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Smith chart applications 111
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j049 Y 062 j069
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112 The Smith chart Zin = (049 j049) (50 ) = 245 j245 [442] [443]
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It is this impedance that must be matched at the generator by a conjugate matching network The admittance represented by the load impedance is the reciprocal of the load impedance, and is found by extending the impedance radius through the center of the VSWR circle until it intersects the circle again This point is found, and represents the admittance Y = 062 j069 Confirming the solution mathematically: Y= 1 Z 072 j080 1 072 + j080 072 j080 072 j080 = 062 j069 116 [444]
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[445]
[446]
The VSWR if found by transferring the impedance radius of the VSWR circle to the radial scales below The radius (072 08) is laid out on the VSWR scale (topmost of the radially scaled parameters) with a pair of dividers from the center mark, and we find that the VSWR is approximately 26:1 The decibel form of VSWR is 83 dB (next scale down from VSWR), and this is confirmed by VSWRdB = 20 log (VSWR) = (20) log (27) = (20) (0431) = 83 dB [447] [448] [449]
The transmission loss coefficient is found in a manner similar to the VSWR, using the radially scaled parameter scales In practice, once you have found the VSWR you need only drop a perpendicular line from the 26:1 VSWR line across the other scales In this case, the line intersects the voltage reflection coefficient at 044 The power reflection coefficient pwr is found from the scale, and is equal to 2 The perpendicular line intersects the power reflection coefficient line at 020 The angle of reflection coefficient is found from the outer circles of the Smith chart The line from the center to the load impedance (Z = 072 + j08) is extended to the angle of reflection coefficient in degrees circle, and intersects it at approximately 84 The reflection coefficient is therefore 044/84 The transmission loss coefficient (TLC) is found from the radially scaled parameter scales also In this case, the impedance radius is laid out on the loss coefficient scale, where it is found to be 15 This value is confirmed from: TLC = 1 + pwr 1 pwr [450]
Smith chart applications 113 1 + (020) 1 (021) 120 = 15 079
[451]
[452]
The return loss is also found by dropping the perpendicular from the VSWR point to the RET N LOSS, dB line, and the value is found to be approximately 7 dB, which is confirmed by Lossret = 10 log ( pwr ) dB = 10 log (021) dB = (10) ( 0677) dB = 677 dB = 69897 dB [453] [454] [455] [456]
The reflection loss is the amount of RF power reflected back down the transmission line from the load The difference between incident power supplied by the generator (15 W in this example), Pinc Pref = P and the reflected power, is the ababs sorbed power (Pa ); or in the case of an antenna, the radiated power The reflection loss is found graphically by dropping a perpendicular from the TLC point (or by laying out the impedance radius on the REFL LOSS, dB scale), and in this example (Fig 4-7) is 105 dB You can check the calculations: The return loss is 7 dB, so 7 dB = 10 log Pref Pinc Pref 15 watts [457]
7 = 10 log
[458]
Pref 7 = log 10 15 watts 10(
7/10)
[459]
Pref 15 W Pref 15 W
[460]
02 =
[461] [462] [463]
(02)(15 W) = Pref 03 W = Pref
114 The Smith chart The power absorbed by the load (Pa ) is the difference between incident power Pinc and reflected power Pref If 03 W is reflected, then that means the absorbed power is (15 03), or 12 W The reflection loss is 105 dB, and can be checked from: 105 dB = 10 log 105 = log 10 10(-105/10) = Pa Pinc Pa 15 W [464]
[465]
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