📚 Internal Resistance | 电源内阻
All practical sources of electrical energy, such as batteries and power supplies, contain an inherent opposition to the flow of current. This hidden resistance, called internal resistance, causes the terminal voltage of a source to drop when a current is drawn. Mastery of internal resistance is essential for A-Level Physics, linking circuit theory with real-world behaviour of cells and batteries.
所有实际的电能来源，如电池和电源，都包含一种固有的对电流流动的阻碍。这种隐藏的电阻称为内阻，导致电源在输出电流时端电压下降。掌握内阻对于 A-Level 物理至关重要，它将电路理论与电池的真实行为联系起来。
1. Introduction: What is Internal Resistance? | 引言：什么是内阻？
Every real voltage source possesses a small resistance inside it. This internal resistance, symbolised by r, arises from the materials of the electrodes, the electrolyte and the physical construction of the cell. When charge flows through the source, work has to be done against this internal opposition, and energy is dissipated as heat inside the source.
每个真实的电压源内部都含有微小的电阻。这个内阻用符号 r 表示，它来源于电极材料、电解液和电池的物理结构。当电荷流过电源时，必须克服这种内部阻碍做功，能量在电源内部以热的形式耗散。
As a result, the actual potential difference available across the terminals of a source, called the terminal p.d., is less than the theoretical maximum voltage the source can provide, known as the electromotive force (EMF). The difference between EMF and terminal voltage is the 'lost volts', equal to Ir.
因此，电源端钮上可用的实际电位差，即端电压，小于电源能提供的理论最大电压——电动势（EMF）。电动势与端电压之差就是“损失电压”，等于 Ir。
2. EMF and Terminal Potential Difference | 电动势与端电压
The electromotive force ε of a source is the energy supplied per unit charge as charge passes through the source. It is measured in volts (V) and represents the maximum possible terminal voltage when no current flows. EMF is a cause that drives charges around a complete circuit, but it is not a force in the Newtonian sense.
电源的电动势 ε 是单位电荷通过电源时被提供的能量，单位是伏特（V），它表示无电流流动时可能达到的最大端电压。电动势是驱动电荷在完整回路中运动的起因，但它并不是牛顿力学中的力。
When a current I flows, the terminal voltage V measured across the source's terminals is always less than ε. The reduction is due to the internal resistance: V = ε – drop. This relationship is the foundation for all analysis of source behaviour under load.
当有电流 I 流动时，在电源端钮上测得的端电压 V 总是小于 ε。这一降低是由内阻导致的：V = ε – Ir。Ir 这一项常被称为内部电位降。这一关系是分析带载下电源行为的全部基础。
V = ε – Ir
3. The Circuit Equation and Internal Resistance Formula | 电路方程与内阻公式
EMF ε and internal resistance r, connected to an external load resistor R. Using Kirchhoff's voltage law, the sum of the p.d.s around the loop is ε – Ir – IR = 0. Hence the current in the circuit is determined by the total resistance R + r.
考虑一个由电动势为 ε、内阻为 r 的电源与外部负载电阻 R 组成的完整电路。利用基尔霍夫电压定律，回路中各部分电位差之和为 ε – Ir – IR = 0。因此，电路中的电流由总电阻 R + r 决定。
ε = I (R + r)
ranging gives I = ε / (R + r), and multiplying by R yields the terminal voltage V = IR = ε – re the two most useful forms of the internal resistance equation for circuit calculations.
该方程表明，电动势是外部电压 IR 与内部电压降 Ir 之和。整理后可得 I = ε / (R + r)，再乘以 R 即得到端电压 V = IR = ε – Ir。这是电路计算中内阻方程最常用的两种形式。
4. V-I Characteristic and Graphical Analysis | 伏安特性和图像分析
The equation V = ε – ed on the y-axis and I on the x-axis, the result is a straight line with negative gradient – . This V-I characteristic is the key graphical tool for investigating internal resistance.
方程 V = ε – Ir 是端电压 V 与电流 I 之间的线性关系。当以 V 为 y 轴、I 为 x 轴作图时，得到一条斜率为 –r、y 轴截距为 ε 的直线。这条伏安特性曲线是研究内阻的关键图像工具。
ternal resistance, r. The intercept on the voltage axis (I = 0) gives the EMF. Some real sources show a slight curve at high currents due to heating and polarisation effects, but for most A-Level experiments the relationship is treated as linear.
