Potential Dividers | 分压器

📚 Potential Dividers | 分压器

A potential divider is one of the most frequently examined circuits in CIE A Level Physics. It uses two or more resistors in series to split a supply voltage into smaller, predictable fractions. Understanding potential dividers is essential for analysing sensor circuits, transistor switching circuits and practical measurement systems.

分压器是 CIE A Level 物理中最常考查的电路之一。它利用两个或两个以上串联的电阻将电源电压分成较小且可预测的比例。理解分压器对于分析传感器电路、晶体管开关电路和实际测量系统至关重要。

1. What is a potential divider? | 什么是分压器

A potential divider is a circuit that uses two or more resistors connected in series to divide a source voltage. The output voltage is usually taken across one of the resistors. Because the same current flows through all series resistors, the voltage across each resistor is proportional to its resistance.

分压器是一种利用两个或两个以上串联电阻来分配电源电压的电路。输出电压通常取自其中一个电阻的两端。由于流过所有串联电阻的电流相同，每个电阻两端的电压与其电阻成正比。

For a simple two-resistor divider, if the output is taken across R₂, then the larger R₂ becomes compared with R₁, the larger the share of the supply voltage that appears across R₂.

对于简单的两个电阻组成的分压器，如果输出取自 R₂ 两端，那么 R₂ 相对于 R₁ 越大，R₂ 两端分得的电源电压比例就越大。

2. The basic circuit and principle | 基本电路与原理

In its simplest form, a potential divider consists of two resistors R₁ and R₂ connected in series to a supply voltage Vₛ. The current I in the circuit is the same through both resistors and can be found from the total resistance.

最简单的分压器由两个电阻 R₁ 和 R₂ 串联后接到电源电压 Vₛ 上。电路中的电流 I 通过两个电阻时相同，可以由总电阻求出。

I = Vₛ / (R₁ + R₂)

The voltage across each resistor is then given by Ohm’s law: V₁ = I R₁ and V₂ = I R₂. This means the supply voltage is shared in the ratio of the resistances.

每个电阻两端的电压由欧姆定律给出：V₁ = I R₁ 以及 V₂ = I R₂。这意味着电源电压按照电阻的比例进行分配。

3. Deriving the output voltage formula | 推导输出电压公式

If the output is taken across R₂, then V₂ = I R₂. Substituting the expression for current I = Vₛ / (R₁ + R₂) gives the standard potential divider equation.

如果输出取自 R₂ 两端，则 V₂ = I R₂。将电流表达式 I = Vₛ / (R₁ + R₂) 代入，就得到标准分压器公式。

V₂ = Vₛ × R₂ / (R₁ + R₂)

This equation is not provided on the CIE formula sheet, so you must be able to derive it from first principles. The numerator is always the resistance across which the output voltage is measured.

这个公式不在 CIE 公式表中给出，因此你必须能够从基本原理出发推导它。分子始终是测量输出电压所对应的那个电阻。

4. The role of a fixed potential divider | 固定分压器的作用

A fixed potential divider made from two fixed resistors provides a steady output voltage that is a known fraction of the supply voltage. It is often used to supply a reference voltage to another part of a circuit, such as the base of a transistor or one input of a comparator.

由两个固定电阻构成的分压器提供稳定的输出电压，该电压是电源电压的一个已知比例。它常被用来为电路的其他部分提供参考电压，例如晶体管基极或比较器的一个输入端。

The output fraction is determined only by the resistor values. If the supply voltage changes, the output voltage changes by the same proportion, which is useful in many control and measurement applications.

输出比例只由电阻值决定。如果电源电压发生变化，输出电压会按相同比例变化，这在许多控制和测量应用中非常有用。

5. Variable potential dividers: potentiometers and rheostats | 可变分压器：电位器和变阻器

A potentiometer is a single resistive track with a sliding contact. When all three terminals are used, it acts as a true variable potential divider. The output voltage can be adjusted continuously from approximately 0 V to nearly the full supply voltage Vₛ.

