📚 Practical Circuits | 实用电路
Practical circuits form the backbone of experimental physics at A‑Level. Whether you are measuring an unknown resistance using a Wheatstone bridge, investigating the charging of a capacitor through a resistor, or using an operational amplifier to process a sensor signal, a clear understanding of how real components behave and how to design reliable measurement setups is essential. This article covers the key practical circuits in the CIE A‑Level Physics syllabus, with an emphasis on how they work, how they are used in the laboratory, and the common pitfalls to avoid.
实用电路是A‑Level物理实验的核心。无论你是用惠斯通电桥测量未知电阻、研究电容通过电阻的充电过程，还是用运算放大器处理传感器信号，真正理解实际元件的行为以及如何设计可靠的测量装置至关重要。本文覆盖了CIE A‑Level物理考纲中关键的实用电路，着重讲解它们的工作原理、在实验室中的使用方法和需要避免的常见问题。
1. Ohm's Law and Kirchhoff's Rules in Real Circuits | 实际电路中的欧姆定律与基尔霍夫定则
tion for many real conductors. For a metallic wire kept at constant temperature, the current is directly proportional to the potential difference across it. In a practical circuit, you must always account for the internal resistance of the power supply and the resistance of connecting leads and contacts.
欧姆定律 V = IR 对许多实际导体只是一种近似。对于保持恒温的金属导线，电流与它两端的电势差成正比。在实用电路中，你必须始终考虑电源内阻以及连接线和接触点的电阻。
Kirchhoff’s current law (KCL) states that the total current entering a junction equals the total current leaving it. Kirchhoff’s voltage law (KVL) states that the sum of the EMFs around any closed loop equals the sum of the potential drops. These laws are fundamental when analysing any multi‑loop practical circuit, such as a potential divider with a load or a bridge network.
基尔霍夫电流定律（KCL）指出，流入某节点的总电流等于流出该节点的总电流。基尔霍夫电压定律（KVL）指出，沿任何闭合回路的电动势总和等于电势降落的总和。这些定律在分析任何多回路实用电路时都是基础，比如带负载的分压器或桥式网络。
In laboratory work, you can verify KVL by placing voltmeters across each resistor in a series loop and checking that the sum of the p.d.s equals the supply EMF. Because real voltmeters have finite resistance, they draw a small current and slightly alter the circuit – an effect that becomes significant when measuring across high‑resistance components.
在实验室中，你可以将电压表并联在每个串联电阻两端，验证各电势差之和等于电源电动势。由于实际电压表具有有限电阻，它会吸取微小电流并轻微改变电路——当测量高阻值元件时，这种效应会变得显著。
2. Potential Dividers and Potentiometers | 分压器与电位器
nd R₂ are the resistors and the total applied voltage is Vᵢₙ, the output voltage Vₒᵤₜ is Vᵢₙ × (R₂/(R₁ + R₂)). This simple arrangement is used to provide a variable voltage from a fixed supply.
分压器由两个串联电阻跨接在电压源上组成，输出电压取自其中一个电阻两端。若 R₁ 与 R₂ 为电阻，总输入电压为 Vᵢₙ，则输出电压 Vₒᵤₜ = Vᵢₙ × (R₂/(R₁ + R₂))。这种简单结构用来从固定电源获得可变电压。
When one of the resistors is replaced by a sensor – a thermistor or an LDR – the output voltage becomes sensitive to temperature or light. In a practical circuit, a buffer amplifier (such as a voltage follower made with an op‑amp) is often added to prevent the load from drawing current and disturbing the divider ratio.
当其中一个电阻用传感器代替时（例如热敏电阻或光敏电阻），输出电压就会对温度或光照敏感。在实用电路中，常常会加入缓冲放大器（如用运放构成的电压跟随器），以防止负载吸取电流而破坏分压比。
A potentiometer is a three‑terminal device with a sliding contact. It can be wired as a variable potential divider. In the laboratory, the slide‑wire potentiometer is used as a null‑measuring instrument to compare EMFs without drawing current from the source under test, giving highly accurate results.
