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Waves and Transmission Lines Wang C. Ng

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Traveling Waves

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Load Envelop of a Standing Wave

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Waves in a transmission line Electrical energy is transmitted as waves in a transmission line. Waves travel from the generator to the load (incident wave). If the resistance of the load does not match the characteristic impedance of the transmission line, part of the energy will be reflected back toward the generator. This is called the reflected wave

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Reflection coefficient The ratio of the amplitude of the incident wave (v + ) and the amplitude the reflective wave (v - ) is called the reflection coefficient:

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Reflection coefficient The reflection coefficient can be determine from the load impedance and the characteristic impedance of the line:

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Short-circuited Load Z L = 0 = -1 v - = - v + at the load As a result, v L = v + + v - = 0

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Load

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Open-circuited Load Z L = = +1 v - = v + at the load As a result, v L = v + + v - = 2 v +

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Load

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Resistive Load Z L = Z 0 = 0 v - = 0 at the load As a result, v L = v +

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Traveling Waves Load

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Resistive Load Z L = 0.5 Z 0 = - 1/3 v - = -0.333 v + at the load As a result, v L = v + + v - = 0.667 v +

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Composite Waves Load

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Resistive Load Z L = 2 Z 0 = + 1/3 v - = 0.333 v + at the load As a result, v L = v + + v - = 1.333 v +

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Composite Waves Load

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Reactive Load (Inductive) Z L = j Z 0 = + j1 v - = v + 90 at the load As a result, v L = v + + v - = (1 + j1) v + = 1.414 v + 45

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Composite Waves Load

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Reactive Load (Capacitive) Z L = -j Z 0 = - j1 v - = v + -90 at the load As a result, v L = v + + v - = (1 - j1) v + = 1.414 v + -45

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Composite Waves Load

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Smith Chart Transmission Line Calculator

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-j2 -j4 -j1 -j0.5 j0.5 j1 j4 j2 j0 0 0.5 1 2 4 Z L / Z 0 = z L = 1 + j 2

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0 0.5 1 0.7 45 = 0.5 + j 0.5 real imaginary ||||

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-j2 -j 4 -j1 -j0.5 j0.5 j1 j4 j2 j 0 0 0.5 1 2 4 z L = 1 + j 2 0.7 45 |||| |||| re im

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-j2 -j4 -j1 -j0.5 j0.5 j1 j4 j2 j0 0 0.5 1 2 4 z L = 1 + j 2 0.7 45 45 0 135 90 180 225 270 315

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-j2 -j4 -j1 -j0.5 j0.5 j1 j4 j2 j0 0 0.5 1 2 4 z L = 0.5- j 0.5 0.45 -120 45 0 135 90 180 225 270 315

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| | 0 0.5 1 -j2 -j4 -j1 -j0.5 j0.5 j1 j4 j2 j0 0 0.5 1 2 4 45 0 135 90 180 225 270 315 D C B E A F G

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Smith Chart Graphically solves the following bi-linear formulas Note: works for admittance too. Just switch sign of

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