Chapter 1

Vector Calculus

1.1 Conversion Between Different Coordinate Systems

Aρ

Aφ

Az

=

cosφ sinφ 0− sinφ cosφ 0

0 0 1

Ax

Ay

Az

(1.1)

Ax

Ay

Az

=

cosφ − sinφ 0sinφ cosφ 00 0 1

Aρ

Aφ

Az

(1.2)

Ar

Aθ

Aφ

=

sin θ 0 cos θcos θ 0 − sin θ0 1 0

Aρ

Aφ

Az

(1.3)

Aρ

Aφ

Az

=

sin θ cos θ 00 0 1

cos θ − sin θ 0

Ar

Aθ

Aφ

(1.4)

Ar

Aθ

Aφ

=

sin θ cosφ sin θ sinφ cos θcos θ cosφ cos θ sinφ − sin θ− sinφ cosφ 0

Ax

Ay

Az

(1.5)

Ax

Ay

Az

=

sin θ cosφ cos θ cosφ − sinφsin θ sinφ cos θ sinφ cosφ

cos θ − sin θ 0

Ar

Aθ

Aφ

(1.6)

1.2 Length, Area, Volume

dl = uxdx+ uydy + uzdz

= uρdρ+ uφρ dφ+ uzdz

= urdr + uθr dθ + uφr sin θ dφ (1.7)

1

2 CHAPTER 1. VECTOR CALCULUS

ds = uxdy dz + uydx dz + uzdx dy

= uρρ dφ dz + uφdρ dz + uzρ dρ dφ

= urr2 sin θ dθ dφ+ uθr sin θ dr dφ + uφr dr dθ (1.8)

dv = dx dy dz

= ρdρ dφ dz

= r2 sin θ dr dθ dφ (1.9)

1.3 Divergence, Curl, Gradient, Laplacian

∇.A =∂Ax

∂x+∂Ay

∂y+∂Az

∂z

=1

ρ

∂ (ρAρ)

∂ρ+

1

ρ

∂Aφ

∂φ+∂Az

∂z

=1

r2∂(

r2Ar

)

∂r+

1

r sin θ

∂ (sin θAθ)

∂θ+

1

r sin θ

∂Aφ

∂φ(1.10)

∇×A = ux

[

∂Az

∂y− ∂Ay

∂z

]

+ uy

[

∂Ax

∂z− ∂Az

∂x

]

+ uz

[

∂Ay

∂x− ∂Ax

∂y

]

= uρ

[

1

ρ

∂Az

∂φ− ∂Aφ

∂z

]

+ uφ

[

∂Aρ

∂z− ∂Az

∂ρ

]

+ uz

[

1

ρ

∂ (ρAφ)

∂ρ− 1

ρ

∂Aρ

∂φ

]

= ur

[

∂ (sin θAφ)

∂θ− ∂Aθ

∂φ

]

1

r sin θ+ uθ

[

1

sin θ

∂Ar

∂φ− ∂ (rAφ)

∂r

]

1

r

+uφ

[

∂ (rAθ)

∂r− ∂Ar

∂θ

]

1

r(1.11)

∇w = ux

∂w

∂x+ uy

∂w

∂y+ uz

∂w

∂z

= uρ

∂w

∂ρ+ uφ

1

ρ

∂w

∂φ+ uz

∂w

∂z

= ur

∂w

∂r+ uθ

∂w

∂θ

1

r+ uφ

∂w

∂φ

1

r sin θ(1.12)

∇2w =∂2w

∂x2+∂2w

∂y2+∂2w

∂z2

=1

ρ

∂

∂ρ

[

ρ∂w

∂ρ

]

+1

ρ2∂2w

∂φ2+∂2w

∂z2

=1

r2∂

∂r

[

r2∂w

∂r

]

+1

r2 sin θ

∂

∂θ

[

sin θ∂w

∂θ

]

+1

r2 sin θ

∂2w

∂φ2(1.13)

1.4. VECTORIAL IDENTITIES 3

1.4 Vectorial Identities

1. A. (B×C) = B. (C×A) = C. (A×B) = det

Ax Ay Az

Bx By Bz

Cx Cy Cz

2. A× (B×C) = (A.C)B− (A.B)C

3. (A×B) . (C×D) = (A.C) (B.D)− (B.C) (A.D)

