View
697
Download
4
Category
Preview:
DESCRIPTION
A property of MVG_OMALLOORMODULE IVElectromagnetic Wave PropagationSyllabusElectromagnetic Wave Propagation: Electromagnetic Waves- Wave Propagation in Lossy Dielectrics- Wave Equations from Maxwell’s Equations- Propagation Constant- Intrinsic Impedance of the Medium- Complex Permittivity- Loss Tangent- Plane Waves in Lossless Dielectrics- Plane Waves in Free Space- Uniform Plane Wave- TEM Wave- Plane Waves in Good Conductors- Skin Effect- Poynting Vector Poynting’s Theorem. Reflection of a Plane Wave
Citation preview
MODULE IV
Electromagnetic Wave Propagation
Compiled by MKP for CEC S5 batch September 2008
SyllabusElectromagnetic wave propagation: Electromagnetic waves - wave propagation in lossy dielectrics - wave equations from Maxwell’s equations - propagation constant - intrinsic impedance of the medium - complex permittivity - loss tangent - plane waves in lossless dielectrics - plane waves in free space - uniform plane wave - TEM wave - plane waves in good conductors - skin effect - Poynting vector -Poynting’s theorem. Reflection of a plane wave at normal incidence - standing waves - Reflection of plane waves at oblique incidence - parallel and perpendicular polarization -Brewster angle. Numerical methods in electromagnetics -finite difference - finite element and moment method ( only concept need to be introduced, detailed study not required)
Compiled by MKP for CEC S5 batch September 2008
ReferencesText Books:
1. Mathew N.O. Sadiku, Elements of Electromagnetics, Oxford University Press
2. Jordan and Balmain, Electromagnetic waves and radiating systems,Pearson Education PHI Ltd.References:
1. Kraus Fleisch, Electromagnetics with applications, McGraw Hill2. William.H.Hayt, Engineering Electromagnetics, Tata McGraw Hill3. N.Narayana Rao, Elements of Engineering Electromagnetics, Pearson
Education PHI Ltd. 4. D.Ganesh Rao, Engineering Electromagnetics, Sanguine Technical
Publishers.5. Joseph.A.Edminister, Electromagnetics, Schaum series-McGraw Hill6. K.D. Prasad, Electromagnetic fields and waves, Sathya Prakashan
Compiled by MKP for CEC S5 batch September 2008
Maxwell’s Equations in final forms
VD ρ∇⋅ =
0B∇⋅ =
BEt
∂∇× = −
∂
VS VD dS dVρ⋅ =∫ ∫
0SB dS⋅ =∫
L S
dE dl B dSdt
⋅ = − ⋅∫ ∫
L S
DH dl J dSt
⎛ ⎞∂⋅ = + ⋅⎜ ⎟∂⎝ ⎠
∫ ∫
Differential form Integral form Derived from
' Gauss s Law
Nonexistance of magneticMonopole
' Faraday s Law
' Ampere s Law DH J
t∂
∇× = +∂ modified by continuity eqn
Compiled by MKP for CEC S5 batch September 2008
Maxwell’s Equations for lossless or non conducting medium
0D∇⋅ =
0B∇⋅ =
BEt
∂∇× = −
∂
0SD dS⋅ =∫
0SB dS⋅ =∫
L S
BE dl dSt
∂⋅ = − ⋅
∂∫ ∫
L S
DH dl dSt
∂⋅ = ⋅
∂∫ ∫
Differential form Integral form
DHt
∂∇× =
∂
In lossless medium, current density J and charge density ρ are zero and Maxwell’s equations are simplified as below.
Eqn
ABC
D
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossless or non conducting medium
Differentiating eq (D) with respect to t
The order of differentiation on the LHS may be changed as the curl operation itself is a differentiation
Taking the curl of eq (C)
( ) (1)DHt t t
⎛ ⎞∂ ∂ ∂∇× = − − −⎜ ⎟∂ ∂ ∂⎝ ⎠
2
0 2 (2)H Et t
ε∂ ∂∇× = − − −
∂ ∂
BEt
∂∇×∇× = −∇×
∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossless or non conducting medium
Substituting eq (2) in eq (3)
(3)HEt
μ⎛ ⎞∂
∇×∇× = − ∇× − − −⎜ ⎟∂⎝ ⎠
2
2 EEt
μ ε⎛ ⎞∂
∇×∇× = − ⎜ ⎟∂⎝ ⎠2
2 (4) EEt
με ∂∇×∇× = − − − −∂
( ) ( ) But A B C A C B A B C× × = ⋅ − ⋅
( ) ( ) And so E E E∇×∇× = ∇⋅ ∇ − ∇⋅∇
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossless or non conducting medium
Substituting in eq (4)
Equation (5) represents wave equation in free space in terms of E.
( ) 2 E E E∇×∇× = ∇⋅ ∇ −∇
0 0 0 t Eu D EB ε∇⋅ = ∇⇒ = ∇⋅ =⇒⋅2 And so E E∇×∇× = −∇
22
2 EEt
με ∂−∇ = −∂
22
2 (5) EEt
με ∂∇ = − − −∂
22
2 EEt
με ∂∇ =∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossless or non conducting medium
Differentiating eq (C) with respect to t
The order of differentiation on the LHS may be changed as the curl operation itself is a differentiation
Taking the curl of eq (D)
( ) (6)BEt t t
⎛ ⎞∂ ∂ ∂∇× = − − − −⎜ ⎟∂ ∂ ∂⎝ ⎠
2
2 (7)E Ht t
μ∂ ∂∇× = − − − −
∂ ∂
DHt
∂∇×∇× = ∇×
∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossless or non conducting medium
Substituting eq (7) in eq (8)
(8)EHt
ε⎛ ⎞∂
∇×∇× = ∇× − − −⎜ ⎟∂⎝ ⎠
2
2 HHt
ε μ⎛ ⎞∂
∇×∇× = −⎜ ⎟∂⎝ ⎠2
2 (9) HHt
με ∂∇×∇× = − − − −∂
( ) ( ) But A B C A C B A B C× × = ⋅ − ⋅
( ) ( ) And so H H H∇×∇× = ∇⋅ ∇ − ∇⋅∇
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossless or non conducting medium
Substituting in eq (9)
Equation (10) represents wave equation in lossless medium in terms of H.
