CEM Workshop Lectures (9/11): Modelling Electromagnetics Field

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Modelling Electromagnetics
Modelling of the field
V3
Prof GR Shevare

2
©ZeusNumerixPvtLtd:ConfidentialDocument
Contents
18-Jul-2020 Modelling Electromagnetics Field: Zeus Numerix
 Vectors, differential and integral operators
 Governing equations and its mathematical properties
 Simplification for space with sinusoidal incident field
 Meaning of variables and their units
 wave equation and properties of wave equations
 Concept of scalar and vector potentials and their simplification in possible in low
frequency region
 Coulomb and Lorentz gauge for uncoupling of potentials, Plane wave solution
 Concept of electric size and its significance in microwave region, storage vs radiation
 Boundary condition at interfaces
2

3
3
Vectors
Dot product:
Definition: A·B=|A||B|cosθAB
Application: Flux passing through a loop
Cross product: (Direction rh screw)
Application: Poynting vector
Φ =
𝑠
𝐁 ⋅ 𝐧 𝑑𝑆
𝐀 × 𝐁 =
𝐱 𝐲 𝐳
𝐴 𝑥 𝐴 𝑦 𝐴 𝑧
𝐵𝑥 𝐵𝑦 𝐵𝑧
𝐒 = 𝐄 × 𝐇
𝐀 ⋅ 𝐁 = 𝐴 𝑥 𝐵𝑥 + 𝐴 𝑦 𝐵𝑦 + 𝐴 𝑧 𝐵𝑧

4
4
Vectors
Gradient, divergence and curl (Cartesian coordinates)
∇𝑓 = 𝐱
𝜕𝑓
𝜕𝑥
+ 𝐲
𝜕𝑓
𝜕𝑦
+ 𝐳
𝜕𝑓
𝜕𝑧 𝐄 = −∇𝑉
∇ ⋅ 𝐃 = 𝜌 𝑣
∇ ⋅ 𝐟 =
𝜕𝑓𝑥
𝜕𝑥
+
𝜕𝑓𝑦
𝜕𝑦
+
𝜕𝑓𝑧
𝜕𝑧
∇ × 𝐇 = 𝐉 +
𝜕𝐃
𝜕𝑡
∇ × 𝐟 =
𝐱 𝐲 𝐳
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
𝑓𝑥 𝑓𝑦 𝑓𝑧

5
5
Vectors
Gauss’s theorem
On any closed surface (one sided surface), S enclosing a volume, V
𝑉
∇ ⋅ 𝐟𝑑𝑉 =
𝑆
𝐧 ⋅ 𝐟𝑑𝑆
V
n
S𝑉
∇𝐹𝑑𝑉 =
𝑆
𝐧𝐹𝑑𝑆
𝑉
∇ × 𝐟𝑑𝑉 =
𝑆
𝐧 × 𝐟𝑑𝑆

6
6
Vectors
Stoke’s theorem
On any open surface (two-sided surface), S with a bounding perimeter, C
𝑆
∇ × 𝐟 ⋅ 𝐧𝑑𝑆 =
𝐶
𝐟 ⋅ 𝐝𝐥
C
n
S

7
7
Vectors
Coordinate free definition of gradient, divergence, curl
Consider a volume, V bounded by six coordinate surfaces,
DSk , k = 1,2…6
∇ ⋅ 𝐟 = 𝐿𝑖𝑚
Δ𝑉→0
1
Δ𝑉
𝑘
𝐧 𝑘 ⋅ 𝐟Δ𝑆 𝑘
∇𝐹 = 𝐿𝑖𝑚
Δ𝑉→0
1
Δ𝑉
𝑘
𝐧 𝑘 𝐹Δ𝑆 𝑘
∇ × 𝐟 = 𝐿𝑖𝑚
Δ𝑉→0
1
Δ𝑉
𝑘
𝐧 𝑘 × 𝐟Δ𝑆 𝑘

