Maxwell Equations Waves Helmholtz Wave Equation Dirac Delta Sources Power Matrices Complex Analysis Optimization. EM & Math - Basics

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1 EM & Math - S. R. Zinka srinivasa_zinka@daiict.ac.in October 16, 2014

2 Outline 1 Maxwell Equations 2 Waves 3 Helmholtz Wave Equation 4 Dirac Delta Sources 5 Power 6 Matrices 7 Complex Analysis 8 Optimization

3 Outline 1 Maxwell Equations 2 Waves 3 Helmholtz Wave Equation 4 Dirac Delta Sources 5 Power 6 Matrices 7 Complex Analysis 8 Optimization

4 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)

5 Outline 1 Maxwell Equations 2 Waves 3 Helmholtz Wave Equation 4 Dirac Delta Sources 5 Power 6 Matrices 7 Complex Analysis 8 Optimization

6 Waves

7 Outline 1 Maxwell Equations 2 Waves 3 Helmholtz Wave Equation 4 Dirac Delta Sources 5 Power 6 Matrices 7 Complex Analysis 8 Optimization

8 Wave Equation Simple 1 - dimensional wave equation is given as 2 F x = 1 2 F 2 v 2 t 2 Using the complex notation, the above equation can be simplified as 2 F s x 2 = ( ) β 2 (jω) 2 F s = β 2 F s ω 2 F s x 2 + β2 F s = 0 (1) Using the theory of linear differential equations, solution for the above equation is given as F s = Ae jβx + Be jβx [( F = Re Ae jβx + Be jβx) e jωt] [ = Re Ae j(ωt+βx) + Be j(ωt βx)]. (2)

9 Helmholtz Wave Equation In a source-less dielectric medium, Taking curl of (4) gives Ds = 0 Bs = 0 Hs = jω Ds = jωε Es (3) Es = jω Bs = jωµ Hs (4) ) ) ( Es = ( jωµ Hs ) ) ( Es 2 Es = jωµ ( Hs ) ) ( Es 2 Es = jωµ (jωε Es ) 2 Es = ( Es ω 2 µε Es 2 Es = 0 ω 2 µε Es (5) Similarly, it can be proved that 2 Hs = ω 2 µε Hs. (6)

10 Finally, Let s Analyze the Helmholtz Wave Equation Let s compare general wave equation (1) and Helmholtz wave equation (5). 2 Fs x 2 + β2 F s = 0 2 Es + ω 2 µε Es = 0 From the above comparison, we get, But, we already knew that β = ω µε. (7) v = ω β. So, from the above equations, we get v = 1 µε = 1 µr ε r c (8) where c is the light velocity.

11 Solution of Helmholtz Equation (in Cartesian System) Vector Helmholtz equation can be decomposed as shown below: 2 E xs + ω 2 µεe xs = 0 2 Es + ω 2 µε Es = 0 2 E ys + ω 2 µεe ys = 0 2 E ys + ω 2 µεe ys = 0 Since all the differential equations are similar, let s solve just one equation using variable-separable method. If E xs can be decomposed into E xc = A (x) B (y) C (z) then substituting the above equation into Helmholtz equation gives 2 E xs x E xs y 2 2 E xs + ω 2 µεe xs = E xs z 2 + ω 2 µεe xs = 0 B (y) C (z) 2 A x + 2 A (x) C (z) 2 B y + 2 A (x) B (y) 2 C z + 2 ω2 µεa (x) B (y) C (z) = A A (x) x B 2 B (y) y C 2 C (z) z 2 γ2 = 0

12 Solution of Helmholtz Equation... Contd 1 2 A A (x) x B 2 B (y) y C 2 C (z) z 2 γ2 = A A (x) x B 2 B (y) y C 2 C (z) z 2 γ2 x γy γ 2 z 2 = 0 (9) The above equation can be decomposed into 3 separate equations: 1 2 A A (x) x 2 γ2 x = B B (y) y 2 γ2 y = C C (z) z 2 γ2 z = 0 It is sufficient to solve only one of the above equations and it s solution is given as 2 A x 2 γ2 xa (x) = 0 A (x) = L 1 e γxx + L 2 e γxx = L e γxx + L + e γxx (10)

