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microwaves Solutions of series 2 Problem 1 A transmission line has the following per unit length parameters : L=0.2 µH/m C=300pF/m R=5Ω/m G=0.01S/m Compute the propagation constant and characteristic impedance of this line at 0.5 GHz. Recompute these quantities in a lossless case. Solution : γ = α + jβ = ( R + jω L )( G + jωC ) = ( 5 + j 628 )( 0.01 + j 0.94 ) = 24.3∠89.465° = 0.23 + j 24.3 Thus α = 0.23 Np / m β = 24.3 rad / m Zo = R + jω L = 25.8 + 0.3 Ω G + jω L Lossless case : R=G=0 Z o = 25.8 Ω α =0 β = 24.3 Problem 2 : A lossless transmissionline is terminated with a 100 Ω load. If the SWR of the line is 1.5, find the two possible values for the characteristic impedance of the line Solution : swr − 1 0.5 = = 0.2 swr + 1 2.5 100 − Z o Z − Zo Γ = L = 100 + Z o Z L + Zo moreover, we know that Zo is real, thus either 100 − Z o = 0.2 ⇒ Z o = 66.7Ω 100 + Z o or 100 − Z o = −0.2 ⇒ Z o = 150Ω 100 + Z o Γ = Problem 3 : A radio transmitter is connected to an antenna having an impedance 80+j40 Ω. with a 50 Ω coaxial cable. If the transmitter can deliver 30W to a 50 Ω load, how much power is delivered to the antenna ? Solution : Z − Z o 30 + j 40 Γ= L = = 0.367 ∠36° Z L + Z o 130 + j 40 ( PLoad = Pin − Preflected = Pin 1 − Γ 2 ) = 30 (1 − ( 0.367) ) = 25.9 2 Problem 4 : Use the Smith chart to fond the following quantities for the transmission line circuit bellow : a) the SWR of the line b) the reflection coefficient at the load c) the load admittance d) the input impedance of the line l=0.4λ Zo=50 Zin ZL=90+j60 Solution : see Smith Chart a) SWR = 2.8 G= 0.45 ∠ 33° c) yL=0.38-j0.25 => YL=7.6-j5 mS d) zin=0.54+j0.63 => Zin=27+j32 Ω b)