8
8
Coordinate Systems
Spherical Polar system (r, q, 𝜑)
Relation with Cartesian system
Y
r
X
Z
f
q
𝑥 = 𝑟 sin 𝜃 cos 𝜑
𝑦 = 𝑟 sin 𝜃 sin 𝜑
𝑧 = 𝑟 cos 𝜃
𝜃 = cos−1
𝑧
𝑥2 + 𝑦2 + 𝑧2
𝜑 = tan−1
𝑦
𝑥
𝑟 = 𝑥2 + 𝑦2 + 𝑧2

9
9
Coordinate Systems
Cylindrical Polar system (r, f, z)
Relation with Cartesian system
Y
r
X
Z
f
𝑥 = 𝜌 cos 𝜑
𝑦 = 𝜌 sin 𝜑
𝑧 = 𝑧
𝑧 = 𝑧
𝜑 = tan−1
𝑦
𝑥
𝜌 = 𝑥2 + 𝑦2
r

10
Maxwell’s Equations
 Coupled 1st order PDE in space and time
 C.E.M – solutions of these equations satisfying prescribed boundary conditions
 Simple geometries – Analytical solutions possible
 Real problems – complex geometries – Inhomogeneities
 Solve coupled eqns. Time/space variables
 Plane wave illumination – freq. domain
 Numerical Methods
 Plane wave illumination – ejwt for all the fields
10

11
11
Maxwell’s Equations
Basic Governing equations Constitutive relations
Continuity equation
Lorentz force equation
∇ × 𝐄 = −
𝜕𝐁
𝜕𝑡
∇ × 𝐇 = 𝐉 +
𝜕𝐃
𝜕𝑡
∇ ⋅ 𝐃 = 𝜌 𝑣
∇ ⋅ 𝐁 = 0
𝐃 = 𝜀𝐄
𝐁 = 𝜇𝐇
𝐉 = 𝜎𝐄
∇ ⋅ 𝐉 +
𝜕𝜌 𝑣
𝜕𝑡
= 0
𝐅 = 𝑄 𝐄 + 𝐯 × 𝐁

12
Field Quantities
 Lines of D vector originate from positive charge and terminate on negative charge
 Between any two electrodes with potential difference between them, the lines of
electric force exist similar to that shown in the above figure at any time instant
12
∇ ⋅ 𝐃 = 𝜌 𝑣
𝐃1 = 𝐃2
𝜀 𝑟1 𝐄1 = 𝜀 𝑟2 𝐄2
𝐄1 = 2𝐄2

13
Field Quantities
 Lines of B vector form closed curves (No monopoles)
 Concept of magnetic circuit
 Flux is constant at all points
 H in air gap = mr times that in iron
13
∇ ⋅ 𝐁 = 0

14
Charge and Current Density
 The origin of the electric and magnetic fields is the Electric Charge
 Static charges: Lead to the electric field
 Moving charges at constant velocity: Lead to steady magnetic field
 Accelerated charges: Lead to time-varying electric and magnetic fields
 Types of charges: Free and bound
 Types of current densities
 Conduction
 Convection
 Displacement
14
𝐉 𝑐𝑜𝑛𝑑 = 𝜎𝐄
𝐉 𝑐𝑜𝑛𝑣 = 𝜌 𝑣 𝐯
𝐉 𝑑𝑖𝑠𝑝 =
𝜕𝐃
𝜕𝑡

15
Wave Equations
 From Maxwell’s equation we have
 This equation satisfied by all vectors, E, H, D, B, P, M
15
∇ × ∇ × 𝐄 = −
𝜕 ∇ × 𝐁
𝜕𝑡
∇ ∇ ⋅ 𝐄 − ∇2 𝐄 = −
𝜕
𝜕𝑡
𝜇𝜎𝐄 + 𝜇𝜀
𝜕𝐄
𝜕𝑡
∇2
𝐄 − 𝜇𝜎
𝜕𝐄
𝜕𝑡
− 𝜇𝜀
𝜕2
𝐄
𝜕𝑡2
= 0
∇2 𝐱 − 𝜇𝜎
𝜕𝐱
𝜕𝑡
− 𝜇𝜀
𝜕2 𝐱
𝜕𝑡2
= 0