( ) 2 H H H∇×∇× = ∇⋅ ∇ −∇
0 0 0 t Hu B HB μ∇⋅ = ∇⇒ = ∇⋅ =⇒⋅2 And so H H∇×∇× = −∇
22
2 HHt
με ∂−∇ = −∂
22
2 (10) HHt
με ∂∇ = − − −∂
22
2 HHt
με ∂∇ =∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossless or non conducting medium
The wave equations for may be obtained by multiplying the wave equation for
D Band E Hε μby and equation for by
22
2 EEt
με ∂∇ =∂
( ) ( )22
2 E
Etε
ε με∂
∇ =∂
22
2 DDt
με ∂∇ =∂
22
2 DDt
με ∂∇ =∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossless or non conducting medium
22
2
HHt
με ∂∇ =∂
( ) ( )22
2
HH
tμ
μ με∂
∇ =∂
22
2
BBt
με ∂∇ =∂
22
2
BBt
με ∂∇ =∂
Similarly,
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossless or non conducting medium
In rectangular coordinate system, the wave equations assumes theform of scalar wave equations in terms of its components
22
2x
xEEt
με ∂∇ =∂
22
2y
y
EE
tμε
∂∇ =
∂2
22
zz
EEt
με ∂∇ =∂
⎫⎪⎪⎬⎪⎪⎭
For E, , For D H B we can obtain similar equations
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossless or non conducting media: sinusoidal time variations
If the electric field intensity is varying harmonically with time
Using these in wave equation for
j ts
E j E et
j Eω ωω∂=
∂=
j tsE E e ω=
22 22
2j t
sE j E et
Eω ωω∂=
∂= −
22
2
EEt
με ∂∇ =∂
E
2 2E Eω με∇ = − 2 2E Eω με∇ = −
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossless or non conducting media: sinusoidal time variations
Similarly we may obtain the following wave equations for the other harmonically varying fields
2 2H Hω με∇ = −
2 2D Dω με∇ = −
2 2B Bω με∇ = −
2 2E Eω με∇ = −2 2D Dω με∇ = −2 2H Hω με∇ = −2 2B Bω με∇ = −
Homogeneous Vector wave equations in complex time harmonic form for free space
Compiled by MKP for CEC S5 batch September 2008
Wave equations for free space
2 20 0E Eω μ ε∇ = −
2 20 0D Dω μ ε∇ = −
2 20 0H Hω μ ε∇ = −
2 20 0B Bω μ ε∇ = −
22
0 0 2
BBt
μ ε ∂∇ =
∂
22
0 0 2 DDt
μ ε ∂∇ =
∂2
20 0 2 HH
tμ ε ∂
∇ =∂
22
0 0 2 EEt
μ ε ∂∇ =
∂
GENERAL SINUSOIDAL TIME VARIATIONS
Compiled by MKP for CEC S5 batch September 2008
Maxwell’s Equations for conducting medium
D ρ∇⋅ =
0B∇⋅ =
BEt
∂∇× = −
∂
VS VD dS dVρ⋅ =∫ ∫
0SB dS⋅ =∫
L S
BE dl dSt
∂⋅ = − ⋅
∂∫ ∫
L S
DH dl J dSt
⎛ ⎞∂⋅ = + ⋅⎜ ⎟∂⎝ ⎠
∫ ∫
Differential form Integral form
DH Jt
∂∇× = +
∂
In a conducting medium with conductivity σ and charge density ρ and Maxwell’s equations are as given below below.
Eqn
ABC
D
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossy or conducting medium
From equation (D)
DH Jt
∂∇× = +
∂
( )H E Et
σ ε∂∇× = +
∂
(1)EH Et
σ ε ∂∇× = + − − −∂
( )2
2
E EHt t t
σ ε∂ ∂ ∂∇× = +
∂ ∂ ∂
2
2 (2)H E Et t t
σ ε∂ ∂ ∂∇× = + − − −
∂ ∂ ∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossy or conducting medium
From equation (C)
Taking curl of this equation
Putting eq (2) in (3)
BEt
∂∇× = −
∂HEt
μ ∂∇× = −∂
(3)HEt
μ⎛ ⎞∂
∇×∇× = − ∇× − − −⎜ ⎟∂⎝ ⎠
2
2 (4)E EEt t
μ σ ε⎛ ⎞∂ ∂
∇×∇× = − + − − −⎜ ⎟∂ ∂⎝ ⎠
( ) 2 (5)But E E E∇×∇× = ∇ ∇⋅ −∇ − − −
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossy or conducting medium
Using eq (5) in (4)
But from equation (A) we have
Putting in equation (6)
( )2
22 (6)E EE E
t tμσ με∂ ∂
∇ = ∇ ∇⋅ + + − − −∂ ∂
( )2
22 (4)E EE E
t tμ σ ε⎛ ⎞∂ ∂
∇ ∇⋅ −∇ = − + − − −⎜ ⎟∂ ∂⎝ ⎠
D ρ∇⋅ = E ρε
∇⋅ =
22
2 (7)E EEt t
ρ μσ μεε
∂ ∂⎛ ⎞∇ = ∇ + + − − −⎜ ⎟ ∂ ∂⎝ ⎠
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossy or conducting medium
There is no charge within a conductor, although it may be there on the surface, the charge density ρ=0. So we can rewrite equation (7) as below.
