📚 IB AQA Physics: Alternating Current Key Points | IB AQA 物理：交流电 考点精讲
Alternating current (AC) is a fundamental topic in IB and AQA physics syllabuses, underpinning everything from household electricity to advanced electronics. This article distills the essential concepts, formulas, and problem-solving strategies you need to master. We cover sinusoidal waveforms, root mean square (RMS) values, reactance and impedance, phase relationships in resistive, inductive and capacitive circuits, power, transformers and rectification, all presented with precise bilingual explanations.
交流电是 IB 和 AQA 物理课程中的核心主题，从家庭用电到高级电子学都离不开它。本文提炼了必须掌握的核心概念、公式和解题策略。我们将覆盖正弦波形、均方根值、电抗与阻抗、电阻性、电感性和电容性电路中的相位关系、功率、变压器和整流，全部配以精确的双语讲解。
1. Understanding Alternating Current | 理解交流电
Alternating current (AC) is the flow of electric charge that periodically reverses direction. In contrast to direct current (DC), where charge moves in a single steady direction, AC voltage varies sinusoidally with time, causing current to oscillate back and forth. This is the form of electricity delivered to homes and industry because it can be efficiently transformed to different voltages.
交流电是指电荷流动方向周期性反转的电流。与电荷沿单一稳定方向运动的直流电不同，交流电压随时间呈正弦变化，导致电流来回振荡。这是输送到家庭和工业的电力形式，因为它可以高效地变换为不同电压。
A standard AC supply has a waveform described by a sine function: the instantaneous voltage v at time t is given by v = V₀ sin(ωt), where V₀ is the peak voltage and ω is the angular frequency. Similarly, the instantaneous current i = I₀ sin(ωt) in a purely resistive circuit. The oscillation repeatedly crosses zero, reaching positive and negative peaks.
标准交流电源的波形由正弦函数描述：t 时刻的瞬时电压 v = V₀ sin(ωt)，其中 V₀ 为峰值电压，ω 为角频率。类似地，在纯电阻电路中瞬时电流 i = I₀ sin(ωt)。振荡反复过零，达到正负峰值。
2. Sinusoidal Waveform Parameters | 正弦波形参数
Key parameters define an AC waveform. The period T is the time for one complete cycle, measured in seconds. The frequency f = 1/T is the number of cycles per second, expressed in hertz (Hz). For mains electricity in many countries, f = 50 Hz, meaning the voltage completes 50 full oscillations each second. The angular frequency ω = 2πf has units of rad s⁻¹.
关键参数定义了交流波形。周期 T 是完成一个完整循环所需的时间，单位为秒。频率 f = 1/T 是每秒的循环数，以赫兹 (Hz) 表示。在许多国家，市电频率为 50 Hz，意味着电压每秒完成 50 个完整振荡。角频率 ω = 2πf，单位为 rad s⁻¹。
The peak value (amplitude) V₀ is the maximum voltage reached. A symmetrical AC waveform oscillates between +V₀ and –V₀. The peak-to-peak voltage Vpp = 2V₀. In practical measurements, the average voltage over a full cycle is zero because positive and negative halves cancel, which is why we use root mean square (RMS) values to describe effective magnitudes.
峰值（振幅）V₀ 是达到的最大电压。对称的交流波形在 +V₀ 和 –V₀ 之间振荡。峰峰值电压 Vpp = 2V₀。在实际测量中，由于正负半周相互抵消，整个周期的平均电压为零，因此我们使用均方根值来描述有效大小。
3. Root Mean Square (RMS) Values | 均方根（有效值）
The RMS value of an AC voltage or current is the equivalent DC value that would deliver the same average power to a resistor. For a pure sinusoidal waveform, the RMS voltage Vrms and current Irms are related to the peak values by a factor of √2:
交流电压或电流的均方根值是指能在电阻上产生相同平均功率的等效直流值。对于纯正弦波形，均方根电压 Vrms 和电流 Irms 与峰值的关系为除以 √2：
Vrms = V₀ / √2 Irms = I₀ / √2
For example, UK mains electricity has an RMS voltage of 230 V. The peak voltage is therefore V₀ = 230 × √2 ≈ 325 V. RMS values are crucial because voltmeters and ammeters designed for AC read RMS, and power calculations in resistive loads use P = Vrms × Irms.
