Transmission Line Impedance Solutions for EEC 130A Students
School
University of California, Davis**We aren't endorsed by this school
Course
EEC 130A
Subject
Electrical Engineering
Date
Dec 10, 2024
Pages
3
Uploaded by JusticeStarJellyfish135
EEC 130A PSS2 Solutions FQ22 1 Problem references: “FAE” refers to the text: Fundamentals of Applied Electromagnetics. Some of the following problems are borrowed from the 7th edition of the textbook. Problem 1 In this problem we will consider the wave impedance of a transmission line as the length of the line is varied. Consider the short-circuited, 50Ωair transmission line shown below. (a)Find the shortest electrical length of the line (that is, the line length in terms of the wavelength) such that (i) 𝑍?𝑛= 𝑗50Ω, (ii) 𝑍?𝑛= −𝑗150Ω, and (iii) 𝑍?𝑛= 0Ω. (b)What type of circuit element does the transmission line behave as in parts (i) and (ii) above? Assuming an operating frequency of 1GHz, what are the values of these circuit elements? (a)Find the 𝑍?𝑛for length 𝑙as 𝑍?𝑛= 𝑍0𝑍𝐿cos(𝛽𝑙) + 𝑗𝑍0sin(𝛽𝑙)𝑍0cos(𝛽𝑙) + 𝑗𝑍𝐿sin(𝛽𝑙)As 𝑍𝐿= 0we have 𝑍?𝑛= 𝑍0𝑗𝑍0sin(𝛽𝑙)𝑍0cos(𝛽𝑙)= 𝑗𝑍0tan(𝛽𝑙)(i)𝑍?𝑛= 𝑗50Ω, thus tan(𝛽𝑙) =𝑍𝑖𝑛?𝑍0= 1, 𝛽𝑙 =2𝜋𝑙𝜆=𝜋4for shortest 𝑙, 𝑙min=𝜆8(ii)𝑍?𝑛= −𝑗150Ω, tan(𝛽𝑙) =𝑍𝑖𝑛?𝑍0= −3, as tan(𝛽𝑙) ≤ 0, 𝑙min=𝜆(tan−1−3)2𝜋+𝜆2= 0.301(iii)𝑍?𝑛= 0Ω, tan(𝛽𝑙) =𝑍𝑖𝑛?𝑍0= 0, similar as (ii), tan(𝛽𝑙) ≤ 0, 𝑙min=𝜆(tan−10)2𝜋+𝜆2=𝜆2(b)For (i), 𝑍?𝑛is imaginary with positive phase, corresponding to inductance. The impedance of inductor is 𝑍𝐿= 𝑗𝜔𝐿 = 𝑗50Ω, 𝜔 = 2𝜋𝑓 = 2𝜋 × 109[rad/s]𝐿 =502𝜋 × 109= 7.96 × 10−9[H] = 7.96 [nH](c)For (ii), 𝑍?𝑛is imaginary with negative phase, corresponding to capacitance. (d)The impedance of capacitor is 𝑍𝐿=1?𝜔𝐶= −𝑗150Ω, 𝜔 = 2𝜋𝑓 = 2𝜋 × 109[rad/s]𝐶 =12𝜋 × 109× 150= 1.06 × 10−12[F] = 1.06 [pF]
EEC 130A PSS2 Solutions FQ22 2 Problem 2 (FAE P2.33) Two half-wave dipole antennas, each with an impedance of 75Ω, are connected in parallel through a pair of transmission lines, and the combination is connected to a feed transmission line, as shown in Fig. 1. All lines are 50Ωand lossless. Figure 1: Circuit for Problem 2. (a)Calculate 𝑍?𝑛1, the input impedance of the antenna-terminated line, at the parallel juncture. (b)Combine 𝑍?𝑛1and 𝑍?𝑛2in parallel to obtain 𝑍𝐿′, the effective load impedance of the feedline. (c)Calculate 𝑍?𝑛of the feedline. (a)Refer to textbook, equation (2.79), 𝑍?𝑛1= 𝑍0𝑍𝐿cos(𝛽𝑙) + 𝑗𝑍0sin(𝛽𝑙)𝑍0cos(𝛽𝑙) + 𝑗𝑍𝐿sin(𝛽𝑙)Substitute 𝑙 = 0.2𝜆, 𝛽 =2𝜋𝜆, 𝑍0= 50Ωand 𝑍𝐿= 75Ω, we have 𝑍?𝑛1= 5075 + 𝑗50 tan (2𝜋𝜆× 0.2𝜆)50 + 𝑗75 tan (2𝜋𝜆× 0.2𝜆)= (35.2 − 𝑗8.62)[Ω](b)Combine impedance in parallel we have 𝑍𝐿′=𝑍?𝑛1𝑍?𝑛2𝑍?𝑛1+ 𝑍?𝑛2=(35.2 − 𝑗8.62)22(35.2 − 𝑗8.62)= (17.60 − 𝑗4.31)[Ω](c)Use 𝑍𝐿′to represent the pair of transmission lines Equivalent circuit Input impedance of feedline is
EEC 130A PSS2 Solutions FQ22 3 𝑍?𝑛= 𝑍0𝑍𝐿′cos(𝛽𝑙) + 𝑗𝑍0sin(𝛽𝑙)𝑍0cos(𝛽𝑙) + 𝑗𝑍𝐿′sin(𝛽𝑙)= 50(17.60 − 𝑗4.31) + 𝑗50 tan (2𝜋𝜆× 0.3𝜆)50 + 𝑗(17.60 − 𝑗4.31) tan (2𝜋𝜆× 0.3𝜆)= (107.57 − 𝑗56.7)[Ω]