Potential Dividers in Electrical Circuits | 电路中的分压器

📚 Potential Dividers in Electrical Circuits | 电路中的分压器

A potential divider is one of the most practical circuit arrangements you will meet in A-Level Combined Science. It allows us to tap off any fraction of a supply voltage by using two or more resistances in series. From volume controls on speakers to light‑sensing streetlamps and temperature‑triggered fan circuits, potential dividers sit at the heart of countless electronic systems. Grasping how the output voltage depends on the ratio of resistances – and what happens when a load is connected – is essential for both theory and practical work.

分压器（电位分压电路）是 A‑Level 综合科学（物理部分）中最常用的电路结构之一。它通过两个或多个串联的电阻，从电源电压中获取一个所需的电压比例。从音量旋钮到光控路灯、再到温控风扇电路，分压器都是核心元件。掌握输出电压如何取决于电阻之比，以及接入负载后会发生什么，是理论学习和实验技能的共同要求。

1. What is a Potential Divider? | 什么是分压器？

A potential divider is simply a chain of resistors connected in series across a voltage source. The voltage between two points in the chain is a fixed fraction of the total supply voltage. The simplest form uses two fixed resistors, but equally common are potentiometers (variable resistors) that allow the output voltage to be adjusted continuously.

分压器本质上是串联在电源两端的一串电阻。链条上两点的电压是电源总电压的一个固定比例。最简单的形式由两个固定电阻构成，但同样常见的是电位器（可变电阻），它能够连续调节输出电压。

2. The Basic Two‑Resistor Divider | 基本的两电阻分压器

Consider two resistors R₁ and R₂ connected in series across a supply voltage Vin. The current I through them is the same, equal to Vin divided by the total resistance R₁ + R₂. The voltage across R₂, which we often call the output voltage Vout, is then I × R₂.

考虑两个电阻 R₁ 和 R₂ 串联跨接在电源电压 Vin 两端。流过它们的电流 I 相同，等于 Vin 除以总电阻 R₁ + R₂。则 R₂ 两端的电压，也就是我们常说的输出电压 Vout，为 I × R₂。

Combining these steps gives the classic potential divider equation, which you should be able to derive from Ohm’s law and Kirchhoff’s voltage law.

将上述步骤合并即可得到经典的分压器方程，你应该能从欧姆定律和基尔霍夫电压定律出发自行推导。

Vout = Vin × R₂ / (R₁ + R₂)

If R₁ = R₂, then Vout = ½ Vin. If R₂ is much larger than R₁, Vout approaches Vin; if R₂ is tiny, Vout is close to zero.

若 R₁ = R₂，则 Vout = ½ Vin。若 R₂ 远大于 R₁，Vout 趋近 Vin；若 R₂ 极小，Vout 趋近于零。

3. The Ratio Rule – Controlling Voltage by Design | 比例规则 —— 通过设计控制电压

The output voltage is determined by the ratio R₂/(R₁ + R₂), not by the absolute values of the resistors alone. This insight allows engineers to set a precise bias voltage, for instance 0.7 V for a transistor base, without worrying about the exact current as long as the ratio is correct.

输出电压由比值 R₂/(R₁ + R₂) 决定，而不是取决于单个电阻的绝对值。这一洞察使工程师能够设定精准的偏置电压，例如为晶体管基极提供 0.7 V，只要比值正确，不必担心具体电流大小。

In a laboratory, if you need a 3.0 V reference from a 9.0 V battery, you would choose R₁ and R₂ such that R₂/(R₁ + R₂) = 1/3. For example, R₁ = 6 kΩ and R₂ = 3 kΩ gives exactly 3.0 V across R₂.

在实验室里，若你需要从 9.0 V 电池获得 3.0 V 的参考电压，可选择 R₁ 与 R₂ 使 R₂/(R₁ + R₂) = 1/3。例如 R₁ = 6 kΩ，R₂ = 3 kΩ，即可在 R₂ 上准确得到 3.0 V。

4. The Potentiometer – A Variable Potential Divider | 电位器 —— 可变分压器

A potentiometer is a three‑terminal device with a resistive track and a sliding wiper. If the outer terminals are connected across a supply, the wiper terminal gives an output voltage that can be varied from 0 V to almost Vin by rotating or sliding the control.

电位器是一种三端器件，有一个电阻轨道和一个滑动触点。若将两个外端接在电源两端，滑动触点就可以输出从 0 V 到接近 Vin 的连续可调电压，只需转动旋钮或推动滑块。

Potentiometers are used as volume controls, dimmer switches and position sensors. In circuit diagrams, the symbol shows an arrow striking a resistor, indicating the variable tap point.

