3. (10%) The input impedance of a short-cireuited transmission line of length / is
A resonant system is formed by connecting two short-circuited transmission lines as follows
At the junction, . Please show that

(a) (5%) Which wave propagates in the z direction?

(b) (5%) Which wave has an electric field that is along the z direction?

(c) (5%) Find βi (in rad/m)?

(d) (5%) Find f2 (in Hz)?

(e) (5%) What is the wavelength of wave ①?

(f) (5%) What is the w elength of wave ②?

(8) (5%) If the time-averaged power flow density of wave① is P1, what is the time-averaged power flow density of wave②?

2. (15%) A parallel-plate transmission line consists of an arrangement of two perfect dielectrics, as shown by the transverse cross section in the figure below. Note that μ1ε1 =μ2ε2, so that the TEM waves propagating in the two dielectrics are in phase at all points along the interface between the dielectrics. Neglect finging of fields and compute the values of L, C, and Z0 of the line. (C is the capacitance per unit length; L is the inductance per unit length.)

(Ⅰ) Please determine the lowest five modes for the following cases: (i) b/a=1 (5%);(i) b/a=1/2 (5%); (iii) b/a=1/3 (5%).

(II) (5%) Given a = 3 cm and free space for the dielectric in the waveguide, please find the propagating modes for f = 9000 MHz in the case (i).

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