Solved Exercise/Numericals - Transmission Lines - 1.

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  =  Z₁  --> Z_in  =  Z_o  =  30 + j60  Ω

b) V_in  =  V_o  =  ( Z_in / (Z_in + Z_o) ) V_g  =  V_g / 2  =  7.5∠0^oV_rms

     I_in  =  I_p  =  V_g / (Z_g + Z_in)  =  V_g / 2Z_o  =  15∠0^o / 2 (30 + j60^o)  =  0.05∠-63.43^o A


c) Since Z_o  =  Z_r, Γ = 0 --> V_o^- = 0, V_o⁺ = V_o

The load voltage is V_L = V_s (z = l) = V_o⁺ e^-γl

e^-γl = V_o⁺ / V_L = 7.5 ∠0^o / 5 ∠ -48^o = 1.5 ∠48^o

e^αl e^jβl = 1.5 ∠48^o

e^αl = 1.5 --> α = (1 / l) ln (1.5) = (1/40) ln (1.5) = 0.0101

e^jβl = e^j48 --> β = (1 / l) (48^o / 180^o) π rad = 0.02094

γ = 0.0101 + j0.2094 / m