电位器广泛用作音量控制、调光开关和位置传感器。在电路图中，它的符号由一个箭头触及电阻表示，指出可变的抽头位置。

5. Loading Effect – When Connecting a Load Changes the Voltage | 负载效应 —— 接入负载改变电压

In an ideal potential divider we assume no current is drawn from Vout. In reality, the output often feeds a load resistor RL, which appears in parallel with R₂. This parallel combination reduces the effective resistance of the lower leg, altering the divider ratio and lowering the output voltage.

在理想分压器中，我们假设从 Vout 吸取的电流为零。但实际中，输出端经常接有负载电阻 RL，它与 R₂ 并联。这一并联组合减小了下臂的有效电阻，改变了分压比，使得输出电压下降。

To minimise loading, the current through the divider (often called the bleeder current) should be much larger than the current drawn by the load. A common rule of thumb is to make the bleeder current at least 10 times the maximum load current.

为减小负载效应，流过分压器本身的电流（常称为泄放电流）应远大于负载电流。一条常用的经验法则是让泄放电流至少是最大负载电流的 10 倍。

6. Sensor Circuits Using a Potential Divider | 用分压器构建传感器电路

One of the most popular applications in the Combined Science syllabus is the sensor circuit, where one of the resistors is replaced by a transducer whose resistance changes with a physical quantity. The two classic transducers are the light‑dependent resistor (LDR) and the thermistor.

综合科学大纲中最常见的应用之一是传感器电路，其中一个电阻被换成了电阻随物理量变化的换能器。两种经典的换能器是光敏电阻（LDR）和热敏电阻。

LDR (Light‑Dependent Resistor): Its resistance falls as light intensity increases. In a dark environment, resistance may be several hundred kΩ; in bright sunlight it can drop to a few hundred Ω.
LDR（光敏电阻）：光照增强时其电阻下降。暗环境下电阻可达数百千欧，强阳光下可降至几百欧。
Thermistor (NTC type): The most common negative temperature coefficient thermistor decreases in resistance as temperature rises. A typical NTC thermistor might have 10 kΩ at 25 °C and around 1 kΩ at 80 °C.
热敏电阻（NTC 型）：最常见的负温度系数热敏电阻随温度升高阻值下降。一只典型 NTC 在 25 °C 时约 10 kΩ，80 °C 时降至约 1 kΩ。

7. Light‑Sensitive Switch – LDR in the Upper or Lower Arm | 光敏开关 —— LDR 在上臂还是下臂

The behaviour of the sensor circuit depends on whether the transducer is placed in the R₁ position (upper arm) or the R₂ position (lower arm). When a LDR is in the R₂ position and light intensity increases, its resistance decreases, so Vout (the voltage across it) falls. A circuit designed to turn on a lamp when it gets dark would therefore place the LDR in the upper arm, so that Vout rises in darkness and triggers a transistor switch.

传感器电路的行为取决于换能器是放在 R₁ 位置（上臂）还是 R₂ 位置（下臂）。当 LDR 放在 R₂ 位置时，光照增强电阻减小，其两端电压 Vout 便下降。要让天黑时点亮路灯，则应把 LDR 放在上臂，使暗环境下 Vout 升高，触发晶体管开关。

In exam problems, you must be able to predict and explain how Vout varies with changing physical conditions by mentally adjusting the resistance in the appropriate arm.

在考题中，你必须能够通过在头脑中调整相应臂的阻值，预测并解释 Vout 随物理条件变化的趋势。

8. The Thermistor in a Temperature‑Controlled Circuit | 热敏电阻在温控电路中的应用

When a thermistor replaces R₂, the output voltage rises as the temperature increases (for an NTC device). This voltage can be fed to a comparator to switch on a cooling fan when a set temperature is exceeded. Conversely, placing the thermistor in the upper arm creates a circuit where Vout falls with rising temperature, suitable for switching on a heater in a cold environment.

当热敏电阻取代 R₂ 时，温度升高（NTC 器件）输出电压上升。将该电压输入比较器，可以在温度超过设定值时开启冷却风扇。反之，若将热敏电阻放在上臂，则 Vout 随温度升高而下降，适合在寒冷环境下接通加热器。

The practical calibration of such circuits involves measuring Vout at several known temperatures and plotting a calibration graph, which usually shows a non‑linear curve due to the exponential nature of the thermistor characteristic.

