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 π 0 , has a length π· and is terminated by an arbitrary load ππΏ . (a) Find the expression of the wave voltage π(π§) and evaluate the ratio π(π§) of the reflecting wave over the transmitting one (reflection coefficient). (b) Determine the ratio π = |π(π§)|πππ₯ /|π(π§)|πππ 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 π 0 is terminated with an inductive load impedance ππΏ = π πΏ + πππΏ . (a) Determine the input impedance ππΌπ by separating its real and imaginary parts. (b) Find the ratio of the magnitude of the voltage at the input ππΌπ to that at the load ππΏ as function of the parameters π 0 , π πΏ , ππΏ . (c) Sketch the graph of the ratio |ππΌπ |/|ππΏ | with respect to the real part π πΏ for several imaginary ones ππΏ and discuss the results. PROBLEM 3 (Optimized solutions) Consider a lossless transmission line. (a) Determine the line’s characteristic resistance π 0 so that it will have a minimum possible standing-wave ratio π€ for a load impedance ππΏ = 40 + π30 πΊ. (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 π 0 = 50 πΊ is to be matched to a load impedance ππΏ = 40 + π10 πΊ through a length π·′ of another transmission line of characteristic impedance π ′0 and unitary wavelength π′0 = 1 π at the operating frequency. (a) Find the required π·′ and π ′0 for matching. (b) Repeat the analysis for arbitrary reactance Im[ππΏ ] = ππΏ and consider three different resistances π 0 = 50, 60, 70 πΊ and represent the solution π ′0 as function of 0 < ππΏ < 20 πΊ . PROBLEM SET C: TRANSMISSION-LINE IMPEDANCES PROBLEM 5 (Inverse problem) Consider a lossless transmission line. Obtain the formulas for finding the electrical length π·/π0 and the terminating resistance π πΏ of a lossless line having a (real) characteristic impedance π 0 such that the input impedance equals ππΌπ = π πΌπ + πππΌπ . PROBLEM 6 (Time-dependent functions) A sinusoidal voltage generator π£πΊ (π‘) = 110 sin ππ‘ π and internal impedance ππΊ = 50 πΊ is connected to a quarter-wave lossless line having a characteristic impedance π 0 = 50 πΊ that is terminated in a purely reactive load ππΏ = π50 πΊ. (a) Obtain voltage and current phasor expressions π(π§) and πΌ(π§). (b) Write the instantaneous voltage and current expressions π£(π§, π‘) and π(π§, π‘). (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 π 0 = 50 πΊ is terminated by a load of impedance ππΏ = 100 − π100 πΊ. At distance π = 0.6π0 from it (π0 is the operating wavelength), a shunt capacitor with variable capacitance 0 < πΆ < 0.3π0 /π£0 (π£0 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 πΆ, for which minimum and maximum reflections are achieved. In the first case (minimum reflection), how is the value of πΆ related to the imaginary part of the terminating load? PROBLEM 8 (Reflection coefficient locus) A lossless transmission line of characteristic impedance π 0 = 50 πΊ is terminated by a load of impedance ππΏ = 15 − π15 πΊ. At parallel position to ππΏ (see figure), a purely imaginary impedance of inductive nature with 0 < Im[ππΈ ] < π 0 has been connected. In addition, an open-circuited series stub of the same transmission line and variable length π0 /4 < π < π0 /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