Alternating Currents | 交流电

📚 Alternating Currents | 交流电

Alternating current (AC) is a type of electrical current whose direction reverses periodically with time. Unlike direct current (DC), which flows steadily in one direction, AC is the standard form of electricity supplied to homes and industries. Understanding AC is essential for studying power generation, transmission, transformers, and rectifier circuits in A-Level Physics.

交流电（Alternating Current，简称 AC）是一种方向随时间周期性变化的电流。与始终沿单一方向流动的直流电（DC）不同，交流电是家庭和工业供电的标准形式。理解交流电对于学习发电、输电、变压器和整流电路等 A-Level 物理内容至关重要。

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1. What is Alternating Current? | 什么是交流电？

An alternating current is one that continuously reverses its direction of flow. The magnitude of the current also changes with time, usually following a sinusoidal pattern. The voltage driving this current likewise alternates. In the UK mains supply, for example, the frequency is 50 Hz, meaning the current changes direction 100 times per second.

交流电是一种流动方向不断反转的电流。电流的大小也随时间变化，通常遵循正弦规律。驱动该电流的电压同样在交替变化。例如，英国的市电频率为 50 Hz，这意味着电流每秒钟改变方向 100 次。

AC is produced by rotating a coil in a magnetic field, which induces an electromotive force (EMF) that varies sinusoidally with time. This makes AC easy to generate and to step up or down using transformers, giving it a huge practical advantage over DC for long-distance power transmission.

交流电通过线圈在磁场中旋转产生，这会感应出随时间呈正弦变化的电动势（EMF）。这使得交流电易于发电，也便于使用变压器升压或降压，因此在远距离输电方面相比直流电具有巨大的实际优势。

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2. The Sinusoidal Waveform | 正弦波形

For a sinusoidal alternating voltage or current, the instantaneous value can be described by the equations:

对于正弦交流电压或电流，其瞬时值可以用以下方程描述：

V = V₀ sin ωt

I = I₀ sin ωt

tity ωt represents the phase angle in radians. This waveform repeats every period T = 1/f.

其中 V₀ 和 I₀ 是峰值，ω 是角频率（ω = 2πf），t 是时间。量 ωt 表示以弧度为单位的相位角。该波形每周期 T = 1/f 重复一次。

The graph of V or I against time is a smooth sine curve that rises to a positive maximum, falls through zero, reaches a negative minimum and returns to zero. The average value over a full cycle is zero, but the heating effect is not zero.

V 或 I 随时间变化的图像是一条平滑的正弦曲线，上升到正最大值，穿过零点，达到负最小值，再回到零。在一个完整周期内的平均值为零，但其热效应并不为零。

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3. Peak, Mean, and RMS Values | 峰值、平均值和均方根值

The peak value (V₀ or I₀) is the maximum magnitude of the voltage or current. The mean (average) value of a pure sinusoidal waveform over a full period is zero, because the positive and negative halves cancel. However, the mean of the absolute value over a half-cycle can be used in rectification.

峰值（V₀ 或 I₀）是电压或电流的最大幅度。一个纯正弦波形在整个周期内的平均值（均值）为零，因为正负半周互相抵消。不过，在半波或全波整流中会用到半个周期内的绝对平均值。

To describe the effective heating ability of AC, we use the root-mean-square (r.m.s.) value. The r.m.s. value of an AC is the equivalent DC value that would produce the same average power dissipation in a resistor.

为了描述交流电的有效加热能力，我们使用均方根值（r.m.s.）。交流电的 r.m.s. 值是一个等效的直流值，它能在电阻中产生相同的平均功率耗散。

For a sinusoidal waveform, the r.m.s. values are related to the peak values by:

对于正弦波形，r.m.s. 值与峰值的关系为：

V_rms = V₀ / √2

I_rms = I₀ / √2

For example, the UK mains supply has a peak voltage of about 325 V, giving an r.m.s. voltage of 325/√2 ≈ 230 V, which is the normally quoted value.

