Voltage Divider: Circuit Principles and Applications | 分压器：电路原理与应用

📚 Voltage Divider: Circuit Principles and Applications | 分压器：电路原理与应用

A voltage divider is one of the most fundamental and versatile circuits in electronics. It consists simply of two or more resistors connected in series across a power supply, producing an output voltage that is a fraction of the input voltage. This concept appears repeatedly in A-Level Physics, from basic DC circuits to sensor-based applications.

分压器是电子学中最基础且用途最广泛的电路之一。它仅由串联在电源两端的两个或多个电阻组成，产生输入电压的一个分数作为输出电压。这一概念在 A-Level 物理中反复出现，从基础直流电路到基于传感器的应用都离不开它。

1. Definition and Circuit Structure | 定义与电路结构

A voltage divider uses the principle that when resistors are connected in series, the same current flows through each resistor, and the voltage across each resistor is proportional to its resistance. The output voltage is typically taken across one of the resistors.

分压器利用的原理是：当电阻串联时，流过每个电阻的电流相同，而每个电阻两端的电压与其阻值成正比。输出电压通常取自其中一个电阻的两端。

The simplest form consists of two fixed resistors, R₁ and R₂, connected in series across an input voltage V_in. The output voltage V_out is measured across R₂.

最简单的形式由两个固定电阻 R₁ 和 R₂ 串联连接在输入电压 V_in 两端构成。输出电压 V_out 在 R₂ 两端测得。

Input terminals: Connected across the power supply (V_in).
Output terminals: Taken across R₂ (or R₁, depending on design).
Circuit type: Pure series circuit — current is identical through both resistors.

输入端：连接在电源两端（V_in）。
输出端：取自 R₂ 两端（或 R₁，视设计而定）。
电路类型：纯串联电路——流过两个电阻的电流完全相同。

2. Deriving the Voltage Divider Equation | 推导分压器公式

Using Ohm’s law, the total resistance in the series circuit is R₁ + R₂. The current flowing through the circuit is given by:

根据欧姆定律，串联电路中的总电阻为 R₁ + R₂。流过电路的电流为：

I = V_in ⁄ (R₁ + R₂)

The voltage across R₂ is then V_out = I × R₂. Substituting the expression for I:

R₂ 两端的电压为 V_out = I × R₂。代入 I 的表达式：

V_out = V_in × R₂ ⁄ (R₁ + R₂)

This is the voltage divider equation. Similarly, if the output is taken across R₁, the equation becomes V_out = V_in × R₁ ⁄ (R₁ + R₂). The output voltage is always less than or equal to the input voltage — it “divides” the input.

这就是分压器公式。类似地，如果输出取自 R₁ 两端，公式变为 V_out = V_in × R₁ ⁄ (R₁ + R₂)。输出电压始终小于或等于输入电压——它“分割”了输入电压。

3. Understanding the Ratio Principle | 理解比例原理

The key insight is that V_out depends only on the ratio of the two resistances, not on their absolute values. Doubling both R₁ and R₂ leaves V_out unchanged, as long as the ratio R₂ ⁄ (R₁ + R₂) remains the same.

关键在于 V_out 仅取决于两个电阻的比值，而不取决于它们的绝对值。只要 R₂ ⁄ (R₁ + R₂) 的比值不变，将 R₁ 和 R₂ 同时加倍，V_out 保持不变。

This ratio property is extremely useful in practical design. However, the absolute resistance values matter when considering the current drawn from the power supply and the output impedance of the divider.

这一比例特性在实际设计中极为有用。然而，在考虑从电源抽取的电流以及分压器的输出阻抗时，电阻的绝对值非常重要。

Scenario \| 场景	R₁ = R₂	R₁ = 3R₂	R₁ = R₂ ⁄ 3
Fraction of V_in across R₂ \| R₂ 两端占 V_in 的比例	½ (50%)	¼ (25%)	¾ (75%)

4. The Loading Effect | 负载效应

In theory, the voltage divider equation assumes that no current flows out of the output terminals. In reality, when a load resistor R_L is connected across the output, it draws current and affects V_out. This is called the loading effect.

理论上，分压器公式假设没有电流从输出端流出。实际上，当负载电阻 R_L 连接在输出端时，它会抽取电流并影响 V_out。这称为负载效应。

The effective resistance of R₂ and R_L in parallel is:

R₂ 和 R_L 并联后的等效电阻为：

R_eff = R₂ × R_L ⁄ (R₂ + R_L)

The output voltage then becomes V_out = V_in × R_eff ⁄ (R₁ + R_eff), which is lower than the unloaded value. To minimise loading, R_L must be much greater than R₂ (typically at least 10 times larger).

此时输出电压变为 V_out = V_in × R_eff ⁄ (R₁ + R_eff)，低于空载时的数值。为将负载效应降至最低，R_L 必须远大于 R₂（通常至少大 10 倍）。

5. Practical Implications of Loading | 负载效应的实际影响

The loading effect has significant practical consequences. When using a voltmeter to measure the output of a voltage divider, the voltmeter’s internal resistance acts as R_L. If the voltmeter has insufficient resistance, it will draw appreciable current and give an inaccurate reading.

