📚 Transformers | 变压器
A transformer is a static electrical device that transfers alternating current (AC) energy from one circuit to another through electromagnetic induction, while usually changing voltage and current levels without altering the frequency. The ability to step voltage up or down efficiently makes the transformer indispensable in power transmission, electronic power supplies, and countless everyday appliances.
变压器是一种静止的电气设备，它通过电磁感应将交流电能从一个电路传输到另一个电路，通常可以改变电压和电流的大小而不改变频率。由于能够高效地升压或降压，变压器在电力输送、电子电源和无数日常电器中不可或缺。
1. Introduction to Transformers | 变压器简介
A transformer consists of two (or more) insulated coils wound on a common laminated soft-iron core. The coil connected to the AC supply is the primary winding, and the coil delivering energy to the load is the secondary winding. By exploiting mutual inductance, the transformer isolates the two circuits electrically while coupling them magnetically.
变压器由两个（或多个）绕在共用叠片软铁芯上的绝缘线圈组成。连接到交流电源的线圈称为初级绕组，向负载输送能量的线圈称为次级绕组。通过利用互感，变压器在电气上将两个电路隔离，同时又通过磁场将它们耦合。
2. Principle of Operation: Electromagnetic Induction | 工作原理：电磁感应
When an alternating voltage is applied to the primary coil, an alternating current flows, producing a time-varying magnetic flux in the core. Because the core provides a closed magnetic path with high permeability, almost all this flux links both the primary and secondary windings. According to Faraday’s law of electromagnetic induction, the changing flux induces an electromotive force (e.m.f.) in each turn of the coils.
当在初级线圈上施加交流电压时，交变电流流过线圈，在铁芯中产生随时间变化的磁通量。由于铁芯提供了一个高磁导率的闭合磁路，几乎所有的磁通都同时交链初级和次级绕组。根据法拉第电磁感应定律，变化的磁通会在每匝线圈中感应出电动势。
ε = −N (ΔΦ/Δt)
The negative sign, described by Lenz’s law, indicates that the induced e.m.f. opposes the change in flux. In a transformer, the same rate of change of flux ΔΦ/Δt is experienced by every turn of both coils, so the induced e.m.f. per turn is identical.
根据楞次定律，负号表示感应电动势抵抗磁通的变化。在变压器中，两个线圈的每一匝都经历相同的磁通变化率 ΔΦ/Δt，因此每匝感应出的电动势相等。
3. Structure of a Transformer: Core and Windings | 变压器结构：铁芯与绕组
The core is built from thin laminations of silicon steel, insulated from one another by varnish or oxide layers. Lamination reduces eddy currents by increasing the electrical resistance of the path through which they would flow. The windings are made of enameled copper wire and are carefully arranged to maximise flux linkage and minimise leakage flux.
铁芯由硅钢薄片叠成，片间通过漆膜或氧化层相互绝缘。叠片通过增大涡流路径的电阻来减小涡流。绕组由漆包铜线制成，并精心布置以最大限度地提高磁通交链并减小漏磁通。
4. The Ideal Transformer Equation: Voltage Ratio | 理想变压器方程：电压比
For an ideal transformer with no energy losses and perfect flux linkage, the induced e.m.f.s are proportional to the number of turns. If Vp and Vs are the primary and secondary voltages, and Np and Ns the respective numbers of turns, the relationship is:
对于无能量损耗且磁通完全交链的理想变压器，感应电动势与匝数成正比。若 Vp 和 Vs 分别为初级和次级电压，Np 和 Ns 为对应的匝数，则有：
Vp / Vs = Np / Ns
This equation shows that the transformer can produce a secondary voltage greater or smaller than the primary voltage simply by choosing the turns ratio. In practice, the actual secondary voltage is slightly less than predicted due to winding resistance and leakage reactance, but the ratio holds very well for unloaded or lightly loaded transformers.