This is the wave equation for conducting medium in terms of E
22
2
E EEt t
μσ με∂ ∂∇ = +
∂ ∂
22
2
E EEt t
μσ με∂ ∂∇ = +
∂ ∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossy or conducting mediumFrom equation (D)
Taking the curl of this equation
DH Jt
∂∇× = +
∂
( )H E Et
σ ε∂∇× = +
∂
(1)EH Et
σ ε ∂∇× = + − − −∂
( ) (2)EH Et
σ ε⎛ ⎞∂
∇×∇× = ∇× + ∇× − − −⎜ ⎟∂⎝ ⎠
( ) 2 (3)But H H H∇×∇× = ∇ ∇⋅ −∇ − − −
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossy or conducting medium
Using eq (3) in (2)
From equation (C)
Putting in (5) and (6) in (4)
( ) ( )2 (4)EH H Et
σ ε⎛ ⎞∂
∇ ∇⋅ −∇ = ∇× + ∇× − − −⎜ ⎟∂⎝ ⎠
(5)BEt
∂∇× = − − − −
∂
( )2
22 (7)B BH H
t tσ ε⎛ ⎞ ⎛ ⎞∂ ∂
∇ ∇⋅ −∇ = − + − − − −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
( )2
2
BEt t∂ ∂
∇× = −∂ ∂
2
2 (6)E Bt t
∂ ∂∇× = − − − −
∂ ∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossy or conducting medium
This is the wave equation for conducting medium in terms of H
( )2
22 (8)H HH H
t tμσ με∂ ∂
∇ ∇⋅ −∇ = − − − − −∂ ∂
( )2
22 (9)H HH H
t tμσ με∂ ∂
∇ = ∇ ∇⋅ + + − − −∂ ∂
0 0But B H⋅ = =⇒∇ ∇⋅
22
2 H HHt t
μσ με∂ ∂∇ = +
∂ ∂
22
2 H HHt t
μσ με∂ ∂∇ = +
∂ ∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossy or conducting medium
22
2
E EEt t
μσ με∂ ∂∇ = +
∂ ∂
22
2 H HHt t
μσ με∂ ∂∇ = +
∂ ∂
22
2
D DDt t
μσ με∂ ∂∇ = +
∂ ∂
22
2 B BBt t
μσ με∂ ∂∇ = +
∂ ∂
22
0 0 2
EEt
μ ε ∂∇ =
∂
22
0 0 2 HHt
μ ε ∂∇ =
∂
22
0 0 2
DDt
μ ε ∂∇ =
∂
22
0 0 2 BBt
μ ε ∂∇ =
∂
⇒⇒⇒⇒
0 00, , σ ε ε μ μ= = =
Compiled by MKP for CEC S5 batch September 2008
Wave equations for lossy or conducting medium – time harmonic form
2 2E j E Eωμσ ω με∇ = −
2 2 H j H Hωμσ ω με∇ = −
2 2D j D Dωμσ ω με∇ = −
2 2 B j B Bωμσ ω με∇ = −
Compiled by MKP for CEC S5 batch September 2008
Uniform plane waveAn electromagnetic wave originates from a point in free space, spreads out uniformly in all directions, and it forms a spherical wave front.An observer at a large distance from the source is able to observe only a small part of the wave and the wave appears to him as a plane wave. For such a wave the electric field and the magnetic field are perpendicular to each other and to the direction of propagation.A uniform plane wave is one in which lie in a plane and have the same value everywhere in that plane at any fixed instant.
E H
E Hand
Compiled by MKP for CEC S5 batch September 2008
Uniform plane wave
Compiled by MKP for CEC S5 batch September 2008
Uniform plane wave
E
E
E
H
H
H
O
Y
X
Z
Compiled by MKP for CEC S5 batch September 2008
Uniform plane waveFor a uniform plane wave travelling in the z direction, the space variations of are zero over a z=constant plane. This implies the fields have neither x nor y dependence.
A plane wave is transverse in nature, that is are both perpendicular to the direction of propagation.So they are called transverse electromagnetic waves (TEM waves)
E Hand
0x y∂ ∂= =
∂ ∂ E Hand
Compiled by MKP for CEC S5 batch September 2008
Wave equations for uniform plane wave traveling in z direction in free space
Consider a uniform plane wave propagating in z direction.It will have Ex and Ey components but no Ez component. Ez = 0There is no variation of the field components along x and y direction
Wave equations for free space is given by
0E Ex y
∂ ∂= =
∂ ∂
22
0 0 2
EEt
μ ε ∂∇ =
∂2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2ˆ ˆ ˆx x y y z zE a E a E ax y z x y z x y z
⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠
( )2
0 0 2 ˆ ˆ ˆ (1)x x y y z zE a E a E at
μ ε ∂= + + − − −
∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for uniform plane wave traveling in z direction in free space
0; 0yxz
EEBut Ex y
∂∂= = =
∂ ∂
22
2 2 0yx EEx y
∂∂= =
∂ ∂
( )22
2 2
2
0 0 2ˆ ˆ ˆ ˆyxx y x x y y
EE a a E a E atz z
μ ε ∂= +
∂∂∂
+∂ ∂
(1)Putting in eq
Hence2
0
2
2 0 2xx E
tEz
μ ε ∂∂=
∂∂
2
0
2
2 0 2yy E
tEz
μ ε∂∂
=∂∂
Similarly for H we can obtain2
0
2
2 0 2xx H
tHz
μ ε ∂∂=
∂∂
2
0
2
2 0 2yy H
tHz
μ ε∂∂
=∂∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for uniform plane wave in free space traveling in z direction
2
0
2
2 0 2xx E
tEz
μ ε ∂∂=
∂∂
2
0
2
2 0 2yy E
tEz
μ ε∂∂
=∂∂
2
0
2
2 0 2xx H
tHz
μ ε ∂∂=
∂∂
2
0
2
2 0 2yy H
tHz
μ ε∂∂
=∂∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for uniform plane wave traveling in z direction in free space – sinusoidal time variations
2 2
0 02 2x xE E
z tμ ε∂ ∂
=∂ ∂2 2
0 02 2y yE E
z tμ ε
∂ ∂=
∂ ∂2 2
0 02 2x xH H
z tμ ε∂ ∂
=∂ ∂2 2
0 02 2y yH H
z tμ ε
∂ ∂=
∂ ∂
22
0 02x
xE Ez
ω μ ε∂= −
∂2
20 02
yy
EE
zω μ ε
∂= −
∂
22
0 02y
y
HH
zω μ ε
∂= −
∂
22
0 02x
xH Hz
ω μ ε∂= −
∂
⇒
⇒
⇒
⇒
Compiled by MKP for CEC S5 batch September 2008
Wave equations for uniform plane wave traveling in z direction in conducting medium
2 2
2 2x x xE E E
z t tμσ με∂ ∂ ∂
= +∂ ∂ ∂
2 2
2 2y y yE E E
z t tμσ με
∂ ∂ ∂= +
∂ ∂ ∂
2 2
2 2x x xH H H
z t tμσ με∂ ∂ ∂
= +∂ ∂ ∂2 2
2 2y y yH H H
z t tμσ με
∂ ∂ ∂= +
∂ ∂ ∂
Compiled by MKP for CEC S5 batch September 2008
Wave equations for uniform plane wave traveling in z direction in conducting medium – sinusoidal time variations
22
2x
x xE j E Ez
ωμσ ω με∂= −
∂
22
2y
y y
Ej E E
zωμσ ω με
∂= −
∂2
22
xx x
H j H Hz
ωμσ ω με∂= −
∂2
22
yy y
Hj H H
zωμσ ω με
∂= −
∂
Compiled by MKP for CEC S5 batch September 2008
Solution of wave equationsStandard partial difference equation for wave motion frequently encountered in engineering has the form
Comparing with the EM wave equation
22
2 2
1 XXv t
∂∇ =
∂ v Velocity of the wave⇒
22
0 0 2 EEt
μ ε ∂∇ =
∂
( )0 0
22
2 2
1 1 /
EEtμ ε
∂∇ =
∂
v Velocity of the wave⇒ =0 0
1μ ε
70 4 10 /Putting F mμ π −= × 12
0 8.854 10 /and H mε −= ×
Compiled by MKP for CEC S5 batch September 2008
Solution of wave equations
This is the velocity of light and when referred to electromagnetic wave it is denoted by c.
