Lecture 3

TEM and Plane Waves

(partially a review)

Homework: From Section 5.10 Exercises – 1, 4(b,c,d,e), 9, 11, 12

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 1

 Why Are We Concerned with TEM and Plane Waves?
• plane waves exist in TEM transmission lines (e.g., coaxial cable)
• far from a source (e.g., antenna) the free-space wave is locally plane

• plane waves illustrate best wave characteristics such as frequency,

propagation constant, wavelength, wave vector, phase constant and
wave number, attenuation constant
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 2
 Some Definitions
• plane wave – a wave whose phase front is a plane (as opposed to a
cylindrical wave or a spherical wave)
plane-wave animation spherical wave animation
Wikimedia Commons

cylindrical wave animation

• uniform plane wave – a plane wave with field vectors (E,H) which
are constant across the phase front

• TEM (transverse electromagnetic) wave – a wave with field

vectors (E,H) which are transverse to the direction of propagation
E
k  kuˆ
H
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 3
 Vector Helmholtz Equation
• assume homogeneous, isotropic and source-free medium
       2 
E   0
E
 
  E   j H Helmholtz’
  j E  or
H  
equations
    H   
2
 H0
Derive the E-field Helmholtz equation from Maxwell’s equations.
• propagation, attenuation and phase constants (wave number)
 2   2 
    j 
    jk

• in a loss-free medium
  0, k    ,   jk
  0 and   H
• since the medium is source-free,   E  0
2    2E
  2E  0
         2
 
H   2 
H0
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 4
 Traveling Waves
• assume propagation along z:  ( x, y , z )  E
E  ( x, y )  e   z  z
T
function of x and y only!
• substitute in the Helmholtz equation
 2E  2E   2E  2  2
E  2
E 2 
2
 2
 2
  E  0  2
 2
 ( 2
z   )E  0
x y z x y
  z2 E

 2E
 2E T  2E T
   ( z
T 0
2   2 )E
x 2 y 2
• consider the case of γz = γ (note that it is also possible that γz ≠ γ!)

 T  2E
 2E T  2 E T,  2 E T,
   0 or   0,   x, y , z ()
x 2 y 2 x 2 y 2

 2E
and 2   2 E 
z
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 5
 TEM Waves
• consider the 2D Laplace equation for the longitudinal z component
of ET(x,y)  2 E T,z  2 E T,z
 0
x 2 y 2

• in a TL, it is complemented by zero BCs: E T,z  0

C
C2

C1 Ez

Ez  0 C
C  C1  C2
reminder about the mean-value theorem: a non-constant harmonic
function cannot assume its maximum value at an interior point
 E T,z ( x, y )  0, ( x, y )  E z ( x, y , z )  0, ( x, y , z )
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 6
 TEM Waves
from Ez = 0 it follows
E E E E E
E x y
 z 0  T, x

T, y
 0 ( )
x y  z x y
0

Using Eqs. () and ( ), prove that H z  0.

 2 E T,x  2 E T,x Hint
 0
x 2 y 2

 2 E T,y  2 E T,y ?E T,y E T,x ?

 0   =0  H T,z  0
x 2 y 2 x y
E T, x E T, y  
 0 ,
x y x y

if the propagation constant of a wave is  z    j 

 , then the
wave is a TEM wave (Hz = Ez = 0) and vice versa
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 7
 TEM Waves and Uniform Plane Waves

• TEM waves feature 2D “static” field distributions in the transverse

(xy) plane obeying the 2D Laplace equation:
• ET(x,y) obeys 2D electrostatic analysis
• HT(x,y) obeys 2D magnetostatic analysis
 2 E T,  2 E T,
 0
x 2 y 2
,   x, y ( )
 2 H T,  2 H T,
 0
x 2 y 2

• the uniform plane wave is a special case of the TEM wave: ET and
HT are simply constant with respect to (x,y) – eqns (***) still hold

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 8

 Traveling Waves in Loss-free Medium

• in a loss-free medium, α = 0    jk , k   
purely imaginary

• assume waves propagating along +z and −z 

E T  ET e j

E( x, y , z )  E T ( x, y )e  jkz  E
 T ( x, y )e jkz  ET  ET e j
E( x, y , z, t )  ET cos(t  kz    )  ET cos(t  kz    )

radian frequency spatial frequency (wave

(angular frequency) number, phase constant)
units: rad/s units: rad/m

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 9

 Plane Wave in Loss-free Medium: Animation
• wave propagating along +z
E x e  jkz  Ex cos(t  kz   x )
z

• wave propagating along –z

E x e jkz  Ex cos(t  kz   x )

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 10

 Phase Velocity
velocity of propagation of the phase front
a phase front is defined by any fixed point on the waveform
cos(arg), arg  t  kz     const.