该方程表明，只需选择匝数比便能使次级电压大于或小于初级电压。实际中，由于绕组电阻和漏电抗，次级电压会略低于理论值，但在空载或轻载时该比值非常准确。
5. Current Ratio and Conservation of Power | 电流比与功率守恒
An ideal transformer does not create or destroy energy; the instantaneous electrical power delivered to the primary equals the instantaneous power taken from the secondary. Thus:
理想变压器既不产生能量也不消灭能量；输入初级的瞬时电功率等于从次级取走的瞬时电功率。因此：
Vp Ip = Vs Is
Combining the power conservation with the voltage ratio yields the current ratio:
将功率守恒与电压比相结合，可得电流比：
Ip / Is = Ns / Np
This inverse relationship means that in a step-up transformer (Ns > Np), the secondary current is smaller than the primary current, while in a step-down transformer (Ns < Np), the secondary current is larger. The current ratio ensures that the power transferred remains constant.
这一反比关系意味着升压变压器（Ns > Np）中，次级电流小于初级电流；而在降压变压器（Ns < Np）中，次级电流则较大。电流比保证了传输的功率不变。
6. Step-Up and Step-Down Transformers | 升压与降压变压器
A step-up transformer has more turns on the secondary than on the primary (Ns > Np). It increases voltage and decreases current proportionally. These are used at power stations to raise voltage for long-distance transmission, reducing I²R losses in the cables.
升压变压器次级匝数多于初级（Ns > Np），它能升高电压并成比例地降低电流。发电厂常用此类变压器将电压提升以进行远距离输送，从而减少电缆中的 I²R 损耗。
A step-down transformer has fewer secondary turns (Ns < Np) and reduces voltage while increasing available current. Such transformers are found in substations to lower transmission voltage to safe distribution levels, and in power adapters for electronic devices.
降压变压器次级匝数较少（Ns < Np），降低电压的同时增大可用电流。变电站使用此类变压器将输电电压降至安全配电水平，电子设备的电源适配器中也是如此。
7. Eddy Current and Core Losses | 涡流与铁芯损耗
In a real transformer, the alternating flux induces circulating currents called eddy currents within the bulk of the iron core. These currents follow closed loops perpendicular to the flux and dissipate energy as heat (I²R loss). Core losses, sometimes called iron losses, also include hysteresis loss—energy wasted as the magnetic domains in the core repeatedly re-align with the alternating magnetisation.
在实际变压器中，交变磁通会在铁芯体内感应出称为涡流的循环电流。这些电流沿垂直于磁通的方向形成闭合回路，并以热的形式（I²R 损耗）消耗能量。铁芯损耗有时也称为铁损，还包括磁滞损耗——铁芯中的磁畴随交变磁化不断重新排列所耗散的能量。
To minimise eddy currents, the core is laminated into thin sheets (typically 0.3–0.5 mm thick) that are insulated from one another. This effectively interrupts the long conducting paths, drastically reducing eddy-current amplitude.
为减小涡流，铁芯被制成相互绝缘的薄片（通常厚 0.3–0.5 mm），这样有效切断了长的导电通路，大幅降低了涡流强度。
8. Copper Losses and Efficiency | 铜损耗与效率
d according to P = I²R. In a well-designed transformer, thick wire with low resistivity is used to keep the winding resistance small, yet copper loss remains the dominant loss under full-load conditions.
铜损耗（或称绕组损耗）源于初级和次级线圈中铜导线的电阻。电流流过时，根据 P = I²R 产生热量。设计良好的变压器采用低电阻率的粗导线以保持绕组电阻很小，但在满载条件下铜损耗仍是主要损耗。
The efficiency of a transformer is defined as the ratio of output power to input power:
变压器的效率定义为输出功率与输入功率之比：
Efficiency = (Vs Is) / (Vp Ip) × 100%
Large power transformers can achieve efficiencies above 99%, whereas small domestic transformers may operate around 85–95%.
大型电力变压器的效率可超过 99%，而小型家用变压器约在 85–95% 之间。
9. Practical Design: Laminated Cores | 实际设计：叠片铁芯
As noted, laminating the core is the primary method of reducing eddy-current loss. The laminations are stamped from silicon steel (typically 3–4% silicon by mass), which increases electrical resistivity and improves magnetic permeability. The sheets are coated with a thin insulating layer and stacked to form a closed magnetic circuit, often in an E-I or C-core shape.