电位器是一种带有滑动触点的三端器件，可以接成可变分压器。在实验室里，滑线电位器用作零位测量仪器，无需从待测源吸取电流即可比较电动势，从而获得高精度结果。
3. The Wheatstone Bridge and Metre Bridge | 惠斯通电桥与米桥
amond shape, with a galvanometer connected between the two midpoints. The bridge is balanced when the ratio of the two resistors in one branch equals the ratio in the other branch: R₁/R₂ = R₃/R₄.
惠斯通电桥是高精度测量未知电阻的经典方法。它由四个电阻排成菱形组成，并在两个中点之间连接一个检流计。当电桥平衡时，其中一个支路两个电阻的比值等于另一支路的比值：R₁/R₂ = R₃/R₄。
In the metre‑bridge form, a one‑metre wire of uniform cross‑ is then calculated from X = R × (l₂/l₁), where l₁ and l₂ are the lengths of wire on either side of the jockey. To minimise errors, the standard resistance R should be chosen so that the balance point lies near the centre of the wire.
在米桥形式中，一段横截面积均匀的一米长导线代替了 R₃ 和 R₄。滑动触头沿导线移动，找到检流计指零的平衡点。未知电阻 X 则由 X = R × (l₂/l₁) 计算得出，其中 l₁ 和 l₂ 是触头两侧的线长。为减小误差，应选择标准电阻 R 使平衡点靠近导线中央。
End errors due to imperfect contacts at the ends of the wire can be eliminated by repeating the measurement after interchanging X and R and taking an average. The bridge method is independent of the supply voltage and galvanometer sensitivity, provided the balance point can be detected sharply.
由导线端点接触不良引起的端误差，可通过互换 X 与 R 重复测量并取平均值来消除。只要平衡点能被灵敏检测，电桥法就与电源电压和检流计灵敏度无关。
4. Measuring Internal Resistance and EMF | 测量内阻与电动势
− Ir. A simple practical circuit to determine ε and r uses a cell, a variable resistor (or resistance box), an ammeter and a voltmeter. By varying the external resistance, you obtain a series of pairs of current I and terminal voltage V.
电池内阻 r 导致在输出电流时端电压下降：V = ε − Ir。测定 ε 和 r 的简单实用电路使用电池、可变电阻（或电阻箱）、一个电流表和一个电压表。通过改变外电阻，可以得到一系列电流 I 与端电压 V 的数据对。
A graph of V (vertical axis) against I (horizontal axis) gives a straight line with gradient −r and intercept ε on the voltage axis. To improve accuracy, use a high‑resistance voltmeter that draws negligible current, or correct for the voltmeter’s loading effect. For cells with very small internal resistance, a four‑terminal (Kelvin) connection can eliminate the effect of lead resistance.
以 V（纵轴）对 I（横轴）作图，可得一条斜率为 −r、在电压轴上截距为 ε 的直线。为提高准确度，应使用吸取电流可忽略的高阻电压表，或对电压表负载效应进行修正。对于内阻极小的电池，可采用四端（开尔文）接法消除引线电阻的影响。
5. RC Circuits and the Time Constant | RC 电路与时间常数
battery of EMF V₀, the voltage across the capacitor rises according to V(t) = V₀(1 − e⁻ᵗ/ᴿᶜ is the time constant of the circuit. After one time constant, the voltage reaches about 63% nal value. During discharge through the same resistor, the voltage falls as V(t) = V₀ e⁻ᵗ/ᴿᶜ.
当一个电容 C 通过电阻 R 由电动势为 V₀ 的电池充电时，电容器两端电压按 V(t) = V₀(1 − e⁻ᵗ/ᴿᶜ) 上升。乘积 τ = RC 是该电路的时间常数。经过一个时间常数后，电压达到最终值的大约 63%。通过同一电阻放电时，电压按 V(t) = V₀ e⁻ᵗ/ᴿᶜ 下降。
ge sensor. By recording the time for the voltage to rise to half its maximum (t₁/₂), the time constant can be found from τ = t₁/₂ / ln2. Using a square‑ d an oscilloscope allows you to observe repeated charging and discharging curves directly.
