| Module 5.7 Standing Wave Pattern (Lossy case) |
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| Time-average power delivered to the input at z, P(z) = (mW) | | |
Introduction: In this module, we discuss the frequency-domain behavior and time-averaged power of the standing wave on a lossy transmission line. By adjusting
the range sliders, you can observe the dynamic behavior of the voltage standing wave ratio ( \(SWR\) ) , and the voltage and current distribution on the
transmission line. Ex.1：When total reflection does not occur at the load ( \(\left|\Gamma_L\right|\) < 1 ), the increase of attenuation constant ( \(\alpha\) ) will cause the decrease of time-averaged
power. Ex.2：As the attenuation constant ( \(\alpha\) ) increases gradually, the voltage and current at the load decrease accordingly. |
| Formula: 1. \(V(z)\) : Total voltage wave at z, defined by
\(\left|V(z)\right|=\left|V_{0}^{+}~{\cdot}~e^{\alpha \left|z\right|} \right|\cdot \left|1+\left|\Gamma (0)\right|~{\cdot}~e^{j(\theta -2\beta \left|z\right|)}~{\cdot}~e^{-2 \alpha \left|z\right|} \right| (V)\)
\(\Gamma (0)=\left|\Gamma (0)\right|\cdot e^{j\theta } =\frac{Z_{L} -Z_{0} }{Z_{L} +Z_{0} } =\frac{(R_{L} -Z_{0} )+jX_{L} }{(R_{L} +Z_{0} )+jX_{L} } ; Z_{L} =R_{L} +jX_{L} (\Omega) \)
\(\Gamma (z)=\left|\Gamma (0)\right|\cdot e^{j\theta } \cdot e^{-j2\beta \left|z\right|}~{\cdot}~e^{-2 \alpha \left|z\right|} \)
2. \(I(z) (A)\) : Total current wave at z, defined by
\(\left|I(z)\right|=\frac{\left|V_{0}^{+} ~{\cdot}~e^{\alpha \left|z\right|} \right|}{Z_{0} } \cdot \left|1-\left|\Gamma (0)\right|~{\cdot}~e^{j(\theta -2\beta \left|z\right|)}~{\cdot}~e^{-2 \alpha \left|z\right|} \right| (A)\)
3. \(P(z) (W)\) : Time-average power delivered to the input at z, defined by
\(P(z)=\frac{1}{2} Re[V(z)I*(z)]=\frac{1}{2} \frac{\left|V_{0}^{+} \right|^{2} }{Z_{0} } (e^{2 \alpha \left|z\right|}-\left|\Gamma (0)\right|^{2}~{\cdot}~e^{-2 \alpha \left|z\right|} ) (W) \)
4. \(SWR\) : 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. \(\alpha\) : Attenuation constant, (Np/m)
5. \(\beta\) : Phase constant, (rad/m)
6. \(f\) : Operating frequency, (GHz)
7. \(d\) : Length of the transmission line, value equal to 10 cm
8. \(V_{0}^{+}\) : Incident voltage at z = 0, (V)
9. \(Z_{0}\) : Characteristic impedance of the transmission line, \((\Omega)\)
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