FINAL EXAM IN FYS-3007

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1 Page 1 of 4 pages + chart FINAL EXAM IN FYS-007 Exam in : Fys-007 Microwave Techniques Date : Tuesday, May 1, 2011 Time : Place : Åsgårdveien 9 Approved remedies : All non-living and non-communicating aides (text book, notes, etc., including calculator ) The exam contains 5 pages including this cover page Contact person: Åshild Fredriksen, (office) (mobile)

2 Page 2 of 4 pages + chart Problem # 1. A load Z L = 200 Ω is to be matched to a lossless transmission line with characteristic impedance Z 0 = 50 Ω, see Fiig. 1. The line is connected to a matched generator. You can use the attached Smith s diagram whenever you find it applicable in order to simplify the calculations. Z 0 = 50 Ω Z L = 200 Ω z = - l z= 0 z Figure 1. Terminated transmission line. (a) Assume that the load is connected directly to the line as shown in Fig. 4. What is the reflection coefficient Γ and the standing wave ratio SWR at the load? Will Γ and SWR be changing along the line? Will there be a maximum and a minimum voltage along the line? Give the reason behind your answers. (b) A λ/4-transformer is used to achieve impedance matching to a 50 Ω transmission line, as shown in Fig. 2. Find the characteristic impedance Z required on the transformer to obtain matching. What are the reflection coefficients Γ 1 and Γ 2 obtained at the connection point between the transmission line and the transformer? Z 0 = 50 Ω Γ 1 Γ 2 Z L = 200 Ω z = - λ/4 z = 0 z Figure 2. Load matched with a λ/4-transformer. (c) Instead of a λ/4-transformer, we now use a series stub, formed as a part of a transmission line with the same characteristic impedance as the main line, as shown in Fig.. Find the position d and the length l of the stub as a function of the wavelength on the line (select the smallest one of the two possible values of d).

3 Page of 4 pages + chart l 50 Ω d Z 0 = 50 Ω 50 Ω Z L = 200 Ω Figure. Impedance matching with a series stub. d) Increase the wavelength to 1.05 λ in the circuit of c). What are the new values of the stub susceptance and the SWR on the line (which is matched to the wavelength λ)? e) What is the fractional bandwidth of this impedance matching, given that 0.05 λ is the maximum deviation from the matched wavelength? A 4-port network has the scattering matrix [ S] Problem # 2. 0 α jβ 0 a 0 0 jβ = jβ 0 0 α 0 jβ α 0 a) What properties of this network can you find from the matrix? Verify your answer by calculations if necessary. What kind of microwave component is described by this matrix? b) If the device is lossless, show that α 2 + β 2 = 1. c) For β = c, write up the [S]-matrix in terms of c. With c = ½, and input power P 1 = 50 W, find the power at port 2,, and 4. d) Perform the operation and calculations necessary in order to transform the component into an anti-symmetric coupler. How would you do it in practice? A waveguide directional coupler is to be constructed by placing two rectangular waveguides on top of each other with the widest walls next to each other and a round hole with radius r in the common side walls to provide the wave coupling between them, as shown in Fig. 4. The coordinates of the hole is given as (x, y, z) = (s, b, 0), and a TE wave mode is assumed. The

4 Page 4 of 4 pages + chart 2r electric and magnetic polarizabilities of the hole is α = the waveguide a = 12 cm, and the height b = 4 cm. e 0 4r and α 0 m =. The width of y x 4 s a 1 b 2 z = 0 Figure 4. A waveguide coupler. z e) What is the cutoff wavelength λ c of the TE mode of the waveguide? Find the wavelength λ 0, wave number k 0, and propagation constant β 0 for this mode. Show that the wave 7μ0c impedance can be written Z = for f 0 = 1.4f c. 2 6 f) Write up the electric and magnetic polarization currents P e and P m, respectively, at the hole of the coupler, and explain the physical principles of wave coupling through the hole. Use the general result that jω AZ πs μ α πs π 2πs sin sin cos ab a Z a a a m 2 = εα 0 e β A jω AZ πs μ α πs π 2 πs sin sin cos ab a Z a a a m 2 = εα 0 e β A where A is the amplitude of the incoming wave, ω = 2π f 0 is the angular frequency, c is the speed of light, ε = ε 0 permittivity, and μ = μ 0 permeability of waveguide. Indicate the ports, + for which the A and A amplitudes are propagating. g) Find the position of s and aperture r 0 of the coupler, for which port is the isolated port and the wave length λ 0 = 2π/k 0.

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