在实际研究中，可使用高输入阻抗电压表或更佳的数据采集器与电压传感器监测电容电压。记录电压升至最大值一半的时间 t₁/₂，可通过 τ = t₁/₂ / ln2 求出时间常数。使用方波输入和示波器，则可以直接观察到重复的充放电曲线。
Leakage current through the dielectric of an electrolytic capacitor can cause the measured voltage to plateau below V₀. For large time constants, the charging time becomes long, and the ambient temperature may affect the results because resistance and capacitance can change with temperature.
电解电容器介质中的漏电流可能导致测得的电压在低于 V₀ 处趋于平缓。对于大的时间常数，充电时间变长，且环境温度可能影响结果，因为电阻和电容会随温度变化。
6. Semiconductor Diodes and Rectification | 半导体二极管与整流
A diode allows current to flow easily in the forward direction but blocks it in reverse. The practical circuit for obtaining the I–V characteristic of a diode uses a variable voltage supply, a protective series resistor, a milliammeter for forward current, and a microammeter for reverse current. The characteristic shows a knee voltage of about 0.6 V for silicon diodes, after which current rises steeply.
二极管允许电流在正向时容易通过，在反向时则阻断。获取二极管 I–V 特性的实用电路采用可变电压源、保护性串联电阻、测量正向电流的毫安表和测量反向电流的微安表。特性曲线显示硅二极管的拐点电压约为 0.6 V，此后电流急剧上升。
Half‑wave rectification uses a single diode to pass only the positive half‑cycles of an AC input. A full‑wave bridge rectifier with four diodes steers both halves of the cycle so that the current through the load always flows in the same direction. The output of a rectifier is a pulsating DC, which can be smoothed by a capacitor connected across the load. The size of the capacitor determines the ripple voltage: a larger capacitance reduces ripple but increases the peak current through the diodes.
半波整流使用单个二极管，仅让交流输入的正半周通过。由四个二极管组成的全波桥式整流器引导两个半周，使流过负载的电流总是朝同一方向。整流器的输出是脉动直流，可通过并接在负载上的电容器进行平滑。电容的大小决定了纹波电压：较大的电容可减小纹波，但会增大通过二极管的峰值电流。
In practical circuits, a Zener diode is used in reverse bias to provide a stable reference voltage. When the applied reverse voltage exceeds the Zener breakdown voltage, the voltage across the diode remains almost constant over a wide range of current, making it useful for voltage regulation.
在实用电路中，齐纳二极管反偏使用以提供稳定的基准电压。当外加反向电压超过齐纳击穿电压时，二极管两端的电压在很大电流范围内几乎保持不变，这使其适合用于稳压。
7. Operational Amplifier Fundamentals | 运算放大器基础
The operational amplifier (op‑amp) is a very high‑gain differential amplifier. In practical circuits it is almost always used with negative feedback to produce stable, predictable gain. The two basic rules for an ideal op‑amp with negative feedback are: (1) no current flows into the input terminals (infinite input impedance), and (2) the voltage difference between the inverting (−) and non‑inverting (+) inputs is zero (virtual short).
运算放大器（运放）是一种增益极高的差分放大器。在实用电路中，它几乎总是与负反馈一起使用，以产生稳定、可预测的增益。带负反馈的理想运放有两条基本规则：(1) 没有电流流入输入端（无穷大输入阻抗）；(2) 反相输入端 (−) 与同相输入端 (+) 之间的电压差为零（虚短）。
Because the open‑ ion of the output to the inverting input, controlling the overall gain and extending the bandwidth. The most common op‑amp package, such as the 741, requires a dual power supply (±15 V) and can output voltages within about 1–2 V of the supply rails.
由于开环增益极高（通常 >10⁵），即使微小的输入差也会使输出饱和。负反馈将一部分输出送回反相输入端，从而控制总体增益并展宽带宽。最常见的运放封装（如 741）需要双电源供电（±15 V），并能输出在电源轨约 1–2 V 范围内的电压。
8. Inverting and Non‑Inverting Amplifiers | 反相与同相放大器
ation, the input signal is applied via a resistor Rᵢ to the inverting terminal, and a feedback resistor Rf is connected between the output and the inverting input. The non‑inverting input is grounded. The closed‑ Rᵢ. The input impedance of this circuit is approximately equal to Rᵢ, which is typically a few kΩ.
