| Module 5.2 Standing Wave Pattern (Lossless 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 lossless 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 occurs at the load ( \(\left|\Gamma_L\right|\) = 1 ), the voltage standing wave ratio will be infinite (∞) and the time-averaged power will be
always zero. Ex.2：When the load impedance is matched ( \(\left|\Gamma_L\right|\) = 0 ), the voltage standing wave ratio will be equal to one and the time-averaged power reaches
its maximum. Ex.3：When the load impedance is infinite, the maximum voltage and minimum current can be observed at the load; otherwise, the minimum
voltage and maximum current can be observed at the load if the load impedance is zero. |
| Formula: 1. \(V(z)\) : Total voltage wave at z, defined by
\(\left|V(z)\right|=\left|V_{0}^{+} \right|\cdot \left|1+\left|\Gamma (0)\right|e^{j(\theta -2\beta \left|z\right|)} \right| (V)\)
\(\Gamma (0)=\left|\Gamma (0)\right|\cdot e^{j\theta } =\frac{Z_{L} -Z_{0} }{Z_{L} +Z_{0} } =\frac{(R_{L} -Z_{0} )+jX_{L} }{(R_{L} +Z_{0} )+jX_{L} } ; Z_{L} =R_{L} +jX_{L} (\Omega) \)
\(\Gamma (z)=\left|\Gamma (0)\right|\cdot e^{j\theta } \cdot e^{-j2\beta \left|z\right|} =\left|\Gamma (0)\right|\cdot e^{j(\theta -2\beta \left|z\right|)} \)
2. \(I(z) (A)\) : Total current wave at z, defined by
\(\left|I(z)\right|=\frac{\left|V_{0}^{+} \right|}{Z_{0} } \cdot \left|1-\left|\Gamma (0)\right|e^{j(\theta -2\beta \left|z\right|)} \right| (A)\)
3. \(P(z) (W)\) : Time-average power delivered to the input at z, defined by
\(P(z)=\frac{1}{2} Re[V(z)I*(z)]=\frac{1}{2} \frac{\left|V_{0}^{+} \right|^{2} }{Z_{0} } (1-\left|\Gamma (0)\right|^{2} ) (W) \)
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. \(\beta\) : Phase constant, (rad/m)
5. \(f\) : Operating frequency, (GHz)
6. \(d\) : Length of the transmission line, value equal to 10 cm
7. \(V_{0}^{+}\) : Incident voltage at z = 0, (V)
8. \(Z_{0}\) : Characteristic impedance of the transmission line, \((\Omega)\)
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