ElectroMagnetic Theory Concepts & Solved Numericals

Q.4) A 70 Ω lossless line has s = 1.6 and θ Γ = 300 o . If the line is 0.6λ long, obtain (a) Γ, Z L , Z in (b) The distance of the first minimum voltage from the load Answer: a) Using the smith chart, locate S at s = 1.6. Draw a circle of radius OS. Locate P where θ Γ = 300 o . At P, | Γ | = OP / OQ = 2.1 cm / 9.2 cm = 0.228 Γ = 0.228 ∠300 o Also at P, Z L = 1.15 – j0.48, Z L = Z o  Z L = 70 (1.15 – j0.48) = 80.5 – j33.6 Ω l = 0.6 λ = 0.6 x 720 o = 432 o = 360 o + 72 o From P, move 432 o to R. At R, Z in = 0.68 – j0.25 Z in = Z o  Z in = 70 (0.68 – j0.25) = 47.6 – j17.5 Ω b) The maximum voltage (the only one) occurs at θ Γ = 180 o ; its distance from the load is      (180 – 60) λ / 720  =   λ / 6  =   0.1667 Ω Q.5) A lossless 60 Ω line is terminated by a 60 + j60 Ω load. (a) Find Γ and s. If Z in = 120 - j60 Ω, how far (in terms of wavelengths) is the load from the generator? Solve this without using the Smith chart

Q.1) A transmission line operating at 500 MHz has Z o = 80Ω, α = 0.04Np/m, β = 1.5 rad/m. Find the line parameters R, L, G and C. Answer: Since Z o is real & α ǂ 0, this is a distortionless line. R o = α Z o = 0.04 x 80 = 3.2 Ω / m G = α / Z o = 0.04 / 80 = 5 x 10-4 Ω / m L = β Z o / ω = 1.5 x 80 / (2 π x 5 x 108 ) = 38.2 nH / m C = L G / R = 38.2 x 10-9 x 5 x 10-4 / 3.2 = 5.97 pF / m Q.2) A telephone line R = 30 Ω /km, L = 100 mH/km, G = 0 , and C = 20 µF/km. At f = 1 kHz, obtain: a) The characteristics impedance of the line. b) The propagation constant. c) The phase velocity. Answer: Q.3) A 40 m long transmission line has V g = 15 ∠0 o V rms , Z o = 30 + j60 Ω, and V L = 5 ∠-48 o V rms . If the line is matched to the load, calculate: a) The input impedance Z in b) The sending end current I in and voltage V in c) The propagation constant γ Answer: a) Z g  =  Z 1  --> Z in  =  Z o  =  30 + j