16
Wave Equations
 In free space or insulators, s = 0 - wave equation
 For low frequency phenomena with non-zero conductivity, we need to solve
diffusion equation - displacement current term negligible
16
∇2 𝐱 − 𝜇𝜎
𝜕𝐱
𝜕𝑡
= 0

17
Scalar and Vector Potentials
 From Maxwell’s equations
 A is called the vector potential
 The quantity in square brackets can be expressed as a gradient of a scalar. To have
consistency with scalar potential in electrostatics we write as follows. V is called the
scalar potential
17
∇ ⋅ 𝐁 = 0 𝐁 = ∇ × 𝐀
∇ × 𝐄 = −
𝜕 ∇ × 𝐀
𝜕𝑡
∇ × 𝐄 +
𝜕𝐀
𝜕𝑡
= 0
𝐄 +
𝜕𝐀
𝜕𝑡
= −∇𝑉 𝐄 = −∇𝑉 −
𝜕𝐀
𝜕𝑡

18
 For static case we get the standard electrostatic potential
 For time varying fields, E depends both on A and V
 accumulation of charge
 Time varying magnetic field
 Electric field split into two parts (Poisson equation solution) under static or quasi-
static conditions (low frequency) at distances << l, (wavelength of wave)
 Due to charge distribution, r
 Due to time varying current, J
18
𝐄 = −∇𝑉
−
𝜕𝐀
𝜕𝑡
∇𝑉
𝑉 𝐫 =
1
4𝜋𝜀0
𝑣′
𝜌 𝐫′
𝐫 − 𝐫′ 𝑑𝑣′ 𝐀 𝐫 =
𝜇0
4𝜋
𝑣′
𝐉 𝐫′
𝐫 − 𝐫′ 𝑑𝑣′

19
 For high frequency at distances no longer << l
 Time retardation effects need to be considered for potentials at field point (r) due to sources
at (r’)
19
𝑉 𝐫 =
1
4𝜋𝜀0
𝑣′
𝜌 𝑡 − 𝐫 − 𝐫′
/𝑐
𝐫 − 𝐫′ 𝑑𝑣′
𝐀 𝐫 =
𝜇0
4𝜋
𝑣′
𝐉 𝑡 − 𝐫 − 𝐫′
/𝑐
𝐫 − 𝐫′
𝑑𝑣′
V
n
SO
rr’

20
Lorentz Gauge
 Maxwell’s equations give
 The definition of a vector requires specifying both its curl and divergence. Since B =
curlA, we are at liberty to choose its divergence. Let,
 This is Lorentz gauge, which is consistent with equation of continuity
20
∇ × 𝐇 = 𝐉 +
𝜕𝐃
𝜕𝑡
∇ × ∇ × 𝐀 = 𝜇𝐉 + 𝜇𝜀
𝜕 −∇𝑉 −
𝜕𝐀
𝜕𝑡
𝜕𝑡
∇2
𝐀 − 𝜇𝜀
𝜕2
𝐀
𝜕𝑡2
= −𝜇𝐉 + ∇ ∇ ⋅ 𝐀 + 𝜇𝜀
𝜕𝑉
𝜕𝑡
∇ ⋅ 𝐀 = −𝜇𝜀
𝜕𝑉
𝜕𝑡

21
Lorentz Gauge
 Lorentz gauge uncouples wave equation for A and V
 For static problems Lorentz gauge reduces to
 This is called the Coulomb gauge
21
∇2
𝐀 − 𝜇𝜀
𝜕2
𝐀
𝜕𝑡2
= −𝜇𝐉 ∇2
𝑉 − 𝜇𝜀
𝜕2
𝑉
𝜕𝑡2
= −
𝜌
𝜀
∇ ⋅ 𝐀 = 0

22
Plane Wave Solution
 As shown before in source free region
 The solution of this equation is given by
 Here k is called the wave vector and magnitude is given by
 This is called the plane wave solution. H also satisfies the same equation but with
different phase
22
∇2
𝐄 − 𝜇𝜀
𝜕2 𝐄
𝜕𝑡2
= 0 𝜇𝜀 = 1/𝑐
𝐄 𝐫 = 𝐄0
+
𝑒 𝑗 𝜔𝑡−𝐤⋅𝐫 + 𝐄0
−
𝑒 𝑗 𝜔𝑡+𝐤⋅𝐫
𝑘 = 𝜔 𝜇𝜀 = 𝜔/𝑐

