Lossles Line

Lossles Line - Special Cases of the Lossless Line...

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Unformatted text preview: Special Cases of the Lossless Line Short-Circuited Line Transmission line terminated in a short Circuit: (a) schematic representation, (b) Normalized voltage on the line, (c) normalized Current, and (d) normalized input impedance V ( z ) = Vo+ ( e − j β z + Γe jβ z ) Vo+ − j β z I ( z) = ( e − Γe j β z ) Zo Γ = -1 ~ Vsc ( z ) = Vo+ ⎡ e − j β z − e j β z ⎤ = 2 jVo+ sin β z ⎣ ⎦ Vo+ − j β z 2Vo+ jβ z ⎡e +e ⎤ = I sc ( z ) = ⎦ Z cos β z Zo ⎣ o ~ ~ sc Z in = Vsc (−l ) ~ I sc ( −l ) = jZ o tan β l Short Circuit If tanβl ≥0, the line appears inductive jω Leq = jZ o tan β l , if tan β l ≥ 0 Leq = Z o tan β l ω (H) If tanβl ≤0, the line appears capacitive 1 = jZ o tan β l if tan β l ≤ 0 jωCeq 1 (F) Ceq = − Z oω tan β l Open Circuit Transmission line terminated in an open circuit: (a) schematic representation, (b) normalized Voltage on the line, (c) normalized current, and (d) Normalized input impedance Γ=1 ~ Vsc ( z ) = Vo+ ⎡ e− j β z + e j β z ⎤ = 2Vo+ cos β z ⎣ ⎦ −2 jVo+ Vo+ − j β z jβ z ⎡e −e ⎤ = I sc ( z ) = sin β z ⎦ Zo ⎣ Zo ~ ~ oc Z in = Voc (−l ) ~ I oc (−l ) = − jZ o cot β l Power Flow on a Lossless Transmission Line Instantaneous Power carried by the incident wave as it arrives at the load ⎡ ~ i jωt ⎤ ⎡ ~i jωt ⎤ P (t ) = v (t ) ⋅ i (t ) = Re ⎢V e ⎥ ⋅ Re ⎢ I e ⎥ ⎣ ⎦ ⎣ ⎦ Assume: + o + o V =V e i jφ + + V ( z ) = Vo+ e jφ e − j β z = Vo+ e jφ I ( z = 0) = i ⎡ Vo+ ⎤ jφ jωt ⎤ jφ + jωt + = Re ⎡ Vo e e ⋅ Re ⎢ ee⎥ ⎣ ⎦ Zo ⎢ ⎥ ⎣ ⎦ + ~ ~ i + o V Zo e + z =0 = Vo+ cos (ωt + φ + ) ⋅ jφ + = +2 o V Zo Vo+ Zo cos (ωt + φ + ) cos 2 (ωt + φ + ) (W) Reflected power carried by the incident wave as it arrives at the load v (t ) = Γ V cos (ωt + φ + θ r ) r i (t ) = − Γ r + o + + o V Zo cos (ωt + φ + θ r ) + P r (t ) = v r (t ) ⋅ i r (t ) = Re ⎡V r e jωt ⎤ ⋅ Re ⎡ I r e jωt ⎤ ⎣ ⎦ ⎣ ⎦ =−Γ 2 Vo+ Zo 2 cos 2 (ωt + φ + + θ r ) (W) Time-Average Power i Pav = = 1 ω i ∫0 P (t )dt = 2π ∫0 T 2π T +2 o ωV 2π Z o ∫ 2π ω 0 ω Vo+ 2 Zo cos 2 (ωt + φ + )dt +2 o cos(2ωt + 2φ ) + 1 ωV dt = 2 2π Z o + 2 + ω Vo ⎧ π 1 ⎫ ⎡sin(4π + 2φ + ) − sin 2φ + ⎤ ⎬ = ⎨− ⎦ 2π Z o ⎩ ω 4ω ⎣ ⎭ i Pav = Vo+ 2 2Z o r Pav = − Γ 2 Vo+ 2 i = − Γ Pav 2 2Z o Net average power delivered to the load i r Pav = Pav + Pav = Vo+ 2 2Zo −Γ 2 Vo+ Zo 2 = Vo+ 2π / ω ⎡ 1 1 sin ( 2ωt + 2φ + ) ⎤ ⎢ t− ⎥ 22 2ω ⎢ ⎥ ⎣ ⎦0 2 ⎡1 − Γ 2 ⎤ (W) ⎦ 2Z o ⎣ ...
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