Department of Radio Science and Engineering
Course S-26.2900: Elements of Electromagnetic Field Theory and Guided Waves
Instructor: Prof. C. R. Simovski
Assistant: Dr. C. A. Valagiannopoulos
PROBLEM SET C: TRANSMISSION-LINE IMPEDANCES
PROBLEM 1 (Basic quantities)
A lossless transmission line with characteristic impedance 𝑅0 , has a length 𝐷 and is terminated by an arbitrary
load 𝑍𝐿 . (a) Find the expression of the wave voltage 𝑉(𝑧) and evaluate the ratio 𝜌(𝑧) of the reflecting wave over
the transmitting one (reflection coefficient). (b) Determine the ratio 𝑆 = |𝑉(𝑧)|𝑚𝑎𝑥 /|𝑉(𝑧)|𝑚𝑖𝑛 of the maximum
over the minimum voltage magnitude (standing wave ratio).
PROBLEM 2 (Voltage transformation ratio)
A lossless quarter-wave line section of characteristic resistance 𝑅0 is terminated with an inductive load
impedance 𝑍𝐿 = 𝑅𝐿 + 𝑗𝑋𝐿 . (a) Determine the input impedance 𝑍𝐼𝑁 by separating its real and imaginary parts. (b)
Find the ratio of the magnitude of the voltage at the input 𝑉𝐼𝑁 to that at the load 𝑉𝐿 as function of the parameters
𝑅0 , 𝑅𝐿 , 𝑋𝐿 . (c) Sketch the graph of the ratio |𝑉𝐼𝑁 |/|𝑉𝐿 | with respect to the real part 𝑅𝐿 for several imaginary ones
𝑋𝐿 and discuss the results.
PROBLEM 3 (Optimized solutions)
Consider a lossless transmission line. (a) Determine the line’s characteristic resistance 𝑅0 so that it will have a
minimum possible standing-wave ratio 𝛤 for a load impedance 𝑍𝐿 = 40 + 𝑗30 𝛺. (b) Find this minimum
standing-wave ratio and the corresponding voltage reflection coefficient. (c) Find the location of the voltage
minimum nearest to the load.
PROBLEM 4 (Impedance matching)
A transmission line characteristic impedance 𝑅0 = 50 𝛺 is to be
matched to a load impedance 𝑍𝐿 = 40 + 𝑗10 𝛺 through a length 𝐷′ of
another transmission line of characteristic impedance 𝑅′0 and
unitary wavelength 𝜆′0 = 1 𝑚 at the operating frequency. (a) Find the
required 𝐷′ and 𝑅′0 for matching. (b) Repeat the analysis for
arbitrary reactance Im[𝑍𝐿 ] = 𝑋𝐿 and consider three different
resistances 𝑅0 = 50, 60, 70 𝛺 and represent the solution 𝑅′0 as function of 0 < 𝑋𝐿 < 20 𝛺 .
PROBLEM SET C: TRANSMISSION-LINE IMPEDANCES
PROBLEM 5 (Inverse problem)
Consider a lossless transmission line. Obtain the formulas for finding the electrical length 𝐷/𝜆0 and the
terminating resistance 𝑅𝐿 of a lossless line having a (real) characteristic impedance 𝑅0 such that the input
impedance equals 𝑍𝐼𝑁 = 𝑅𝐼𝑁 + 𝑗𝑋𝐼𝑁 .
PROBLEM 6 (Time-dependent functions)
A sinusoidal voltage generator 𝑣𝐺 (𝑡) = 110 sin 𝜔𝑡 𝑉 and internal impedance 𝑍𝐺 = 50 𝛺 is connected to a
quarter-wave lossless line having a characteristic impedance 𝑅0 = 50 𝛺 that is terminated in a purely reactive
load 𝑍𝐿 = 𝑗50 𝛺. (a) Obtain voltage and current phasor expressions 𝑉(𝑧) and 𝐼(𝑧). (b) Write the instantaneous
voltage and current expressions 𝑣(𝑧, 𝑡) and 𝑖(𝑧, 𝑡). (c) Obtain the instantaneous power and the average power
delivered to the load.
PROBLEM 7 (Minimum reflection coefficient)
A lossless transmission line of characteristic impedance
𝑅0 = 50 𝛺 is terminated by a load of impedance 𝑍𝐿 = 100 −
𝑗100 𝛺. At distance 𝑊 = 0.6𝜆0 from it (𝜆0 is the operating
wavelength), a shunt capacitor with variable capacitance
0 < 𝐶 < 0.3𝜆0 /𝑣0 (𝑣0 is the speed of vacuum into the host material), has been installed. (a) Determine the line
onto which the input reflection coefficient is moving across the complex plane of the Smith chart. (b) Find the
capacitance 𝐶, for which minimum and maximum reflections are achieved. In the first case (minimum reflection),
how is the value of 𝐶 related to the imaginary part of the terminating load?
PROBLEM 8 (Reflection coefficient locus)
A lossless transmission line of characteristic impedance
𝑅0 = 50 𝛺 is terminated by a load of impedance 𝑍𝐿 = 15 −
𝑗15 𝛺. At parallel position to 𝑍𝐿 (see figure), a purely imaginary
impedance of inductive nature with 0 < Im[𝑍𝐸 ] < 𝑅0 has been
connected. In addition, an open-circuited series stub of the
same transmission line and variable length 𝜆0 /4 < 𝑊 < 𝜆0 /3, has been employed in order to modify the input
impedance. Using the Smith chart, determine the region containing all the possible values of the input reflection
coefficient.
PROBLEM SET C: TRANSMISSION-LINE IMPEDANCES