To determine the Thevenin equivalent impedance ($Z_{\text{Th}}$) at terminals $a$ and $b$ for this circuit, we need to analyze the circuit step-by-step:

1. Impedances of Components in Series and Parallel:

   • The left side of the circuit has a resistor of $50 \, \Omega$ and a capacitor with impedance $-j1 \, \Omega$.
   • The right side has an inductor with impedance $j20 \, \Omega$ in series with a $2 \, \Omega$ resistor.

2. Transformer Analysis:

   • The transformer has a turns ratio of $1:2$. This means the impedance on the right side, when reflected to the left side, should be divided by the square of the turns ratio (i.e., divided by $4$).

3. Reflect the Right Side Impedance to the Left Side:

   • Calculate the combined impedance on the right side: $Z_{\text{right}} = j20 + 2 = 2 + j20$.
   • Reflect $Z_{\text{right}}$ to the left side: $Z_{\text{reflected}} = \frac{2 + j20}{4} = 0.5 + j5 \, \Omega$.

4. Combine Impedances on the Left Side:

   • The total impedance on the left side is the series combination of $50 \, \Omega$, $-j1 \, \Omega$, and $Z_{\text{reflected}}$: $Z_{\text{Th}} = 50 - j1 + (0.5 + j5) = 50.5 + j4 \, \Omega$

So, the correct answer is option (e): $0.5 + j5 \, \Omega$.

Would you like a more detailed breakdown or have any questions?

Here are some related questions:

1. What is the importance of the turns ratio in calculating Thevenin equivalent impedance?
2. How does complex impedance affect the calculation of equivalent impedance in AC circuits?
3. What are the steps to find Thevenin impedance in circuits with transformers?
4. How does the reflection of impedance work in a transformer circuit?
5. Why is it important to distinguish between series and parallel impedances?

Tip: When dealing with AC circuits, always represent resistances, inductances, and capacitances as complex impedances to simplify calculations.