Exercise 19 - OLD EXAM, FDTD

Transcription

1 Exercise 19 - OLD EXAM, FDTD A 1D wave propagation may be considered by te coupled differential equations u x + a v t v x + b u t a) 2 points: Derive te decoupled differential equation and give c in terms of a,b. = 0 (1) = 0 (2) c 2 2 v x 2 2 v t 2 = 0 (3) b) 3 points: One possible solution of (3) as te form { v (x,t) = Re V 0 e i(kx ωt)} (4) were V 0 denotes te complex amplitude, k te wavenumber and ω te angular frequency of te wave. Demonstrate tat te equations (1) - (3) are fulfilled by (4) and compute k for a given ω. c) 2 points: Compute u(x,t), wen te solution (4) olds. d) 3 points: Derive a FDTD sceme tat discretises (3) using central differences and give te stability criteria for a discretisation x, t in x and t. Total: 10 points 1

2 Exercise 20 - OLD EXAM, Electrostatics + FD Figure 1: A linear triangular element is presented. A coaxial cable is given in te figure above. Te total carge per lengt on te inner conductor wit radius a is +σ and te total carge per lengt on te outer conductor wit radius b is σ. Te gap between te conductors is filled wit a dielectric material wit te permittivity value of ε = ε r ε 0. a) 2 points: State te Gauss Law in Electrostatics and describe its pysical meaning. b) 2 points: Determine te electrical field in regions 1 and 2. c) 1 points: By using te result in part b, find te potential difference between te two conductors. a) 2 points: Derive te governing equation for te radial dependence of te potential distribution between te conductors and find an update sceme by using central difference formula. Set te first boundary value as V (r = a) = V 0 and use te result from c) to compute te second boundary value V (r = b). Total: 10 points 2

3 Exercise 21 - OLD EXAM, FEM Solutions A orn antenna wit an aperture of 1 m is feed by an air filled waveguide of l.5 m lengt, 0.4 m eigt and 0.1 m widt. Te TE 02 mode is simulated wit COMSOL Multipysics as depicted in Fig. 6.1(a)-(f). Te figures sow te E z fields wit a constant color scale. a) 2 points: Compute te cutoff frequencies for te first two modes. b) 3 points: Estimate te frequency of te depicted TE 02 mode from Fig. 6.1(a)-(f) c) 1 point: Wic of te six depicted results in figures 6.1(a)-(f) models te described problem most accurately? d) 5 points: Te oter five models contain one mistake eac - eiter by definition of boundary condition or material properties. Name te mistakes! e) 2 points: Sketc a suitable mes for te problem and explain it. f) 1 point: How can te computational effort be reduced witout canging te mes density? Give details. Total: 14 points 3

4 Exercise 22 - OLD EXAM, Multiple Coice Part Answer te following questions eiter in sort notes or by circling te rigt statement. a) 2 points: Wic of te following statements is not a correct finite difference approximation to dv dx at x 0 if = x? Name te oters. a) V (x 0+) V (x 0 ) b) V (x 0) V (x 0 ) c) V (x 0+) V (x 0 ) d) V (x 0+/2) V (x 0 /2) e) V (x 0+) V (x 0 ) 2 b) 2 points: Using te difference equation V n = V n 1 + V n+1 wit V 0 = V 5 = 1 and starting wit initial values V n = 0 for 1 n 4, te value of V 2 after te fourt iteration is: a) 1 b) 3 c) 4 d) 9 e) 25 c) 1 point: Te triangular element of Figure 3 is in free space. Te approximate value of te potential at te center of te triangle is a) 10 V b) 7.5 V c) 5 V d) 2.5 V d) 2 point: Te area of te element in Figure 3 is: a) 14 b) 8 c) 7 d) 4 e) 3 e) 1 point:1 Wic of tese statements is not true about finite element sape functions? a) Tey are interpolatory in nature. b) Tey must be continuous across te elements. c) Teir sum is identically equal to unity at every point witin te element. d) Te sape function associated wit a given node vanises at any oter node. e) Te sape function associated wit a node is zero at tat node. f) 1 point: A major difference between te finite difference and te finite element metods is tat: a) Using one, a sparse matrix results in te solution. b) In one, te solution is known at all points in te domain. c) One applies to solving partial differential equation. d) One is limited to time-invariant problems. g) 4 point: State two pysical problems, wic can be described by te Poisson equation, and two pysical problems for te Laplace equation. Total: 14 points 4

5 (a) Simulation 1 (b) Simulation 2 (c) Simulation 3 (d) Simulation 4 (e) Simulation 5 (f) Simulation 6 Figure 2: Comsol Multipysics simulation results in te xy plane. All simulation results depicted wit te same pase. (Scale in meter and V/m) 5

6 Figure 3: triangle in free space 6