EM & Math - Basics · Basics CT531, DA-IICT. Maxwell EquationsWavesHelmholtz Wave EquationDirac...

Maxwell Equations Waves Helmholtz Wave Equation Dirac Delta Sources Power Matrices Complex Analysis Optimization

EM & Math - Basics

S. R. Zinka

[email protected]

October 16, 2014

Basics CT531, DA-IICT

mailto:[email protected]


Outline

1 Maxwell Equations

2 Waves

3 Helmholtz Wave Equation

4 Dirac Delta Sources

5 Power

6 Matrices

7 Complex Analysis

8 Optimization



Summary of Maxwell’s Equations

Coulomb's Law (or it's dual)Biot-Savart's Law (or it's dual)Farday's Law (or it's dual)



Outline

1 Maxwell Equations

2 Waves



5 Power

6 Matrices

7 Complex Analysis

8 Optimization



Waves



Outline

1 Maxwell Equations

2 Waves



5 Power

6 Matrices

7 Complex Analysis

8 Optimization



Wave Equation

Simple 1 - dimensional wave equation is given as

∂2F∂x2 =

1v2

∂2F∂t2

Using the complex notation, the above equation can be simplified as

∂2Fs

∂x2 =

(β

ω

)2

(jω)2 Fs = −β2Fs

⇒ ∂2Fs

∂x2 + β2Fs = 0 (1)

Using the theory of linear differential equations, solution for the above equation is given as

Fs = Aejβx + Be−jβx

⇒ F = Re[(

Aejβx + Be−jβx)

ejωt]= Re

[Aej(ωt+βx) + Bej(ωt−βx)

]. (2)



Helmholtz Wave EquationIn a source-less dielectric medium,

∇ · ~Ds = 0

∇ ·~Bs = 0

∇× ~Hs = jω~Ds = jωε~Es (3)

∇×~Es = −jω~Bs = −jωµ~Hs (4)

Taking curl of (4) gives

∇×(∇×~Es

)= ∇×

(−jωµ~Hs

)⇒ ∇

(∇ ·~Es

)−∇2~Es = −jωµ

(∇× ~Hs

)⇒ ∇

(∇ ·~Es

)−∇2~Es = −jωµ

(jωε~Es

)⇒ ∇2~Es = ∇

(∇ ·~Es

)−ω2µε~Es

⇒ ∇2~Es =~0−ω2µε~Es (5)

Similarly, it can be proved that

∇2~Hs = −ω2µε~Hs. (6)



Finally, Let’s Analyze the Helmholtz Wave Equation

Let’s compare general wave equation (1) and Helmholtz wave equation (5).

∂2Fs∂x2 + β2Fs = 0 ∇2~Es + ω2µε~Es = 0

From the above comparison, we get,

β = ω√

µε. (7)

But, we already knew that

v =ω

β.

So, from the above equations, we get

v =1√

µε=

1√

µrεrc (8)

where c is the light velocity.



Solution of Helmholtz Equation (in Cartesian System)Vector Helmholtz equation can be decomposed as shown below:

∇2Exs + ω2µεExs = 0

∇2~Es + ω2µε~Es = 0 ∇2Eys + ω2µεEys = 0

∇2Eys + ω2µεEys = 0

Since all the differential equations are similar, let’s solve just one equation using variable-separablemethod. If Exs can be decomposed into

Exc = A (x)B (y)C (z)

then substituting the above equation into Helmholtz equation gives

∇2Exs + ω2µεExs = 0

⇒ ∂2Exs

∂x2 +∂2Exs

∂y2 +∂2Exs

∂z2 + ω2µεExs = 0

⇒ B (y)C (z)∂2A∂x2 + A (x)C (z)

∂2B∂y2 + A (x)B (y)

∂2C∂z2 + ω2µεA (x)B (y)C (z) = 0

⇒ 1A (x)

∂2A∂x2 +

1B (y)

∂2B∂y2 +

1C (z)

