# The Smith Chart

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## Abstract

Connexions module: m11388 1 The Smith Chart ∗ Bill Wilson This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License † Abstract Introduction of Smith Chart. Now let's see how we can use The Bilinear Transform 1 to get the co-ordinates on the Z(s) Z0 plane transferred over onto the r (s) plane. The Bilinear Transform2 tells us how to take anyZ(s)Z0 and generate an r (s) from it. Let's start with an easy one. We will assume that Z(s) Z0 = 1 + iX, which is a vertical line, which passes through 1, and can take on whatever imaginary part it wants Figure 1 (Complex Impedence With Real Part = +1). Complex Impedence With Real Part = +1 Figure 1 According to The Bilinear Transform 3 , the first thing we should do is add 1 to Z(s) Z0 . This gives us the line 2 + iXFigure 2 (Adding 1). ∗ Version 1.2: Jun 23, 2003 12:00 am -0500 † http://creativecommons.org/licenses/by/1.0 1 "Bilinear Transform", (5) <http://cnx.org/content/m11387/latest/#eqn5> 2 "Bilinear Transform", (5) <http://cnx.org/content/m11387/latest/#eqn5> 3 "Bilinear Transform", (5) <http://cnx.org/content/m11387/latest/#eqn5> http://cnx.org/content/m11388/1.2/ Connexions module: m11388 2 Adding 1 Figure 2 Now, we take the inverse of this, which will give us a circle, of diameter 1/2 Figure 3 (Inverting). Now, according to The Bilinear Transform 4 we take this circle and multiply by -2 Figure 4 (Multiplying by -2). Inverting Figure 3 4 "Bilinear Transform", (5) <http://cnx.org/content/m11387/latest/#eqn5> http://cnx.org/content/m11388/1.2/ Connexions module: m11388 3 Multiplying by -2 Figure 4 And finally, we take the circle and add +1 to it: as shown here (Figure 5: Adding 1 Once Again). There, we are done with the transform. The vertical line on the Z(s) Z0 plane that represents an impedance with a real part of +1 and an imaginary part with any value from − (i∞) to i∞ has been reduced to

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