Wave Guide 012ma

The Relation between Maxwell’s Equation and Amperes Circuital Law

I

D  J   gda Ñ S  t  

The net current (Conduction and Displacement ) passing through the area enclosed by the loop

dl

Ñ  H gdl

H

C

Total magneto motive force along a closed path around a total current density

D  Ñ C H gdl  Ñ S  J  t  gda 

Integral form Differential form

Applying Stoke’s Theorem

D  Ñ S (  H )gda  Ñ S  J  t  gda 

D  H  J  t

The Relation between Maxwell’s Equation and Faraday's Law Negative rate of change of magnetic flux Φ enclosed by the loop

 

d   Ñ Bgda dt t S

dl

E

Ñ  E gdl C

Total electro motive force along a closed path around a total current density

 Ñ C E gdl   t Ñ S Bgda

Integral form differential form

Applying Stoke’s Theorem

 Ñ S (  E )gda   t Ñ S Bgda

 E  

B t

The Relation between Maxwell’s Equation and Gauss Law of Electrostatics Total charge in side a volume enclosed by the closed surface

D

da nˆ

 Dgda   dV  Ñ

nˆ

V

da q

S

Total outward electric displacement through the closed surface

Ñ  Dgda    dV S

V

Divergence Theorem

 (gD)dV    dV V

V

Integral form differential form

gD  

The Relation between Maxwell’s Equation and Gauss Law of Magnetostatics B D

da nˆ

Total magnetic dipole charge in side a volume enclosed by the closed surface

 Bgda  (dp)dV Ñ

nˆ

V

da d p

Total magnetic flux through the closed surface

Ñ  Bgda   (dp)dV S

V

Divergence Theorem

 (gB)dV   (dp)dV V

S

V

Integral form differential form

gB  0

SUMMARY OF MAXWELL’S EQUATION IN DIFFERENTIAL FORM

B  E   t EMF induced in a closed loop

Time rate of change of the magnetic flux

D  H  J  t

Closed path in the magnetic field around the charge movement

Conduction current + displacement current

gD  

Div of Electric displacement density

gB  0 Stored charge

Div of Magnetic flux density

Involves space derivatives – useful only for a continuous media

SUMMARY OF MAXWELL’S EQUATION IN INTEGRAL FORM

 Ñ C E gdl   t Ñ S Bgda

Negative rate of change of magnetic flux Φ enclosed by the loop

d   dt

Total electro motive force induced in a closed loop 

Ñ  Bgda S

B  Ñ C H gdl  Ñ S  J  t  gda 

The net current , which is the combination of conduction and displacement current

Total magneto motive force along a closed loop

Ñ  Bgda  0 S

Non existence of Monopole

Total Stored charge

Ñ  Dgda    dV S

V

Total outward electric displacement through the closed surface

SPACE AND TIME VARIATION FORMS FOR THE FIELD QUANTITY E, D,B and H

Sinusoidal oscillation in time

e

jt

Propagation in z direction - Sinusoidal oscillation in the z direction Exponential decay with distance z Combining all the three variations

e

 z

X  X 0e

Combining the two space forms into a general propagation constant Field variables will take the form

e

 j z

jt  z  j  z

    j

X  X 0e

jt  z

    j

Is the general propagation constant

For the propagation without attenuation α =0

  

  j

2

    2

2

2

  2   2 

   

Thus if γ is imaginary the electromagnetic eave is traveling without any change in the amplitude at the constant phase velocity

 1 V   

If γ is real then the field suffers on exponential decay with distance

B  E   t

D  H  J  t gD  

D E

B  H

J E

gB  0 H   E   t E  H   E  t

 gE   gH  0

E  E0 e jt   j t

H   E   t E  H   E  t

 gE   gH  0

  E   j H   H   E  j E

 gE  

 gH  0

H   E   t E  H   E  t

 gE   gH  0

    E  (.E )   2 E     H  (.H )   2 H

2  E  E 2  E     2 t t

2  H  H 2  H     2 t t

2  E  E 2  E     2 t t

2  H  H 2  H     2 t t

 E  E   2 t

For free space

2

2

 0

 H  H   2 t 2

2

For TE mode

 E0

For TM mode

2

 H  H   2 t 2

2

2 H  0

 E  E   2 t 2

2

2  E 2  E   2 t

2  H 2  H   2 t

E  E0 e jt

  j t

 2    t 2 2

 2 E   2  E  H    H 2

2 H   2 H

2 E   2 E

 2   2 

2

Conditions at the boundary surface  Maxwell's equations in the differential form express the relationship that must exits between the four field vectors E, D, H, B at any point in the continuous medium  In this form because they involve space derivatives , they cannot be expected to yield information at points of discontinuity in the medium  However, the integral forms can always be used to determine what happens at the boundary surface between different media  The following statements regarding electric and magnetic field are valid at any surface of discontinuity  The following statements regarding electric and magnetic field are valid at any surface of discontinuity (boundary surface (BS)) 1.) ET and Bn are always continuous at the boundary surface 2.) HT is discontinuous by JS per unit width at the perfectly conducting boundary surface. If the surface is a nonconductor, then HT is continuous at the boundary surface 3.) Dn is always discontinuous by ρS at the boundary surface