∂2C∂z2 −γ2 = 0



Solution of Helmholtz Equation ... Contd

⇒ 1A (x)

∂2A∂x2 +

1B (y)

∂2B∂y2 +

1C (z)

∂2C∂z2 −γ2 = 0

⇒ 1A (x)

∂2A∂x2 +

1B (y)

∂2B∂y2 +

1C (z)

∂2C∂z2 −γ2

x−γ2y−γ2

z = 0 (9)

The above equation can be decomposed into 3 separate equations:

1A (x)

∂2A∂x2 − γ2

x = 0

1B (y)

∂2B∂y2 − γ2

y = 0

1C (z)

∂2C∂z2 − γ2

z = 0

It is sufficient to solve only one of the above equations and it’s solution is given as

⇒ ∂2A∂x2 − γ2

xA (x) = 0

⇒ A (x) = L1eγxx + L2e−γxx = L−eγxx + L+e−γxx (10)



Solution of Helmholtz Equation ... Contd

So, finally Exc is given as

Exs =(L−eγxx + L+e−γxx) (M−eγyy + M+e−γyy) (N−eγzz + N+e−γzz) (11)

⇒ Ex = Re[(

L−eγxx + L+e−γxx) (M−eγyy + M+e−γyy) (N−eγzz + N+e−γzz) ejωt]

(12)

with the conditionγ2

x + γ2y + γ2

z = γ2. (13)



Outline

1 Maxwell Equations

2 Waves



5 Power

6 Matrices

7 Complex Analysis

8 Optimization



Dirac Delta Function - Heuristic Description

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-2 -1 0 1 2

The Dirac delta can be loosely thought of as a function on the real line which is zeroeverywhere except at the origin, where it is infinite,

δ (x) =

{+∞, x = 00, x 6= 0

and which is also constrained to satisfy the identityˆ +∞

−∞δ (x) dx = 1.



Dirac Delta Function - A Few Properties

• δ (−x) = δ (x) (Symmetry Property)

• ´ +∞−∞ δ (αx) dx =

´ +∞−∞ δ (u) du

|α| =1|α| (Scaling Property)

• ´ +∞−∞ f (x) δ (x− x0) dx = f (x0) (Translation or Sifting Property)

• δ (x)⇔ 1



Volume Charge Densities of Point, Line, and SheetCharges

X′ , Y′ , Z′ ⇒ Source coordinates

X, Y, Z ⇒ Test charge coordinates



Outline

1 Maxwell Equations

2 Waves



5 Power

6 Matrices

7 Complex Analysis

8 Optimization



Instantaneous & Time Average Power

Instantaneous power corresponding to the above set of voltage & current is defined as

Pinst (t) = v0i0 cos (ωt + φ1) cos (ωt + φ2)

=v0i0

2[cos (ωt + φ1 + ωt + φ2) + cos (ωt + φ1 −ωt− φ2)]

=v0i0

2[cos (2ωt + φ1 + φ2) + cos (φ1 − φ2)] (14)

Time average power is defined as

Pavg =1

T0

ˆ T0

0Pinst dt =

v0i02

cos (φ1 − φ2) (15)



Time Average Power - Complex Notation

vreal = v0 cos (ωt + φ1) vcomplex = V = v0ej(ωt+φ1)

ireal = i0 cos (ωt + φ2) icomplex = I = i0ej(ωt+φ2)

Pavg =v0 i0

2 cos (φ1 − φ2) Pavg = 12 Re (VI∗) = v0 i0

2 cos (φ1 − φ2)

�

�In the above, does the equation 1

2 Re (VI∗) remind you of some thing ? ... Isn’t it very similar to the

complex Poynting vector 12 Re

(~E× ~H∗

)that you study in EMT course ?!