8
0 0
1 3 10 /v m sμ ε
= = ×
8
0 0
1 3 10 /v c m sμ ε
= = = ×
Compiled by MKP for CEC S5 batch September 2008
Solution of wave equation: Uniform plane wave traveling in z direction in free space
In this case the wave equations in E are
The general solution of such a differential equation has the form
2 2
0 02 2x xE E
z tμ ε∂ ∂
=∂ ∂
2 2
0 02 2y yE E
z tμ ε
∂ ∂=
∂ ∂2 2
2 2 2
1x xE Ez c t
∂ ∂=
∂ ∂
2 2
2 2 2
1y yE Ez c t
∂ ∂=
∂ ∂8
0 0
1 3 10 /Where c m sμ ε
= = ×
1 2( ) ( )E f z ct f z ct= − + +
Compiled by MKP for CEC S5 batch September 2008
Uniform plane wave traveling in z direction in free space
Here f1 and f2 are any arbitrary functions of (z-ct) and (z+ct)The functions of (z-ct) and (z+ct) may assume any form as,
The first function represents a wave traveling in positive z direction while the second term represents a wave traveling in negative z direction.
( )Trigonometrical A sin z ctβ⇒ −( ) z ctExponential A e β −⇒
Compiled by MKP for CEC S5 batch September 2008
Solution of wave equation: Uniform plane wave traveling in z direction in lossless or non conducting medium- sinusoidal time variations
In this case the wave equations in E are 2
22
xx
E Ez
ω με∂= −
∂
22
2y
y
EE
zω με
∂= −
∂2
22 0y
y
EE
zω με
∂+ =
∂2 2 0y yD E Eω με+ =
22
2 Putting Dz∂
≡∂
( )2 2 0yD Eω με+ =
Characteristic equation is ( )2 2 0m ω με+ =
Compiled by MKP for CEC S5 batch September 2008
Uniform plane wave traveling in z direction in lossless or non conducting medium- sinusoidal time variations
( )2 2 0m ω με+ =2 2m ω με= −
m jω με= ±jβ= ± Where β ω με=
Hence the solution is
1 2j z j z
y m mE E e E eβ β−= +
2 and are arbitrary constants mm1E E
Compiled by MKP for CEC S5 batch September 2008
Uniform plane wave traveling in z direction in lossless or non conducting medium- sinusoidal time variations
The above equation is the phasor form of the electric field. To find the time domain form we multiply the phasor form by and then take the real part of it.
1 2j z j z
y m mE E e E eβ β−= +
j te ω
( ){ }1 2( , ) j z j zy m
tm
jE z t Re E e E e eβ β ω−= +
{ }1 2j z j j z
mt j t
mRe eE e Ee eωβ β ω−= +
{ }( ) ( )1 2
j t z j t zm mRe E e E eω β ω β− += +
1 2( ) ( )m mE cos t z E cos t zω β ω β= − + +
Compiled by MKP for CEC S5 batch September 2008
Uniform plane wave traveling in z direction in lossless or non conducting medium- sinusoidal time variations
1 2( ) ( )y m mE E cos t z E cos t zω β ω β= − + + Forward traveling wave
Backward traveling wave
β ω με=
1 cωβ με
= = c ωβ
=
Compiled by MKP for CEC S5 batch September 2008
Uniform plane wave traveling in z direction in lossless or non conducting medium- sinusoidal time variations
For the phase of the forward traveling wave to remain constant,
As t increases z must also increase. This means that the wave travels in the z direction with a constant phase.Similarly for the backward traveling component
As t increases z must decrease in order to keep the phase constant. This means that the wave travels in the –z direction with a constant phase.