23
Plane Electromagnetic Waves
 Plane wave given by
 For plane waves no boundary conditions for any direction
23
𝑒 𝑗 𝜔𝑡±𝐤⋅𝐫
Q
P
P
Q
𝜔𝑡 − 𝐤 ⋅ 𝐫 = constant
t = t1
t = t2>t1
Q
P
Q
P
t = t2>t1
𝜔𝑡 + 𝐤 ⋅ 𝐫 = constant

24
Electromagnetic Waves
 Nature of plane electromagnetic waves in lossless media
 This shows k is perpendicular to E. Similarly k is perpendicular to H
 This shows k, E and H are mutually perpendicular. This type of plane wave is called
TEM wave
24
∇ ⋅ 𝐄0 𝑒 𝑗(𝜔𝑡−𝐤⋅𝐫)
= 0 = 𝐤 ⋅ 𝐄0
∇ × 𝐄0 𝑒 𝑗(𝜔𝑡−𝐤⋅𝐫)
= −𝜇0
𝜕
𝜕𝑡
𝐇0 𝑒 𝑗(𝜔𝑡−𝐤⋅𝐫)
−𝑗 𝐤 𝜀0 𝜇0 × 𝐄0 𝑒 𝑗(𝜔𝑡−𝐤⋅𝐫)
= −𝜇0 𝑗𝜔𝐇0 𝑒 𝑗(𝜔𝑡−𝐤⋅𝐫)
𝐤 × 𝐄0 =
𝜇0
𝜀0
𝐇0

25
 Assume k along Z axis and E0 along X axis The direction and magnitude of H0 can be
calculated from
 Consider the plane wave
 The corresponding magnetic field is (k changes sign for backward wave)
 h is the intrinsic impedance of free space = 377 ohms
25
𝐤 × 𝐄0 =
𝜇0
𝜀0
𝐇0 = 𝜂𝐇0
𝐇0𝑦 =
1
𝜂
𝐄0𝑥
𝐄 𝐫 = 𝐄0
+
𝑒 𝑗 𝜔𝑡−𝐤⋅𝐫
+ 𝐄0
−
𝐇 𝐫 = 𝐇0
+
+ 𝐇0
−
=
1
𝜂
𝐄0
+
− 𝐄0
−

26
 Electromagnetic waves propagate from sources
 In electromagnetic waves E and H vary periodically in space and time
 What is wavefront?
 Surfaces of constant phase at fixed time called wavefronts
 Point source - spherical waves (wavefront is sphere)
 Line source - cylindrical waves (wavefront is cylinder)
 All waves - spherical, cylindrical or any other shape become a plane wave at
distances very large from source
26

27
 Transverse electromagnetic wave (TEM)
 These are plane waves
 To uniquely specify the wave we need
 Wave vector - k
 Electric field - E
 Magnetic field - H is not specified as it can be obtained from Maxwell’s equation
 If only k is specified, E can be oriented along any direction in the plane perpendicular
to k. If we specify a particular orientation within this plane, we specify the
polarization of the wave
27

28
 Polarization
 If direction of E is same at all times we call such wave as plane polarized
 If direction rotates continuously in the plane perpendicular to the plane of k the wave is said
to be circularly polarized
 Both r h and l h circular polarization are possible
 If in a circular polarized wave the tip of the amplitude vectors traces an ellipse the wave is
said to be elliptically polarized
28

29
Electric Size of Object
 Relative size of object w.r.t wavelength of EM wave
 l at 3 kHz 105 m
 l at 3 MHz 102 m
 l at 3 GHz 1 cm
 l at optical 10-6 m
 Physical size - a
 Electrical size - ka
29

30
30
Wave Propagation
Conceptually different from low f devices
VQ ≠ VP
Voltage drop or gain between P and Q
Impedance between P and Q depends on VQ - VP
P
VP
Q
VQ

