standing wave

standing wave - Standing Waves Standing Waves V ( z ) = Vo+...

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Unformatted text preview: Standing Waves Standing Waves V ( z ) = Vo+ ( e − jβ z + Γe jβ z ) Vo+ − jβ z I ( z) = ( e − Γe jβ z ) Zo Let's find V ( z ) and replace Γ = Γ e jθr { V ( z ) = V ( z ) ⋅ V ( z ) = ⎡V ⎣ * + o (e − jβ z +Γe jθ r ⋅e jβ z 1/ 2 2 j θ +2 β z − j θ +2 β z V ( z ) = V ⎡1 + Γ e ( r ) + Γ e ( r ) + Γ ⎤ ⎣ ⎦ Since e jx + e − jx = 2 cos x + o 1/ 2 2 V ( z ) = V ⎡1 + Γ + 2 Γ cos(θ r + 2 β z ) ⎤ ⎣ ⎦ Similarly, + o I ( z) = Vo+ )⎤ ⋅ ⎡(V ) ( e ⎦⎢ ⎣ 1/ 2 ⎡1 − Γ − 2 Γ cos(θ r + 2 β z ) ⎤ ⎦ Zo ⎣ 2 +* o jβ z +Γe − jθ r ⋅e − jβ z )⎤} ⎥ ⎦ 1/ 2 Standing Waves Standing-wave pattern for (a) V ( z ) and (b) I ( z ) for a lossless transmission line of characteristic impedance Zo = 50 Ω, terminated in a load with a reflection coefficient o Γ = 0.3e j 30 . The magnitude of the Vo+ = 1V . The incident wave standing-wave ratio is S= V V max min = 1.3/ 0.7 = 1.86 Standing Waves Voltage standing-wave patterns for (a) a matched load, (b) a short-circuited line, and (c) an open-circuited line Note: i) Repetition of waves in λ, then the repetition of standing waves is λ/2. ~ ii) ~ V&I are opposite in phase (evident by + and -) iii) Peak-to-peak pattern depends on Γ iv) | Γ| =0 no reflection no standing waves v) | Γ| = ±1 Minima = 0 (open or short circuits) Maxima and Minima Maxima when reflected and incident waves are in phase or ~ ~ + o = V ⎡1 + Γ ⎤ ⎣ ⎦ V ( z) 1/ 2 2 V ( z ) = V ⎡1 + Γ + 2 Γ cos ( 2 β z + θ r ) ⎤ ⎣ ⎦ Maxima when incident and reflected waves are opposite phase + o ~ min max or cos(2 β z + θ r ) = 1 2 β z + θ r = −2β lmax + θ r = −2nπ θ r + 2nπ θ r λ nλ = + lmax = 2β 4π 2 θ r λ nλ ⎧ ⎪ lmax = 4π + 2 ⎪ ⎨ n = 1, 2,...... if θ r < 0 ⎪n = 0,1, 2,...... if θ ≥ 0 and − π ≤ θ ≤ π r r ⎪ ⎩ lmin ⎧ lmax + λ if lmax < λ ⎪ 4 4 =⎨ ⎪lmax − λ 4 if lmax ≥ λ 4 ⎩ = Vo+ 1 − Γ V ( z) or cos(2β z + θ r ) = −1 2β z + θ r = (2n + 1)π −2β lmin + θ r = −(2n + 1)π (2n + 1)π + θ r θ r λ (2n + 1)λ = + 2β 4π 4 −π ≤ θ r ≤ π lmin = First min is n=0 ~ V ( z) S = VSRWR = SWR = max ~ V ( z) min Voltage standing wave ratio = 1+ Γ 1− Γ Slotted Line ...
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This note was uploaded on 11/02/2009 for the course EEC 130A taught by Professor Pham during the Spring '08 term at UC Davis.

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