在反相放大器组态中，输入信号通过电阻 Rᵢ 加到反相端，反馈电阻 Rf 连接在输出端与反相输入端之间。同相输入端接地。闭环电压增益为 Aᵥ = −Rf / Rᵢ。该电路的输入阻抗约等于 Rᵢ，通常为几千欧。
For the non‑inverting amplifier, the input is applied directly to the non‑ The gain is Aᵥ = 1 + (Rf / R₁). This configuration has a very high input impedance (the op‑amp input impedance itself), making it ideal for buffering high‑resistance sources like pH probes or condenser microphones.
对于同相放大器，输入信号直接加到同相端，由 Rf 与 R₁ 组成的反馈网络将输出电压分压后送到反相端。增益为 Aᵥ = 1 + (Rf / R₁)。这种组态具有极高的输入阻抗（即运放本身的输入阻抗），使其非常适合用于缓冲高阻抗源，如 pH 探头或电容式麦克风。
Both circuits can be built on a breadboard using a standard op‑amp IC and ±9 dual supply. When measuring the gain, a sinusoidal input of a few hundred millivolts at 1 must be monitored with an oscilloscope to check that it is not clipping due to saturation.
这两种电路都可以用标准运放 IC 和 ±9 V 电池作为双电源在面包板上搭建。在测量增益时，通常使用频率为 1 kHz、幅值为几百毫伏的正弦信号。必须用示波器监视输出端，确保不会因饱和而削波。
9. Op‑Amp as a Comparator and Schmitt Trigger | 运放作为比较器与施密特触发器
When an op‑amp is used without feedback, it acts as a comparator. The output saturates at the positive supply rail if V⁺ > V⁻, and at the negative rail if V⁺ < V⁻. A practical comparator circuit can be used to convert an analogue sensor signal into a digital on/off output, for example to switch on a light when the ambient brightness falls below a set threshold.
当运放无反馈使用时，它相当于一个比较器。若 V⁺ > V⁻，输出饱和于正电源轨；若 V⁺ < V⁻，则饱和于负电源轨。实用的比较器电路可用于将模拟传感器信号转换为数字开关输出，例如当环境亮度低于设定阈值时开启灯光。
A simple comparator may suffer from noise‑induced multiple transitions when the input is near the switching point. The Schmitt trigger solves this by adding positive feedback, which creates two different threshold voltages: an upper threshold Vᵤₜ and a lower threshold Vₗₜ. This hysteresis gives a clean transition and prevents chatter in the output.
简单的比较器在输入靠近切换点时可能会因噪声引发多次跳变。施密特触发器通过加入正反馈解决了这一问题，它创建了两个不同的阈值电压：上阈值 Vᵤₜ 和下阈值 Vₗₜ。这种迟滞作用提供了清晰无疑的跳变，并防止输出产生振荡。
In a non‑inverting Schmitt trigger, the input is applied to the non‑inverting terminal and the feedback is returned through a voltage divider to the same terminal. The thresholds are set by the supply rails and the resistor ratio. This circuit is widely used in sensor signal conditioning and oscillators.
在同相施密特触发器中，输入加到同相端，反馈通过分压器引回同一端。阈值由电源轨和电阻比值设定。这种电路广泛用于传感器信号调理和振荡器中。
10. Relaxation Oscillator Using a Comparator or 555 Timer | 用比较器或 555 定时器构成的弛豫振荡器
A common practical project is the relaxation oscillator, where a capacitor alternately charges towards one threshold and discharges to another, producing a square or triangular wave. Using an op‑amp Schmitt trigger with an RC network, the frequency of oscillation is determined by the RC time constant and the threshold voltages set by hysteresis.
一个常见的实用项目是弛豫振荡器，其中的电容朝着一个阈值充电、然后向着另一个阈值放电，从而产生方波或三角波。利用带 RC 网络的运放施密特触发器，振荡频率由 RC 时间常数和迟滞所设定的阈值电压决定。
rging intervals. The output frequency is f = 1.44 / ((Rₐ + 2R₆) C) and the duty cycle can be adjusted. This oscillator can drive LEDs, produce audio tones, or serve as a clock for digital circuits.
