Internal Resistance | 内阻

📚 Internal Resistance | 内阻

In many circuit calculations, a battery or cell is treated as an ideal source with a fixed potential difference. In reality, every real source has an internal resistance that reduces the terminal potential difference when current flows. This topic explains how internal resistance arises, how it is measured, and why it matters in practical circuits.

在许多电路计算中，电池或电池组被当作具有固定电势差的理想电源。实际上，每个真实电源都有内阻，当电流流过时，内阻会降低端电压。本主题阐释内阻是如何产生的、如何测量内阻，以及它在实际电路中为何重要。

1. Electromotive Force and Terminal Potential Difference | 电动势与端电压

The electromotive force (emf) of a source is the energy transferred per unit charge when chemical, mechanical or other forms of energy are converted into electrical energy. It is measured in volts and is often represented by the symbol E or ε.

电源的电动势（emf）是单位电荷通过电源时，由化学能、机械能或其他形式能量转化为电能所转移的能量。它以伏特为单位，通常用符号 E 或 ε 表示。

The terminal potential difference V is the voltage measured across the terminals of the source when current is flowing. It is smaller than the emf because some energy is transferred to the internal resistance of the source.

端电压 V 是电流流动时在电源两端测得的电压。它小于电动势，因为部分能量转移到了电源的内阻上。

2. Defining Internal Resistance | 内阻的定义

Internal resistance is the resistance offered by the source itself to the flow of charge. It is usually denoted by r and measured in ohms (Ω). It arises from the resistance of the electrolyte in a chemical cell, the resistance of the electrodes, and other internal components of the source.

内阻是电源本身对电荷流动所表现出的电阻。它通常用 r 表示，单位为欧姆（Ω）。它来自化学电池中电解液的电阻、电极的电阻以及电源的其他内部元件。

Since the internal resistance r is in series with the external circuit, the same current I flows through both r and the external resistance R. The total resistance of the whole circuit is therefore R + r.

由于内阻 r 与外部电路串联，相同的电流 I 同时流过 r 和外电阻 R。因此，整个电路的总电阻为 R + r。

R(total) = R + r

3. The EMF Equation | 电动势方程

For a source of emf E connected to an external resistance R, the current in the circuit is given by Ohm’s law using the total resistance R + r. Therefore:

对于电动势为 E、外接电阻为 R 的电源，根据欧姆定律，电路中的电流由总电阻 R + r 决定。因此：

I = E / (R + r)

The terminal potential difference V across the external resistance is V = IR. Combining this with the above equation gives the key relationship for any real source:

外电阻两端的端电压 V 为 V = IR。将上式与此关系结合，可以得到适用于任何真实电源的关键关系式：

E = I(R + r) = V + Ir

V = E − Ir

This equation shows that the terminal potential difference is equal to the emf minus the potential difference lost across the internal resistance.

这个方程表明，端电压等于电动势减去内阻上损失的电压。

4. Measuring Internal Resistance | 测量内阻

A common experiment uses a cell, an ammeter, a variable resistor and a voltmeter. The voltmeter is connected directly across the cell terminals to measure the terminal pd V, while the ammeter measures the current I in the circuit.

一个常见实验使用电池、电流表、可变电阻和电压表。电压表直接连接在电池两端以测量端电压 V，而电流表测量电路中的电流 I。

The variable resistor R is changed, and pairs of V and I are recorded. A graph of V against I is then plotted. Since V = E − Ir, the graph should be a straight line with a negative gradient.

改变可变电阻 R，记录一组组 V 和 I。然后绘制 V 对 I 的图像。由于 V = E − Ir，该图像应是一条斜率为负的直线。

V = E − Ir

By comparing with the straight-line equation y = mx + c, the gradient is −r and the y-intercept is E.

与直线方程 y = mx + c 比较可知，斜率为 −r，纵截距为 E。

5. The V-I Characteristic | 端电压-电流特性

The equation V = E − Ir shows that as the current increases, the terminal potential difference decreases linearly. The term Ir is often called the lost volts, because it represents the energy per coulomb lost inside the source.