31
Microwave Propagation
 Voltage difference between two points on the line in absence of load. No load
impedance not zero
 Capacitance between line and nearby conductors. Can create spurious interference if
single wire used to carry power as in low frequency devices
 Need to use shielded lines- concept of transmission lines
 Phase of the wave (space dependence)
 Voltage/current → E/H
31

32
Propagation
 Length of the line important (Electrical/ Physical)
 Inductance per unit length and capacitance per unit length (Impedance important)
 Distributed vs lumped parameters
 Layout of subcomponents critical
 Conventional method of analysis not valid
 Free space – launched through antenna e.g. Radar, cellphones (antenna)
 Transmission lines e.g waveguide, coaxial line, stripline, microstripline
 Key design criteria – Impedance matching
32

33
Radiation
 Moving charges produce static electric and magnetic fields.
 Accelerating charges produce radiation.
 Radiation is a phenomenon related to time varying currents since they have
accelerating and decelerating charges present
 The higher the frequency, the more is the radiation.
 It also depends on the spatial variation.
 Circuits though having time varying currents may not always radiate effectively if
there is symmetry in them
33

34
Without flare standing wave current
distribution is that for an O.C
transmission line. Current pairs equal
and opposite, no radiation. With
small flare, L, C and hence Z change
with position. To first order wave
number unchanged. Little change in
current distribution. Same
assumption valid for 3rd case.
Currents start reinforcing, instead of
canceling - Center fed dipole
34
Radiation l
X=0

35
35
Radiation
Storage versus radiation fields (Ref: R. Schmitt, Electromagnetics Explained, Elsevier Science, 2002)
No Property Storage fields Radiation fields
1 Region of operation
In the vicinity Energy is stored or it can be
transferred through L or C coupling
They are far reaching and propagating fields
2 Longevity Disappear when source turned off No effect
3 Components
Can be any of DC, AC having only E, only H, or
any combination
Should travel in the form of waves. Contain E x H
Direction of propagation is perpendicular to E and
H. Ratio of E to H is 3×108 in free space
4 Rate of decay 1/r2 or 1/r3 dependence 1/r dependence
5 Interaction React when energy is extracted or measured Affected by number of loads
6 Wave impedance Depends on source circuit and medium Depends solely on medium

36
Antenna Fields
 Near fields
 Electrostatic fields
 Inversely proportional to frequency
 1/r3 dependence
 Induction field
 Independent of frequency
 1/r2 dependence
 Far field
 Radiation fields
 Proportional to frequency
 1/r dependence
36
The three fields become equal when distance
~ l/6 for Hertz dipole

37
Vector Wave Equation
 The general problem that we are interested in can contain
 Sources (both volumetric and surface)
 Scatterers made up of perfect electric conductors (PEC)
 Scatterers made up of in-homogenous dielectric and/ or magnetic materials
 The solution of Maxwell’s equation for this type of problem is the subject of
computational electromagnetics
 One way to tackle the problem is assume that source generates plane wave of single
sinusoidal frequency, w. The problem then is to determine the E and H fields at any
arbitrary point in space - Frequency domain method
37

38
Vector Wave Equation
 The governing equation for the frequency domain method is derived from Maxwell’s
equations
 These are inhomogeneous second order differential equations. There may be several
functions that satisfy this differential equation, only one of these is the real solution.
 For this we need to know the boundary conditions associated with the domain.
These can be derived from Maxwell’s equation in integral form
38
∇ ×
1
𝜇
∇ × 𝐄 − 𝜔2
𝜀𝐄 = −𝑗𝜔𝐉
∇ ×
1
𝜀
∇ × 𝐇 − 𝜔2
𝜇𝐇 = ∇ ×
1
𝜀
𝐉

39
Boundary Conditions
 At interface between two media
 Here Js is surface current density and rs is surface change density at the interface
 Note only two of the four boundary conditions are independent
39
𝐧 × 𝐇1 − 𝐇2 = 𝐉 𝑠
Medium 2
n
m2, e2
Medium 1
m1, e1
𝐧 × 𝐄1 − 𝐄2 = 0
𝐧 ⋅ 𝐃1 − 𝐃2 = 𝜌𝑠
𝐧 ⋅ 𝐁1 − 𝐁2 = 0

