过去问-电路分析-东京大学-2021

https://www.i.u-tokyo.ac.jp/edu/course/ice/pdf/denjo2021-e.pdf

Based on the image provided, here is the extracted text for Problem 1.

---

Problem 1

Suppose a circuit, shown in Fig. 1, composed of a sine-wave AC voltage source ($E$: effective voltage value) and a load. The load is composed of a resistor ($R$: resistance) and a reactor ($X_L$: inductive reactance). Assume $E = 200\text{ V}$, $R = 3\ \Omega$ and $X_L = 4\ \Omega$. Answer the following questions regarding this circuit.

1. Calculate the effective current value $I$ shown as an arrow in Fig. 1.
2. Calculate the active power $P$ consumed by the load and the apparent power $S$ supplied to the load.
3. Suppose that a capacitor ($X_C$: capacitive reactance) is connected in parallel to the load as shown in Fig. 2. Describe the synthetic impedance $\dot{Z}$ between the terminals a and b as a function of $X_C$. Use $j$ as the imaginary unit.
4. Find the value of $X_C$ to bring the power factor of the load to $100\%$ in the case of (3).
5. Describe qualitatively a benefit of improving a power factor of a load by the use of a capacitor as shown in Fig. 2.
6. Find all the values of $X_C$ to bring the power factor of the load to $90\%$ in the case of (3). The answer must be rounded off to three significant figures. Use the following approximate values if necessary. $\sqrt{2} \simeq 1.41$, $\sqrt{3} \simeq 1.73$, $\sqrt{5} \simeq 2.24$, $\sqrt{7} \simeq 2.65$, $\sqrt{11} \simeq 3.32$, $\sqrt{13} \simeq 3.61$, $\sqrt{17} \simeq 4.12$, $\sqrt{19} \simeq 4.36$, $\sqrt{23} \simeq 4.80$ and $\sqrt{29} \simeq 5.39$.