例如，英国市电的均方根电压为 230 V。因此峰值电压 V₀ = 230 × √2 ≈ 325 V。均方根值至关重要，因为用于交流的电压表和电流表读取的是 RMS 值，且电阻性负载中的功率计算使用 P = Vrms × Irms。
Multimeters have an AC mode that directly displays RMS values (often assuming a sinusoidal input). Always remember that insulation and component ratings must withstand the peak voltage, not just the RMS value, to avoid breakdown.
万用表的交流模式直接显示有效值（通常假设正弦输入）。始终记住，绝缘和元件额定值必须能够承受峰值电压，而不仅仅是有效值，以避免击穿。
4. AC in Resistive Circuits | 纯电阻交流电路
When an AC source is connected to a pure resistor, the voltage and current are perfectly in phase. This means their waveforms reach zero, positive maximum, and negative maximum at the same instants. Ohm’s law applies instantaneously: v = iR, and for RMS values Vrms = Irms R.
当交流电源连接到纯电阻时，电压与电流完全同相。这意味着它们的波形在同一时刻过零、达到正最大值和负最大值。欧姆定律瞬时成立：v = iR，对于均方根值 Vrms = Irms R。
The instantaneous power p = vi = (V₀ sin ωt)(I₀ sin ωt) = V₀I₀ sin² ωt. This power is never negative; it always flows from source to resistor, dissipating energy as heat. The average power over a cycle is Pavg = Vrms Irms = I²rms R = V²rms / R.
瞬时功率 p = vi = (V₀ sin ωt)(I₀ sin ωt) = V₀I₀ sin² ωt。该功率永不为负；始终从电源流向电阻器，以热的形式耗散能量。一个周期内的平均功率 Pavg = Vrms Irms = I²rms R = V²rms / R。
In phasor diagrams, a pure resistor’s voltage and current phasors rotate together, with zero phase angle between them. No phase shift simplifies circuit analysis tremendously for heating elements and filament lamps.
在相量图中，纯电阻的电压和电流相量一同旋转，它们之间的相位角为零。无相位差极大地简化了加热元件和白炽灯等电路的分析。
5. AC in Inductive Circuits | 纯电感交流电路
An inductor opposes changes in current due to its self-inductance L. When AC is applied, the current lags behind the voltage by a phase angle of 90° (π/2 rad). The voltage leads the current, or equivalently, the current reaches its peak a quarter-cycle after the voltage peak.
电感器由于其自感 L 阻碍电流的变化。施加交流电时，电流滞后于电压 90°（π/2 rad）的相位角。电压超前于电流，或者说电流在电压峰值之后四分之一周期达到峰值。
The opposition to AC in an inductor is called inductive reactance XL, measured in ohms. It increases linearly with frequency and inductance:
在电感器中对交流的阻碍称为感抗 XL，单位为欧姆。它随频率和电感线性增加：
XL = ωL = 2πfL
Thus, an inductor presents low reactance at low frequencies (approaching a short circuit for DC, where f=0) and very high reactance at high frequencies. The rms current Irms = Vrms / XL.
因此，电感在低频时呈现低电抗（对于直流 f=0 近似短路），而在高频时呈现很高的电抗。均方根电流 Irms = Vrms / XL。
Energy is stored in the magnetic field of the inductor and returned to the circuit, so the average power over a full cycle in a pure inductor is zero. Real inductors have some resistance, leading to a phase shift less than 90° and power loss.
能量存储在电感器的磁场中并返回电路，因此纯电感在一个完整周期内的平均功率为零。实际电感器具有一定的电阻，导致相位差小于 90° 并产生功率损耗。
6. AC in Capacitive Circuits | 纯电容交流电路
A capacitor stores charge and opposes changes in voltage. In an AC circuit, the current leads the voltage by 90° (π/2 rad). The voltage reaches its peak a quarter-cycle after the current peak, because it takes time for the capacitor to charge and discharge through the alternating source.
电容器储存电荷并阻碍电压的变化。在交流电路中，电流超前电压 90°（π/2 rad）。电压在电流峰值之后四分之一周期才达到峰值，因为电容器通过交流电源充放电需要时间。
The capacitive reactance XC is given by:
容抗 XC 由下式给出：
XC = 1 / (ωC) = 1 / (2πfC)
Notice that XC decreases as frequency increases – a capacitor is essentially an open circuit to DC (f=0, infinite reactance) and becomes a short circuit at very high frequencies. The rms current Irms = Vrms / XC.