如前所述，叠片铁芯是减少涡流损耗的主要方法。叠片用硅钢冲制（硅含量质量比通常为 3–4%），以提高电阻率和改善磁导率。薄板涂有薄绝缘层，并堆叠形成闭合磁路，常采用 E-I 型或 C 型铁芯。
In addition, the core is designed to operate below magnetic saturation to keep the magnetising current small and to avoid distortion of the voltage waveform. The cross-sectional area is chosen so that the peak flux density remains within the linear portion of the B-H curve.
此外，铁芯被设计在磁饱和以下工作，以保持励磁电流较小并避免电压波形畸变。铁芯截面积的选择使得峰值磁通密度保持在 B-H 曲线的线性区。
10. Applications in Power Transmission | 电力传输中的应用
The most consequential use of transformers is in the transmission and distribution of electric power. At a generating station, a step-up transformer raises the generator voltage (typically 11–33 kV) to several hundred kilovolts (e.g., 400 kV or 765 kV). This high voltage reduces the current for a given amount of transmitted power, greatly reducing heat losses in overhead lines.
变压器最重要的应用是电力的输送与分配。在发电站，升压变压器将发电机电压（通常 11–33 kV）升至数百千伏（如 400 kV 或 765 kV）。这种高电压在传输给定功率时降低电流，从而大大减少架空线的热损耗。
At the receiving end, sub-station transformers step the voltage down in stages (e.g., 132 kV, 33 kV, 11 kV) until it reaches the 230 V (or 110 V) supply suitable for domestic and commercial use. Without transformers, AC power transmission over such distances would be impractical.
在接受端，变电站变压器逐级降压（如 132 kV, 33 kV, 11 kV），直至达到适合家庭和商业使用的 230 V（或 110 V）电源。没有变压器，长距离的交流输电将无法实现。
11. Transformer Rating and Real-World Considerations | 变压器额定值及实际考量
Transformers are rated in volt-amperes (VA) or kilovolt-amperes (kVA) rather than watts, because the load power factor varies. The rating specifies the apparent power that the transformer can deliver continuously without overheating. Important parameters include maximum rated voltage, current, and the allowable temperature rise.
变压器的额定值用伏安（VA）或千伏安（kVA）表示，而非瓦特，因为负载的功率因数会变化。该额定值规定了变压器在不过热的情况下能连续输送的视在功率。重要参数包括最大额定电压、电流和允许温升。
Real transformers also exhibit leakage inductance—magnetic flux that links only one winding and not the other. This causes a voltage drop under load and is modelled as an equivalent series reactance. In addition, the magnetising current required to sustain the core flux causes a small, constant core loss even when the secondary is open-circuited.
实际变压器还存在漏电感——仅交链一个绕组而不交链另一个的磁通量。这会导致负载下的电压降，通常用等效串联电抗来建模。此外，维持铁芯磁通所需的励磁电流即使在次级开路时也会引起小而恒定的铁芯损耗。
12. Summary of Key Points | 要点总结
A transformer operates on the principle of electromagnetic induction between two magnetically coupled coils. In the ideal case, the voltage ratio equals the turns ratio, and power is conserved, so the current ratio is the inverse of the turns ratio. Real transformers suffer from copper losses, eddy-current and hysteresis losses, yet efficiencies above 95% are typical. Laminated silicon-steel cores, high-conductivity windings, and careful design minimise these losses, making the transformer an essential, highly efficient component of modern electrical infrastructure.
变压器利用两个磁耦合线圈之间的电磁感应原理工作。在理想情况下，电压比等于匝数比，且功率守恒，因此电流比为匝数比的反比。实际变压器存在铜损耗、涡流损耗和磁滞损耗，但效率通常仍在 95% 以上。叠片硅钢铁芯、高导电率绕组和精心设计将这些损耗降到最低，使得变压器成为现代电力基础设施中必不可少且高效能的核心部件。
Published by TutorHao | Physics Revision Series | aleveler.com
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