Outline

1 Maxwell Equations

2 Waves



5 Power

6 Matrices

7 Complex Analysis

8 Optimization



Eigenvalues & Eigenvectors

An eigenvector of a square matrix A is a non-zero vector x that, when the matrix multiplies x, yieldsa constant multiple of x, the latter multiplier being commonly denoted by λ. That is:

Ax = λx

The number λ is called the eigenvalue of A corresponding to x.

• tr (A) = λ1 + λ2 + · · ·+ λn

• det (A) = λ1λ2 · · · λn

• Eigenvalues of Ak are λk1, λk

2, · · · , λkn.



Conjugate/Hermitian Transpose

For a given matrix A, it’s conjugate/Hermitian transpose AH is defined such that

AHij = Aij.

• The eigenvalues of AH are the complex conjugates of the eigenvalues of A.• If AH = A, then A is known as Hermitian matrix. Eigenvalues of a Hermitian matrix are real.• If AH = −A, then A is known as skew Hermitian matrix. Eigenvalues of a skew Hermitian

matrix are imaginary.• If AAH = AHA, then A is known as normal matrix.• If AAH = AHA = I, then A is known as unitary matrix.

• (A + B)H = AH + BH

• (rA)H = r∗AH

• det(AH)= (det A)∗

• tr(AH)= (tr A)∗

• (AH)−1

=(A−1

)H



Matrix Multiplication - Properties

• (AB)T = BTAT

• (AB)∗ = A∗B∗

• (AB)H = BHAH

• For square matrices, det (ABC) = det (A)det (B)det (C)

• tr (ABC) = tr (BCA) = tr (CAB)



Inner & Outer ProductsThe inner product of two vectors in matrix form is equivalent to a column vector multiplied on theleft by a row vector:

a · b = aTb

= [a1a2 · · · an]

b1b2

...bn

=N

∑i=1

aibi.

The outer product of two vectors in matrix form is equivalent to a row vector multiplied on the leftby a column vector:

a⊗ b = abT

=

a1a2

...am

[b1b2 · · · bn] =

a1b1 a1b2 · · · a1bna2b1 a2b2 · · · a2bn

......

......

amb1 amb2 · · · ambn

.

a · b = aTb = tr (a⊗ b) = tr(abT)



Norm

The length or norm of a vector x is

‖x‖ =√

∑i|xi|2 =

√xHx =

√tr (xxH).

The norm of a matrix is in turn defined by

‖A‖ =√

∑i,j

∣∣aij∣∣2 =

√tr (AAH).

Cauchy–Schwarz inequality:

∣∣tr (xyH)∣∣2 =∣∣xHy

∣∣2 ≤ ‖x‖2 ‖y‖2 =(xHx

) (yHy

)= tr

(xxH) tr

(yyH)

∣∣tr (ABH)∣∣2 ≤ tr(AAH) tr

(BBH)



Positive Definiteness

A is a positive definite Hermitian matrix, if 〈x, Ax〉 > 0 for all x 6= 0, where 〈·, ·〉 is the Hermitianinner product, i.e.,

xHAx > 0

All the eigenvalues of a positive definite Hermitian matrix are positive.



Sherman–Morrison formula (A Special Case ofWoodbury Formula)

(A + uvT)−1

= A−1 − A−1uvTA−1

1 + vTA−1u



Outline

1 Maxwell Equations

2 Waves



5 Power

6 Matrices

7 Complex Analysis

8 Optimization



Complex Derivative

Given a complex-valued function f of a single complex variable, the derivative of f at a point z0 inits domain is defined by the limit

f ′ (z0) = limz→z0

f (z)− f (z0)

z− z0= lim

h→0

f (z0 + h)− f (z0)

h,

where h ∈ C.



Existence of Complex Derivative



Holomorphic Functions & Cauchy-RiemannEquations

If complex derivative exists, then it may be computed by taking the limit as h→ 0 along the real axisor imaginary axis; in either case it should give the same result.