注意 XC 随频率增加而减小——电容器对直流 (f=0) 基本为开路（无限电抗），而在极高频率下变为短路。均方根电流 Irms = Vrms / XC。
Like a pure inductor, a pure capacitor dissipates no net energy; energy is alternately stored in the electric field and returned to the source. Phasor diagrams clearly show the 90° leading relationship of current with respect to voltage.
与纯电感一样，纯电容器不消耗净能量；能量交替地存储在电场中并返回电源。相量图清楚地显示了电流相对于电压的 90° 超前关系。
7. Reactance and Impedance | 电抗与阻抗
Impedance Z is the total opposition to AC in a circuit containing resistance, inductance and capacitance. It is the AC analogue of DC resistance, but includes both resistive and reactive components. The reactance X is the combined effect of XL and XC, with opposite signs because of the opposite phase shifts.
阻抗 Z 是包含电阻、电感和电容的电路中对交流的总阻碍。它是直流电阻的交流类比，但包含电阻和电抗分量。电抗 X 是 XL 和 XC 的综合效应，由于相位差相反，符号相反。
For a series RLC circuit, the impedance is calculated using a phasor approach, giving the magnitude:
对于串联 RLC 电路，阻抗通过相量法计算，其大小为：
Z = √(R² + (XL – XC)²)
The phase angle φ between the total voltage and the total current is given by tan φ = (XL – XC) / R. A positive φ means the circuit is net inductive (voltage leads current), while a negative φ indicates net capacitive behavior (current leads voltage). The rms current is Irms = Vrms / Z.
总电压与总电流之间的相位角 φ 由 tan φ = (XL – XC) / R 给出。φ 为正表示电路净呈感性（电压超前电流），φ 为负则表示净呈容性（电流超前电压）。均方根电流 Irms = Vrms / Z。
Resonance occurs in an RLC circuit when XL = XC, making the impedance purely resistive (Z = R) and φ = 0. The resonant frequency f₀ = 1/(2π√(LC)). At resonance, current is maximized and the circuit can produce large voltage oscillations across L and C (important for radio tuning).
当 XL = XC 时，RLC 电路发生谐振，使其阻抗为纯电阻性 (Z = R) 且 φ = 0。谐振频率 f₀ = 1/(2π√(LC))。在谐振时，电流达到最大，电路可在 L 和 C 上产生大幅电压振荡（这对无线电调谐很重要）。
8. Power in AC Circuits | 交流电功率
The power dissipated in an AC circuit depends on the phase angle. Only the resistive component consumes real power. The average power is given by Pavg = Vrms Irms cos φ, where cos φ is the power factor. For a pure resistor, cos φ = 1; for a pure inductor or capacitor, cos φ = 0 and average power is zero.
交流电路中的功耗取决于相位角。只有电阻分量消耗有功功率。平均功率由 Pavg = Vrms Irms cos φ 给出，其中 cos φ 是功率因数。对于纯电阻，cos φ = 1；对于纯电感或纯电容，cos φ = 0，平均功率为零。
The product Vrms Irms is called the apparent power S (measured in volt-amperes, VA). The real power P (in watts) is always less than or equal to apparent power. Reactive power Q (in VAR) flows back and forth without doing net work. Improving the power factor, e.g., by adding capacitors to inductive loads, reduces wasted current in transmission lines.
乘积 Vrms Irms 称为视在功率 S（单位为伏安，VA）。有功功率 P（单位为瓦特）始终小于或等于视在功率。无功功率 Q（单位为乏，VAR）来回流动而不做净功。提高功率因数，例如通过向感性负载添加电容器，可减少传输线中的无功电流浪费。
Power factor correction is an important real-world application; electricity suppliers often charge industrial users for low power factor because it requires higher currents for the same real power, increasing I²R losses in cables.
功率因数校正是一个重要的实际应用；电力供应商通常会对低功率因数的工业用户收费，因为相同的实际功率需要更大的电流，增加电缆中的 I²R 损耗。
9. The Transformer | 变压器
A transformer is a device that changes an alternating voltage from one value to another using electromagnetic induction. It consists of two coils, the primary and secondary, wound on a common laminated iron core. An alternating current in the primary creates a changing magnetic flux, which links the secondary and induces an emf.
