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SMG-8306 Transmission lines and waveguides

Small Exam I, February 3rd 2012. Answer to three of the four guestions.
Each guestion gives in maximum 4 points
Jari Kangas

1. (a) It can be shown that the propagation of signals can be analyzed using the 2nd order wave
eguation
PV PV
Vo) VG) m
02? ot?
i. Consider a reasonable starting point and derive the wave eguation.

 

 

ii. Assume a lossless line, find a solution to the eguation.

(b) Consider how to approximate the derivatives in egn. (1) and present a numerical scheme
in view of the FDTD-codes (similarly as was done with the leap-frog scheme).

(4p.)
2. (a) Express the Poynting theorem and give interpretation of it and its terms.
(b) Correct or incorrect? To get points, support your answer by an argument or an example.
i. Surface current is an idealization that helps in analyzing cases that include very good
conductors at high freguencies.
ii. Consider a transmission line (whose characteristic impedance is Zo) of length 1 con-
nected to a load (characterized as Z). The impedance seen at the generator end

is
ZL + Zotanhyl

Zi=20>—1m1
ä Va — Zi tanh yl
(4 p.)
3. (a) Prepare a concise description about the core assumptions inherent in analysis of transmis-
sion lines.

+ (b) Outline a procedure to find out the parameters of a lossless transmission line.
(4 p.)
4. (a) Define in a few words:
i. phase velocity
ii. guasi-TEM wave
(b) Lets consider a plane wave that is travelling along 2-axis and whose electric field is E(x,t) =

Eo cos(wt — kz)j. Let it travel in medium whose relative permittivity is 4 and relative
permeability 1. Let freguency be 3.0GHz and Eo = 30V/m.
2

i. Find the amplitude and direction of magnetic field. .
ii. Find the phase velocity and wavelength.

iii. Find the phase shift (in degrees) between positions 21 = 0.5m and 22 = 1.7m.

(4 p.)