555 定时器 IC 以非稳态模式连接时，提供了一个简洁的解决方案。两个外部电阻 Rₐ、R₆ 和一个电容 C 设定了充电与放电间隔。输出频率为 f = 1.44 / ((Rₐ + 2R₆) C)，且可调节占空比。这种振荡器可以驱动 LED、产生音频音调，或作为数字电路的时钟。
In the lab, students can build the circuit on a breadboard and use an oscilloscope to observe the capacitor waveform and the output square wave. Changing C or the resistors alters the frequency, confirming the relationship given in the formula. Care must be taken with decoupling capacitors near the IC power pins to avoid erratic behaviour.
在实验室中，学生可以在面包板上搭建此电路并用示波器观察电容波形和输出方波。改变 C 或电阻值会改变频率，从而验证公式给出的关系。必须在 IC 电源引脚附近放置去耦电容，以避免不稳定行为。
11. Practical Measurement Techniques and Error Reduction | 实用测量技巧与误差减小
Every measurement in a practical circuit involves uncertainty. When using analogue meters, it is good practice to choose a range that gives a reading in the upper third of the scale. Digital meters offer higher input impedance and better resolution, but the number of digits displayed does not guarantee accuracy – the specifications for basic accuracy and precision must be checked.
实用电路中的每次测量都涉及不确定度。使用模拟电表时，好的做法是选择能使读数落在刻度上三分之一处的量程。数字电表提供更高的输入阻抗和更好的分辨率，但显示的位数并不保证准确度——必须检查基本准确度和精度的规格说明。
Systematic errors, such as contact resistance, lead resistance, and meter loading, can be identified and largely eliminated by using proper wiring techniques (e.g. four‑wire sensing for low resistances). Random errors are minimised by taking several readings and calculating the mean. Where a graph is plotted, the line of best fit should be drawn with an aim to balance points above and below the line, and outliers should be examined but not automatically discarded.
系统误差，例如接触电阻、引线电阻和电表负载效应，可通过合适的接线技术（如低电阻测量中的四线法）来识别并基本消除。随机误差则通过多次读数并计算平均值来最小化。在绘制图线时，应画出最佳拟合线，使点在线上下两侧均匀分布，对异常点应加以分析，但不应自动舍弃。
Safety in practical circuits is paramount. Even low‑voltage circuits (<50 V) can cause burns if short‑circuited through a metal ring or watch strap. Capacitors can store dangerous charge long after the power is removed; they should be discharged via a suitable resistor before handling. Mains‑powered circuits must be isolated, and no live adjustments should be made.
实用电路中的安全至关重要。即使是低压电路（<50 V），若通过金属戒指或表带短路也可能造成灼伤。电容器在断电后仍可长时间储存危险电荷；操作前应通过合适电阻放电。交流供电电路必须隔离，并且不得进行带电调节。
12. Building Circuits and Troubleshooting | 搭建电路与故障排查
When constructing practical circuits on a prototyping board, keep leads short and tidy to minimise stray capacitance and inductive coupling. Colour‑code the wiring where possible: red for positive supply, black for ground, and other colours for signals. Before applying power, double‑check the polarity of electrolytic capacitors and diodes, and verify that the ICs are inserted the right way round.
在面包板上搭建实用电路时，应使连线短且整齐，以尽量减少寄生电容和感性耦合。尽可能对导线进行颜色编码：红色为电源正极，黑色为地线，其他颜色为信号线。通电前应再次检查电解电容和二极管的极性，并确认集成电路的插入方向正确无误。
If a circuit does not work, use a systematic approach: check the supply voltages at the IC pins with a voltmeter, trace the signal from input to output using an oscilloscope, and measure the DC voltages at key nodes to see if they agree with the expected values. Common faults include dry joints, unintended short circuits, and overheating components. A thermal camera or a gentle touch can locate a component that is dissipating excessive power.
如果电路不工作，应采用系统方法：用电压表检查集成电路引脚上的供电电压，用示波器从输入到输出追踪信号，并测量关键节点的直流电压是否与预期值相符。常见故障包括虚焊、意外短路和元件过热。热成像仪或小心触摸可定位功耗过高的元件。
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