STEM Problem Detail

Problem Statement

Prompt

Provide step by step solution of the question number 1.

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## Image 1 Solution

**(Sol)**
\[ Z_{\text{line}} = 0.1 \angle 85^\circ; |V_1| = 1.05, |V_2| = 0.95 \]

\begin{align*}
S_{12} &= \frac{|V_1|^2}{|Z|} e^{j\theta_z} - \frac{|V_1||V_2|}{|Z|} e^{j\theta_z} e^{j\theta_{12}} = \frac{1.05^2}{0.1} \angle 85^\circ - \frac{1.05 \cdot 0.95}{0.1} \angle 85^\circ \angle \theta_{12} \\
&= 11.025 \angle 85^\circ - 9.975 \angle(85^\circ + \theta_{12}) \\
P_{12} &= 11.025 \cos 85^\circ - 9.975 \cos(85^\circ + \theta_{12})
\end{align*}

\begin{align*}
-S_{21} &= -\frac{|V_2|^2}{|Z|} \angle \theta_z + \frac{|V_1||V_2|}{|Z|} \angle(\theta_z - \theta_{12}) = -\frac{0.95^2}{0.1} \angle 85^\circ + \frac{1.05 \cdot 0.95}{0.1} \angle(85^\circ - \theta_{12}) \\
&= -9.025 \angle 85^\circ + 9.975 \angle(85^\circ - \theta_{12}) \\
-P_{21} &= -9.025 \cos 85^\circ + 9.975 \cos(85^\circ - \theta_{12})
\end{align*}

(b) \(\theta_{12} = 180 - \theta_z = 180 - 85^\circ = 95^\circ \Rightarrow P_{12\text{maximum}}\)
(a) \(P_{12\text{max}} = 11.025 \cos 85^\circ - 9.975 \cos(85^\circ + 95^\circ) = 10.936\)
(d) \(\theta_{12} = \theta_z = 85^\circ \Rightarrow -P_{21} \text{ maximum}\)
(c) \(-P_{12\text{max}} = -9.025 \cos 85^\circ + 9.975 \cos(85^\circ - 85^\circ) = 9.1884\)
(e) \(P_{\text{loss}} = P_{12} - (-P_{21}); \theta_{12} = 10^\circ\)

\begin{align*}
P_{12} &= 11.025 \cos 85^\circ - 9.975 \cos(85^\circ + 10^\circ) = 1.8303 \\
-P_{21} &= -9.025 \cos 85^\circ + 9.975 \cos(85^\circ - 10^\circ) = 1.7951 \\
P_{\text{loss}} &= P_{12} - (-P_{21}) = 1.8303 - 1.7951 = 0.0352
\end{align*}

(a) \(P_{12\text{max}} = 10.9 \sim 11.0 (4\%)\); 10.936
(b) \(\theta_{12} = 95^\circ (\text{完全正確才給分}(4\%))\)
(c) \(-P_{12\text{max}} = 9.0 \sim 9.2 (4\%)\); 9.1884
(d) \(\theta_{12} = 85^\circ (\text{完全正確才給分}(4\%))\)
(e) \(P_{\text{loss}} = 0.0350 \sim 0.0354 (4\%)\); 0.0352

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Problem #44