arg  k  t  z      const.
k 
phase velocity vp
  1
vp  ,k     v p 
k vp 
1
in vacuum v p 0  c   2.998... 108 m/s *
 0 0
Express the wavelength λd of a plane wave in a loss-free dielectric
medium of relative permittivity εr in terms of the respective wave-
length in vacuum λ0 and εr.
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 11
 Waves in Lossy Medium
 T ( x, y )e  z  E
 ( x, y , z )  E
E  T ( x, y )e z where
  j 
 
  jk
0

  jk  j (    j  )[   j (    /  )]

general solution for attenuation and phase constants

    
   1  tan  m  tan  d  Q tan  m 
2 
       / 
k   1  tan  m  tan  d  Q tan  d 
2 
where Q  1  tan 2  m  tan 2  d  (tan  m  tan  d )2

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 12

 Waves in Very Good Conductors
metals are very good conductors for which the following holds
   ,    0

Prove that the attenuation constant α and the wave number k of a

good conductor (assume μ is real) are approximately the same and
equal to

 k 
2
or

  (1  j )
2
j 45
   e

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 13

 Waves in Lossy Medium: Penetration (Skin) Depth
• consider x-component only
 
E x ( z )  E x e  z  E x e z , E x  Ex e j x and E x  Ex e j x
 Ex ( z , t )  Ex e  z cos(t  kz   x )  Ex e z cos(t  kz   x )

• skin (or penetration) depth δ: the distance a wave travels into the
lossy medium until its magnitude reduces e times (e ≈ 2.71828… )
1 2 1
 for good conductors:    (m)
   f 
*

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 14

 Uniform Plane Wave in Lossy Medium: Animation
WAVE ATTENUATION AS IT TRAVELS INTO A LOSSY MEDIUM

e z

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 15

 Attenuation Constant α
• the attenuation constant α is characterized by the ratio of the
signal strength at two points a distance ΔL apart
Em ( z2 )  Em ( z1 )e  ( z2  z1 )  Em ( z1 )e L L  z2  z1 , m
 Em ( z1 ) 
  ln   / L, Np/m
 Em ( z2 ) 

• attenuation is often given in dB/m

 Em ( z1 ) 
 dB  20log10   / L, dB/m
 Em
 ( z
2) 
e
• dB to Np conversion
20
 dB  20log10 e      dB  8.6858896381  *
ln10

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 16

 TEM Wave Impedance η
• the E and H field vectors of a traveling TEM wave are related
through Maxwell’s equations
1  z 1

H 
  (E T e )  (e  z )  E
 T  e  z   E
T

 j  j
E T, y   T, x
E    T, y E T, x 
E
E T   xˆ  yˆ    T 0
 zˆ   
E
 z  z x y 
0 0
  conservative
2D field
0
 z
  e     z 1    j 

H zˆ  ET  zˆ  ET e  zˆ  E
 j   

E

• by definition the intrinsic impedance of the


medium is the impedance of a TEM wave in  ()
this medium 
• in a loss-free medium, η is real
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 17
 Intrinsic Impedance of a Good Conductor

• assume the conductor has no magnetic loss, µꞌꞌ = 0, (Cu, Al)

    
 
   j / 
 1   f  1 j
   (1  j )  (1  j )  ,
 j 2  

• the Re and Im parts of the intrinsic impedance η of a good conductor

are equal (η has phase of 45°)

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 18

 Wave Traveling in a General Direction
• Helmholtz equation in rectangular coordinates

2E 2E  2E  2

2
 2
 2
  E0
x y z 3 scalar equations
 2 E i  2 E i  2 E i 2 
 2
 2
 2
  Ei  0, i  x, y , z
x y z

• each scalar equation solved by separation of variables

2 2 2
 d X d Y d Z
Ex  X ( x)Y ( y ) Z ( z )  YZ 2  XZ 2  XY 2   2 XYZ
dx dy dz
1 d 2 X 1 d 2Y 1 d 2 Z eigenvalue equation:
 2
 2
 2
  2
X
 dx
 Y dy
   Z dz  x2   y2   z2   2
2 2
x  y2 z

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 19

 Wave Vector k
1 d2X 2 1 d Y
2
1 d 2Z
  x ,   2
y ,   2
z  x2   y2   z2   2
X dx 2 Y dy 2 Z dz 2
 x x  y y
 X ( x)  e , Y ( y)  e , Z ( z )  e z z
• wave propagates along a direction û such that it advances along +x,
+y, and +z simultaneously:
 (  x x   y y  z z )

Ex  X ( x)Y ( y ) Z ( z )  e  e  γr
 x2   y2   z2   2  γ   uˆ   x xˆ   y yˆ   z zˆ   j 

r  xxˆ  yyˆ  zzˆ propagation vector
position vector
• loss-free case
 j (k xk yk z )
E x  e x y z  e  jk r
k  k x xˆ  k y yˆ  k z zˆ  kuˆ wave vector k    *
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 20
 Properties of TEM Wave Field Vectors: Summary
• both field vectors are transverse to the direction of propagation
k  E  0, k  H  0