变压器是一种利用电磁感应将交流电压从一个值变为另一个值的设备。它由两个线圈（初级和次级）缠绕在一个共用的叠片铁芯上构成。初级中的交流电产生变化的磁通量，该磁通量穿过次级并感应出电动势。
For an ideal transformer (no energy loss), the ratio of the voltages equals the ratio of the number of turns:
对于理想变压器（无能量损耗），电压比等于匝数比：
Vs / Vp = Ns / Np
Because power is conserved, the current ratio is inversely proportional to the turns ratio: Is / Ip = Np / Ns. Step-up transformers (Ns > Np) increase voltage and decrease current; step-down transformers do the opposite.
由于功率守恒，电流比与匝数比成反比：Is / Ip = Np / Ns。升压变压器 (Ns > Np) 提高电压、降低电流；降压变压器则相反。
Real transformers have losses due to winding resistance (copper losses), eddy currents and hysteresis in the core, but efficiencies are typically above 95%. The laminated core reduces eddy currents. Transformers only work with AC; a steady DC would not produce a changing flux, so no emf is induced in the secondary.
实际变压器存在由绕组电阻（铜损）、涡流和磁滞引起的损耗，但效率通常高于 95%。叠片铁芯可减少涡流。变压器仅适用于交流电；恒定的直流电不会产生变化的磁通量，因此次级中不会感应出电动势。
10. Rectification of AC | 交流电的整流
Rectification is the conversion of AC into DC using diodes. A half-wave rectifier uses a single diode to allow current during positive half-cycles only. The output is a pulsating DC with a large ripple. The average DC voltage for half-wave rectification is Vavg = V₀ / π ≈ 0.318 V₀.
整流是利用二极管将交流电转换为直流电。半波整流器使用单个二极管，仅在正半周允许电流通过。输出为脉动直流，纹波较大。半波整流的平均直流电压 Vavg = V₀ / π ≈ 0.318 V₀。
A full-wave bridge rectifier uses four diodes arranged to flip the negative half-cycle, making both halves contribute to a unidirectional output. The average voltage is doubled: Vavg = 2V₀ / π ≈ 0.637 V₀. The ripple frequency is twice the supply frequency.
全波桥式整流器使用四个二极管排列，将负半周翻转，使两个半周都对单向输出有贡献。平均电压翻倍：Vavg = 2V₀ / π ≈ 0.637 V₀。纹波频率为电源频率的两倍。
Smoothing is achieved by connecting a capacitor across the rectifier output. The capacitor charges to the peak voltage and discharges slowly through the load, reducing the ripple. With a large enough capacitor, the output approximates a steady DC voltage. The ripple can be further reduced with voltage regulators.
平滑处理是通过在整流器输出端并联电容器实现的。电容器充电至峰值电压，并通过负载缓慢放电，从而减小纹波。使用足够大的电容器，输出可近似为稳定的直流电压。通过稳压器可以进一步减小纹波。
In IB and AQA physics exams, you may be asked to sketch output waveforms for half-wave and full-wave rectification, both with and without smoothing, and to calculate ripple voltages.
在 IB 和 AQA 物理考试中，你可能会被要求绘制半波和全波整流输出波形（有平滑和无平滑），并计算纹波电压。
11. Summary of Key Formulas | 关键公式总结
The table below compiles the essential equations for AC circuit analysis. Keep them on your fingertips for quick application.
下表汇编了交流电路分析的基本方程。牢牢掌握它们以便快速应用。
| Quantity | Formula | Notes |
|---|---|---|
| Angular frequency | ω = 2πf | f in Hz |
| RMS value | Vrms = V₀/√2, Irms = I₀/√2 | For sinusoidal waves only |
| Inductive reactance | XL = ωL = 2πfL | Current lags voltage by 90° |
| Capacitive reactance | XC = 1/(ωC) = 1/(2πfC) | Current leads voltage by 90° |
| Impedance (series RLC) | Z = √(R² + (XL – XC)²) | Phase angle φ, tan φ = (XL–XC)/R |
| Resonant frequency | f₀ = 1/(2π√(LC)) | At resonance, Z = R |
| Average ac power | Pavg = Vrms Irms cos φ | cos φ is power factor |
| Ideal transformer | Vs/Vp = Ns/Np = Ip/Is | Conservation of power |
| Half-wave avg DC | Vavg = V₀/π | Single diode |
| Full-wave avg DC | Vavg = 2V₀/π | Bridge rectifier |
Mastering these relationships and understanding the physical meaning behind them will enable you to solve a wide variety of AC circuit problems confidently.
掌握这些关系并理解其背后的物理意义，你将能够自信地解决各种交流电路问题。
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