The Operational Amplifier (Op-Amp) | 运算放大器（Op-Amp）

📚 The Operational Amplifier (Op-Amp) | 运算放大器（Op-Amp）

The operational amplifier, commonly known as the op-amp, is one of the most versatile and widely used building blocks in analogue electronics. It is essentially a high-gain differential voltage amplifier that can be configured to perform a vast array of functions—from basic amplification and signal conditioning to mathematical operations like addition, subtraction, integration, and differentiation.

运算放大器（常简称为运放）是模拟电子学中最通用、应用最广泛的基础模块之一。它本质上是一个高增益差分电压放大器，可以通过不同的外部电路配置，实现从基本放大、信号调理到加法、减法、积分、微分等各种数学运算功能。

1. What Is an Op-Amp? | 什么是运算放大器？

An op-amp is an integrated circuit (IC) with a differential input stage—meaning it amplifies the difference between the voltages applied to its two input terminals: the non-inverting input (+) and the inverting input (−). The device has a single output and requires dual power supplies, typically +V_s and −V_s, to allow the output to swing both positive and negative with respect to ground.

运放是一种集成电路，具有差分输入级——即它放大的是两个输入端之间的电压差。这两个输入端分别为同相输入端（+）和反相输入端（−）。运放有一个单端输出，并且通常需要双电源供电（如 +V_s 和 −V_s），使输出电压能够相对于地线正向或负向摆动。

The standard symbol for an op-amp is a triangle with the two inputs on the left, the output on the right, and the power supply connections at the top and bottom (often omitted in circuit diagrams). The gain of the op-amp without any external feedback is called the open-loop gain, A_OL.

运放的标准符号是一个三角形，左侧有两个输入端，右侧为输出端，上下分别接正负电源（电路图中常省略）。在没有任何外部反馈时，运放自身的增益称为开环增益，记作 A_OL。

Op-amps are analogue components, but they also serve as the interface between analogue and digital systems in converters and sensor circuits. For A-level physics, you are expected to understand the behaviour of an ideal op-amp and how feedback networks determine its closed-loop performance.

运放是模拟器件，同时在转换器和传感器电路中扮演着模拟与数字系统之间的桥梁。在 A-level 物理课程中，你需要理解理想运放的行为以及反馈网络如何决定其闭环性能。

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2. Ideal Op-Amp Characteristics | 理想运放的特性

To simplify circuit analysis, we assume the op-amp is ‘ideal’. An ideal op-amp possesses a set of extreme characteristics that make calculations straightforward. These are:

为简化电路分析，我们通常假设运放是“理想”的。理想运放具有一系列极端特性，使得分析和计算变得简单。这些特性包括：

• Infinite open-loop gain (A_OL → ∞): Even the smallest difference between the input terminals would drive the output to the supply rails. | 无限大的开环增益（A_OL → ∞）：输入端之间即使存在极微小的电压差，也会使输出达到电源轨电压。
• Infinite input impedance (Z_in → ∞): No current flows into either the inverting or non-inverting input. | 无限大的输入阻抗（Z_in → ∞）：没有电流流入反相或同相输入端。
• Zero output impedance (Z_out = 0): The output voltage is independent of the load; the op-amp can supply any current without voltage drop. | 零输出阻抗（Z_out = 0）：输出电压不受负载影响；运放能提供任意大小的电流而输出电压不出现压降。
• Infinite bandwidth: The gain remains constant for all frequencies. | 无限带宽：所有频率下的增益保持不变。
• Zero offset voltage: When the input difference is zero, the output is exactly zero. | 零输入失调电压：当输入电压差为零时，输出也为零。

These ideal assumptions lead to the two ‘golden rules’ used in analysing op-amp circuits with negative feedback: (1) No current flows into the input terminals; (2) The voltage at the inverting input equals the voltage at the non-inverting input (virtual equality).

这些理想化假设引出分析含负反馈运放电路的两条“黄金法则”：①没有电流流入输入端；②反相输入端的电压等于同相输入端的电压（虚短）。

Real op-amps approximate these ideals: typical open-loop gains exceed 10⁵, input impedances are in the megaohm range, and output impedances are tens of ohms. The ideal model is sufficient for most A-level problems.