方程 V = E − Ir 表明，随着电流增大，端电压线性下降。Ir 这一项通常被称为损失电压，因为它表示单位电荷在电源内部损失的能量。

For example, if a battery has emf 1.5 V and internal resistance 0.40 Ω, then when a current of 1.0 A flows, the terminal pd is V = 1.5 − (1.0 × 0.40) = 1.1 V. The drop of 0.40 V is the lost volts across the internal resistance.

例如，如果一个电池的电动势为 1.5 V，内阻为 0.40 Ω，那么当电流为 1.0 A 时，端电压为 V = 1.5 − (1.0 × 0.40) = 1.1 V。下降的 0.40 V 就是内阻上的损失电压。

6. Interpreting the V-I Graph | 解读 V-I 图

On a V-I graph, the y-intercept gives the emf E, because when I = 0, no current flows and the terminal pd equals the emf. The gradient of the line is −r, so the internal resistance is the magnitude of the gradient.

在 V-I 图上，纵截距给出电动势 E，因为当 I = 0 时，没有电流流动，端电压等于电动势。图线的斜率为 −r，因此内阻等于斜率的大小。

y-intercept / 纵截距	emf E / 电动势 E
gradient / 斜率	−r / 内阻的负值
x-intercept / 横截距	I(max) = E / r / 短路电流

If the line has equation V = 6.0 − 2.5I, then E = 6.0 V and r = 2.5 Ω. Deviations from a straight line may occur if r changes significantly with current or temperature.

如果直线方程为 V = 6.0 − 2.5I，则 E = 6.0 V，r = 2.5 Ω。如果 r 随电流或温度发生显著变化，图线可能会偏离直线。

7. Short-Circuit Current | 短路电流

The maximum current a source can deliver occurs when the external resistance is zero, which is called a short circuit. In this case, the terminal potential difference is zero, and the current is limited only by the internal resistance.

电源能提供的最大电流出现在外电阻为零时，这种情况称为短路。此时端电压为零，电流仅由内阻限制。

I(max) = E / r

This short-circuit current is usually large, so it can damage the source or cause overheating. On the V-I graph, it is the point where the line crosses the current axis.

短路电流通常很大，因此可能损坏电源或导致过热。在 V-I 图上，短路电流是直线与电流轴的交点。

8. Energy and Power in the Circuit | 电路中的能量与功率

The total power produced by the source is P = EI. The useful power delivered to the external circuit is P = VI. The power wasted as heat in the internal resistance is P = I²r.

电源产生的总功率为 P = EI。输送到外部电路的有用功率为 P = VI。内阻上以热形式浪费的功率为 P = I²r。

EI = VI + I²r

This equation expresses the conservation of energy in the circuit. If the source were ideal with r = 0, all the produced power would be delivered externally.

这个方程表达了电路中的能量守恒。如果电源是理想的，即 r = 0，那么所有产生的功率都会被输送到外部电路。

9. Maximum Power Transfer | 最大功率传输

The power delivered to the external resistance R is P = I²R = E²R / (R + r)². By differentiating this expression with respect to R, maximum power transfer occurs when the external resistance equals the internal resistance.

输送到外电阻 R 的功率为 P = I²R = E²R / (R + r)²。对该表达式关于 R 求导可知，当外电阻等于内阻时，功率传输达到最大。

P(max) occurs when R = r

Under this condition, the efficiency is only 50%, because half of the total power is dissipated in the internal resistance. This result is useful for matching loads to sources, but it is not desirable for power distribution systems where high efficiency is needed.

在这种情况下，效率仅为 50%，因为总功率的一半会耗散在内阻上。这一结果对于负载与电源的匹配很有用，但在需要高效率的电力传输系统中并不理想。

10. Effects of Internal Resistance in Real Sources | 实际电源内阻的影响

Internal resistance explains why a battery’s terminal voltage drops when a heavy current is drawn. A car battery has a very low

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Internal Resistance | 内阻