4. (A×B)× (C×D) = (A.B×D)C− (A.B×C)D

1.5 Differential Identities (DCG Rule & DG=L)

1. ∇. (wA) = w∇.A +A.∇w

2. ∇× (wA) = w∇×A−A×∇w

3. ∇. (A×B) = B.∇×A−A.∇×B

4. ∇× (A×B) = [A∇.B−B∇.A]− [(A.∇)B− (B.∇)A]

5. ∇×∇×A = ∇ (∇.A)−∇2A

6. ∇ (A.B) = (A.∇)B+ (B.∇)A+A× (∇×B) +B× (∇×A)

1.6 Integral Identities

1.∫

∇.Adv =∮

A.ds

2.∫

∇×Adv = −∮

A× ds

3.∫

∇w dv =∮

w ds

4.∫

∇×A. ds =∮

A. dl

5.∫

n×∇w ds =∮

w dl

6.∫ [

∇u.∇w + u∇2w]

dv =∮

(u∇w) .ds .... Green’s1st identity (from the di-vergence theorm)

7.∫ [

u∇2w − w∇2u]

dv =∮

(u∇w − w∇u) .ds .... Green’s2nd identity (fromthe above theorm) ... This identity can explain thereciprocity theorm.

8.∫

[∇×A.∇×B−A.∇×∇×B] dv =∫

∇. [A×∇×B] dv =∮

[A×∇×B] .ds... (from the divergence theorm)

9.∫

[B.∇×∇×A−A.∇×∇×B] dv =∮

[A×∇×B−B×∇×A] .ds ...(from the above theorm)

4 CHAPTER 1. VECTOR CALCULUS

1.7 Scalar and Vector Fields

For any scalar functionΦ(x, y, z),

1.∫ p2

p1

∇Φ.dl =Φ(p2)− Φ(p1) (1.14)

So,∮

C

∇Φ.dl =0 (1.15)

ThusΦ is called aconservative potential field.

2. If ∇2Φ = 0 (many fields obey this rule), then from Green’s First identity,∫

(∇Φ)2dv =

∮

Φ∂Φ

∂nds (1.16)

If Φ is constant on surface, them∇Φ = 0 also (from integral identities 1, 7).ThusΦ is constant through out volume also.

For any vector functionP(x, y, z),

1. If ∇.P = 0 =⇒ P is calledsolenoidalor rotational.

So,Ps = ∇× A = ∇× A′, whereA is a vector of the formA = A′ +∇Φ.

This is called agauge transformation. So,Φ is basically arbitrary.

2. If ∇× P = 0 =⇒ P is calledlamellar or irrotational. So,

Pl = −∇ψ

3. So, basicallyP can be divided into lamellar and solenoidal parts as shown below(from Helmholtz’s theorm).

P = Ps + Pl = ∇×(

A′ +∇Φ)

−∇ψ

Ps andPl can be thought of as sources, for example, charge density andcurrentdensity.

∇× Ps = J

∇.Pl = ρ

Chapter 2

Electro-Magnetic Basics

In EM theory,S= E × H∗

In Circuit theory,P = V I∗

Z =V I∗

II∗=

P

| I |2

Y =(V I∗)

∗

V V ∗=

P ∗

| V |2

Reaction Concept,

Zi,j = −〈j, i〉IiIj

where〈j, i〉 =∫ (

Ej .Ji − Hj .M i)

dv.

5

6 CHAPTER 2. ELECTRO-MAGNETIC BASICS

Chapter 3

Wave Equations and VectorPotentials

3.1 The Maxwell’s Equations

∇× E = −Jm − ∂B∂t

∇× H = Je +∂D∂t

∇.D = ρe

∇.B = ρm

and equations of continuity is given by

∇.Je = −∂ρe∂t

∇.Jm = −∂ρm∂t

3.2 E and H Wave Equations

∇2E+ k2E = jωµJe −

∇∇.Je

jωǫ+ {∇ × Jm}

∇2H+ k2H = jωǫJm − ∇∇.Jm

jωµ− {∇× Je}

7

8 CHAPTER 3. WAVE EQUATIONS AND VECTOR POTENTIALS

3.3 A- Wave Equation

WhenJm = 0 andJe 6= 0, then∇.B = 0. So,B can be then written as

B = ∇× A

A is related toscalar potential, φe as

∇.A = −jωǫµφe

Then wave equation in terms ofA is given by

∇2A+ k2A = −µJe

and electric fieldE is given by

E = −∇φe − jωA = E =1

jωǫµ∇ (∇.A)− jωA

3.4 F- Wave Equation

WhenJe = 0 andJm 6= 0, then∇.D = 0. So,D can be then written as

D = −∇× F

F is related toscalar potential, φm as

∇.F = −jωǫµφm

Then wave equation in terms ofA is given by

∇2F+ k2F = −ǫJm

and magnetic fieldH is given by

H = −∇φm − jωF = E =1

jωǫµ∇ (∇.F)− jωF

Finally, A andF are given by,

A =µ

4π

∫

Je

e−j(k.R)

Rdv

F =ǫ

4π

∫

Jm

e−j(k.R)

Rdv

3.5. ALTERNATIVE NOTATIONS 9

3.5 Alternative Notations

These equations are written using “R.F.Harrington’s Notation” ... I will not use thisnotation here ...