实际运放近似实现这些理想条件：典型开环增益超过 10⁵，输入阻抗在兆欧级，输出阻抗为几十欧姆。在 A-level 的绝大多数问题中，理想模型已足够准确。

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3. Open-Loop and Closed-Loop Configurations | 开环与闭环结构

When an op-amp is used without any feedback connection—meaning the output is not linked back to the input—it operates in open-loop mode. Because of the extremely high gain, the output voltage is essentially either at the positive saturation voltage (+V_sat) or the negative saturation voltage (−V_sat) as soon as a tiny differential voltage is present. This behaviour makes the open-loop op-amp suitable for comparator applications, where the goal is to determine which of two voltages is larger.

当运放没有任何反馈连接（即输出端不与输入端相连）时，它工作于开环模式。由于增益极高，只要输入端出现微小的电压差，输出电压就会迅速地等于正饱和电压（+V_sat）或负饱和电压（−V_sat）。这种特性使开环运放非常适合用作比较器，用来判断两个电压中哪一个更大。

In most linear amplification circuits, however, we apply negative feedback—a connection from the output back to the inverting input through a resistive network. Negative feedback reduces the overall gain but greatly improves stability, bandwidth, and linearity. The circuit’s gain then becomes predictable and is set entirely by the external component values, not by the op-amp’s own gain.

然而，在大多数线性放大电路中，我们会引入负反馈——通过电阻网络将输出端连接到反相输入端。负反馈虽然降低了总体增益，但大大改善了电路的稳定性、带宽和线性度。此时，电路的增益变得可预测，并且完全由外部元件数值决定，而不再取决于运放自身的开环增益。

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4. The Inverting Amplifier | 反相放大器

The inverting amplifier is one of the most fundamental op-amp circuits. An input voltage is applied through a resistor R_in to the inverting input, while a feedback resistor R_f connects the output back to the same inverting input. The non-inverting input is connected directly to ground (0 V).

反相放大器是最基本的运放电路之一。输入电压通过电阻 R_in 加到反相输入端，同时反馈电阻 R_f 将输出端连接到该反相输入端。同相输入端则直接接地（0 V）。

Using the ideal op-amp rules: since the non-inverting input is at 0 V, the inverting input must also be at 0 V (a ‘virtual earth’). No current flows into the op-amp input, so the current through R_in equals the current through R_f. Applying Ohm’s law gives:

运用理想运放的法则：因为同相输入端为 0 V，所以反相输入端也必定为 0 V（即“虚地”）。由于没有电流流入运放输入端，流过 R_in 的电流等于流过 R_f 的电流。由欧姆定律可得：

I = V_in / R_in = –V_out / R_f

Rearranging, the voltage gain A_V for the inverting amplifier is:

经整理，反相放大器的电压增益 A_V 为：

A_V = V_out / V_in = –R_f / R_in

The negative sign indicates a 180° phase shift: when the input rises, the output falls, and vice versa. The magnitude of the gain is simply the resistance ratio. By choosing suitable resistors, we can design amplifiers with any desired gain less than, equal to, or greater than 1.

负号表示输出与输入之间有 180° 的相位偏移：当输入电压上升时，输出电压下降，反之亦然。增益的大小仅由电阻比值决定。通过选择合适的电阻，我们可以设计出增益小于、等于或大于 1 的放大器。

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5. The Non-Inverting Amplifier | 同相放大器

The non-inverting amplifier provides a positive voltage gain without signal inversion. The input signal is applied directly to the non-inverting input. A voltage divider formed by R₁ and R_f connects the output back to the inverting input, creating negative feedback.

同相放大器能提供正的电压增益，且不会反转信号相位。输入信号直接加到同相输入端。由 R₁ 和 R_f 组成的分压器将输出反馈至反相输入端，形成负反馈。

In this configuration, the voltage at the inverting input is forced to match the input voltage V_in due to the virtual short. The current through R₁ equals the current through R_f, which gives:

在此结构中，由于虚短，反相输入端的电压被迫等于输入电压 V_in。流过 R₁ 的电流等于流过 R_f 的电流，因此有：

V_in / R₁ = (V_out – V_in) / R_f

Rearranging, the closed-loop voltage gain becomes:

移项整理后，闭环电压增益为：

A_V = 1 + (R_f / R₁)

Unlike the inverting amplifier, the non-inverting gain is always greater than or equal to 1, and the output voltage is in phase with the input. This circuit also offers a much higher input impedance because the signal is fed directly into the non-inverting terminal, making it ideal for buffering weak signals from high-impedance sources.

