos

2 2

2 '

             

b n a

m u f_c_mn

  ' /

   ' /u'

f u'/ ' 

 



/ ' 1/

'  f 

u

2

1 '

       

f f_c

TE

 

2

1 '

       

f fc

 

  



/ 1

'

2 

       

f f u

c p

2

1

' _

      

f f_c

 

(6)

Example: Consider a length of air-filled copper X-band waveguide, with dimensions a=2.286cm, b=1.016cm operating at 10GHz. Find the cutoff frequencies of all possible propagating modes.

Solution: From the formula for the cut-off frequency

Example: An air-filled 5-by 2-cm waveguide has

at 15GHz

 What mode is being propagated?  Find 

 Determine Ey/Ex

Power transmission

 The average Poynting vector for the waveguide fields is

 where TE or TM depending on the mode

[W]

Waveguide Cavities

 Cavities, or resonators, are used for storing energy

 Used in klystron tubes, band-pass filters and frequency meters  It’s equivalent to a RLC circuit at high frequency

 Their shape is that of a cavity, either cylindrical or cubical.

Cavity TM Mode to z

Resonant frequency

The resonant frequency is the same for TM or TE modes, except that the lowest-order TM is TM111 and the lowest-order in TE is TE101.

Cavity TE Mode to z

2 2

2 '

             

b n a

m u fc_mn



40

 

sin50



V/m sin

20 j z

z x y e

E    









z E E

H E H E H

E

y x

x y y x ave

ˆ 2

Re 2 1 Re

2 1

2 2

* *

*



 

 

 

P

 



 

 



 a

x b

y

y x ave

ave dydx

E E dS

P

0 0

2 2

2

P

) ( ) ( ) ( ) , , ( : Variables of

Separation by

Solving E_z x y z  X x Y y Z z

z k c z k c z Z

y k c y k c Y(y)

x k c x k c X(x)

z z

y y

x x

sin cos

) (

sin cos

6 5

4 3

2 1

 

2 2 2 2

z y x k k

k k

where   

2 2

2

2 '

                    

c p b

n a

m u fr

) ( ) ( ) ( ) , , ( : Variables of

Separation by

Solving Hz x y z  X xY y Z z

z k c z k c z Z

y k c y k c Y(y)

x k c x k c X(x)

z z

y y

x x

sin cos

) (

sin cos

6 5

4 3

2 1

 

2 2 2 2

z y x k k

k k

(7)

TE

mnp

Boundary Conditions

From these, we conclude: kx=mp/a

ky=np/b

kz=pp/c

where c is the dimension in z-axis

Quality Factor

, Q

 The cavity has walls with finite conductivity and is therefore losing stored energy.  The quality factor, Q, characterized the loss and also the bandwidth of the cavity

resonator.

 Dielectric cavities are used for resonators, amplifiers and oscillators at microwave frequencies.

Example:For a cavity of dimensions; 3cm x 2cm x 7cm filled with air and made of copper (sc=5.8 x 107)

 Find the resonant frequency and the quality factor for the dominant mode.

Answer:

GHz

fr 9

7 0 2

1 3

1 2

10

3 10 2 2 2

110  

                   



,b y E

,a x E

,c z H

x y z

0 at , 0

0 at 0

 

                  

c y p b

y n a

x m H

H_z _ocos  cos  sin 

L

P W

lation e of oscil y per cycl

loss energ

stored ge energy Time avera

π Q

 2

2

 







 





3 3 2 2



2 2 101

2 TE

mode dominant For the

101

c a ac c a b

abc c a Q_TE

 

  



c o

f where

   

101

1



GHz

f_r 5.44

7 1 2

0 3

1 2

10

3 10 2 2 2 _

                    



6

9 1.6 10

) 10 44 . 5 (

1 _ _ 

 

c o

 

















568,378

7 3 7 3 7 3 2 2

7 2 3 7 3

2 2 3

3 2 2

101      

   



TE