4.5: Determining \(Z_{0}\) of a Line from the Smith Chart

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It was shown in Section 3.5.2 that the reflection coefficient locus with respect to frequency of a terminated line is a circle on the Smith chart even if the characteristic impedance of the line is not the same as the reference impedance of the Smith chart. The one caveat here is that the reflection coefficient of the termination must be independent of frequency so a resistive termination is sufficient. If the characteristic impedance of the line is \(Z_{01}\) and the Smith chart is referenced to \(Z_{02}\), which is usually the same as the system impedance of the measurement system, then \(Z_{01}\) can be determined from the center, \(C_{Z02}\), and radius, \(R_{Z02}\), of the reflection coefficient circle. Thus measurements can be used to determine the unknown impedance \(Z_{01}\). Another situation where this is useful is in design where a transmission line circle can be drawn to complete a design problem and from this the characteristic impedance of the line found. In both situations \(C_{Z02},\: R_{Z02}\), and \(Z_{02}\) are known and \(Z_{01}\) must be determined.

A simple closed-form solution for the unknown characteristic impedance \(Z_{01}\) cannot be obtained from Equations (3.5.14) and (3.5.15). However, by substituting Equation (3.5.15) in Equation (3.5.14), and if \(Z_{01}\) and \(Z_{02}\) are close (so that \(B\) is small), then

\[\label{eq:1}C_{Z-2}\approx B-\frac{1}{B}+\frac{1}{B}\approx B \]

The approximation is better for smaller \(|\Gamma_{L,\:Z01}|\). Also

\[\label{eq:2}R_{Z02}\approx |\Gamma_{L,\: Z01}| \]

So provided that the characteristic impedance of the line, \(Z_{01}\), is close to the system reference impedance, \(Z_{02}\),

\[\label{eq:3}\frac{Z_{01}}{Z_{02}}=\frac{1+B}{1-B}\approx\frac{1+C_{Z02}}{1-C_{Z02}} \]

and this is just the normalized impedance reading at the center of the circle. For example, if a line with a characteristic impedance \((Z_{01})\) of \(55\:\Omega\) is terminated in a \(45\:\Omega\) load, then in a \((Z_{02}\: =)\: 50\:\Omega\) system, \(C_{Z02} = 0.0939\) and \(R_{Z02} = 0.0996\). Using Equation \(\eqref{eq:3}\) the derived \(Z_{01} = 54.7\:\Omega\) and, using Equation \(\eqref{eq:2}\), \(\Gamma_{L,\: Z01} = 0.0996\) compared to the ideal \(0.1000\).

Table \(\PageIndex{1}\) presents the actual characteristic impedance of the line as the ratio \(Z_{Z01}/Z_{Z02}\) for particular center and radius values measured on the polar plot referenced to \(Z_{02}\). The actual value of impedance is compared to the approximate value for \(Z_{Z01}/Z_{Z02}\approx 1 + C_{Z02}/1 − C_{Z02}\). It is seen that the approximation in Equation \(\eqref{eq:3}\) provides a good estimate of the unknown characteristic impedance \((Z_{01})\) improving as the center of the locus is closer to the origin.

