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Outer circle parameters
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The standard Smith chart shown in Fig 4-4C contains three concentric calibrated circles on the outer perimeter of the chart Circle A has already been covered and it is the pure reactance circle The other two circles define the wavelength distance (B) relative to either the load or generator end of the transmission line, and either the transmission or reflection coefficient angle in degrees (C) There are two scales on the wavelength circle (B in Fig 4-4C), and both have their zero origin on the left-hand extreme of the pure resistance line Both scales represent one-half wavelength for one entire revolution, and are calibrated from 0 through 050 such that these two points are identical with each other on the circle In other words, starting at the zero point and travelling 360 around the circle brings one back to zero, which represents one-half wavelength, or 05 Although both wavelength scales are of the same magnitude (0 050), they are opposite in direction The outer scale is calibrated clockwise and it represents wavelengths toward the generator; the inner scale is calibrated counterclockwise and represents wavelengths toward the load These two scales are complementary at all points Thus, 012 on the outer scale corresponds to (050 012) or 038 on the inner scale
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106 The Smith chart The angle of transmission coefficient and angle of reflection coefficient scales are shown in circle C in Fig 4-4C These scales are the relative phase angle between reflected and incident waves Recall from transmission line theory (see Chap 3), that a short circuit at the load end of the line reflects the signal back toward the generator 180 out of phase with the incident signal; an open line (ie, infinite impedance) reflects the signal back to the generator in phase (ie, 0 ) with the incident signal These facts are shown on the Smith chart by the fact that both scales start at 0 on the right-hand end of the pure resistance line, which corresponds to an infinite resistance, and it goes halfway around the circle to 180 at the 0 end of the pure resistance line Note that the upper half-circle is calibrated 0 to +180 , and the bottom half-circle is calibrated 0 to 180 , reflecting indictive or capacitive reactance situations, respectively
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There are six scales laid out on five lines (D through G in Fig 4-4C and in expanded form in Fig 4-5) at the bottom of the Smith chart These scales are called the radially scaled parameters and they are both very important, and often overlooked With these scales, we can determine such factors as VSWR (both as a ratio and in decibels), return loss in decibels, voltage or current reflection coefficient, and the power reflection coefficient The reflection coefficient is defined as the ratio of the reflected signal to the incident signal For voltage or current: = and = Iref Iinc [416] Eref Einc [415]
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Power is proportional to the square of voltage or current, so: Ppwr = 2 or pwr = Pref Pinc [418] [417]
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Example 10 W of microwave RF power is applied to a lossless transmission line, of which 28 W is reflected from the mismatched load Calculate the reflection coefficient Pref pwr = [419] Pinc
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Smith chart components 107
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4-5 Radially scaled parameters
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108 The Smith chart pwr = 28 W 10 W
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[420] [421]
= 028
The voltage reflection coefficient is found by taking the square root of the power reflection coefficient, so in this example it is equal to 0529 These points are plotted at A and B in Fig 4-5 Standing wave ratio (SWR) can be defined in terms of reflection coefficient: VSWR = or 1+ VSWR = 1 or, in our example, 1+ VSWR = 1 028 028 [424] 1+ 1 pwr pwr [423] [422]
1 + 0529 1 0529 1529 = 325:1 0471
[425]
= or, in decibel form,
[426]
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