Analytical Methods

Full Wave Analysis "Full-wave" means that the solution is obtained by Maxwell's equations for fields in time or frequency domain. This type of rigorous analysis, when performed for general 3D case, makes no simplifying assumptions about nature of the EM problem. Other EM analysis methods, such as those based on ray-tracing, usually make an assumption about the problem, such as the object being modeled is large compared to a wavelength. Hybrid codes combine a full-wave method with another method (sometimes called high-frequency or asymptotic) to increase the range of problems that can be solved. Full wave analysis means: "exact analytical (not numerical) solution without any approximation“ For example, when we solve an aperture antenna approximately by means of infinite ground concept we don't make a full wave solution. 

Separation of Variables Method Separation of variables sometimes called the method of Fourier for solving a PDE: Separation of variables in rectangular coordinates for Laplace’s Equations: Dirichlet boundary conditions: After applying boundary condition: Using superposition:

Separation of Variables Method Separation of variables in rectangular coordinates for wave equation: After applying boundary condition:

Separation of Variables Method Another Example: Method of separation of variables for a problem with four inhomogeneous boundary conditions: Using superposition: = + + +

Separation of Variables Method Separation of Variables in Cylindrical Coordinates: Laplace’s Equation in Cylindrical Coordinates : After ... We have: by the superposition principle to form a complete series solution:

Separation of Variables Method Wave Equation in Cylindrical Coordinates: Dividing both sides by ρ2 : Using: Assuming x=λρ and R=y is Bessel’s equation

Separation of Variables Method Bessel’s equation has a general solution: We may replace x by jx, modified Bessel’s equation is generated: Modified Bessel’s equation has a general solution: Jn & Yn are first and second kinds of order n Yn is also called the Neumann function In & Kn are first and second kinds of order n

Separation of Variables Method Gamma function Final solution is:

Separation of Variables Method Asymptotic expressions:

Separation of Variables Method Example: A plane wave E=az Eo e−jkx is incident on an infinitely long conducting cylinder: Determine the scattered field. Solution: Integrating over 0≤φ≤2π gives: Taking the m-th derivative of both sides: Since Esz must consist of outgoing waves that vanish at infinity, it contains: Hence:

Separation of Variables Method Solution (cont.): The total field in medium 2 is: While the total field in medium 1 is zero, At ρ=a, the boundary condition requires that the tangential components of E1 and E2 be equal. Hence:

Separation of Variables in Spherical Coordinates Laplace’s equation for finding potential due to an uncharged conducting sphere located in an external uniform electric field: To solve (1): To solve (2): is Cauchy-Euler equation (1) (2)

Separation of Variables in Spherical Coordinates Making these substitutions: Its solution is obtained by the method of Frobenius as: Legendre differential equation Legendre functions of the first and second kind

Separation of Variables in Spherical Coordinates Qn are not useful since they are singular at θ=0, π We use Qn in problems having conical boundaries that exclude the axis from the solution region. For this problem, θ=0,π is included so that: After determining An and Bn , using boundary conditions:

Separation of Variables in Spherical Coordinates Wave Equation: is one of: Using: spherical Bessel functions ordinary Bessel functions

Separation of Variables in Spherical Coordinates By replacing H=y & cosθ=x, the second equation, which is Legendre’s associated differential equation, can be rewritten: General solution is: Using ordinary Legendre functions: Legendre functions of the first and second kind

Separation of Variables in Spherical Coordinates Example: A thin ring of radius a carries charge of density ρ. Find V(P(r, θ,φ) at: (a) point P(0, 0, z) on the axis of the ring, (b) point P(r, θ,φ) in space. Solution: From elementary electrostatics, at P(0, 0, z): Using ∇2V=0 having boundary-value solution in P(0, 0, z): Since Qn is singular at θ=0,π,Bn=0, Thus: For 0≤r≤a, Dn=0 since V must be finite at r=0: Using boundary-value:

Separation of Variables in Spherical Coordinates Solution (cont.): For r≥a, Cn=0 since V must be finite as r→∞: Therefore: ?

Separation of Variables in Spherical Coordinates Example: A PEC spherical shell of radius 𝑎 is maintained at potential Vo cos 2φ Determine V at any point inside the sphere. Solution: Solution is similar to that of previous problem Using the boundary condition:

Orthogonal Functions Orthogonal functions, including Bessel, Legendre, Hermite, Laguerre, and Chebyshev, are useful in solution of PDE. They are very useful in series expansion of functions such as Fourier- Bessel series, Legendre series, etc. Bessel and Legendre functions are importance in EM problems. A system of real functions φn(n=0,1,2,…) is said to be orthogonal with weight w(x) on (a, b) if: For example, the system of functions cos(nx) is orthogonal:

Orthogonal Functions An arbitrary function f(x), defined over interval (a, b), can be expressed in terms of any complete, orthogonal set of functions: Where: Others are in Table 2.5

Orthogonal Functions Example: Expand f(x) in a series of Chebyshev polynomials. Let: Since f(x) is an even function, odd terms in expansion vanish: By multiply both sides by: and integrate over −1≤ x≤1:

Series Expansion As noticed, PDE can be solved with the aid of infinite series and generally, with series of orthogonal functions. An example: Let the solution be of the form: by substituting this equation into PDE:

Practical Applications Scattering by Dielectric Sphere: It is illuminated by a plane wave propagating in z direction and E polarized in x direction. As a similar way: Using: = …

Practical Applications Scattering by Dielectric Sphere (cont.): Form of the incident fields: Form of the scattered fields:

Practical Applications Scattering by Dielectric Sphere (cont.): Similarly, transmitted field inside the sphere: Continuity of tangential components at surface: Radar Cross Section (RCS): By using asymptotic expression (R=∞) for spherical Bessel functions: Where amplitude functions S1(θ) and S2(θ) are given as:

Practical Applications Scattering by Dielectric Sphere (cont.): Using: Similarly, forward-scattering cross section:

Practical Applications Attenuation Due to Raindrops: At f>10GHz, attenuation caused by atmospheric particles (oxygen, ice crystals, rain, fog, and snow) can reduce the reliability and performance of radar and space communication links. We will examine propagating through rain drops: By Mie solution, the attenuation and phase shift of an EM wave by raindrops having spherical shape (for low rate intensity) is investigated. For high rain intensity, an oblate spheroidal model would be more realistic as: EM attenuation due to travel a wave through a homogeneous medium (with N identical spherical particles/m3) in a distance 𝑙 is given by e−γ 𝑙. γ is attenuation coefficient as [11]: To relate attenuation and phase shift to a realistic rainfall, drop-size distribution (D) for a given rate intensity N(D) must be known. Total attenuation and phase shift over the entire volume: [11] H.C. Van de Hulst, “Light Scattering of Small Particles”. New York: JohnWiley,1957, pp. 28–37, 114–136, 284.

Practical Applications Attenuation Due to Raindrops (cont.): Laws and parsons drop-size distributions for various rain rates Raindrop terminal velocity