这类电路的实际标定需要在多个已知温度下测量 Vout，并绘制标定曲线。由于热敏电阻特性呈指数关系，标定图通常是一条非线性曲线。

9. Using a Potential Divider as an Input to a Comparator | 用分压器作为比较器的输入端

A potential divider is frequently paired with an operational amplifier (op‑amp) or a voltage comparator to create a decision‑making circuit. One divider provides a variable voltage from a sensor, while a second divider sets a fixed reference voltage. When the sensor voltage crosses the reference, the comparator output swings high or low, turning on an actuator such as an LED or a relay.

分压器常与运算放大器（运放）或电压比较器配合，构成决策电路。一个分压器提供来自传感器的可变电压，另一个分压器则设置为固定的参考电压。当传感器电压越过参考值时，比较器输出跳变为高电平或低电平，从而启动执行器，如发光二极管或继电器。

This arrangement appears in automatic nightlights, fire alarms and incubators. Understanding how to set the reference voltage by choosing R₁ and R₂ is a key design skill.

这种结构出现在自动夜灯、火灾报警器和恒温箱中。理解如何通过选择 R₁ 和 R₂ 来设定参考电压，是一项关键的设计技能。

10. Practical Investigation – Building and Testing a Divider | 实验探究 —— 搭建并测试分压器

In an A‑Level practical, you may be asked to construct a potential divider and investigate its behaviour with a changing resistance. Using a 6 V battery, a fixed resistor (say 1 kΩ) and a 10 kΩ potentiometer, you can measure Vout with a digital voltmeter while altering the potentiometer’s setting.

在 A‑Level 实验中，你可能会被要求搭建一个分压器，并研究它在电阻变化时的行为。利用一节 6 V 电池、一个固定电阻（如 1 kΩ）和一个 10 kΩ 电位器，你可以一边调节电位器，一边用数字电压表测量 Vout。

Record data for Vout as a function of the potentiometer’s resistance between wiper and earth. Plot a graph of Vout against the variable resistance R₂. The curve should follow the expression Vout = Vin × R₂/(R₁ + R₂). Any deviation can be attributed to loading by the voltmeter or internal resistance of the battery.

记录 Vout 随电位器滑动端与地之间阻值变化的数据。绘制 Vout 随可变电阻 R₂ 变化的图像。曲线应遵循 Vout = Vin × R₂/(R₁ + R₂)。任何偏离都可归因于电压表的负载效应或电池内阻。

11. Common Misconceptions and Exam Pitfalls | 常见误区与考试陷阱

Some students think the output voltage is fixed by the value of a single resistor, instead of the ratio. Remember: doubling both resistors (keeping the ratio constant) does not change Vout; it only reduces the current drawn from the supply.

有些学生以为输出电压取决于单一电阻的大小，而非比值。请记住：两个电阻同时加倍（比值不变）并不改变 Vout，只会减小从电源汲取的电流。

Another pitfall is forgetting that the voltmeter has a finite impedance, typically 10 MΩ for a digital meter. When measuring across a high‑resistance divider (e.g. 1 MΩ resistors), the meter can shunt the lower resistor enough to noticeably alter the voltage. Always consider the meter’s loading effect in precision experiments.

另一个陷阱是忘记了电压表本身有有限阻抗，数字表通常为 10 MΩ。在测量高阻分压器（如 1 MΩ 电阻）时，电表会分流下臂电阻，足以明显改变电压值。精密实验中必须始终考虑电表的负载效应。

12. Summary and Revision Checklist | 总结与复习清单

To master potential dividers, ensure you can:

Derive Vout = Vin × R₂/(R₁ + R₂) from first principles.
Explain what happens to Vout when R₂ increases or decreases.
Describe the difference between a fixed divider and a potentiometer.
Analyse sensor circuits with LDRs and thermistors, predicting output direction.
Quantify the loading effect when an external resistor is connected across Vout.
Sketch and interpret Vout–R₂ graphs and calibration curves for transducer circuits.
Apply the bleeder current rule to design a stiff voltage divider.

要掌握分压器，请确保你能：

从基本原理推导 Vout = Vin × R₂/(R₁ + R₂)。
解释 R₂ 增大或减小时 Vout 如何变化。
描述固定分压器与电位器的区别。
分析含 LDR 和热敏电阻的传感器电路，预测输出变化方向。
量化在 Vout 两端接入外部电阻引起的负载效应。
绘制并解读换能器电路的 Vout–R₂ 图像和标定曲线。
运用泄放电流规则设计一个“硬”分压器。

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Potential Dividers in Electrical Circuits | 电路中的分压器