13 Solution of Helmholtz Equation... Contd So, finally E xc is given as E xs = ( L e γxx + L + e γxx) ( M e γyy + M + e γyy) ( N e γzz + N + e γzz) (11) E x = Re [ (L e γxx + L + e γxx) ( M e γyy + M + e γyy) ( N e γzz + N + e γzz) e jωt] (12) with the condition γ 2 x + γ 2 y + γ 2 z = γ 2. (13)

14 Outline 1 Maxwell Equations 2 Waves 3 Helmholtz Wave Equation 4 Dirac Delta Sources 5 Power 6 Matrices 7 Complex Analysis 8 Optimization

15 Dirac Delta Function - Heuristic Description The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, { +, x = 0 δ (x) = 0, x = 0 and which is also constrained to satisfy the identity ˆ + δ (x) dx = 1.

16 Dirac Delta Function - A Few Properties δ ( x) = δ (x) (Symmetry Property) + δ (αx) dx = + du δ (u) = 1 (Scaling Property) α α + f (x) δ (x x 0) dx = f (x 0 ) (Translation or Sifting Property) δ (x) 1

17 Volume Charge Densities of Point, Line, and Sheet Charges X, Y, Z Source coordinates X, Y, Z Test charge coordinates

18 Outline 1 Maxwell Equations 2 Waves 3 Helmholtz Wave Equation 4 Dirac Delta Sources 5 Power 6 Matrices 7 Complex Analysis 8 Optimization

19 Instantaneous & Time Average Power Instantaneous power corresponding to the above set of voltage & current is defined as P inst (t) = v 0 i 0 cos (ωt + φ 1) cos (ωt + φ 2) Time average power is defined as = v 0i 0 2 [cos (ωt + φ 1 + ωt + φ 2) + cos (ωt + φ 1 ωt φ 2)] = v 0i 0 2 [cos (2ωt + φ 1 + φ 2) + cos (φ 1 φ 2)] (14) P avg = 1 T 0 ˆ T0 0 P inst dt = v 0i 0 2 cos (φ 1 φ 2) (15)

20 Time Average Power - Complex Notation v real = v 0 cos (ωt + φ 1) v complex = V = v 0 e j(ωt+φ 1) i real = i 0 cos (ωt + φ 2) i complex = I = i 0 e j(ωt+φ 2) P avg = v 0 i 0 2 cos (φ 1 φ 2) P avg = 1 2 Re (VI ) = v 0 i 0 2 cos (φ 1 φ 2) 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 1 2 Re E H that you study in EMT course?!

21 Outline 1 Maxwell Equations 2 Waves 3 Helmholtz Wave Equation 4 Dirac Delta Sources 5 Power 6 Matrices 7 Complex Analysis 8 Optimization

22 Eigenvalues & Eigenvectors An eigenvector of a square matrix A is a non-zero vector x that, when the matrix multiplies x, yields a 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) = λ 1 + λ λ n det (A) = λ 1 λ 2 λ n Eigenvalues of A k are λ k 1, λk 2,, λk n.

23 Conjugate/Hermitian Transpose For a given matrix A, it s conjugate/hermitian transpose A H is defined such that A H ij = A ij. The eigenvalues of A H are the complex conjugates of the eigenvalues of A. If A H = A, then A is known as Hermitian matrix. Eigenvalues of a Hermitian matrix are real. If A H = A, then A is known as skew Hermitian matrix. Eigenvalues of a skew Hermitian matrix are imaginary. If AA H = A H A, then A is known as normal matrix. If AA H = A H A = I, then A is known as unitary matrix. (A + B) H = A H + B H (ra) H = r A H det ( A H) = (det A) tr ( A H) = (tr A) ( A H) 1 = ( A 1) H

24 Matrix Multiplication - Properties (AB) T = B T A T (AB) = A B (AB) H = B H A H For square matrices, det (ABC) = det (A) det (B) det (C) tr (ABC) = tr (BCA) = tr (CAB)

