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The isoresistance circles, also called the constant resistance circles, represent points of equal resistance Several of these circles are shown highlighted in Fig 4-3 These circles are all tangent to the point at the righthand extreme of the pure resis*According to the standard sign convention the inductive reactance (XL ) is positive (+) and the capacitive reactance (Xc ) is negative ( ) The term X in Eq 41 above is the difference between the two reactances (X = XL Xc )
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98 The Smith chart
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4-2 Normalized impedance line
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Smith chart components 99
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4-3 Constant resistance circles
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100 The Smith chart tance line, and are bisected by that line When you construct complex impedances (for which X = nonzero) on the Smith chart, the points on these circles will all have the same resistive component Circle A, for example, passes through the center of the chart, so it has a normalized constant resistance of 10 Note that impedances that are pure resistances (ie, Z = R + j0) will fall at the intersection of a constant resistance circle and the pure resistance line, and complex impedances (ie, X not equal to zero) will appear at any other points on the circle In Fig 4-3, circle A passes through the center of the chart, so it represents all points on the chart with a normalized resistance of 10 This particular circle is sometimes called the unity resistance circle
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The constant reactance circles
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Constant reactance circles are highlighted in Fig 4-4 The circles (or circle segments) above the pure resistance line (Fig 4-4A) represent the inductive reactance (+X ), and those circles (or segments) below the pure resistance line (Fig 4-4B) represent capacitive reactance ( X) In both cases, circle A represents a normalized reactance of 080 One of the outer circles (ie, circle A in Fig 4-4C) is called the pure reactance circle Points along circle A represent reactance only; in other words, an impedance of Z = 0 jX (R = 0) Figure 4-4D shows how to plot impedance and admittance on the Smith chart Consider an example in which system impedance Zo is 50 , and the load impedance is ZL = 95 + j55 This load impedance is normalized to Z= ZL Zo 95 + j55 50 [46]
[47] [48]
= 19 + j11
An impedance radius is constructed by drawing a line from the point represented by the normalized load impedance, 19 + j11, to the point represented by the normalized system impedance (10) in the center of the chart A circle is constructed from this radius, and is called the VSWR circle Admittance is the reciprocal of impedance, so it is found from Y= 1 Z [49]
Because impedances in transmission lines are rarely pure resistive, but rather contain a reactive component also, impedances are expressed by complex notation: Z = R jX [410]
Smith chart components 101
4-4A Constant inductive reactance lines
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102 The Smith chart
4-4B Constant capacitive reactance lines (Courtesy of Kay Elementrics)
Smith chart components 103
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4-4C Angle of transmission coefficient circle
104 The Smith chart
Z VSWR circle Impedance radius
j11
j023
4-4D VSWR circles
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Smith chart components 105 where Z is the complex impedance R is the resistive component X is the reactive component In order to find the complex admittance, take the reciprocal of the complex impedance by multiplying the simple reciprocal by the complex conjugate of the impedance For example, when the normalized impedance is 19 + j11, the normalized admittance will be: 1 Y= [411] Z = 1 19 j11 19 + j11 19 j11 19 j11 36 + 12 19 j11 = 039 j023 48 [412]
[413]
[414]
One of the delights of the Smith chart is that this calculation is reduced to a quick graphical interpretation! Simply extend the impedance radius through the 10 center point until it intersects the VSWR circle again This point of intersection represents the normalized admittance of the load
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