A =1

4π

∫

Je

e−j(k.R)

Rdv

F =1

4π

∫

Jm

e−j(k.R)

Rdv

where,

H = ∇× A

∇.A = −jωǫφe

∇2A+ k2A = −Je

E = −∇φe − jωµA

and

E = −∇× F

∇.F = −jωµφm

∇2F+ k2F = −Jm

H = −∇φm − jωµF

3.6 Πe- Wave Equation

In the case ofJe = 0, Jm = 0, ρe = 0, ρm = 0 and∇.B = 0,H can be written as

H = jωǫ∇×Πe

Πe is related toscalar potential,Φe as

∇.Πe = Φe

Then wave equation forΠe is given by

∇2Πe + k2Πe = − Pǫ0

and electric fieldE is given by

E = ∇×∇×Πe −Pǫ0

10 CHAPTER 3. WAVE EQUATIONS AND VECTOR POTENTIALS

3.7 Πm- Wave Equation

In the case ofJe = 0, Jm = 0, ρe = 0, ρm = 0 and∇.D = 0,E can be written as

E = −jωµ∇×Πm

Πm is related toscalar potential, Φm as

∇.Πm = Φm

Then wave equation forΠm is given by

∇2Πm + k2Πm = −M1

and magnetic fieldH is given by

H = ∇×∇×Πm − M

1P andM are electric and magnetic polarizations, respectievely.

Chapter 4

Radiation Equations

4.1 Farfield Region

If A andF are given by,

A =µ

4π

∫

Je

e−j(k.R)

Rdv ≃ µe−jkr

4πr

∫

Jee+j(kxx

′+kyy′+kzz

′)dv

F =ǫ

4π

∫

Jm

e−j(k.R)

Rdv ≃ ǫe−jkr

4πr

∫

Jme+j(kxx

′+kyy′+kzz

′)dv

EAfar ≃ −jωAθθ − jωAφφ

HAfar ≃ +j

ω

ηAφθ − j

ω

ηAθφ

HFfar ≃ −jωFθθ − jωFφφ

EAfar ≃ −jωηFφθ + jωηFθφ

4.2 Further Simplification

If N, L are definied as below,

N =

∫

(

Jex′ x′ + Jey′ y′ + Jez′ z′)

e+j(kxx′+kyy

′+kzz′)dv

=

∫

(Jex′ x+ Jey′ y + Jez′ z) e+j(kxx′+kyy

′+kzz′)dv

11

12 CHAPTER 4. RADIATION EQUATIONS

L =

∫

(

Jmx′ x′ + Jmy′ y′ + Jmz′ z′)

e+j(kxx′+kyy

′+kzz′)dv

=

∫

(Jmx′ x+ Jmy′ y + Jmz′ z) e+j(kxx′+kyy

′+kzz′)dv

Then from eq. (1.5),

Nθ =

∫

[Jex′ cos θ cosφ+ Jey′ cos θ sinφ− Jez′ sin θ] e+j(kxx′+kyy

′+kzz′)dv′

Nφ =

∫

[−Jex′ sinφ+ Jey′ cosφ] e+j(kxx′+kyy

′+kzz′)dv′

Lθ =

∫

[Jmx′ cos θ cosφ+ Jmy′ cos θ sinφ− Jmz′ sin θ] e+j(kxx′+kyy

′+kzz′)dv′

Lφ =

∫

[−Jmx′ sinφ+ Jmy′ cosφ] e+j(kxx′+kyy

′+kzz′)dv′

”Here include generalized equations for spherical and cylendrical co-ordinates”Fianlly, E and H feilds are given by,

Eθ ≃ − jke−jkr

4πr(Lφ + ηNθ)

Eφ ≃ +jke−jkr

4πr(Lθ − ηNφ)

Hθ ≃ +jke−jkr

4πr

(

Nφ − Lθ

η

)