1 2( ) ( )y m mE E cos t z E cos t zω β ω β= − + +
= t z A constantω β−
t z A constantω β+ =
1 2( ) ( ), m myE sin tO z sin t zr E Eω β ω β= − + +
Compiled by MKP for CEC S5 batch September 2008
Wave motion
zλλ−
t
TT−
Compiled by MKP for CEC S5 batch September 2008
Phase velocityDue to the variation of E with both time and space we may plot E as a function of t by keeping z constant or vice versa.The plots of E(t,z=constant) and E(z,t=constant) are shown in figure.The wave takes distance λ to repeats itself, and hence λ is called the wave length.Also the wave takes time T to repeat itself, and hence T is called the period of the wave.Since it takes time T for the wave to travel a distance λ at the speed v, λ=vTSince T=1/f we may write v f λ=
2 fω π=1 2Tf
πω
= =vωβ = 2, Substituting πβ
λ=
Compiled by MKP for CEC S5 batch September 2008
Phase velocity
For every wavelength of distance traveled, the wave undergoes a phase change of 2π radians.Now consider the forward waveTo prove that this wave travels with a velocity v in the z direction, consider a fixed point P on the wave.Sketch the above wave equation at times t=T/4 and t=T/2 as in figure.As the wave advances with time, point P moves along the z direction.Point P is a point of constant phase,
2 πβλ
=
1 ( )y mE E sin t zω β= −
= t z A constantω β−
Compiled by MKP for CEC S5 batch September 2008
Phase velocity
This proves that the wave is traveling in the z direction with velocity v
= t z A constantω β−
dz vdt
ωβ
= =
Compiled by MKP for CEC S5 batch September 2008
Wave motion
E
zλ 2λ
E
tT 2T
Compiled by MKP for CEC S5 batch September 2008
Wave motion
E
zλ 2λ
E
tT 2T
Compiled by MKP for CEC S5 batch September 2008
Phase velocityE
E
E
zβ
zβ
zβ
0t =
/ 4t T=
/ 2t T=
P
P
P
Compiled by MKP for CEC S5 batch September 2008
Relationship between E and H and Intrinsic impedance in perfect dielectric
Maxwell’s equation derived from Faraday’s law is BEt
∂∇× = −
∂
. ., Hi e Et
μ ∂∇× = −∂
ˆ ˆ ˆ
ˆ ˆ ˆ
x y z
x x y y z z
x y z
a a a
H a H a H ax y z t
E E E
μ∂ ∂ ∂ ∂ ⎡ ⎤= − + +⎣ ⎦∂ ∂ ∂ ∂
Compiled by MKP for CEC S5 batch September 2008
Relationship between E and H and Intrinsic impedance in perfect dielectric
For a uniform plane wave traveling in the z direction,
0x y∂ ∂= =
∂ ∂0zH =
ˆ ˆ ˆ
ˆ ˆ0 0 0
0
x y z
x x y y
x y
a a a
H a H az t
E E
μ∂ ∂ ⎡ ⎤= − + +⎣ ⎦∂ ∂
ˆ ˆ ˆ ˆ (1)yxy x x y x y
HHE a E a a az t t
μ μ∂∂ ∂⎡ ⎤− − = − − − −⎣ ⎦∂ ∂ ∂
Compiled by MKP for CEC S5 batch September 2008
Relationship between E and H and Intrinsic impedance in perfect dielectric
From equation (1) we have
From Maxwell’s curl equation for H
By proceeding in a similar way as we did for the first curl equation we get
( )y xE H az t
μ∂ ∂
= − − −∂ ∂
( )yx HE bz t
μ∂∂
= − − − −∂ ∂
DHt
∂∇× = −
∂. ., Ei e H
tε ∂∇× =∂
( )y xH E cz t
ε∂ ∂
= − − − −∂ ∂
( )yx EH dz t
ε∂∂
= − − −∂ ∂
Compiled by MKP for CEC S5 batch September 2008
Relationship between E and H and Intrinsic impedance in perfect dielectric
Let the equation of the plane wave be
Differentiating the above equation wrt t
From equation (d)
11( ) yE f z ct where cμε
= − =
1 '( ) ( )yEf z ct z ct
t t∂ ∂
= − −∂ ∂
1 '( ) c f z ct= − −
yx EHz t
ε∂∂
=∂ ∂
1 '( ) x c f z ctHz
ε− −∂
=∂
Compiled by MKP for CEC S5 batch September 2008
Relationship between E and H and Intrinsic impedance in perfect dielectric
Integrating both sides,
1 ( ) xH c f z ct kε= − − + yc E kε= − +1
yEεμε
= −yEμ
ε= −
y
x
EH
με
= −
The constant k is taken to be zero as it forms only a static part of the
the solution which is not importanthere
y
x
EH
με
= − y xE Hμ
ε= −
x yH Eεμ
= −
Compiled by MKP for CEC S5 batch September 2008
Relationship between E and H and Intrinsic impedance in perfect dielectric
Similarly let the equation of the plane wave be
Differentiating the above equation wrt t
From equation (c)
11( ) xE f z ct where cμε
= − =
1 '( ) ( )xE f z ct z ctt t
∂ ∂= − −
∂ ∂ 1 '( ) c f z ct= − −
y xH Ez t
ε∂ ∂
= −∂ ∂
1 '( ) y c f ctHz
zε −∂
=∂
Compiled by MKP for CEC S5 batch September 2008
Relationship between E and H and Intrinsic impedance in perfect dielectric
Integrating both sides,
1 ( ) yH c f z ct kε= − + xc E kε= +1
xEεμε
=xEμ
ε=
x
y
EH
με
=
The constant k is taken to be zero as it forms only a static part of the
the solution which is not importanthere
x
y
EH
με
= x yE Hμ
ε=
y xH Eεμ
=
Compiled by MKP for CEC S5 batch September 2008
Relationship between E and H and Intrinsic impedance in perfect dielectric
2 2 x yE E E= +
2 2y xE H Hμ μ
ε ε= +
2 2x yE H Hμ
ε= + Hμ
ε=
x yE Hμε
=
y xH Eεμ
=
y xE Hμε
= −
x yH Eεμ
= − E Total electric field⇒
H Total magnetic field⇒
EH
μηε
= = EH
μηε
= =
Compiled by MKP for CEC S5 batch September 2008
Relationship between E and H and Intrinsic impedance in perfect dielectric
This equation is similar to ohms law R=V/I by the analogy
So it is called intrinsic impedance or characteristic impedance of the medium.For free space, intrinsic impedance is For any other medium
EH
μηε
= =
V E⇔ H I⇔
00
0
=377μηε
= Ω
0
0
r
r
μ μηε ε
= 377 r
r
μηε
= Ω
Compiled by MKP for CEC S5 batch September 2008
Solution of wave equation: Uniform plane wave traveling in z direction in lossy or conducting medium- sinusoidal time variations
In this case the wave equations in E are
The wave equation for Ey may be written as
22
2x
x xE j E Ez
ωμσ ω με∂= −
∂
22
2y
y y
Ej E E
zωμσ ω με
∂= −
∂
( )2
2y
y
Ej j E
zωμ σ ωε
∂= +
∂2
22
yy
EE
zγ
∂=
∂ ( ) jWhere jγ ωμ σ ωε= +
Compiled by MKP for CEC S5 batch September 2008
Uniform plane wave traveling in z direction in lossy or conducting medium- sinusoidal time variations
Here is defined as the propagation constant.
