| Module 6.3 Input impedance |
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Introduction: 本模組我們將探討有終端負載之無損傳輸線的輸入阻抗特性，並藉由調整 toolbar 以觀察輸入阻抗於該傳輸線上之動態行為與分布。 Ex.1：負載阻抗為匹配時 ( \(\left|\Gamma_L\right|\) = 0 )，輸入阻抗於傳輸線上任一觀察點皆為系統阻抗 ( \(Z_0\) ) ; 反之，若負載阻抗不為匹配時 ( \(\left|\Gamma_L\right|\) ≠ 0 )， 輸入阻抗將會隨觀察點的移動而變化！ Ex.2：若固定負載阻抗值 ( \(\left|\Gamma_L\right|\) ≠ 0 )，反射係數之大小於傳輸線上任一觀察點皆為相同，而僅有相位上的改變。 Ex.3：將觀察的週期拉長或將頻率 ( \(f\) ) 升高時，從阻抗的響應分佈可明顯發現輸入阻抗的週期為半波長！ |
| Formula: 1. \(Z_{in}(z) (\Omega)\) : Input impedance looking into the transmission line at z.
\( Z_{in} \left(z\right) = Z_{0} \cdot \frac { Z_{L} + j Z_{0} { tan\beta }\left|z\right|}{Z_{0} + j Z_{L} { tan\beta }\left|z\right|} (\Omega )\)
2. \(\Gamma(z)\) : Voltage reflection coefficient at z, defined by
\(\Gamma (z)=\left|\Gamma (0)\right|\cdot e^{j\theta } \cdot e^{-j2\beta \left|z\right|} \)
3. \(SWR\) : Voltage standing wave ratio, defined by
\(SWR=\frac{V_{\max } }{V_{\min } } =\frac{1+\left|\Gamma (0)\right|}{1-\left|\Gamma (0)\right|} \)
| Parameters: 1. \(\left|\Gamma (0)\right|\) : Voltage reflection coefficient at z = 0, defined by
\(\Gamma (0)=\left|\Gamma (0)\right|\cdot e^{j\theta } =\frac{Z_{L} -Z_{0} }{Z_{L} +Z_{0} } \)
2. \(\theta\) : Phase angle of the voltage reflection coefficient, (rad)
3. \(Z_{L}\) : Load impedance, defined by
\(Z_{L} =R_{L} +jX_{L} =Z_{0} \cdot \frac{1+\Gamma (0)}{1-\Gamma (0)} (\Omega) \)
a. \(R_{L}\) : Real part of \(Z_{L}\),
b. \(X_{L}\) : Imaginary part of \(Z_{L}\).
4. \(\beta\) : Phase constant, (rad/m)
5. \(f\) : Operating frequency, (GHz)
6. \(d\) : Length of the transmission line, value equal to 10 cm
7. \(Z_{0}\) : Characteristic impedance of the transmission line, \((\Omega)\)
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