• the E and H field vectors are mutually orthogonal

H  E, and H  k , E  k

• the E, H and k vectors form a right-hand triplet

E
E H
uˆ  k  kuˆ
| E H | H

• |H| and |E| are related through the intrinsic impedance of the medium
1 |E|
H  uˆ  E  | H |
 
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 21
 Time-dependent Poynting Vector of TEM Waves

• consider field components at a given position in space (assume z = 0)

E(t )  eˆ mE cos(t ), V/m   eˆ m
E
 E
  hˆ m e  j
H (t )  hˆ mH cos(t   ), A/m H H

• the time-dependent Poynting vector (power-flow density) is

S(t )  E(t )  H (t )
 S(t )  0.5(eˆ  hˆ ) mE mH  cos   cos(2t   )  , W/m 2
uˆ constant term double-frequency term

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 22

 Average Power Flow Density and Complex Poynting Vector
• average power flow density
S(t )  0.5uˆ mE mH  cos   cos(2t   ) , W/m 2
T
1

 S av   S(t )dt  uˆ 0.5mE mH cos(  )  0.5 Re E
T 0
 H 
  , W/m 2

constant in time!

• Sav describes the direction and amount of the average power flux
density carried by the wave

• the complex Poynting vector

S  0.5E    S av  Re S
 H

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 23

 Poynting Vector as a Function of Distance
• consider linearly polarized wave propagating along +z

 ( z )  xˆ E 0 e  z E 0
 ( z )  yˆ e  z     jk  j 
• field vectors:E ,H 
 
0
E 
0
H

1 1 | 
E |2
1
   
• Poynting vector: S( z )  E  H  zˆ 0
e 2 z
 ˆ
z  |  |2 e 2 z
H 0
2 2  2
Prove these formulas!
if the medium is loss-free (α = 0, η is real), Poynting’s vector is
real and independent of z: average power is transferred in the +z
direction (Sav,z > 0 and it is the same everywhere along z)

if the medium is dissipative (η is complex), the Poynting vector

is complex and Sav,z decays along z as exp(−2αz)
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 24
 Total Power Carried by a Plane Wave
E
a
S
H Vcyl
The region inside the cylinder in the figure does not contain sources
and is loss-free. A plane wave propagates through it as shown. What is
the total power Pav carried by the wave through the cylinder’s cross-
section of radius a = 10 mm, if the E-field magnitude is 1 mV/m and
the medium intrinsic impedance η is 120 Ω?
Note: Pav   S av  ds, W
S

The E-field of a plane wave attenuates in a lossy medium so that its

magnitude drops by a factor of 2 with every meter. How quickly
does the power flow attenuate? What is the attenuation constant α in
Np/m and in dB/m?
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 25
 Poynting Vector in a Very Good Conductor: Surface Resistance

 |2 e 2 z where   (1  j )   1  j
S( z )  zˆ 1  | H (see slide 18)
0
2 2 
• since the power entering the conductor is all dissipated, Sav is a
measure of the dissipated power flux density
1   2 2 z 
Re S  Im S  S av ( z )  zˆ | H0 | e  k 
22 2
1  |2 Rs
(see slide 13)
 Sav( z 0)  zˆ Rs | H 0
2
dissipated power per unit area
 1
• surface resistance of a conductor: Rs  Re   ,
2 
• the surface resistance Rs is a measure of the power loss due to the
metallic leads of a transmission line (waveguide)
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 26
 Power Loss in a Very Good Conductor
• consider a wave entering a conductor along z and
assume interface is at z = 0
1  |2 e 2 z  S e 2 z
S av ( z )  zˆ Rs | H 0 av,0
2

Sav,0

• apply Poynting’s theorem to the volume vs

total power inflow = loss power

 Sav  ds  P

S on the surface of a very
good conductor
 P    Sav,0  (  zˆ )ds   Sav( z  L )  zˆ ds   nˆ  H

s0 s
 J s 0
0 like on a PEC
1 1
 P  Rs | H0 | s0  Rs | J s |2 s0 , W
 2
2 2
Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 27
 Power Loss in a Conductor
• generally the Poynting vector may not be exactly along z
P    Sav,0  (  zˆ )ds   Sav( z  L )  zˆ ds
s0 s 0

 
 P  0.5 Re   H
E    zˆ ds 
s0 0  E  e
 H

zˆ  nˆ

0

zˆ ds
s 0

0
2( zˆ uˆ ) L

 P  0.5 Re   H

  zˆ   E
 ds 
0 0
0
s0

• in very good conductors, the wave tends to propagate along the

interface unit normal regardless of the angle of incidence, in which
case the above reduces to
P  0.5 Re  J s  E
 ds    zˆ  nˆ  H
H    J
0 0 0 s
s0

P  0.5 Re  J s  J s ds  0.5 Rs | J s |2 ds
s0 s0

Nikolova 2012 LECTURE 03: PLANE WAVES: REVIEW 28