(c) Please find out the time-average power flow per unit area normal to the direction of propagation. (6%)

5. (8%) For a transmission line system shown in Fig. 2, please find out the input impedance .

(a) For a uniform waveguide with arbitrary cross section as shown in Fig. 3(a), please derive the two- dimensional (2-D) vector phasor expression of transverse magnetic field components in terms of are 2-D vector phasors for Ez and Hz. (10%)

(b) For an air-filled metallic rectangular waveguide with the transverse dimensions a = 5.0cm and b = 3.0cm as shown in Fig. 3(b), if the operation wavelength is λ = 5.0cm, find out the modes which can appear on this waveguide. (10%)

(a) (10%) Let A be a square matrix of order n and A is normal. w is an arbitrary non-zero 1 X n column vector, proof that λ1is the largest eigenvalue corresponding to eigenvector where

(b) (10%) Let λ2 is an eigenvalue of A such that |λ2 -λ1| ＞|λp-λ1  whereλp is any eigenvalue of A other than λ2 and λ1. Describe a method to find λ2 and its associated eigenvector and prove that method in details.

(a) (10%) Calculate the Fourier transform of the function, You are supposed to show detailed calculations.

(b) (10%) Solve y from the differential equation under the condition that y(±∞) = 0

3. (20%) Three particles of identical mass m are connected by springs and attached to non-moving walls as shown in Fig. 1. All the springs are identical and have a spring constant k. The forces exerted on particles obey hooks law. Gravitation is neglected. Find the equation of motion for the three particles deviating from their equilibrium positions (x1, X2, x3 in Fig. 1), respectively and find the eigenfunctions and corresponding eigenvalues of the oscillator system.

(a) (10%) Evaluate the integral

(b) (10%) Evaluate the Fourier transform of the chirp functions,

115年 - 115 台灣聯合大學系統_碩士班招生考試_電機類：電磁學#139672

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