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Unformatted text preview: NETWORK MODELS OF TRANSMISSION LINES AND WAVE BEHAVIOR M OTIVATION : Ideal junction elements are powercontinuous. Power out = power out instantaneously In reality, power transmission takes finite time. Power out ≠ power in Consider a lossless, continuous uniform beam. Model it as a number of segments. In the limit as the number of segments approaches infinity, the model competently describes wave behavior (e.g. wave speed, characteristic impedance). What if number of segments is finite? • How do you choose the parameters of each segment? • What wave speed and characteristic impedance are predicted by this finitesegment model? Mod. Sim. Dyn. Sys. Transmission Lines page 1 APPROACH: Consider the transmission line and each of its segments as 2port elements relating 2 pairs of 2 variables. There are 4 possible forms (choices of input and output). Two of them are causal , the impedance form: ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ e a e b = ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ Z ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ f a f b Z and the admittance form: ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ f a f b = ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ Y ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ e a e b Y Mod. Sim. Dyn. Sys. Transmission Lines page 2 The remaining two forms are acausal : ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ e a f a = ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ M ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ e b f b ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ e b f b = ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ M ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ e a f a M is called a transmission matrix. M The benefit of the acausal forms is that segments may be concatenated by matrix multiplication. Mod. Sim. Dyn. Sys. Transmission Lines page 3 H OW IS M STRUCTURED ? Consider the elements of a linear, lossless transmission line. Using the Laplace variable, s, we may describe the capacitor as an impedance: C a b e a = 1 Cs (f b – f a ) ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ e a e b = ⎣ ⎢ ⎡ ⎦ ⎥ ⎤1/Cs 1/Cs1/Cs 1/Cs ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ f a f b Mod. Sim. Dyn. Sys. Transmission Lines page 4 We may describe the inertia as an admittance: Instead, describe them as transmission matrices. The capacitor: e a = e b f a = f b Cs e a ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ e a f a = ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ 1Cs 1 ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ e b f b The inertia: f c = f d e c = e d Is f d ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ e c f c = ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ 1 Is 0 1 ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ e d f d These may be concatenated easily by multiplication. Mod. Sim. Dyn. Sys. Transmission Lines page 6 N OTE : • The determinant of the transmission matrix, M, is unity. • C (or I) can be replaced by any 1port system. Mod. Sim. Dyn. Sys. Transmission Lines page 7 G ENERAL FORM To find the general form of the transmission matrix, consider the properties of an idealized lossless transmission line....
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 Fall '06
 NevilleHogan
 Transmission Lines, Transmission line, Mod. Sim, lim Zo

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