Notes 17 Spring 2005

Notes 17 Spring 2005 - Notes 17 Spring 2005.doc Smith Chart...

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Notes 17 Spring 2005.doc Smith Chart The Smith chart is the most widely used graphical technique for analyzing transmission line circuits and designing impedance matching circuits. Today much is computerized, however, it is important to understand the basis for the computations. The Smith chart lies in the complex plane of the reflection coefficient 1 i r j j e Γ + Γ = Γ = Γ φ . i Γ r Γ All impedances are normalized to the impedance of the line before plotting, i.e., L L 0 L L 0 L L jx r Z jX R Z Z z + = + = = Since 1 z 1 z Z Z Z Z L L 0 L 0 L L + = + = Γ we can rearrange to get L L L 1 1 z Γ Γ + = Then, making the appropriate substitutions for L L and z Γ into the boxed equation we can write () () i r i r L L j 1 j 1 jx r Γ + Γ Γ + Γ + = + which can be solved separately for and in terms of L r L x r Γ and i Γ : () 2 i 2 r 2 i 2 r L 1 1 r Γ + Γ Γ Γ = (a) () 2 i 2 r i L 1 2 x Γ + Γ Γ = (b) Now rearranging equation (a) we get 2 L 2 i 2 L L r r 1 1 r 1 r + = Γ + + Γ
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Notes 17 Spring 2005.doc If we identify this equation term for term with the standard equation of a circle of radius a that is centered at , i.e., () 0 0 y , x () () 2 2 o 2 o a y y x x = + we find that it describes a circle in the complex Γ plane that is centered at = Γ + = Γ 0 , r 1 r i L L r with a radius of L r 1 1 + . Note that the smallest possible resistive component of the load’s impedance is obviously 0 r L = which yields a greatest radius of 1. We, therefore, will only have to utilize the unit circle on the complex plane in our formulation. Γ We can similarly find for equation (b) () 2 L 2 L i 2 r x 1 x 1 1 = Γ + Γ describing a circle in the complex plane only part of which is contained within the unit circle. Its center is located at Γ = Γ = Γ L i r x 1 , 1 . Since the load reactance, , can be either negative or positive
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This note was uploaded on 01/20/2012 for the course EE 4460 taught by Professor Czarnecki during the Fall '10 term at LSU.

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Notes 17 Spring 2005 - Notes 17 Spring 2005.doc Smith Chart...

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