40
Boundary Conditions
 At a perfectly conducting surface (PEC)
 A PEC cannot have internal fields. Thus we have
 The boundary can support surface current density Js and surface change density rs at
the interface
40
𝐧 × 𝐄 = 0 𝐧 ⋅ 𝐁 = 0
𝐧 × 𝐇 = 𝐉 𝑠 𝐧 ⋅ 𝐃 = 𝜌𝑠

41
Boundary Conditions
 At an imperfectly conducting surface
 Applicable for a thin coating on PEC - Impedance boundary condition
 Alternate form of condition
 This can be written in standard form amenable to numerical solution (boundary
condition of third kind)
41
𝐧 × 𝐧 × 𝐄 = −𝜂𝑍0 𝐧 × 𝐇
𝜂𝑍0 𝐧 × 𝐧 × 𝐇 = 𝐧 × 𝐄 𝜂 =
𝜇 𝑟2
𝜀 𝑟2
𝑍0 =
𝜇0
𝜀0
1
𝜇 𝑟1
𝐧 × ∇ × 𝐄 −
𝑗𝑘0
𝜂
𝐧 × 𝐄 = 0
1
𝜀 𝑟1
𝐧 × ∇ × 𝐇 − 𝑗𝑘0 𝜂 𝐧 × 𝐧 × 𝐇 = 0

42
Boundary Conditions
 Across a resistive and conductive sheet
 A thin sheet with current density proportional to tangential electric field at surface
(thickness t)
 For thin conducting layer
 For thin dielectric layer
 For thin magnetic layer
42
𝐧 × 𝐧 × 𝐄 = −
𝐉 𝑠
𝜎𝜏
= −
1
𝜎𝜏
𝐧 × 𝐇1 − 𝐇2
𝐧 × 𝐧 × 𝐄 = −
𝑍0
𝑗𝑘0 𝜀 𝑟 − 1 𝜏
𝐧 × 𝐇1 − 𝐇2
𝐧 × 𝐧 × 𝐇 = −
𝑌0
𝑗𝑘0 𝜇 𝑟 − 1 𝜏
𝐧 × 𝐄1 − 𝐄2

43
Boundary Conditions
 Radiation conditions
 When domain is unbounded, a condition must be imposed at an artificial outer
boundary. Such a condition is referred as a radiation condition. Assuming all sources
located at finite distance from origin:
 This is also called as Sommerfield radiation condition
 These radiation conditions are lowest order with limited accuracy
 Higher order conditions give better accuracy
43
𝐿𝑖𝑚
𝑟→∞
𝑟 ∇ ×
𝐄
𝐇
+ 𝑗𝑘0 𝐫 ×
𝐄
𝐇
= 0

44
44
Electromagnetic Scattering
Source radiating plane wave at
location of target
Incident wave direction
specified by angles qinc, finc in
terms of coordinate system
localised at source
Z
ksc(qs,fs)
X
Y
kinc(qinc,finc)
Z
X
Y
Source
Target
Scattered wave direction specified by
angles qs, fs in terms of coordinate system
localised at target
Both coordinate systems related only by
translation

45
Electromagnetic Scattering
 Can define incident wave direction specified by angles qinc, finc in
terms of target coordinate system
 All parameters defined in target coordinate system
 To uniquely specify the incident and scattered wave, need to
specify the polarization (direction of E)
 Cartesian components not convenient for arbitrary orientation of
k vectors of incident and scattered fields
 Spherical polar components most convenient
45
Z
ksc(qs,fs)
X
Y
kinc(qinc,finc)
Target

46
46
Spherical Polar Components
Spherical Polar system (r, q, f)
Unit vectors
Unit vectors vary in direction as angles vary
Unit vector f always lies in XY plane
If incident or scattered wave direction specified in terms of
these unit vectors
Direction of k always points in direction of r
Direction of E and H always lies in q, f plane
𝐫 = sin 𝜃 cos 𝜑 𝐱 + sin 𝜃 cos 𝜑 𝐲 + cos 𝜃 𝐳
𝛉 = cos 𝜃 cos 𝜑 𝐱 + cos 𝜃 sin 𝜑 𝐲 − sin 𝜃 𝐳
𝛗 = − sin 𝜑 𝐱 + cos 𝜑 𝐲
Y
r
X
Z
f
q

