📚 AC Circuits: Key Points for IB & CIE Physics | 交流电考点精讲
Alternating current (AC) forms the backbone of modern electrical power systems and is a core topic in both IB Higher Level and CIE A-Level Physics. Understanding AC circuits involves grappling with sinusoidal functions, phase relationships, impedance, and power factor. This guide covers essential concepts, formulas, and problem-solving techniques for AC circuits.
交流电是现代电力系统的基石，也是IB高级物理和CIE A-Level物理的核心主题。理解交流电电路需要掌握正弦函数、相位关系、阻抗和功率因数。本文涵盖交流电电路的基本概念、公式和解题技巧。
1. Introduction to Alternating Current | 交流电概述
Alternating current (AC) is an electric current that reverses direction periodically, in contrast to direct current (DC) which flows only in one direction. AC is generated by rotating coils in a magnetic field, producing a sinusoidal voltage. The standard mains electricity supplies AC at 50 Hz or 60 Hz, with typical RMS voltages of 230 V or 120 V.
交流电是方向周期性反转的电流，与单向流动的直流电不同。交流电由磁场中旋转的线圈产生，产生正弦电压。市电通常提供 50 Hz 或 60 Hz 的交流电，典型有效值电压为 230 V 或 120 V。
2. Sinusoidal AC: Instantaneous Values | 正弦交流电的瞬时值
For a sinusoidal AC signal, the instantaneous voltage v and current i can be expressed as:
对于正弦交流信号，瞬时电压 v 和电流 i 可表示为：
v = V₀ sin(ωt) = V₀ sin(2πft)
i = I₀ sin(ωt)
Where V₀ and I₀ are the peak values, ω is the angular frequency in rad/s, and f is the frequency in hertz. The period T = 1/f relates to the time for one complete cycle. The instantaneous values vary sinusoidally between positive and negative peaks.
其中 V₀ 和 I₀ 是峰值，ω 是角频率（rad/s），f 是频率（Hz）。周期 T = 1/f 对应一个完整循环的时间。瞬时值在正负峰值之间正弦变化。
3. Root Mean Square (RMS) Value | 方均根值
The RMS value of an AC is the equivalent DC value that would produce the same heating effect in a resistor. For sinusoidal waveforms, it is calculated as:
交流电的有效值（RMS）是产生相同热效应的等效直流值。对于正弦波形，计算公式为：
V_rms = V₀ / √2, I_rms = I₀ / √2
RMS is the standard measure used for household mains. Average power in a resistive load can be expressed as P_avg = I_rms² R = V_rms I_rms. Meters and specifications typically refer to RMS values unless otherwise stated.
RMS 是家庭用电的标准测量值。电阻负载中的平均功率可表示为 P_avg = I_rms² R = V_rms I_rms。除非另行说明，仪表和规格通常引用有效值。
4. Phase Difference in AC Circuits | 交流电路中的相位差
In AC circuits, the voltage and current may not reach their peaks simultaneously. The phase difference φ quantifies this shift. It is measured in radians or degrees, ranging from -π/2 to +π/2 for passive components.
在交流电路中，电压和电流可能不同时达到峰值。相位差 φ 量化这一偏移，用弧度或度衡量，对于无源元器件范围在 -π/2 至 +π/2 之间。
- Resistor: φ = 0° (V and I in phase)
- Inductor: φ = +90° (voltage leads current)
- Capacitor: φ = -90° (current leads voltage)
电阻器：φ = 0°（电压与电流同相）；电感器：φ = +90°（电压超前电流）；电容器：φ = -90°（电流超前电压）。
5. Pure Resistive Circuit | 纯电阻电路
A pure resistor simply obeys Ohm’s law at every instant: V_R = I_R R. Both voltage and current phasors are in phase, so the instantaneous power p = v i is always positive. The energy is completely dissipated as heat.
纯电阻在任何时刻都遵循欧姆定律：V_R = I_R R。电压和电流相量同相，瞬时功率 p = v i 始终为正，能量完全以热量耗散。
P_avg = V_rms I_rms = I_rms² R = V_rms² / R
The power delivered to a resistance is purely active power, with power factor equal to 1.
电阻消耗的功率为纯有功功率，功率因数为 1。
6. Pure Inductive Circuit | 纯电感电路
An inductor opposes changes in current through its self-inductance L. The induced emf causes the current to lag the voltage by 90°. The opposition is called inductive reactance X_L:
电感通过自感 L 阻碍电流变化。感应电动势使电流滞后电压 90°。这种阻碍称为感抗 X_L：
X_L = ωL = 2πfL (unit: ohm, Ω)
The peak voltage and current relate as V_L = I_L X_L. No net power is dissipated over a full cycle; energy is temporarily stored in the magnetic field and then returned to the circuit.