It is a complex number and can be represented as
It has a real part α called attenuation constant and an imaginary part β called phase shift constant.
( ) j jγ ωμ σ ωε= +
jγ α β= +
22
2y
y
EE
zγ
∂=
∂2 2 0y yD E Eγ− =
( )2 2 0yD Eγ− =
Compiled by MKP for CEC S5 batch September 2008
Uniform plane wave traveling in z direction in lossy or conducting medium- sinusoidal time variations
Characteristic equation is
Hence the solution of the partial differential equation is
2 2 0m γ− = 2 2m γ=m γ= ±
1 2z z
y m mE E e E eγ γ− += +
( ) ( )1 2
j z j zm mE e E eα β α β− + + += +
1 2z z z z
m mE e e E e eα β α β− −= +
1 2z z z z
y m mE E e e E e eα β α β− −= +
Compiled by MKP for CEC S5 batch September 2008
Uniform plane wave traveling in z direction in lossy or conducting medium- sinusoidal time variations
Similarly by taking the wave equation in terms of Hx and then proceeding we get
The terms are the amplitudes of the forward-traveling and backward-traveling waves.The terms cause the amplitudes of the forward and backward waves to decay as they propagate through the medium. Hence α is termed as the attenuation constant.
1 2z z z z
x m mH H e e H e eα β α β− −= +
ze α±
1 2 z zm mE e and E eα α−
Compiled by MKP for CEC S5 batch September 2008
Uniform plane wave traveling in z direction in lossy or conducting medium- sinusoidal time variations
The time domain fields are obtained by multiplying Ey by ejωt and taking the real part of the result.
( )1 2( , ) z z z z j ty m mE x t Re E e e E e e eα β α β ω− −= +
1 2cos( ) cos( )z zm mE e t z E e t zα αω β ω β−= − + +
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in lossy or conducting medium:Expression for α and β
We know that
Equating (1) and (2)
Equating the real and imaginary parts of (3)
( )j jγ ωμ σ ωε= +
2 2 (1)jγ ωμσ ω με= − − − −
jα β= +
( )22 jγ α β= + 2 2 2 (2)jα β αβ= − + − − −
2 (5)ωμσ αβ= − − −
2 22 = (4)α βω με −− − − −
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in lossy or conducting medium:Expression for α and β
From eq(5) we have
Putting eq (7) in (4)
(6)2ωμσαβ
= − − − (7)2ωμσβα
= − − −
22 2 = (8)
2ωμσω με αα
⎛ ⎞− − − − −⎜ ⎟⎝ ⎠
( )22 2
2 =4ωμσ
ω με αα
− −
2 2 4 2 2 2 4 =4ω με α α ω μ σ− −
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in lossy or conducting medium:Expression for α and β
( )22 2 4 4 =4ω με α α ωμσ− −
( )24 2 24 4 0α ω με α ωμσ+ − =2
4 2 2 02
ωμσα ω με α ⎛ ⎞+ − =⎜ ⎟⎝ ⎠
2 P t pu α =2
2 2 0 (9)2
p p ωμσω με ⎛ ⎞+ − = − − −⎜ ⎟⎝ ⎠
(9 ) Solving the quadratic equation for p we get
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in lossy or conducting medium:Expression for α and β
Similarly by putting eq (6) in (4) we get another quadratic equation in β2, solving which we get
⎫⎪⎬⎪⎭
Taking the positive root only
22
1 1 (11)2
ω με σαωε
⎧ ⎫⎪ ⎪⎛ ⎞= ± + − − − −⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭
22
1 1 2
p ω με σωε⎛ ⎞= + −⎜ ⎟⎝ ⎠
22
1 1 (12)2
ω με σβωε
⎧ ⎫⎪ ⎪⎛ ⎞= ± + + − − −⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in lossy or conducting medium:Expression for α and β
22
1 1 2
Attenuation Constant ω με σαωε
⎧ ⎫⎪ ⎪⎛ ⎞= ± + −⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭
22
1 1 2
Phase shift Constant ω με σβωε
⎧ ⎫⎪ ⎪⎛ ⎞= ± + +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in lossy or conducting medium:Expression for vp and λ
Phase velocity pv ωβ
=
22
1 1 2
pv ω
ω με σωε
=⎧ ⎫⎪ ⎪⎛ ⎞+ +⎨ ⎬⎜ ⎟
⎝ ⎠⎪ ⎪⎩ ⎭
2
1
1 1 2
pvμε σ
ω ε
=⎧ ⎫⎪ ⎪⎛ ⎞+ +⎨ ⎬⎜ ⎟
⎝ ⎠⎪ ⎪⎩ ⎭
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in lossy or conducting medium:Expression for vp and λ
Wave length2πλβ
=
22
2
1 12
πλω με σ
ωε
=⎧ ⎫⎪ ⎪⎛ ⎞+ +⎨ ⎬⎜ ⎟
⎝ ⎠⎪ ⎪⎩ ⎭
Compiled by MKP for CEC S5 batch September 2008
Intrinsic impedance of lossy dielectric
The phasor equation for a wave travelling in z direction in a loosydielectric is
Since the fields vary sinusoidally with time
Putting the above values in
1z z
y mE E e eα β− −= 1z
mE e γ−=
1y z
m
EE e
zγγ −∂
= −∂
yEγ= −
( )y xE H az t
μ∂ ∂
= − − −∂ ∂
jt
ω∂=
∂
y xE j Hγ ωμ− = −
y
x
E jH
ωμηγ
= =
Compiled by MKP for CEC S5 batch September 2008
Intrinsic impedance of lossy dielectric
jωμηγ
=
( ) But j jγ ωμ σ ωε= +
( )j
j jωμη
ωμ σ ωε=
+
jj
ωμσ ωε
=+
jj
ωμησ ωε
=+
Compiled by MKP for CEC S5 batch September 2008
Conductors and dielectricsElectromagnetic materials are roughly classified as conductors and dielectrics.Maxwell’s curl equation for sinusoidally varying quantities is given by
H E j Eσ ωε∇× = +
C DH J J∇× = +
C Conduction cJ urrent density⇒
DJ Displacement current density⇒
C
D
J E
J j E
σ
ωσωεε
= =
C
D
determines the natJ
The rat ure of materiioJ
alσωε
=
Compiled by MKP for CEC S5 batch September 2008
Conductors and dielectrics
. 1 1 D CJ J Good dielectrics σ ωε σωε
⇒ ⇒ ⇒
2. 1 D C Semi conductor J Jσ ωε σωε
≅ ≅⇒ ⇒ ⇒≅
. 3 1 D C Good conductor J Jσ ωε σωε
⇒ ⇒ ⇒
When the displacement current is much greater than conduction currentthe medium behaves like a dielectric. σIf = 0 the medium is a perfect or lossless dielectric.σ ≠If 0 the medium is a lossy dielectric.