47
 Polarization
 Direction of k always points in direction of r
 Direction of E and H always lies in q, f plane
 Polarization (direction of E ) specified by specifying its (q, f) components
 For the target, let us define the coordinate system such that XY plane is the
horizontal plane. Since the unit vector f always lies in XY plane, we can specify
 E parallel to f direction corresponds to horizontal polarization
 E parallel to q direction corresponds to vertical polarization
for both the incident and scattered wave
47

48
48
Specifying field components in Cartesian coordinates
For computational purposes E and H must be specified in terms
of Cartesian components
𝐄 𝜔, 𝑡 = 𝐄0 𝑒 𝑗(𝜔𝑡−𝑘 𝐤⋅𝐫′)
𝐤 = sin 𝜃 cos 𝜑 𝐱 + sin 𝜃 cos 𝜑 𝐲 + cos 𝜃 𝐳
𝐫′
= 𝑥′
𝐱 + 𝑦′
𝐲 + 𝑧′
𝐳
𝐄0 = 𝐸 𝜃 𝛉 + 𝐸 𝜑 𝛗
𝛉 = cos 𝜃 cos 𝜑 𝐱 + cos 𝜃 sin 𝜑 𝐲 − sin 𝜃 𝐳
𝛗 = − sin 𝜑 𝐱 + cos 𝜑 𝐲
𝐸 𝑥 = 𝐸 𝜃 cos 𝜃 cos 𝜑 − 𝐸 𝜑 sin 𝜑
𝐸 𝑦 = 𝐸 𝜃 cos 𝜃 sin 𝜑 + 𝐸 𝜑 cos 𝜑
𝐸𝑧 = −𝐸 𝜃 sin 𝜃

49
Radar Fundamentals
 Bistatic Radar: Radar in which the receiver and transmitter have separate antennas
and some distance apart
 Radar cross section: The area of a fictitious perfect reflector of electromagnetic
waves that would reflect the same amount of energy as the actual target
 Polarization: A description( vertical, horizontal, circular etc.) of the angular variation
with time of electric field at a fixed point
49

50
Radar Cross Section (RCS)
 Quantitatively, the RCS is an effective surface area that intercepts the incident wave
and that scatters the energy isotropically in space
 The RCS, s is defined in three-dimensions as
 Pinc is the incident power density measured at the target
 Ps is the scattered power density seen at a distance R away from the target
 Ei and Es are the incident and scattered electric field intensities, respectively
50
𝜎 = 𝐿𝑖𝑚
𝑅→∞
4𝜋𝑅2
𝑃𝑠
𝑃𝑖𝑛𝑐

51
 Units of RCS
 RCS is measured in m2
 Alternatively it is measured in decibel relative to 1 m2 (dBsm)
 RCS (dBsm) = 10 x log10(RCS/1m2)
 What is the dBsm of a stealth fighter with RCS of 0.01 m2?
 10 x log10(0.01/1m2) = -20 dBsm
51

52
 RCS and polarization
 The above equation shows that RCS is dependent on the polarizations of incident
and scattered wave
 Four special cases are possible (horizontal and vertical polarization)
52
𝜎 = 𝐿𝑖𝑚
𝑅→∞
4𝜋𝑅2
𝐸𝑠 𝜃𝑠, 𝜑𝑠
2
𝐸𝑖𝑛𝑐 𝜃𝑖𝑛𝑐, 𝜑𝑖𝑛𝑐
2
𝜎 𝐻𝐻 = 𝐿𝑖𝑚
𝑅→∞
4𝜋𝑅2
𝐸𝑠 𝜃𝑠 = 𝜋/2, 𝜑𝑠
2
𝐸𝑖𝑛𝑐 𝜃𝑖𝑛𝑐 = 𝜋/2, 𝜑𝑖𝑛𝑐
2
𝜎 𝐻𝑉 = 𝐿𝑖𝑚
𝑅→∞
4𝜋𝑅2
𝐸𝑠 𝜃𝑠 = 0, 𝜑𝑠
2
𝐸𝑖𝑛𝑐 𝜃𝑖𝑛𝑐 = 𝜋/2, 𝜑𝑖𝑛𝑐
2
𝜎 𝑉𝐻 = 𝐿𝑖𝑚
𝑅→∞
4𝜋𝑅2
𝐸𝑠 𝜃𝑠 = 𝜋/2, 𝜑𝑠
2
𝐸𝑖𝑛𝑐 𝜃𝑖𝑛𝑐 = 0, 𝜑𝑖𝑛𝑐
2 𝜎 𝑉𝑉 = 𝐿𝑖𝑚
𝑅→∞
4𝜋𝑅2
𝐸𝑠 𝜃𝑠 = 0, 𝜑𝑠
2
𝐸𝑖𝑛𝑐 𝜃𝑖𝑛𝑐 = 0, 𝜑𝑖𝑛𝑐
2

