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Unformatted text preview: City University of Hong Kong Electronic Engineering Semester B 2007-2008 EE3108 Engineering Analysis Tutorial 6 Numerical differentiation is often needed to treat problems that involve iterative solutions. Consider the following characteristic equation that governs a propagation parameter x of optical waveguides: x V 2 − x2 , tan = 2 x
where V is a design parameter related to the wavelength, the waveguide thickness, and the index of refraction. The propagation parameter x is related to the velocity of wave propagation. Its derivative dx/dV is related to the dispersion of waves. This tutorial aims to arrive at a plot of dx/dV. In doing so, we need to first find x over a range of values of V. I. Iteration solution of x The solution of x for any given V can be obtained iteratively. Modify the following files in order to plot x against V. The function file wg_x(V) should be an user-defined function based on the Newton’s method. %Function for waveguide equation function out=wg_x(V) x=1e-2; d=5e-4; % fx = -sqrt(V^2-x^2)/x+tan(x/2); df_dx = x = error = out=x % %file: t0601 clear; V=[1:0.01:3]; % plot(V,x); xlabel('V'); ylabel('x'); EE 3108 Semster B 2007/2008 T6-1 S C Chan II. Numerical differentiation The first-order derivative can be performed using the centered difference technique. Modify the following script file to plot dx/dV against V. %file: t0602 clear; V=[1:0.01:3]; % m=length(V); h=(V(m)-V(1))/(m-1); %Centered difference for i=1:m if elseif else end end %Plot plot(V,Dx); xlabel('V'); ylabel('dx/dV'); % 1.9 0.75 1.8 0.7 1.7 0.65 1.6 0.6 1.5 0.55 dx/dV 1.4 0.5 x 1.3 0.45 1.2 0.4 1.1 0.35 1 0.3 0.9 0.25
1 1.2 1.4 1.6 1.8 2 V 2.2 2.4 2.6 2.8 3 1 1.2 1.4 1.6 1.8 2 V 2.2 2.4 2.6 2.8 3 T6-2 ...
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This note was uploaded on 01/11/2011 for the course EE 3108 taught by Professor Nelsonszechunchan during the Spring '07 term at City University of Hong Kong.
- Spring '07