例如，英国市电的峰值电压约为 325 V，其 r.m.s. 电压为 325/√2 ≈ 230 V，这也就是通常所给出的电压值。

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4. Deriving the RMS Value for Sinusoidal AC | 推导正弦交流电的 RMS 值

rent, take the mean over one full cycle, and then take the square root. For I = I₀ sin ωt:

为了得到 r.m.s.值，我们先将瞬时电流平方，求一个完整周期内的平均值，再开方。对于 I = I₀ sin ωt：

I² = I₀² sin² ωt

The key step is recognising that the average of sin² ωt over a full period is ½. Because the sin² curve is always positive and symmetrical, its mean value is ½ of the peak square value.

关键的一步是认识到在一个完整周期内 sin² ωt 的平均值为 ½。因为 sin² 曲线始终为正且对称，其均值是峰值平方的一半。

Therefore, the mean square current = I₀²/2. Taking the square root gives I_rms = √ = I₀ / √2. The same steps apply to voltage. This derivation emphasizes why the √2 factor appears and is a typical exam problem.

因此，均方电流 = I₀²/2。开方后得到 I_rms = √ = I₀ / √2。对于电压也是相同的过程。这个推导强调了 √2 因子出现的原因，也是典型的考试内容。

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5. Power in AC Circuits | 交流电路中的功率

it, the instantaneous power is P = I V = I₀ V₀ sin² ωt. The average power over a cycle is:

在纯电阻交流电路中，瞬时功率为 P = I V = I₀ V₀ sin² ωt。一个周期内的平均功率为：

= ½ I₀ V₀ = I_rms V_rms

Thus, average power can be calculated directly from r.m.s. values just as in DC circuits. This shows why r.m.s. values are so useful: they allow us to apply familiar DC power formulas to AC circuits.

因此，平均功率可以直接由 r.m.s. 值计算，就像在直流电路中那样。这表明了 r.m.s. 值的巨大用处：它让我们能将熟悉的直流功率公式直接应用于交流电路。

The relationship P_ave = I_rms × V_rms holds for resistive loads. For components like capacitors or inductors where voltage and current are out of phase, the average power is lower, but that topic belongs to reactance studies.

对于电阻性负载，P_ave = I_rms × V_rms 关系成立。对于电容或电感这类电压与电流存在相位差的元件，平均功率会较低，但该内容属于电抗部分的学习范围。

Quantity	Symbol	Sinusoidal relation
Peak current	I₀	-
r.m.s. current	Irms	Irms = I₀/√2
Average power	Pave	Pave = Irms Vrms

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6. Half-Wave Rectification | 半波整流

Rectification is the process of converting AC into DC. A single diode can achieve half-wave rectification. The diode conducts only when the AC voltage is in the forward-bias direction, allowing current to pass during one half of the cycle and blocking it during the other half.

整流是将交流电转变为直流电的过程。单个二极管即可实现半波整流。只有在交流电压处于正向偏置方向时二极管才导通，仅在周期的一半时间内允许电流通过，而在另一半时间内截止。

The output voltage V_out against time is a series of positive half-cycles separated by gaps where the voltage is zero. The average voltage of this output is V_avg han the r.m.s. of the original AC. Half-wave rectification is simple but very inefficient.

输出电压 V_out 随时间的变化是一系列被零电压间隔开来的正半周波形。该输出波形的平均电压为 V_avg = V₀/π，远低于原交流电的 r.m.s. 值。半波整流虽然简单，但效率极低。

• The diode conducts for one half-cycle only.
• 输出中只有半个周期导通。
• The output is unidirectional but highly fluctuating.
• 输出虽为单向，但波动极大。
• Average voltage V_avg = V₀/π ≈ 0.318 V₀.
• 平均电压 V_avg = V₀/π ≈ 0.318 V₀。

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7. Full-Wave Rectification: The Bridge Rectifier | 全波整流：桥式整流器

Full-wave rectification utilises both halves of the AC input. The most common method uses four diodes arranged in a bridge configuration. During the positive half-cycle, a pair of diodes conducts; during the negative half-cycle, the other pair conducts. This directs the current through the load resistor in the same direction during both halves.