WAVE GUIDE AND ITS PROPERTIES

 At frequencies higher than 3 GHz ,transmission of electromagnetic waves through two line transmission line and cable become difficult.  This is mainly due to the losses that occur both in the solid dielectric needed to support the conductor and in the conductor themselves  A metallic tube can be used to transmit electromagnetic wave at these frequencies  A hallow metallic tube of uniform cross section for transmitting electromagnetic wave by successive reflections from the inner walls of the tube is called wave guide Properties of wave guide  Wave traveling in a wave guide has a phase velocity and attenuation  When the wave reaches the end of the wave guide it is reflected unless the load impedance is adjusted to absorb the wave  Any irregularity in a wave guide produce reflection . The reflected wave can be eliminated by proper impedance matching  When both incident and reflected waves are present in a wave guide , a standing wave pattern results as in a transmission.

 There is a cut-off value for the frequency of transmission (f) depending upon the dimensions and shape of the wave guide.  Only waves having frequencies greater than cut-off frequency fc will be propagated.  Hence wave guide act as a high pass filter with fc as the cut off frequency  Entire body of the wave guide will act as ground and the wave propagate through multiple reflections from the walls of the wave guide  In wave guide we define a quantity called wave impedance (a function of frequency)  Wave propagation inside the cavity is in accordance with Field theory  If one end of the wave guide is closed using a shorting plate , there will be reflections and hence standing wave will be formed.  If other end is also closed , then the hallow box so formed can support a signal which can bounce back and fourth between the two shorting plated resulting in resonance. This is the principle of cavity resonator.  This is the principle of cavity resonator.

General guidelines for solving wave guide problem  Since the various components of the electric and magnetic fields are related through the Maxwell’s equations, all the field components are not independent (coupled to each other by Maxwell’s Equation)  We can select some field components as an independent components and express the remaining field components as a function of the chosen independent components  If we assume that the wave is moving in z-direction , the field components which are along the z-direction (Hz and Ez) have special significance as they represent longitudinal field components  The other four field components (Ex, Ey, Hx and Hy), are in a plane transverse to the direction of the wave propagation and hence can be called as transverse components  First step In solving wave guide problems is , Assume Ez and Hz as an independent variables and express the transverse components (Ex, Ey, Hx and Hy) in terms of Ez and Hz components using Maxwell's curl equations

  E   j H and   H  j E

 In the first step you will end up with the following general equations for the transverse components In rectangular coordinate

j H z  Ez Ex   2  2 h y h x j H z  Ez Ey  2  2 h x h y j Ez  H z H x  2  2 h y h x

j Ez  H z Hy   2  2 h x h y

In cylindrical coordinate

j 1 H z  Ez E   2  2 h   h  j H z  1 Ez E  2  2 h  h   j 1 Ez  H z H  2  2 h   h  j Ez  1 H z H   2  2 h  h  

Note: you can get the expressions for cylindrical coordinate from the expressions for rectangular coordinate using the following substitutions

x   , y  ,

 1      and y   x 

 Also this general equations for the transverse components will take the following reduction depending on TE or TM modes TE waves – Transverse electric

Ez  0 and H z  0 Rectangular

Cylindrical

TM waves – Transverse magnetic

H z  0 and Ez  0 Rectangular

Cylindrical

 Ez Ex   2 h x  Ez

 Ez h 2   1 Ez E   2 h  

j H z Ex   2 h y

j 1 H z E   2 h  

j H z Ey  2 h x  H z Hx   2 h x

j H z E  2 h 

Ey  

 H z H   2 h 

j Ez H x  2 h y

H 

 H z Hy   2 h y

 1 H z H   2 h  

j Ez Hy  2 h x

H  

h 2 y

E  

j 1 Ez h 2  

j Ez h 2 

 Ez and Hz satisfy the wave equation for source free homogeneous medium

 2 Ez   2  Ez  0

 2 H z   2  H z  0

 In the second step, Solve the wave equations to obtain the general solution for Ez and Hz

 The general solutions geometry) is

for wave equation (in case of rectangular

Ez   C1 cos Bx  C2 sin Bx   C3 cos Ay  C4 sin Ay 

H z   C5 cos Bx  C6 sin Bx   C7 cos Ay  C8 sin Ay 

 In the third step, apply appropriate boundary conditions (shown in the table given bellow) to resolve the unknown constants C1 to C8 in the general solution and get the Ez and Hz solutions for a particular wave guide geometry.