53
 RCS depends on
 Target size Target aspect
 Material of target Shape of target
 Frequency Direction of illumination
 Polarization wave
53
Mono-static radar

54
54
Bi-static radar
Receiving antennaTransmitting antenna

55
55

56
56
Typical head-on radar echo at microwave frequencies
Object RCS (m2)
Pickup truck 200
Automobile 100
Jumbo jet airliner 100
Large bomber or commercial jet 40
Cabin cruiser boat 10
Large fighter aircraft 6
Small fighter aircraft 2
Adult male human 1
Missile 0.5
Bird 0.01
Insect 0.00001
Source: Merill I Skolnick: Introduction to Radar systems

57
RCS Reduction
 Shape designing – usually conflicts with aerodynamic requirements
 Use of radar absorbing materials
 Use of onboard avionics – jamming enemy radars
57
Advanced stealth aircraft
RCS reduction to 1% of bomber
~ 0.4 to 0.5 m2

58
Computational Electromagnetics
 FREQUENCY DOMAIN SOLUTION
 Sinusoidal steady state solution of Maxwell’s equations
 Pre-1960: Analytical (closed form or infinite series)
 Post-1960: Era of digital computers
 High frequency asymptotic methods
 Well suited for metallic structures
 Difficulty in treating non-metallic and volumetric complexity
 Integral equation methods (Method of Moments)
 Can deal with complexity of above nature
 Limits electrical size of structures – need to invert large matrices
58

59
Computational Electromagnetics
 Frequency Domain Solution
 Finite element method
 Generates sparse matrices
 Can deal with any complex geometry
 Incorporation of material and device non-linearities into frequency domain solutions poses
significant problems
 Leads To Time Domain Solution Techniques
 Finite difference time domain (FDTD)
 Finite volume time domain (FVTD)
 Explicit methods – No matrix inversion
 Virtually no limit on size of problem
59

60
Frequency Domain Method
 Surface Equivalence Principle
60
n
J1, K1
J2, K2S
G2 G1
e1, m1
e2, m2
S
n
S
e1, m1
Js
Ks
e1, m1
𝐉 𝑠 = − 𝐧 × 𝐇2
𝐊 𝑠 = −𝐄2 × 𝐧
E, H in
region 1
J1, K1

61
Frequency Domain Method
 Surface Equivalence Principle
61
S
E2 H2
e2, m2
pec
e1, m1
E1 H1
n
S e1, m1
e1, m1
E1 H1
n
𝐉 𝑠 = 𝐧 × 𝐇1
𝐊 𝑠 = 𝐄1 × 𝐧
Null fields

62
62
Useful Mathematical Identities

63
References
 R. Schmitt, Electromagnetics Explained, Elsevier Science,2002.
 N. Narayana Rao, Elements of Engineering Electromagnetics, Prentice Hall of India,
New Delhi, 2002.
 D. K. Cheng, Field and Wave Electromagnetics, Addison Wesley, 1999.
 J Jin, The Finite element method in electromagnetics, Wiley, 2002
63

64
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+91 72760 31511
Abhishek Jain
contact@zeusnumerix.com
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CEM Workshop Lectures (9/11): Modelling Electromagnetics Field

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