25 Inner & Outer Products The inner product of two vectors in matrix form is equivalent to a column vector multiplied on the left by a row vector: a b = a T b = [a 1 a 2 a n] b 1 b 2. b n N = a i b i. i=1 The outer product of two vectors in matrix form is equivalent to a row vector multiplied on the left by a column vector: a b = ab T = a 1 a 2. a m [b 1b 2 b n] = a 1 b 1 a 1 b 2 a 1 b n a 2 b 1 a 2 b 2 a 2 b n.... a m b 1 a m b 2 a m b n. a b = a T b = tr (a b) = tr ( ab T)

26 Norm The length or norm of a vector x is x = x i 2 = x H x = tr (xx H ). i The norm of a matrix is in turn defined by A = i,j a ij 2 = tr (AA H ). Cauchy Schwarz inequality: tr ( xy H) 2 = x H y 2 x 2 y 2 = ( x H x ) ( y H y ) = tr ( xx H) tr ( yy H) tr ( AB H) 2 tr ( AA H) tr ( BB H)

27 Positive Definiteness A is a positive definite Hermitian matrix, if x, Ax > 0 for all x = 0, where, is the Hermitian inner product, i.e., x H Ax > 0 All the eigenvalues of a positive definite Hermitian matrix are positive.

28 Sherman Morrison formula (A Special Case of Woodbury Formula) ( A + uv T ) 1 = A 1 A 1 uv T A v T A 1 u

29 Outline 1 Maxwell Equations 2 Waves 3 Helmholtz Wave Equation 4 Dirac Delta Sources 5 Power 6 Matrices 7 Complex Analysis 8 Optimization

30 Complex Derivative Given a complex-valued function f of a single complex variable, the derivative of f at a point z 0 in its domain is defined by the limit where h C. f f (z) f (z 0) f (z 0 + h) f (z 0) (z 0) = lim = lim, z z0 z z 0 h 0 h

31 Existence of Complex Derivative

32 Holomorphic Functions & Cauchy-Riemann Equations If complex derivative exists, then it may be computed by taking the limit as h 0 along the real axis or imaginary axis; in either case it should give the same result. Approaching along the real axis, one finds f (z 0) = lim h 0 & h R f (z 0 + h) f (z 0) h On the other hand, approaching along the imaginary axis, f (z 0) = lim h 0 & h R f (z 0 + ih) f (z 0) ih The equality of the derivative of f taken along the two axes is = f x (z 0). (16) = 1 i f y (z 0). (17) i f x = f y u x = v u and y y = v x. (18)

33 Wirtinger Derivatives Let g (z, z ), a function of a complex number z and its conjugate z. Then there exists a function f (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 Rearranging the above set of equations gives 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) g z = 1 ( f 2 x i f ) y g z = 1 ( f 2 x + i f ). (20) y

34 Real Valued Functions of Complex Variables Most of the functions that we deal in smart antennas are real valued functions. A few examples that we encounter are given below: P(w) = w H Rw. SINR = w H R S w w H R I w + w H R n w.

35 Stationary Points

36 Saddle Points 10 x

37 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 must have v(x, y) = 0 for all x, y R. Then g z = 1 ( f 2 x i f y g z = 1 ( f 2 x + i f y ) = 1 ( u 2 ) = 1 2 x i u ) y ( u x + i u y ). (21) When g z = 0, Similarly, when g z = 0, u u = 0, and x y = 0. u u = 0, and x y = 0. So, either of the conditions g g z = 0 or z = 0 is necessary and sufficient condition to give a stationary point of a real valued function of complex variables.

38 Gradient w = w w ˆx + x y ŷ + w z ẑ

39 Complex Gradient If g is a real valued function of complex variables z 1, z1, z 2, z2,, etc, then the necessary and sufficient condition to determine the stationary point of g is: z g = z g = z 1 z 2.. z N z 1 z 2.. z N g = (or) g = = 0 = 0

40 Properties of Complex Gradient = z = z ( a H z ) = 0 ( a H z ) = a ( z H a ) = a ( z H a ) = 0 ( z H Rz ) = Rz ( z H Rz ) = R T z

41 Outline 1 Maxwell Equations 2 Waves 3 Helmholtz Wave Equation 4 Dirac Delta Sources 5 Power 6 Matrices 7 Complex Analysis 8 Optimization

42 Level Sets Versus the Gradient

43 Method of Lagrange Multipliers Λ (x, y, λ) = f (x, y) + λ [g (x, y) c]

44 Method of Lagrange Multipliers - Example