 m  a

Ez  C sin 



 n  b



x sin 

 

jt  z ye 

 m  a

H z  C  cos 



 n  b



x cos 

 

jt  z ye 

y

z

b a

x

First boundary bottom wall or plane Second boundary Left wall or plane Third boundary Top wall or plane

Fourth boundary Right wall or plane

TE Boundary conditions TM Boundary conditions Ez= 0 for all the four Surface will act as a short boundaries but we have x circuit for E, Ez =0 all along and y components the boundary wall

Ex  0 at y  0

Ez  0 at y  0

and for every x (0,a)

and for every x (0,a)

Ex  0 at y  b

Ez  0 at x  0

and for every x (0,a)

and for every y (0,b)

E y  0 at x=0

Ez  0 at y=b

and for every y (0,b)

and for every x (0,a)

E y  0 at x  a

Ez  0 at x  a

and for every y (0,b)

and for every y (0,b)

 m   n  jt  z Ez  C sin  x sin ye    a b     m  n    jt  z H z  C  cos  x cos ye    a b      In the fourth step, substitute this final solution for Ez and Hz in the general equations to get find all the transverse components Ex, Ey, Hx and Hy in terms Hz and Ez H z  0 and Ez  0 Ez  0 and H z  0 TM waves TE waves Rectangular

j H z Ex   2 h y

Cylindrical

E  

j 1 H z h 2  

j H z h 2 

Rectangular

Cylindrical

 Ez Ex   2 h x  Ez

 Ez h 2   1 Ez E   2 h   E  

j H z Ey  2 h x  H z Hx   2 h x

 H z H   2 h 

j Ez H x  2 h y

H 

 H z Hy   2 h y

 1 H z H   2 h  

j Ez Hy  2 h x

j Ez H   2 h 

E 

Ey  

h 2 y

j 1 Ez h 2  

  m   C cos    h2  a     n   Ey   2 C  sin   h  b   j  n   Ex  

m a 

 n  jt  z ye   b 

x sin  

m   x cos   a   m   Hx  2 C  sin x cos     h  b   a   Hy 

n b n b 

 ye

jt  z





jt  z ye 



Transverse components Ex, Ey, Hx and Hy in terms Hz and Ez with TE boundary conditions

j  m   m   n  jt  z C cos x sin ye       h2  a   a   b  E  x

Transverse components Ex, Ey, Hx and Hy in terms Hz and Ez with TM boundary conditions

j  n   m   n  jt   z C  cos x sin ye      2 b a b       h

j  m   m   n  jt  z  C sin x cos ye       h2  a   a   b    m   m   n  j t  z H x  2 C   sin x cos ye      h  a   a   b    n   m   n  jt  z H y   2 C  cos x sin ye      h  b   a   b 

Ey  

Finding the expressions for Hz and Ez inside a rectangular cavity For a wave propagating along the positive z axis inside the cavity we can write

 m H  C cos   a  z



 n x cos    b 



jt  z ye 



For a wave propagating along the negative z axis inside the cavity we can write

 m H  C cos   a  z



 n x cos    b 

Adding the above two traveling wave

Hz   C e

  z

  z

C e



jt  z ye 

 H z  H z  H z

 m  cos  a

 n x cos    b 



jt ye 



To make Ey vanish at Z=0 and z=d the constants C+ and C- must satisfy the following condition

C   C  or C   C 

For TE wave , Ez=0 therefore

j H z Ey  2 h x

j    m     z   z Ey  2 C e  C e  cos   x cos   h x   a   j   m    z    z  m 

Ey 

h

2



  C e

C e

 

a 

sin  

n b 

 ye



jt



 

 n  jt  x cos   ye  a   b  

j   m   n    z    z  m  jt  0  2   C e  C e   sin  x cos   ye  h   a   a   b     z   z   z  z C e  C e  C e  e     0

2 jC  sin  z  0  sin  z  0 for z=d  p   =  d  

 m   n  jt H z  2 jC sin  z cos  x cos ye     a   b  

 m   n  jt H z  C sin  z cos  x cos ye     a   b   m   n  jt H z  C cos  x cos y sin  ze     a   b 

Thus for rectangular cavity resonator field equation is given by for

 m   n   p  jt H z  C cos  x cos y sin ze       a   b   b  Ina similar way the equation for

 m Ez  C sin   a

TM mnp

 n x sin    b 

 p  y cos    b 

 

jt ze 

TEmnp

EXPANSION OF



AND  2 IN DIFFERENT COORDINATE

    xˆ  yˆ  zˆ x y z  1  ˆ   ˆ    zˆ    z

Rectangular coordinate Cylindrical coordinate

 1  ˆ 1  ˆ   rˆ    r r  r sin   Rectangular coordinate

2 2 2    2  2  2  2 x y z

2 2 2  1   2   2 2 2 2    z

Cylindrical coordinate

Spherical coordinate

Spherical coordinate

1     1        2 2  2        z 2

2

2