与反相放大器不同，同相放大器的增益始终大于或等于 1，且输出电压与输入同相。该电路还提供了极高的输入阻抗，因为信号直接送入同相输入端，因此非常适合对来自高阻抗源的微弱信号进行缓冲放大。

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6. The Voltage Follower (Buffer) | 电压跟随器（缓冲器）

A special case of the non-inverting amplifier occurs when R_f = 0 and R₁ → ∞ (i.e. an open circuit). The gain becomes exactly 1. This circuit is called a voltage follower or unity-gain buffer.

当 R_f = 0 且 R₁ → ∞（即开路）时，同相放大器成为一个特例。此时增益恰好为 1。该电路称为电压跟随器或单位增益缓冲器。

The output voltage equals the input voltage both in magnitude and phase. The key advantage is that the buffer presents an extremely high input impedance so that it does not load the previous stage, while offering a very low output impedance to drive subsequent stages. This isolates signal sources from loads and is widely used in measurement systems.

输出电压在大小和相位上都与输入电压相同。该电路最大的优势在于其极高的输入阻抗（几乎不向前级索取电流），同时输出阻抗极低，能有效驱动后级电路。这实现了信号源与负载之间的隔离，在测量系统中被广泛使用。

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7. Op-Amp Saturation | 运放饱和

Op-amps cannot produce an output voltage beyond their supply rails. In practice, the maximum output voltage—called the saturation voltage V_sat—is typically about 1–2 V less than the supply voltages. When the input signal is too large, or the gain is too high, the op-amp output ‘hits the rail’ and saturates, clipping the waveform.

运放输出的电压不可能超过其供电电压的范围。实际上，最大输出电压——称为饱和电压 V_sat——通常比电源电压低约 1–2 V。当输入信号过大或增益设置过高时，运放输出会“撞轨”并饱和，导致波形被削顶（限幅）。

Saturation is particularly important in open-loop and comparator circuits. A tiny input voltage difference drives the output to either +V_sat or −V_sat almost instantaneously. In linear circuits with negative feedback, however, saturation is usually avoided by keeping the intended output swing within the linear range defined by the supply rails.

饱和现象在开环和比较器电路中尤为重要。微小的输入电压差就能几乎瞬间将输出推到 +V_sat 或 −V_sat。然而，在具有负反馈的线性电路中，通常通过设计使期望的输出摆幅保持在电源轨所限定的线性范围内，从而避免饱和。

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8. The Op-Amp as a Comparator | 用作比较器的运放

An op-amp without negative feedback acts as a comparator. The two voltage inputs V₊ and V₋ are applied, and the output saturates depending on which input is larger: if V₊ > V₋, the output goes to +V_sat; if V₊ < V₋, it goes to −V_sat.

无负反馈的运放即可充当比较器。将两个电压 V₊ 和 V₋ 分别接到同相和反相输入端，输出将根据哪个输入电压更大而饱和：若 V₊ > V₋，输出为 +V_sat；若 V₊ < V₋，输出为 −V_sat。

Comparators are used in many sensor circuits, such as light/dark detectors using LDRs, temperature alarms, and zero-crossing detectors. A reference voltage is often applied to one input while the fluctuating signal is connected to the other. The sharp output transitions are ideal for triggering digital logic or lighting an LED indicator.

比较器被广泛应用于各种传感器电路中，例如使用光敏电阻的亮暗检测器、温度报警器以及过零检测器等。通常将一个参考电压接到一个输入端，而变化信号接入另一个输入端。输出端的急剧跳变非常适合触发数字逻辑或点亮指示灯。

Dedicated comparator ICs exist for this purpose, but a general-purpose op-amp can serve the same function when high speed or precise thresholds are not critical.

