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329fall08hw13 - ECE 329 Homework 13 Due Thu Dec 4 2008 5PM...

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ECE 329 Homework 13 Due: Thu, Dec 4, 2008, 5PM 1. Smith Chart derivation: Since Γ = z - 1 z + 1 = r + jx - 1 r + jx + 1 = [( r - 1) + jx ][( r + 1) - jx ] ( r + 1) 2 + x 2 = ( r 2 + x 2 - 1) + j 2 x ( r + 1) 2 + x 2 Γ r + j Γ i , it follows that Γ r = ( r 2 + x 2 - 1) ( r + 1) 2 + x 2 and Γ i = 2 x ( r + 1) 2 + x 2 , where r and x are normalized line resistance and reactance, respectively, and Γ r and Γ i denote the real and imaginary parts of reflection coe ffi cient Γ . Using the above expressions for Γ r and Γ i , it can be shown that the following relations are valid: ( Γ r - r r + 1 ) 2 + Γ 2 i = ( 1 r + 1 ) 2 constant r circles ( Γ r - 1) 2 + ( Γ i - 1 x ) 2 = ( 1 x ) 2 constant x circles a) Smith Chart construction: The equations above describe circles on the complex Γ -plane with r and x dependent centers and radii, respectively. Using a compass, draw constant r circles (on a plane with Γ r and Γ i axes) for the values of r = 0 , 1, 2, — set the scales of your axes such that your circles comfortably fill a 8.5”X11” page, and mark each circle by the r -value. Also draw and mark portions of constant x
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