全波整流利用交流输入的两个半周。最常见的方法是使用四个二极管组成桥式电路。在正半周，一对二极管导通；在负半周，另一对二极管导通。这样在两个半周期间通过负载电阻的电流方向始终保持一致。

The resulting V_out–t graph shows a series of unidirectional humps with no gaps. The output frequency is twice the input frequency (e.g. 100 Hz ripple for a 50 Hz supply). The average voltage is doubled compared with half-wave: V_avg = 2V₀/π.

输出的 V_out–t 图显示为一系列没有间隔的单向波形凸起。输出频率是输入频率的两倍（例如，对于 50 Hz 市电，纹波频率为 100 Hz）。平均电压相比半波整流翻倍：V_avg = 2V₀/π。

• Four diodes are used; a centre-tapped transformer is an alternative.
• 使用四只二极管；也可以采用中心抽头变压器方案。
• Output frequency is doubled, making smoothing easier.
• 输出频率加倍，使得滤波平滑更容易。
• The bridge rectifier is widely used in power supplies.
• 桥式整流器广泛用于电源电路中。

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8. Smoothing with a Capacitor | 电容滤波平滑

The output of a rectifier still varies considerably and is not suitable as a steady DC supply. Smoothing is achieved by connecting a large capacitor in parallel with the load. The capacitor charges when the rectified voltage rises and discharges slowly through the load resistor when the voltage falls.

整流后的输出仍然波动很大，不适合作为稳定的直流电源。通过在负载两端并联一个大电容可实现平滑。当整流电压上升时电容充电，当电压下降时电容通过负载电阻缓慢放电。

This reduces the variation in output voltage, leaving a small 'ripple'. The ripple voltage depends on the capacitance C, the load resistance R, and the supply frequency f. A larger capacitance stores more charge, and a higher load resistance reduces discharge current, both leading to a smoother output.

这样可以减小输出电压的变化，只留下较小的“纹波”。纹波电压取决于电容 C、负载电阻 R 以及电源频率 f。电容越大储存的电荷越多，负载电阻越大放电电流越小，这两者均能使输出更平滑。

Approximately, the peak-to-peak ripple voltage ΔV can be estimated by:

输入电压纹波的峰-峰值 ΔV 可近似估算为：

ΔV = I_load / (f × C)

where I_load ≈ V_avg / R. For effective smoothing, the time constant RC should be much larger than the period of the ripple.

其中 I_load ≈ V_avg / R。为了有效平滑，时间常数 RC 应当远大于纹波的周期。

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9. The Transformer Principle | 变压器原理

A transformer consists of two coils (primary and secondary) wound on a common soft-iron core. An alternating current in the primary coil creates a changing magnetic flux in the core, which links the secondary coil. By Faraday's law, this changing flux induces an alternating EMF in the secondary coil.

变压器由绕在同一软铁芯上的两个线圈（初级和次级）组成。初级线圈中的交流电在铁芯中产生变化的磁通，该磁通穿过次级线圈。根据法拉第定律，这一变化的磁通会在次级线圈中感应出交变电动势。

For an ideal transformer with no flux leakage and no energy losses, the induced EMF per turn is the same in both coils. Hence the ratio of secondary voltage V_s to primary voltage V_p equals the turns ratio:

对于无磁通泄漏且无能量损耗的理想变压器，两个线圈中每匝的感应电动势相同。因此次级电压 V_s 与初级电压 V_p 之比等于匝数比：

V_s / V_p = N_s / N_p

If N_s > N_p, the transformer is a step-up transformer; if N_s < N_p, it is a step-down transformer.

