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94 Transmission lines 1019 0974
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= 10 log
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= 10 log (1046) = (10) (002) = 02 dB Compare the matched line loss (A = 015 dB) with the total loss (Loss = 02 dB), which includes mismatch loss and line loss The difference (ie, Loss A) is only 005 dB If the VSWR was considerably larger, however, the loss would rise
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CHAPTER
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The Smith chart
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THE MATHEMATICS OF TRANSMISSION LINES, AND CERTAIN OTHER DEVICES, BECOMES
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cumbersome at times, especially in dealing with complex impedances and nonstandard situations In 1939, Phillip H Smith published a graphical device for solving these problems, and an improved version of the chart followed in 1945 That graphic aid, somewhat modified over time, is still in constant use in microwave electronics, and other fields where complex impedances and transmission line problems are found The Smith chart is indeed a powerful tool for the RF designer
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The modern Smith chart is shown in Fig 4-1 It consists of a series of overlapping orthogonal circles (ie, circles that intersect each other at right angles) This chapter will dissect the Smith chart, so that the origin and use of these circles is apparent The set of orthogonal circles makes up the basic structure of the Smith chart
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The normalized impedance line
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A baseline is highlighted in Fig 4-2, and bisects the Smith chart outer circle This line is called the pure resistance line, and forms the reference for measurements made on the chart Recall that a complex impedance contains both resistance and reactance, and is expressed in the mathematical form: Z = R jX [41]
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96 The Smith chart
4-1 The Smith chart (Courtesy of Kay Elementrics)
Smith chart components 97 where Z is the complex impedance R is the resistive component of the impedance X is the reactive component of the impedance* The pure resistance line represents the situation where X = 0, and the impedance is therefore equal to the resistive component only In order to make the Smith chart universal, the impedances along the pure resistance line are normalized with reference to system impedance (eg, Zo in transmission lines); for most microwave RF systems the system impedance is standardized at 50 In order to normalize the actual impedance, divide it by the system impedance For example, if the load impedance of a transmission line is ZL, and the characteristic impedance of the line is Zo, then Z = ZL/Zo In other words, Z= R jX Zo [42]
The pure resistance line is structured such that the system standard impedance is in the center of the chart, and has a normalized value of 10 (see point A in Fig 42) This value derives from the fact that Zo/Zo = 10 To the left of the 10 point are decimal fraction values used to denote impedances less than the system impedance For example, in a 50- transmission line system with a 25- load impedance, the normalized value of impedance is 25 /50 or 050 ( B in Fig 4-2) Similarly, points to the right of 10 are greater than 1 and denote impedances that are higher than the system impedance For example, in a 50 system connected to a 100- resistive load, the normalized impedance is 100 /50 , or 20; this value is shown as point C in Fig 4-2 By using normalized impedances, you can use the Smith chart for almost any practical combination of system, and load and/or source, impedances, whether resistive, reactive, or complex Reconversion of the normalized impedance to actual impedance values is done by multiplying the normalized impedance by the system impedance For example, if the resistive component of a normalized impedance is 045, then the actual impedance is Z = (Znormal ) (Zo ) [43] = (045) (50 ) = 225 [44] [45]
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