负载效应具有重要的实际影响。当使用电压表测量分压器的输出电压时，电压表的内阻充当 R_L。如果电压表的内阻不足，它将抽取可观的电流，导致读数不准确。

Digital multimeters typically have very high input resistance (around 10 MΩ), making them suitable for measuring voltage divider outputs. However, older analogue meters with lower resistance can significantly disturb the circuit being measured.

数字万用表通常具有很高的输入电阻（约 10 MΩ），适合测量分压器输出。然而，内阻较低的旧式模拟仪表可能会显著干扰被测电路。

To reduce loading effects in design, choose resistor values that are small compared to the expected load. However, smaller resistors draw more current from the power supply, increasing power dissipation. A balance must be struck.

为减小设计中的负载效应，应选择远小于预期负载的电阻值。然而，较小的电阻会从电源抽取更多电流，增大功耗。必须在这两者之间取得平衡。

6. The Potentiometer as a Variable Divider | 电位器作为可变分压器

A potentiometer is a three-terminal variable resistor that acts as an adjustable voltage divider. It has a resistive track with a sliding contact (wiper). By moving the wiper, the ratio of the two effective resistances changes continuously.

电位器是一种三端可变电阻，充当可调分压器。它有一条电阻轨道和一个滑动触点（滑臂）。通过移动滑臂，两个等效电阻的比值连续变化。

When the wiper is at the top, V_out ≈ V_in; at the bottom, V_out ≈ 0. In between, V_out varies linearly with the wiper position. This makes potentiometers ideal for volume controls, dimmer switches, and tuning circuits.

当滑臂位于顶部时，V_out ≈ V_in；位于底部时，V_out ≈ 0。在中间位置，V_out 随滑臂位置线性变化。这使得电位器非常适合用于音量控制、调光开关和调谐电路。

Wiper Position \| 滑臂位置	Resistance Ratio \| 电阻比值	V_out (V_in = 9 V)
Top \| 顶端	R_upper = 0, R_lower = R	9 V
Middle \| 中间	R_upper = R⁄2, R_lower = R⁄2	4.5 V
Bottom \| 底端	R_upper = R, R_lower = 0	0 V

7. Voltage Divider with an LDR | 光敏电阻分压器

One of the most common A-Level applications is replacing one resistor with a light-dependent resistor (LDR). An LDR’s resistance decreases as light intensity increases. This creates a light-sensing circuit.

A-Level 中最常见的应用之一是将其中一个电阻替换为光敏电阻（LDR）。光敏电阻的阻值随光照强度增加而减小。这样就构成了一个光感应电路。

Consider an LDR in the position of R₂ and a fixed resistor R₁ connected to V_in. In bright light, the LDR’s resistance is low, so V_out is low. In darkness, the LDR’s resistance is high, so V_out approaches V_in. The output voltage therefore indicates light level.

考虑将 LDR 放在 R₂ 的位置，固定电阻 R₁ 连接到 V_in。在强光下，LDR 阻值低，因此 V_out 低。在黑暗中，LDR 阻值高，因此 V_out 接近 V_in。输出电压因此反映光照水平。

Bright light: R_LDR small → V_out small
Darkness: R_LDR large → V_out large

强光：R_LDR 小 → V_out 小
黑暗：R_LDR 大 → V_out 大

This configuration is used in automatic street lights, burglar alarms, and camera light meters.

这种配置用于自动路灯、防盗报警器和相机测光表。

8. Voltage Divider with a Thermistor | 热敏电阻分压器

A thermistor is a resistor whose resistance changes significantly with temperature. In A-Level Physics, the NTC (negative temperature coefficient) thermistor is commonly used — its resistance decreases as temperature rises.

热敏电阻是一种阻值随温度显著变化的电阻。在 A-Level 物理中，通常使用 NTC（负温度系数）热敏电阻——其阻值随温度升高而减小。

When an NTC thermistor is placed in the R₂ position with a fixed resistor R₁ above it, the output voltage behaves as follows:

当 NTC 热敏电阻置于 R₂ 位置，上方连接固定电阻 R₁ 时，输出电压的变化如下：

V_out = V_in × R_thermistor ⁄ (R₁ + R_thermistor)

High temperature: R_thermistor small → V_out small
Low temperature: R_thermistor large → V_out large

高温：R_热敏电阻小 → V_out 小
低温：R_热敏电阻大 → V_out 大

This principle is used in temperature sensors, fire alarms, and electronic thermostats. The output can be connected to a comparator circuit to trigger an alarm when a threshold temperature is exceeded.

这一原理用于温度传感器、火灾报警器和电子恒温器。输出可连接到比较器电路，当超过阈值温度时触发警报。

9. Connecting to a Comparator Circuit | 连接比较器电路

A voltage divider alone produces an analogue output. In many practical applications, we need a digital on/off signal. This is achieved by connecting the divider output to a comparator (often an operational amplifier, or op-amp).