Compiled by MKP for CEC S5 batch September 2008
Conductors and dielectricsWhen the displacement current is much smaller than conduction currentthe medium behaves like a conductor. σ ωεIf the medium is a good conductor.
When the displacement current is comparable to conduction currentthe medium behaves like a semi conductor.
σ
ωεThe term is called loss tangent
DJ j Eωε=
CJ Eσ=
C DJ J J= +
( )j Eσ ωε= +θ Lossta tangenn tσ
ωεθ = ⇒
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good dielectrics
This can be considered as a special case of the wave motion in lossy or conducting medium.All the equations derived for this case is applicable in the case of good dielectrics with appropriate modification of parameters.For a good dielectric σ/ωε<<1 so that we can approximate the value of α and β as follows
( )j jγ ωμ σ ωε= +
( )2 j jγ ωμ σ ωε= +2 1j j
jσγ ωμ ωεωε
⎛ ⎞= ⋅ +⎜ ⎟
⎝ ⎠2 2 2 1j j σγ ω με
ωε⎛ ⎞= −⎜ ⎟⎝ ⎠
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good dielectrics
1j j σγ ω μεωε
= −
Expanding the second radical using Binomial theorem
( ) 2 3( 1) ( 1)( 2)1 12 3
n n n n n nx nx x x− − −− = − + − + ⋅⋅ ⋅ ⋅ ⋅
1/2 21 1 1
1 2 21 12 2
j jj σ σ σωε ωε ωε
⎛ ⎞−⎜ ⎟⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎝ ⎠− = − + + ⋅⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
2112 8jσ σωε ωε
⎛ ⎞= − + + ⋅⋅⋅⎜ ⎟⎝ ⎠
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good dielectrics211
2 8jj σγ ω μωε ω
ε σε
⎡ ⎤⎛ ⎞− + + ⋅⋅ ⋅⎢ ⎥⎜ ⎟⎝ ⎠⎢
=⎥⎣ ⎦
jα β= +
Equating the real and imaginary parts
2jj σα ω μεωε
= ⋅−2σω μεωε
= ⋅2σ μ
ε=
2σ μα
ε=211
8j jβ ω
εμε σ
ω⎡ ⎤⎛ ⎞+⎢ ⎥⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦=
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good dielectrics211
8σω
β ω μεε
⎡ ⎤⎛ ⎞+⎢ ⎥⎜ ⎟⎝ ⎠⎢⎣ ⎦
=⎥
β ω με= 1Since σωε
β ω με=
, , 2
For a perfect dielectric σ μα β ω μεε
= =
2 = 118
pVelocity of propagation v ωωσω με
βωε
⎡ ⎤⎛ ⎞+⎢ ⎥⎜ ⎟⎝ ⎠⎢⎣ ⎦
=
⎥
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good dielectrics
2
1= 118
pVelocity of propagation vσμεωε
⎡ ⎤⎛ ⎞+⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦
1 2z z z z
y m mE E e e E e eα β α β− −= +
1 2( , ) cos( ) cos( )z zy m mE z t E e t z E e t zα αω β ω β−= − + +
:And the wave equations are
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good conductors
This can be considered as a special case of the wave motion in lossy or conducting medium.All the equations derived for this case is applicable in the case of good dielectrics with appropriate modification of parameters.For a good conductor σ/ωε>>1 so that we can approximate the value of α and β as follows
( )j jγ ωμ σ ωε= +
1 jj ωεωμσσ
⎛ ⎞= +⎜ ⎟⎝ ⎠
1 Since orσ σ ωεωε
jωμσ=
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good conductors
jγ ωμσ=
1 2 12
jγ ωμσ+ −⎛ ⎞= ⎜ ⎟⎝ ⎠
( )212
j ωμσ= +
( )12
j ωμσ= + jα β= +
Equating the real and imaginary parts
2ωμσα β= =
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good conductorsThe wave equation in this case is of the form
When we consider the forward traveling wave only
A high frequency uniform plane wave suffers attenuation as it passes through a lossy medium.Its amplitude gets multiplied by the factor, e -αx where α is the attenuation constant.
1 2 orz z z zy m mE E e e E e eα β α β− −= +
1 2( , ) cos( ) cos( )z zy m mE z t E e t z E e t zα αω β ω β−= − + +
1( , ) cos( )zy mE z t E e t zα ω β−= −
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good conductors
ze α−
z
1mE
yE
y
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good conductorsSkin depth is defined as the distance the wave must travel to have its amplitude reduced by a factor of e-1. The exponential multiplying factor is unity at z=0 and decreases to 1/e when z=1/αSo skin depth is
Thus we see that the skin depth decreases with an increase in frequency.The intrinsic impedance of a medium may be expressed in terms ofskin depth.
1 = Skin depth δα
2=δωμσ
1=fπ μσ
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good conductors
= jj
ωμησ ωε+
for a conductoj rωμησ
≅
45 ωμσ
= ∠1But, =
2ωμσ
δ
2
2 45 2σωμσ
= ∠ 22
2 45 σδ
= ∠
2 45 δσ
= ∠ ( )1 1 jδσ
= + Ω ( )1 1 jηδσ
= + Ω
Compiled by MKP for CEC S5 batch September 2008
Wave propagation in good conductorsThe phenomenon by which field intensity in a conductor rapidly decreases with increase in frequency is called skin effect.At high frequencies the fields and the associated currents are confined to a very thin layer of the conductor surface.