\(C_{Z02}\)	\(Z_{01}/Z_{02}\approx\) \(\frac{(1+C_{Z02})}{(1-C_{Z02})}\)	\(R_{Z02}=0.2\)		\(R_{Z02}=0.3\)		\(R_{Z02}=0.4\)		\(R_{Z02}=0.5\)		\(R_{Z02}=0.6\)		\(R_{Z02}=0.7\)
\(C_{Z02}\)			error	\(Z_{01}/Z_{02}\)	error	\(Z_{01}/Z_{02}\)	error	\(Z_{01}/Z_{02}\)	error	\(Z_{01}/Z_{02}\)	error	\(Z_{01}/Z_{02}\)	error
\(0.00\)	\(1.000\)	\(1.000\)	\(0\%\)	\(1.000\)	\(0 \%\)	\(1.000\)	\(0 \%\)	\(1.000\)	\(0 \%\)	\(1.000\)	\(0 \%\)	\(1.000\)	\(0 \%\)
\(0.02\)	\(1.041\)	\(1.043\)	\(<1\%\)	\(1.045\)	\(<1 \%\)	\(1.050\)	\(1 \%\)	\(1.050\)	\(1 \%\)	\(1.065\)	\(2 \%\)	\(1.080\)	\(4 \%\)
\(0.04\)	\(1.083\)	\(1.088\)	\(<1\%\)	\(1.093\)	\(1 \%\)	\(1.100\)	\(2 \%\)	\(1.113\)	\(3 \%\)	\(1.135\)	\(5 \%\)	\(1.170\)	\(7 \%\)
\(0.06\)	\(1.128\)	\(1.130\)	\(<1\%\)	\(1.140\)	\(1 \%\)	\(1.155\)	\(2 \%\)	\(1.175\)	\(4 \%\)	\(1.208\)	\(7 \%\)	\(1.265\)	\(11 \%\)
\(0.08\)	\(1.174\)	\(1.183\)	\(<1\%\)	\(1.193\)	\(2 \%\)	\(1.210\)	\(3 \%\)	\(1.250\)	\(6 \%\)	\(1.285\)	\(9 \%\)	\(1.375\)	\(15 \%\)
\(0.10\)	\(1.222\)	\(1.230\)	\(<1\%\)	\(1.245\)	\(2 \%\)	\(1.270\)	\(4 \%\)	\(1.310\)	\(7 \%\)	\(1.370\)	\(11 \%\)	\(1.500\)	\(19 \%\)
\(0.12\)	\(1.273\)	\(1.285\)	\(1\%\)	\(1.305\)	\(2 \%\)	\(1.335\)	\(5 \%\)	\(1.385\)	\(8 \%\)	\(1.465\)	\(13 \%\)	\(1.640\)	\(22 \%\)
\(0.14\)	\(1.326\)	\(1.340\)	\(1\%\)	\(1.365\)	\(3 \%\)	\(1.408\)	\(6 \%\)	\(1.465\)	\(10 \%\)	\(1.565\)	\(15 \%\)	\(1.800\)	\(26 \%\)
\(0.16\)	\(1.381\)	\(1.400\)	\(1\%\)	\(1.428\)	\(3 \%\)	\(1.475\)	\(6 \%\)	\(1.550\)	\(11 \%\)	\(1.685\)	\(18 \%\)	\(1.990\)	\(31 \%\)
\(0.18\)	\(1.439\)	\(1.460\)	\(1\%\)	\(1.495\)	\(4 \%\)	\(1.550\)	\(7 \%\)	\(1.645\)	\(13 \%\)	\(1.820\)	\(21 \%\)	\(2.230\)	\(35 \%\)
\(0.20\)	\(1.500\)	\(1.528\)	\(2\%\)	\(1.565\)	\(4 \%\)	\(1.635\)	\(8 \%\)	\(1.745\)	\(14 \%\)	\(1.965\)	\(24 \%\)	\(2.515\)	\(40 \%\)
\(0.22\)	\(1.564\)	\(1.595\)	\(2\%\)	\(1.645\)	\(5 \%\)	\(1.720\)	\(9 \%\)	\(1.860\)	\(16 \%\)	\(2.130\)	\(27 \%\)	\(2.915\)	\(46 \%\)
\(0.24\)	\(1.632\)	\(1.670\)	\(2\%\)	\(1.725\)	\(5 \%\)	\(1.815\)	\(10 \%\)	\(1.980\)	\(18 \%\)	\(2.330\)	\(30 \%\)	\(3.440\)	\(53 \%\)
\(0.26\)	\(1.703\)	\(1.745\)	\(2\%\)	\(1.810\)	\(6 \%\)	\(1.920\)	\(11 \%\)	\(2.120\)	\(20 \%\)	\(2.555\)	\(33 \%\)	\(4.380\)	\(61 \%\)
\(0.28\)	\(1.778\)	\(1.830\)	\(3\%\)	\(1.905\)	\(7 \%\)	\(2.030\)	\(12 \%\)	\(2.270\)	\(22 \%\)	\(2.830\)	\(37 \%\)	\(-\)	\(-\)
\(0.30\)	\(1.857\)	\(1.915\)	\(3\%\)	\(2.005\)	\(7 \%\)	\(2.150\)	\(14 \%\)	\(2.450\)	\(24 \%\)	\(3.205\)	\(42 \%\)	\(-\)	\(-\)
\(0.32\)	\(1.941\)	\(2.008\)	\(3\%\)	\(2.110\)	\(8 \%\)	\(2.285\)	\(15 \%\)	\(2.655\)	\(27 \%\)	\(-\)	\(-\)	\(-\)	\(-\)
\(0.34\)	\(2.030\)	\(2.105\)	\(4\%\)	\(2.225\)	\(9 \%\)	\(2.435\)	\(16 \%\)	\(2.885\)	\(30 \%\)	\(-\)	\(-\)	\(-\)	\(-\)
\(0.36\)	\(2.125\)	\(2.215\)	\(4\%\)	\(2.350\)	\(10 \%\)	\(2.600\)	\(18 \%\)	\(3.125\)	\(32 \%\)	\(-\)	\(-\)	\(-\)	\(-\)
\(0.38\)	\(2.226\)	\(2.325\)	\(4\%\)	\(2.490\)	\(11 \%\)	\(2.785\)	\(20 \%\)	\(3.515\)	\(37 \%\)	\(-\)	\(-\)	\(-\)	\(-\)
\(0.40\)	\(2.333\)	\(2.455\)	\(5\%\)	\(2.640\)	\(12 \%\)	\(3.000\)	\(22 \%\)	\(3.925\)	\(40 \%\)	\(-\)	\(-\)	\(-\)	\(-\)
\(0.42\)	\(2.585\)	\(2.585\)	\(5\%\)	\(2.805\)	\(13 \%\)	\(3.240\)	\(24 \%\)	\(-\)	\(-\)	\(-\)	\(-\)	\(-\)	\(-\)
\(0.44\)	\(2.730\)	\(2.730\)	\(6\%\)	\(2.985\)	\(14 \%\)	\(3.535\)	\(27 \%\)	\(-\)	\(-\)	\(-\)	\(-\)	\(-\)	\(-\)
\(0.46\)	\(2.704\)	\(2.885\)	\(6\%\)	\(3.180\)	\(15 \%\)	\(3.865\)	\(30 \%\)	\(-\)	\(-\)	\(-\)	\(-\)	\(-\)	\(-\)

Table \(\PageIndex{1}\): Table of the normalized characteristic impedance \(Z_{01}/Z_{02}\) of a terminated transmission line having characteristic impedance \(Z_{01}\) plotted on a Smith chart with a reference impedance of \(Z_{02}\) in terms of the center, \(C_{Z02}\), of the circular locus (with respect to line length) of the line for various radii, \(R_{Z02}\), of the circular locus. \(C_{Z02}\) and \(R_{Z02}\) are in terms of reflection coefficient measured on the Smith chart. Also shown is the approximation, \((1 + C_{Z02})/(1 − C_{Z02}),\) of \(Z_{01}/Z_{02}\) (see Equation \(\eqref{eq:3}\)).