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The Smith chart in lossy circuits
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Thus far, you have dealt with situations in which loss is either zero (ie, ideal transmission lines), or so small as to be negligible In situations where there is appreciable loss in the circuit or line, however, you see a slightly modified situation The VSWR circle, in that case, is actually a spiral, rather than a circle Figure 4-10 shows a typical situation Assume that the transmission line is 060 long, and is connected to a normalized load impedance of Z = 12 + j12 An ideal VSWR circle is constructed on the impedance radius represented by 12 + j12 A line (A) is drawn, from the point where this circle intersects the pure resistance baseline (B), perpendicularly to the ATTEN, 1 dB/MAJ DIV line on the radially scaled parameters A distance representing the loss (3 dB) is stepped off on this scale A second perpendicular line is drawn, from the 3 dB point, back to the pure resistance line (C) The point where line C intersects the pure resistance line becomes the radius for a new circle that contains the actual input impedance of the line The length of the line is 060 , so you must step back (060 050) or 01 This point is located on the WAVELENGTHS TOWARD GENERATOR outer circle A line is drawn from this point to the 10 center point The point where this new line intersects the new circle is the actual input impedance (Zin )The intersection occurs at 076 + j04, which (when denormalized) represents an input impedance of 38 + j20
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Frequency on the Smith chart
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A complex network may contain resistance, inductive reactance, and capacitive reactance components Because the reactance component of such impedances is a function of frequency, the network or component tends to also be frequency-sensitive
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118 The Smith chart
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Z VSWR circle
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4-10 Solution
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(Courtesy of Kay Elementrics)
Smith chart applications 119 You can use the Smith chart to plot the performance of such a network with respect to various frequencies Consider the load impedance connected to a 50- transmission line in Fig 4-11 In this case, the resistance is in series with a 22-pF capacitor, which will exhibit a different reactance at each frequency The impedance of this network is 1 Z=R j [475] C or 1 [476] Z = 50 j 2 FC And, in normalized form, Z' = 10 j (2 F C) 50 [477]
R ZS Zo ZS ZL XC
1 XC
50 50 50
j0 j0 jXC 723 FGHz XC /50
ZL C 22 pF
1 2 FC XC /Zo
Freq (GHz) 1 2 3 4 5 6
XC j723 j362 j241 j18 j145 j12 j145 j072 j048 j036 j029 j024
4-11 Load and source impedance transmission line circuit
120 The Smith chart
j024 j048 j072
j029 j036
j145
4-12 Solution (Courtesy of Kay Elementrics)
Smith chart applications 121 j 69 10 10 F j 723 1010 F j723 FGHz
= 10
[478]
= 10 Or, converted to gigahertz,
[479]
Z' = 10
[480]
The normalized impedances for the sweep of frequencies from 1 to 6 GHz are therefore Z = 10 j145 [481] Z = 10 j072 Z = 10 j048 Z = 10 j036 Z = 10 j029 Z = 10 j024 [482] [483] [484] [485] [486]
These points are plotted on the Smith chart in Fig 4-12 For complex networks, in which both inductive and capacitive reactance exist, take the difference between the two reactances (ie, X = XL XC )