Approaching along the real axis, one finds

f ′ (z0) = limh→0 & h∈R

f (z0 + h)− f (z0)

h=

∂f∂x

(z0) . (16)

On the other hand, approaching along the imaginary axis,

f ′ (z0) = limh→0 & h∈R

f (z0 + ih)− f (z0)

ih=

1i

∂f∂y

(z0) . (17)

The equality of the derivative of f taken along the two axes is

i∂f∂x

=∂f∂y⇒ ∂u

∂x=

∂v∂y

and∂u∂y

= − ∂v∂x

. (18)



Wirtinger Derivatives

Let g (z, z∗), a function of a complex number z and its conjugate z∗ . Then there exists a functionf (x, y) of the real variables x and y such that g (z, z∗) = f (x, y), where z = x + iy.

So, differentiating with respect to x and y, and using the chain rule, we have

∂f∂x

=∂g∂z

∂z∂x

+∂g∂z∗

∂z∗

∂x=

∂g∂z

+∂g∂z∗

∂f∂y

=∂g∂z

∂z∂y

+∂g∂z∗

∂z∗

∂y= i

∂g∂z− i

∂g∂z∗

. (19)

Rearranging the above set of equations gives

∂g∂z

=12

(∂f∂x− i

∂f∂y

)∂g∂z∗

=12

(∂f∂x

+ i∂f∂y

). (20)



Real Valued Functions of Complex Variables

Most of the functions that we deal in smart antennas are real valued functions. A few examples thatwe encounter are given below:

P(w) = wHRw.

SINR =wHRSw

wHRIw + wHRnw.



Stationary Points

- 4 - 2 2 4

- 4

- 2

2

4



Saddle Points

x3

−10

−5

0

5

10

−4 −3 −2 −1 0 1 2 3 4



Stationary Points of a Real Valued Function

Let g (z, z∗) = u (x, y) + iv (x, y), where u and v are real functions. If g is real valued, then we musthave v(x, y) = 0 for all x, y ∈ R. Then

∂g∂z

=12

(∂f∂x− i

∂f∂y

)=

12

(∂u∂x− i

∂u∂y

)∂g∂z∗

=12

(∂f∂x

+ i∂f∂y

)=

12

(∂u∂x

+ i∂u∂y

). (21)

When ∂g∂z = 0,

∂u∂x

= 0, and∂u∂y

= 0.

Similarly, when ∂g∂z∗ = 0,

∂u∂x

= 0, and∂u∂y

= 0.

So, either of the conditions ∂g∂z = 0 or ∂g

∂z∗ = 0 is necessary and sufficient condition to give a stationary

point of a real valued function of complex variables.



Gradient

∇w =∂w∂x

x +∂w∂y

y +∂w∂z

z



Complex Gradient

If g is a real valued function of complex variables z1, z∗1 , z2, z∗2 , · · · , etc, then the necessary and suffi-cient condition to determine the stationary point of g is:

∇zg =

∂

∂z1∂

∂z2...∂

∂zN

g =

00...0

= 0

(or)

∇z∗ g =

∂∂z∗1

∂∂z∗2...∂

∂z∗N

g =

00...0

= 0



Properties of Complex Gradient

∇ = ∇z∗ ∇ = ∇z

∇(aHz)= 0 ∇

(aHz)= a∗

∇(zHa)= a ∇

(zHa)= 0

∇(zHRz

)= Rz ∇

(zHRz

)= RTz∗



Outline

1 Maxwell Equations

2 Waves



5 Power

6 Matrices

7 Complex Analysis

8 Optimization



Level Sets Versus the Gradient



Method of Lagrange Multipliers

Λ (x, y, λ) = f (x, y) + λ [g (x, y)− c]



Method of Lagrange Multipliers - Example


EM & Math - Basics · Basics CT531, DA-IICT. Maxwell EquationsWavesHelmholtz Wave EquationDirac...

Documents

Transcript of EM & Math - Basics · Basics CT531, DA-IICT. Maxwell EquationsWavesHelmholtz Wave EquationDirac...