单独的分压器产生模拟输出。在许多实际应用中，我们需要数字化的开/关信号。这通过将分压器输出连接到比较器（通常是运算放大器，即运放）来实现。

The comparator compares V_out from the sensor divider against a fixed reference voltage V_ref, produced by a second voltage divider. When V_out crosses V_ref, the comparator output switches between its two saturation levels.

比较器将传感器分压器的 V_out 与由第二个分压器产生的固定参考电压 V_ref 进行比较。当 V_out 越过 V_ref 时，比较器输出在其两个饱和电平之间切换。

For example, in a fire alarm: the thermistor divider output V_out decreases when temperature rises. When V_out falls below V_ref, the comparator output goes high, activating the alarm.

例如，在火灾报警器中：当温度升高时，热敏电阻分压器输出 V_out 减小。当 V_out 降至 V_ref 以下时，比较器输出变为高电平，触发警报。

V_out (sensor) vs V_ref → Comparator output → Alarm

V_out（传感器）与 V_ref 比较 → 比较器输出 → 警报

10. Worked Example | 例题详解

Question: A voltage divider consists of a 2 kΩ resistor (R₁) connected in series with a 3 kΩ resistor (R₂). The input voltage is 12 V. Calculate: (a) the output voltage across R₂; (b) the current through the circuit; (c) the new output voltage if a 6 kΩ load is connected across R₂.

题目：一个分压器由 2 kΩ 电阻（R₁）与 3 kΩ 电阻（R₂）串联组成。输入电压为 12 V。求：（a）R₂ 两端的输出电压；（b）流过电路的电流；（c）若在 R₂ 两端连接一个 6 kΩ 负载，新的输出电压。

(a) Output voltage:

（a）输出电压：

V_out = 12 × 3000 ⁄ (2000 + 3000) = 12 × 3⁄5 = 7.2 V

(b) Current:

（b）电流：

I = 12 ⁄ (2000 + 3000) = 12 ⁄ 5000 = 2.4 mA

(c) With 6 kΩ load: R₂ and R_L in parallel give R_eff = (3 × 6) ⁄ (3 + 6) = 2 kΩ. Then:

（c）连接 6 kΩ 负载时：R₂ 与 R_L 并联得 R_eff = (3 × 6) ⁄ (3 + 6) = 2 kΩ。则：

V_out = 12 × 2000 ⁄ (2000 + 2000) = 6 V

The load reduces the output from 7.2 V to 6.0 V — a clear demonstration of the loading effect.

负载将输出电压从 7.2 V 降至 6.0 V——清晰地展示了负载效应。

11. Common Mistakes and Exam Tips | 常见错误与备考提示

Mistake 1: Using the wrong resistor in the numerator. The output voltage is taken across the resistor in the denominator position — identify which resistor V_out is measured across.

错误一：在分子中使用了错误的电阻。输出电压取自特定电阻两端——务必确认 V_out 测量的是哪个电阻。

Mistake 2: Forgetting unit conversion. Resistance values are often given in kΩ while currents are required in mA or A.

错误二：忘记单位换算。电阻值通常以 kΩ 给出，而电流可能需要用 mA 或 A 表示。

Mistake 3: Ignoring the loading effect when asked about real measurements.

错误三：在涉及实际测量时忽略负载效应。

Exam tip: When drawing a voltage divider with a sensor, always label which component is the sensor and state its resistance behaviour. Examiners award marks for showing the direction of resistance change and the resulting output change.

备考提示：在绘制带传感器的分压器时，务必标注哪个元件是传感器，并说明其阻值变化规律。考官会根据阻值变化方向及其导致的输出变化给分。

12. Summary | 总结

The voltage divider is a simple yet powerful circuit based on Ohm’s law and the series circuit property of shared current. The output voltage is determined by the ratio of the resistors, not their absolute values. The loading effect must be considered when a real load is connected. By replacing one resistor with a sensor such as an LDR or thermistor, the divider becomes a versatile transducer circuit for measuring light, temperature, and many other physical quantities.

分压器是一个简单而功能强大的电路，其基础是欧姆定律和串联电路电流相同的特性。输出电压由电阻的比值决定，而非绝对值。当连接真实负载时，必须考虑负载效应。通过将其中一个电阻替换为 LDR 或热敏电阻等传感器，分压器就成为一个多功能的传感器电路，用于测量光照、温度及许多其他物理量。

Mastering the voltage divider equation, the loading effect, and sensor applications will ensure success in both theoretical questions and practical circuit analysis in your CIE A-Level Physics examination.

掌握分压器公式、负载效应和传感器应用，将确保你在 CIE A-Level 物理考试中在理论题和实际电路分析方面都能取得成功。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Voltage Divider: Circuit Principles and Applications | 分压器：电路原理与应用