市面上有专门的比较器集成芯片，但当对速度或阈值精度要求不高时，通用运放也能实现相同的功能。

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9. Understanding Gain and Bandwidth | 理解增益与带宽

In an ideal op-amp, gain is independent of frequency, but real devices exhibit a gain-bandwidth product limitation. The open-loop gain falls off with increasing frequency, typically at a rate of −20 dB per decade, so that the product (gain × bandwidth) remains constant. When negative feedback is applied, the closed-loop bandwidth increases as the closed-loop gain decreases.

理想运放的增益与频率无关，但实际器件存在增益带宽积的限制。开环增益随频率升高而下降，通常以每十倍频程 −20 dB 的速率滚降，使得增益与带宽的乘积保持恒定。引入负反馈后，闭环增益降低的同时闭环带宽会相应增加。

For A-level problems, this concept explains why a high-gain amplifier has a narrower useful frequency range, while a unity-gain buffer can operate at much higher frequencies.

在 A-level 题目中，这一概念解释了为何高增益放大器的可用频率范围较窄，而单位增益缓冲器能在高得多的频率下工作。

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10. Input and Output Impedance Effects | 输入与输出阻抗的影响

The assumption of infinite input impedance and zero output impedance simplifies calculations, but real effects become important when interfacing with sensors or loads. A finite input impedance can cause loading on the signal source, altering the actual voltage seen at the input. A non-zero output impedance means that the output voltage drops slightly when current is drawn by the load.

无限输入阻抗和零输出阻抗的假设简化了计算，但在连接传感器或负载时，实际效应就显得重要了。有限的输入阻抗可能对信号源产生加载效应，改变输入端实际得到的电压。非零的输出阻抗则意味着当负载索取电流时，输出电压会有轻微下降。

Using a voltage follower between a high-impedance source and a low-impedance load overcomes both problems, which is why buffer circuits are ubiquitous in analogue design.

在高阻抗源和低阻抗负载之间使用电压跟随器可同时解决这两个问题，这便是缓冲电路在模拟设计中无处不在的原因。

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11. Key Formulae Summary | 关键公式总结

For quick reference, the essential equations are collected below:

为便于参考，重要公式集中如下：

Inverting Amplifier	AV = –Rf / Rin
Non-Inverting Amplifier	AV = 1 + Rf / R1
Voltage Follower	AV = 1
Virtual Earth / Virtual Short	V+ ≈ V− (with negative feedback)
Comparator Output	Vout = +Vsat if V+ > V− , else −Vsat

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12. Worked Example | 计算示例

An inverting amplifier uses R_in = 10 kΩ and R_f = 120 kΩ. If the input voltage is a sine wave with a peak value of ±100 mV, determine the peak output voltage before saturation occurs, assuming the supply rails give V_sat = ±12 V.

一个反相放大器使用 R_in = 10 kΩ、R_f = 120 kΩ。若输入电压为峰值 ±100 mV 的正弦波，并假设电源轨提供的 V_sat = ±12 V，求出未发生饱和时的峰值输出电压。

Gain A_V = –R_f / R_in = –120 kΩ / 10 kΩ = –12. The peak output voltage would be ±100 mV × 12 = ±1.2 V. This is comfortably within the ±12 V linear range, so the output remains an undistorted sine wave with peak 1.2 V.

增益 A_V = –R_f / R_in = –120 kΩ / 10 kΩ = –12。峰值输出电压应为 ±100 mV × 12 = ±1.2 V。该值远在 ±12 V 的线性范围之内，因此输出仍为无失真的正弦波，峰值为 1.2 V。

If R_f were increased to 1.5 MΩ with the same R_in, the gain would be –150, and an input of ±100 mV would theoretically produce ±15 V, exceeding the supply rails. The output would then saturate at approximately ±12 V, clipping the peaks.

若保持 R_in 不变而将 R_f 增大至 1.5 MΩ，增益将变为 –150，±100 mV 的输入理论上会产生 ±15 V 输出，超过电源轨。此时输出将在约 ±12 V 处饱和，削去波峰的顶部。

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Published by TutorHao | Physics Revision Series | aleveler.com

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