峰值电压与电流的关系为 V_L = I_L X_L。整个周期内无净功率耗散；能量暂时储存在磁场中并返回电路。
7. Pure Capacitive Circuit | 纯电容电路
A capacitor stores charge on its plates, leading to a current that leads the voltage by 90°. Capacitive reactance X_C is given by:
电容器在极板上储存电荷，导致电流超前电压 90°。容抗 X_C 的表达式为：
X_C = 1 / (ωC) = 1 / (2πfC) (Ω)
The voltage amplitude is V_C = I_C X_C. Like an inductor, a capacitor does not dissipate net energy; it stores energy in the electric field and releases it each cycle.
电压幅值为 V_C = I_C X_C。与电感类似，电容器不耗散净能量；它在电场中储存能量并在每个周期释放。
8. Reactance and Impedance | 电抗与阻抗
Impedance Z is the total opposition to current in an AC circuit, combining resistance R and reactance X. For a series RLC circuit, the net reactance is X = X_L – X_C, and the impedance magnitude is:
阻抗 Z 是交流电路对电流的总阻碍，由电阻 R 和电抗 X 组成。对于串联 RLC 电路，净电抗 X = X_L – X_C，阻抗大小为：
Z = √(R² + (X_L – X_C)²)
The phase angle φ between the supply voltage and current satisfies tan φ = (X_L – X_C) / R. Ohm’s law for AC becomes V_rms = I_rms Z. The table below summarises the characteristics of pure components.
电源电压与电流之间的相位角 φ 满足 tan φ = (X_L – X_C) / R。交流欧姆定律为 V_rms = I_rms Z。下表总结了纯元器件的特性。
| Component | Reactance / Resistance | Phase φ | Phasor relation |
|---|---|---|---|
| Resistor | R | 0° | V_R in phase with I |
| Inductor | X_L = ωL | +90° (V leads I) | V_L leads I |
| Capacitor | X_C = 1/(ωC) | -90° (I leads V) | V_C lags I |
相应地：电阻器：R，同相；电感器：感抗 X_L = ωL，电压超前电流 90°；电容器：容抗 X_C = 1/(ωC)，电流超前电压 90°。
9. Series RLC Circuit and Phasor Diagrams | 串联RLC电路与相量图
In a series RLC circuit, the same current flows through all components. Phasor diagrams help visualise the addition of voltages. The resistor voltage V_R is in phase with I, V_L leads by 90°, and V_C lags by 90°. The supply voltage V is the phasor sum:
在串联 RLC 电路中，同一电流流过所有元件。相量图有助于可视化电压相加。电阻电压 V_R 与 I 同相，V_L 超前 90°，V_C 滞后 90°。电源电压 V 为相量和：
V = √(V_R² + (V_L – V_C)²)
Resonance occurs when X_L = X_C, meaning V_L = V_C and the circuit behaves purely resistive. At resonance, impedance is minimum Z = R, and current is maximum. The resonant frequency is f₀ = 1/(2π√(LC)).
当 X_L = X_C 时发生谐振，即 V_L = V_C，电路呈纯阻性。谐振时阻抗最小 Z = R，电流最大。谐振频率为 f₀ = 1/(2π√(LC))。
10. Power in AC Circuits | 交流电路中的功率
The average power delivered to an AC circuit is given by:
交流电路的平均功率为：
P = V_rms I_rms cos φ
where cos φ is the power factor. For purely resistive loads, cos φ = 1, and all power is active. For pure inductors or capacitors, cos φ = 0, indicating no real power dissipation—only reactive power. In mixed circuits, the power factor lies between 0 and 1, and improving it (e.g., by adding capacitors) reduces wasted current in power lines.
其中 cos φ 是功率因数。纯电阻负载 cos φ = 1，所有功率为有功功率。纯电感或电容 cos φ = 0，表明无实际功率耗散，只有无功功率。在混合电路中，功率因数在 0 到 1 之间，提高功率因数（如添加电容器）可减少输电线路中的无功电流浪费。
11. Transformers | 变压器
A transformer uses two coils wound on a common iron core to change AC voltages. It operates on Faraday’s law of electromagnetic induction. For an ideal transformer with no energy losses:
变压器使用绕在同一铁芯上的两个线圈来改变交流电压，其工作原理基于法拉第电磁感应定律。对于无能量损耗的理想变压器：
V_s / V_p = N_s / N_p = I_p / I_s
where V_p, V_s are primary and secondary voltages; N_p, N_s are turns; I_p, I_s are currents. Power is conserved: P_p = V_p I_p = P_s = V_s I_s. A step-up transformer has N_s > N_p (increases voltage, decreases current); a step-down has N_s < N
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