如果电源是理想电源，图线将是水平的（r = 0）。在现实中，图线向下倾斜。斜率的大小就等于内阻 r。电压轴上的截距（I = 0）给出电动势 ε。一些真实电源在大电流时会由于发热和极化效应出现轻微弯曲，但在大多数 A-Level 实验中将这一关系视为线性。
5. Measuring Internal Resistance Experimentally | 实验测量内阻
A standard laboratory method to find ε and r involves a voltmeter connected across the source terminals, an ammeter in series, and a variable resistor or resistance box as the external load. The circuit is switched on briefly for each reading to minimise heating.
实验室测定 ε 和 r 的标准方法是，在电源端钮上并联一只电压表，串联一只电流表，并用变阻器或电阻箱作为外部负载。每次读数时短暂接通电路，以尽量减少发热。
is plotted, and a best-fit straight line is drawn. The internal resistance is given by r = – hould be used so that negligible current is drawn from the source by the voltmeter itself.
通过改变 R，可收集一系列 (V, I) 数据对。绘制 V-I 图，并画出最佳拟合直线。内阻 r 由斜率的负值给出，电动势 ε 从 y 轴截距读取。应使用高内阻的电压表（最好为数字式），以便电压表本身从电源汲取的电流可忽略不计。
Uncertainty analysis can be carried out by drawing error bars and finding maximum and minimum slopes. The final values of ε and r are quoted with their absolute uncertainties. This approach reinforces both practical skills and data analysis for the CIE practical papers.
可通过绘制误差棒、求出最大斜率和最小斜率来进行不确定度分析。最终 ε 和 r 的值应附有绝对不确定度。这一方法同时强化了实验技能和数据分析能力，正是 CIE 实验考试所要求的。
6. Factors Affecting Internal Resistance | 影响内阻的因素
The internal resistance of a cell is not a fixed constant; it depends on several practical factors. Understanding these helps explain why battery performance changes with age, temperature and load.
电池的内阻并非固定不变的常数；它取决于若干实际因素。理解这些因素有助于解释为何电池性能会随使用时间、温度和负载而变化。
- Electrode surface area and material: Larger electrode plates and porous structures reduce r. Corrosion or sulphation on plates increases r.
- Electrolyte concentration: In many wet cells, higher electrolyte concentration reduces internal resistance, up to a point.
- Temperature: Increasing temperature generally lowers r because ion mobility increases, but high temperatures accelerate degradation.
- State of charge: As a battery discharges, chemical changes build up internal resistance. A nearly flat battery exhibits significantly higher r.
- Battery chemistry: Different types (alkaline, NiMH, lithium, lead-acid) have inherently different internal resistance ranges.
中文对应要点：
- 电极表面积与材料：较大的电极板和多孔结构可降低内阻。极板上的腐蚀或硫酸盐化会增加内阻。
- 电解液浓度：在许多湿电池中，较高的电解液浓度可在一定程度上降低内阻。
- 温度：升高温度通常会降低内阻，因为离子迁移率增加，但高温会加速电池老化。
- 充电状态：电池放电时，化学变化会使内阻累积增加。快没电的电池内阻会显著升高。
- 电池化学体系：不同类型（碱性、镍氢、锂电、铅酸）固有内阻范围各不相同。
The table below provides typical internal resistance values for new, good-quality cells:
下表给出了全新优质电池的典型内阻值：
| Battery type | Typical internal resistance (new) |
|---|---|
| Alkaline AA | ~0.15 Ω |
| NiMH rechargeable AA | ~0.02 Ω |
| Lead-acid car battery | ~0.005 Ω |
| Lithium-ion (18650) | ~0.05–0.1 Ω |
These values clarify why a car battery can supply hundreds of amps for cranking, whereas an alkaline AA cell would overheat and lose voltage rapidly under the same demand.
这些数值清楚地表明，为何汽车电池能提供数百安培的启动电流，而一节碱性 AA 电池在同样需求下会迅速发热并掉压。
7. Power Transfer and Maximum Power Theorem | 功率传输与最大功率定理
= I²R. Substituting I = ε / (R + r) gives a power expression entirely in terms of R and r:
传递到外部负载电阻 R 的功率为 P = I²R。将 I = ε / (R + r) 代入，可得到完全用 R 和 r 表示的功率表达式： Published by TutorHao | A-Level Physics Revision Series | aleveler.com
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