Hφ ≃ − jke−jkr

4πr

(

Nθ +Lφ

η

)

Chapter 5

Plane Waves

5.1 TEz and TMz Modes

Free space parameters:

k =2π

λ=

ω

vp=

2πf√ǫµ

For unguided plane waves,kc = 0 andkz = k0.In case, if waves are guided by rectangular wave guide,Guided parameters:

kz =

{

√

k2 − k2c , k ≥ kc

−j√

k2c − k2, k ≤ kc

kz =2π

λg=

ω

vg

For rectangular wave guide,

kc =

√

(mπ

a

)2

+(nπ

b

)2

whena = ∞, rectangular wave guide becomesparallel plate wave guide.If wave impedances are defined as,

Z =Ex

Hy

= −Ey

Hx

Then wave impedances for both TE and TM modes are given as,

ZTE =ωµ

kzand ZTM =

kz

ωǫ

Wave impedances in terms of propagation constants,

ZTE =η

√

1−(

kc

k

)2=

(

λg

λ

)

η,For Propagating modes

13

14 CHAPTER 5. PLANE WAVES

ZTM = η

√

1−(

kc

k

)2

=

(

λ

λg

)

η,For Propagating modes

and one can notice that,ZTEZTM = η2 (Babinet’s principle!!!) ....Wave impedances are real for propagating modes and imaginary for evanescent modes

Chapter 6

EM Fields in MultilayerStructures

6.1 Introduction

ForTM z fields to exist,

∂ρe

∂z− ρe

∂ (lnǫ)

∂z+ z.∇× (ǫJm) + jωµǫ (z.Je) 6= 0

15

16 CHAPTER 6. EM FIELDS IN MULTILAYER STRUCTURES

ForTEz fields to exist,

∂ρm

∂z− ρm

∂ (lnµ)

∂z− z.∇× (µJe) + jωµǫ (z.Jm) 6= 0

6.2 Fourier Transormation

Dz (kx, ky, z) =1

4π2

∫ ∫

Dz (x, y, z) exp (jkxx+ jkyy) dxdy

Dz (x, y, z) =

∫ ∫

Dz (kx, ky, z) exp (−jkxx− jkyy) dkxdky

6.3 Boundary Conditions

Electrical wall parallel toz-axis:Dz = 0

∂Bz

∂n= 0

Magnetic wall parallel toz-axis:∂Dz

∂n= 0

Bz = 0

Interface parallel toz-axis:1

ǫ1

∂D1z

∂z=

1

ǫ2

∂D2z

∂z

1

µ1

∂B1z

∂z=

1

µ2

∂B2z

∂z

6.4 TMz-fields

Az (x, y, z) = jωµ

∫ ∫

Dz

(k2 − k2zi)exp (−jkxx− jkyy) dkxdky

Bx =∂Az

∂y= ωµ

∫ ∫

ky

(k2 − k2zi)Dzexp (−jkxx− jkyy) dkxdky

By = −∂Az

∂x= −ωµ

∫ ∫

kx

(k2 − k2zi)Dzexp (−jkxx− jkyy) dkxdky

Dx =1

jωµ

∂2Az

∂x∂z= −j

∫ ∫

kx

(k2 − k2zi)

∂Dz

∂zexp (−jkxx− jkyy)dkxdky

Dy =1

jωµ

∂2Az

∂y∂z= −j

∫ ∫

ky

(k2 − k2zi)

∂Dz

∂zexp (−jkxx− jkyy) dkxdky

6.5. TEZ -FIELDS 17

6.5 TEz-fields

Fz (x, y, z) = jωǫ

∫ ∫

Bz

(k2 − k2zi)exp (−jkxx− jkyy) dkxdky

Bx (x, y, z) =1

jωǫ

∂2Fz

∂x∂z= −j

∫ ∫

kx

(k2 − k2zi)

∂Bz

∂zexp (−jkxx− jkyy) dkxdky

By (x, y, z) =1

jωǫ

∂2Fz

∂y∂z= −j

∫ ∫

ky

(k2 − k2zi)

∂Bz

∂zexp (−jkxx− jkyy) dkxdky

Dx (x, y, z) = −∂Fz

∂y= −ωǫ

∫ ∫

ky

(k2 − k2zi)Bzexp (−jkxx− jkyy) dkxdky

Dy (x, y, z) =∂Fz

∂x= ωǫ

∫ ∫

kx

(k2 − k2zi)Bzexp (−jkxx− jkyy) dkxdky