电位器是一条带有滑动触点的单一电阻轨道。当三个端子全部使用时，它就是一个真正的可变分压器。输出电压可以在接近 0 V 到几乎满电源电压 Vₛ 之间连续调节。

Potentiometer: three terminals are used, giving continuously variable output voltage.
电位器：使用三个端子，提供连续可变的输出电压。
Rheostat: usually only two terminals are used, so it acts as a variable series resistor rather than a full potential divider.
变阻器：通常只使用两个端子，因此它相当于一个可变串联电阻，而不是完整的分压器。

CIE questions often ask you to identify whether a component is wired as a potentiometer or a rheostat, and to predict how the output voltage changes as the slider moves.

CIE 题目经常会要求你判断某个元件是接成电位器还是变阻器，并预测当滑片移动时输出电压如何变化。

6. Sensor circuits: thermistor | 传感器电路：热敏电阻

A potential divider becomes especially useful when one of the resistors is replaced by a sensing component, such as a thermistor. An NTC thermistor has a resistance that decreases as its temperature increases.

当其中一个电阻被热敏电阻等传感元件替代时，分压器变得特别有用。NTC 热敏电阻的阻值随着温度升高而减小。

If the thermistor is placed as R₂, then when the temperature rises, its resistance falls and the output voltage V₂ also falls. If the thermistor is placed as R₁, then when the temperature rises, R₁ falls and the share of voltage across R₂ increases, so V₂ rises.

如果热敏电阻放在 R₂ 位置，那么当温度升高时，它的阻值减小，输出电压 V₂ 也减小。如果热敏电阻放在 R₁ 位置，那么当温度升高时，R₁ 减小，R₂ 两端分得的电压比例增大，因此 V₂ 升高。

This means the same thermistor can produce either a rising or a falling output voltage with temperature, depending on its position in the divider.

这意味着同一个热敏电阻可以根据它在分压器中的位置，使输出电压随温度上升或下降。

7. Sensor circuits: light-dependent resistor (LDR) | 传感器电路：光敏电阻

A light-dependent resistor, or LDR, has a high resistance in the dark and a low resistance in bright light. It is commonly used in light-sensing potential divider circuits.

光敏电阻在黑暗环境中电阻很高，在明亮环境中电阻很低。它通常用于光传感分压器电路中。

In a typical light-sensing circuit, the LDR is placed as R₁ and a fixed resistor is placed as R₂. When the light intensity increases, the LDR resistance falls, so the voltage across the fixed resistor R₂ rises. When the light intensity decreases, the LDR resistance increases and V₂ falls.

在典型的光传感电路中，LDR 放在 R₁ 位置，固定电阻放在 R₂ 位置。当光照强度增大时，LDR 的阻值减小，固定电阻 R₂ 两端的电压升高。当光照强度减小时，LDR 的阻值增大，V₂ 下降。

If the LDR were placed as R₂ instead, the output voltage would fall as the light intensity increases. Always check which resistor the output is taken across.

如果 LDR 放在 R₂ 位置，那么输出随光照强度增大而下降。一定要检查输出是取自哪个电阻的两端。

8. The effect of loading | 负载效应

In an ideal potential divider, no current is drawn from the output terminal. In practice, if a load resistor Rₗ is connected across the output, it forms a parallel combination with one of the divider resistors. This lowers the effective resistance of that branch and changes the output voltage.

在理想分压器中，输出端不分出电流。实际上，如果在输出端连接负载电阻 Rₗ，它会与其中一个分压电阻并联。这会降低该支路的等效电阻并改变输出电压。

If the load is connected across R₂, the effective resistance of the output branch becomes:

如果负载连接在 R₂ 两端，输出支路的等效电阻变为：

R = R₂ × Rₗ / (R₂ + Rₗ)

This new effective resistance is smaller than R₂ alone, so the output voltage is usually less than the unloaded value. CIE questions often ask you to calculate the new output voltage when a load is added.

这个新的等效电阻小于单独的 R₂，因此输出电压通常小于空载时的数值。CIE 题目经常要求你在加入负载后计算新的输出电压。

9. Solving problems: step-by-step method | 解题方法：分步法

To solve potential divider problems accurately, use a systematic method. First identify which resistor, or which parallel branch, the output voltage is taken across. Then calculate the effective resistance of that branch if a load is connected.