若 N_s > N_p，变压器为升压变压器；若 N_s < N_p，则为降压变压器。

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10. Ideal Transformer Equations and Efficiency | 理想变压器方程与效率

For an ideal transformer, the power input to the primary equals the power output from the secondary, assuming 100% efficiency. This gives the current relationship:

对于理想变压器，在假设效率为 100% 的情况下，初级输入功率等于次级输出功率。由此可得电流关系：

I_p V_p = I_s V_s

Combining with the turns ratio, we get I_s / I_p = N_p / N_s. This shows that a step-up transformer (increasing voltage) simultaneously steps down the current, and vice versa, conserving power.

结合匝数比，可得 I_s / I_p = N_p / N_s。这表明升压变压器在升高电压的同时会成比例降低电流，降压变压器则相反，以此保持功率守恒。

Efficiency of real transformers is defined as output power / input power. Typical power transformer efficiencies exceed 95%. Exam questions often ask students to calculate efficiency or to identify reasons for energy losses.

实际变压器的效率定义为输出功率与输入功率之比。典型的电力变压器效率超过 95%。考题常要求学生计算效率或指出能量损耗的原因。

Type	Turns ratio Ns / Np	Effect
Step-up	> 1	Vs > Vp, Is < Ip
Step-down	< 1	Vs < Vp, Is > Ip

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11. Energy Losses in Real Transformers | 实际变压器的能量损耗

Real transformers are not ideal and suffer from several sources of power loss. Understanding these losses is key to improving design and efficiency.

实际变压器并非理想，存在多种功率损耗。理解这些损耗是改进设计和提高效率的关键。

Copper losses (I²R loss): The primary and secondary coils have resistance, and current flowing through them generates heat. This loss increases with the square of the current, so it is significant under heavy loads.

铜损耗（I²R 损耗）：初级和次级线圈有电阻，电流流过时产生焦耳热。该损耗与电流平方成正比，因此在重负载下尤为显著。

Eddy current losses: The changing magnetic flux also induces circulating currents in the iron core itself. These eddy currents produce heat and waste energy. They are minimised by laminating the core – building it from thin, insulated sheets of iron.

涡流损耗：变化的磁通也会在铁芯内部感应出环流，即涡流。涡流产生热量，浪费能量。通过将铁芯分成彼此绝缘的薄片（层叠）可以大大减小涡流。

Hysteresis loss: The core's magnetisation constantly reverses with the AC, and the work done in cycling the magnetic domains appears as heat. Using soft-iron materials with narrow hysteresis loops reduces this loss.

磁滞损耗：铁芯的磁化状态随交流电不断反转，磁畴循环取向所作的功以热的形式散失。使用磁滞回线狭窄的软铁材料可减少这种损耗。

Flux leakage: Not all the flux produced by the primary coil links through the secondary. This reduces the induced EMF and effective power transfer. Good core design brings leakage to a minimum.

磁通泄漏：并非所有由初级线圈产生的磁通都穿过次级线圈，这降低了感应电动势和有效的能量传输。优良的铁芯设计可使泄漏降至最低。

• Copper loss ∝ I², reduced by using thicker wire.
• 铜损耗 ∝ I²，使用更粗的导线可减小该损耗。
• Eddy currents are reduced by laminating the core.
• 涡流通过层叠铁芯来减小。
• Hysteresis loss is minimised with soft-iron cores.
• 磁滞损耗通过采用软铁芯来降低。
• Flux leakage is kept small through efficient core design.
• 通过高效的铁芯设计使磁通泄漏保持较小。

Efficiency = (output power / input power) × 100%. A well-designed transformer can reach 99% efficiency.

效率 = （输出功率 / 输入功率）× 100%。一个设计良好的变压器效率可达 99%。

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Published by TutorHao | Physics Revision Series | aleveler.com

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