1 = Skin depth δα
Compiled by MKP for CEC S5 batch September 2008
Poynting theoremThe vector product at any point is a measure of the rate of energy flow per unit area at that point ; the direction of energy flow is perpendicular to E and H in the direction of the vector
P E H= ×
E H×
2/P E H W atts m= ×
Compiled by MKP for CEC S5 batch September 2008
Poynting theorem- ProofMaxwell’s first curl equation states that
This equation can be rewritten as
Pre dotting the equation (2) with we get,
We have the vector identity
DH Jt
∂∇ × = +
∂
D EJ H Ht t
ε∂ ∂= ∇ × − = ∇ × −
∂ ∂Ei
( ) (1)D E EE J E H E E Ht t
ε∂ ∂= ∇ × − = ∇ × − −−−
∂ ∂ii i i i
( ) ( ) ( ) (2)E H H E E H⋅ ∇ × = ⋅ ∇ × − ∇ ⋅ × − − −
( ) ( ) ( )E H H E E H∇⋅ × = ⋅ ∇ × − ⋅ ∇ ×
Compiled by MKP for CEC S5 batch September 2008
Poynting theorem - ProofPutting eq (2) in (1)
( ) ( ) EE J H E E H Et
ε ∂= ⋅ ∇ × − ∇⋅ × −
∂i i
( )B EE J H E H Et t
ε⎛ ⎞∂ ∂
= ⋅ − − ∇ ⋅ × −⎜ ⎟∂ ∂⎝ ⎠i i
( )H EH E H Et t
μ ε∂ ∂= ⋅ − − ∇ ⋅ × −
∂ ∂i
( ) (3)H EH E E Ht t
μ ε∂ ∂= − ⋅ − −∇ ⋅ × − − −
∂ ∂i
2 21 1 (4)2 2
H EBut H H and E Et t t t
∂ ∂ ∂ ∂⋅ = ⋅ = − − −∂ ∂ ∂ ∂
Compiled by MKP for CEC S5 batch September 2008
Poynting theorem - ProofSubstituting equation (4) in equation (3), we get
Integrating both sides of equation (5) over a volume V, we get
( )2 21 12 2
E J H E E Ht t
μ ε∂ ∂= − − − ∇⋅ ×
∂ ∂i
( )2 2
2 2H E E H
t tμ ε⎛ ⎞ ⎛ ⎞∂ ∂
= − − − ∇⋅ ×⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
( )2 2
(5)2 2H E E H
tμ ε⎛ ⎞∂
= − + − ∇⋅ × − − −⎜ ⎟∂ ⎝ ⎠
( ) ( )2 2
2 2v v
H EE J d d E H dt ν
μ εν ν ν⎛ ⎞∂
= − + − ∇⋅ ×⎜ ⎟∂ ⎝ ⎠∫ ∫ ∫i
Compiled by MKP for CEC S5 batch September 2008
Poynting theorem - ProofWe apply divergence theorem to the last term on the LHS, convert the volume integral into a surface integral and get
( ) ( )2 2
2 2v S
H EE J d d E H dSt ν
μ εν ν⎛ ⎞∂
= − + − ×⎜ ⎟∂ ⎝ ⎠∫ ∫ ∫i
( ) ( )2 2
2 2S v
H EE H dS E J d dt ν
μ εν ν⎛ ⎞∂
− × = + +⎜ ⎟∂ ⎝ ⎠∫ ∫ ∫i
( ) ( ) 2 2S v
B H E DE H dS E J d dt ν
ν ν⎛ ⎞∂ ⋅ ⋅
− × = + +⎜ ⎟∂ ⎝ ⎠∫ ∫ ∫i
Compiled by MKP for CEC S5 batch September 2008
Poynting theorem
Term 1 Term 2 Term 3
( ) ( )2 2
2 2S vEE H E d
tdH dS J
ν
μ εν ν⎛ ⎞∂
+⎜ ⎟× +⎝ ⎠
− =∂∫ ∫∫ i
Ingoing power flux over the surface S
Total dissipated power within the volume V at any instant due to ohmiclosses
Rate of decrease due to total electromagnetic energy stored within the volume V
Magnetic energy stored within the volume V
Electrostatic energy stored within the volume V
Compiled by MKP for CEC S5 batch September 2008
Poynting theorem - Interpretation
2
2E d
ν
ε ν∫
Stored electrical energy
Stored magnetic energy
Ohmic losses
Power out
Power in
( )S
E H dS×∫
( )v
E J dν∫ i
2
2H d
ν
μ ν∫
Compiled by MKP for CEC S5 batch September 2008
Poynting theorem - Interpretation
The first term represents the rate of flow of energy inward through the surface of the volume or ingoing power flux over the surface S
The second term represents total dissipated power within the volume V at any instant. For a conductor of cross sectional area A carrying a current I and having a voltage drop E per unit length, the power loss is EI watts per unit length.The power dissipated per unit volume is
Total power dissipated in a volume is
( )S
E H dS− ×∫
( )v
E J dν∫ i
( )v
E J dν∫ i
.EI EJ watts per unit volumeA
=
Compiled by MKP for CEC S5 batch September 2008
Poynting theorem - Interpretation
The first part of the third term is the stored electric energy per unit volume ( electrostatic energy density ) of the electric field.
The second part of the third term is the stored magneticenergy stored per unit volume ( magnetic energy density ) of the magnetic fieldThe volume integral of the sum represents total electromagnetic energy stored within the volume V
212
Eε
212
Hμ
Compiled by MKP for CEC S5 batch September 2008
Poynting theorem - InterpretationThe net inward power flux supplied by the field over the surface S must equal the time rate of increase of electromagnetic energy inside the volume V plus the total losses in volume V, assuming the volume contains no generators. If represent the total ingoing instantaneous power flux, then represents the total power flowing out of the volume.
( )S
E H dS− ×∫( )
SE H dS×∫
Compiled by MKP for CEC S5 batch September 2008
Poynting vectorFrom Poynting theorem it can be seen that the vector product of electric field intensity and magnetic field intensity is another vector which is denoted by
The vector P is called Poynting vector.It measures the rate of flow of energy, and its direction is thedirection of power flow and it is perpendicular to the plane containing vectors.P is the energy per unit area passing per unit time through the surface of the volume v in watts/meter square.For plane waves the direction of energy flow is the direction ofpropagation.The Poynting vector offers a useful coordinate free way of specifying the direction of propagation of plane waves.
2 /P E H Watts m= ×
E H and
E H
Recommended