要准确解决分压器问题，需要使用系统的方法。首先要确定输出电压是取自哪个电阻或哪条并联支路。然后，如果接有负载，计算该支路的等效电阻。

Identify R₁ and R₂ in the series circuit.
确定串联电路中的 R₁ 和 R₂。
Include any load resistor in parallel with the output resistor.
把任何负载电阻与输出电阻并联计算。
Apply V₂ = Vₛ × R₂ / (R₁ + R₂) using the effective resistance.
使用等效电阻代入 V₂ = Vₛ × R₂ / (R₁ + R₂)。
Check the direction of the output change for sensor questions.
对于传感器问题，检查输出变化的方向。

This method works for fixed dividers, variable dividers and sensor circuits with loading.

这个方法适用于固定分压器、可变分压器以及带负载的传感器电路。

10. Worked example | 例题解析

A 12 V battery is connected in series with a fixed 4 kΩ resistor R₁ and a 6 kΩ resistor R₂. Calculate the voltage across R₂. Then a 3 kΩ load resistor is connected across R₂. Find the new output voltage.

一个 12 V 电池与一个固定 4 kΩ 电阻 R₁ 和一个 6 kΩ 电阻 R₂ 串联。计算 R₂ 两端的电压。然后在 R₂ 两端并联一个 3 kΩ 的负载电阻。求新的输出电压。

Without the load, the output voltage is:

空载时，输出电压为：

V₂ = 12 × 6 / (4 + 6) = 72 / 10 = 7.2 V

When the 3 kΩ load is connected across R₂, the effective resistance of the output branch is:

当 3 kΩ 负载并联在 R₂ 两端时，输出支路的等效电阻为：

R = 6 × 3 / (6 + 3) = 18 / 9 = 2 kΩ

The new total series resistance is 4 kΩ + 2 kΩ = 6 kΩ. The new output voltage is:

新的总串联电阻为 4 kΩ + 2 kΩ = 6 kΩ。新的输出电压为：

V₂ = 12 × 2 / 6 = 4 V

This example shows that adding a load can significantly reduce the output voltage.

这个例子表明，接入负载会显著降低输出电压。

11. Common exam pitfalls | 常见考试误区

Many students lose marks because they place the wrong resistor in the numerator of the potential divider formula. Remember that the numerator must be the resistance across which the output voltage is measured.

许多学生因为把错误的电阻放在分压器公式的分子中而丢分。要记住，分子必须是测量输出电压所对应的那个电阻。

Another common error is treating parallel resistors as if they were in series when a load is added. Always calculate the parallel equivalent resistance before applying the potential divider equation.

另一个常见错误是在加入负载后把并联电阻当作串联处理。一定要先计算并联等效电阻，再应用分压器公式。

With thermistor and LDR questions, students sometimes confuse whether the output voltage rises or falls. The direction depends on the position of the sensor in the divider, so always identify R₁ and R₂ carefully.

在热敏电阻和光敏电阻的题目中，学生有时会混淆输出电压是上升还是下降。变化方向取决于传感器在分压器中的位置，因此一定要仔细确定 R₁ 和 R₂。

12. Summary and key points | 总结与要点

A potential divider splits voltage in proportion to resistance. The output voltage measured across R₂ is given by V₂ = Vₛ × R₂ / (R₁ + R₂). Replacing one resistor with a thermistor or an LDR converts the divider into a sensing circuit.

分压器按电阻比例分配电压。测量 R₂ 两端的输出电压为 V₂ = Vₛ × R₂ / (R₁ + R₂)。用热敏电阻或光敏电阻代替其中一个电阻，就可以将分压器转变为传感器电路。

Adding a load draws current from the output and usually reduces the output voltage because the load forms a parallel branch with the output resistor. Master these ideas and you will be well prepared for CIE potential divider questions.

接入负载会从输出端吸取电流，通常使输出电压降低，因为负载与输出电阻构成并联支路。掌握这些要点后，你将为 CIE 分压器相关题目做好充分准备。

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Potential Dividers | 分压器