📚 The Inverting Amplifier | 反相放大器
The inverting amplifier is one of the most fundamental configurations using an operational amplifier (op-amp). It provides a stable voltage gain that is set by two external resistors, making it widely used in signal processing, audio mixing, and measurement circuits. This article examines the principle, derivation, and practical behaviour of the inverting amplifier as required for A-Level Physics under the CIE specification.
反相放大器是使用运算放大器（运放）的最基本组态之一。它通过两个外部电阻提供稳定的电压增益，因此被广泛应用于信号处理、音频混音和测量电路中。本文将根据 CIE 课程大纲，分析反相放大器的原理、推导过程及实际特性。
1. Introduction to Operational Amplifiers | 运算放大器简介
An operational amplifier is a high-gain differential voltage amplifier. Typically housed in an 8-pin integrated circuit, it has two input terminals – the inverting input (labelled –) and the non-inverting input (labelled +) – and a single output. In open-loop mode the voltage gain is enormous (often >105), but the amplifier is rarely used without feedback. Negative feedback allows precise control of gain, bandwidth, and impedance.
运算放大器是一种高增益差动电压放大器。它通常封装在 8 引脚的集成电路中，有两个输入端——反相输入端（标为 –）和同相输入端（标为 +）——以及一个输出端。开环模式下电压增益极大（常大于 105），但放大器很少不施加反馈就使用。负反馈可以精确控制增益、带宽与阻抗。
2. Ideal Op-Amp Characteristics | 理想运算放大器的特性
To simplify circuit analysis, we first assume an ideal op-amp with the following properties:
为了简化电路分析，我们首先假定一种理想运放，其特性如下：
| Property / 特性 | Ideal value / 理想值 | Typical real op-amp (e.g. TL081) / 典型实际运放 |
|---|---|---|
| Open-loop voltage gain / 开环电压增益 | ∞ | ~2 × 105 (106 dB) |
| Input impedance / 输入阻抗 | ∞ | ~1012 Ω |
| Output impedance / 输出阻抗 | 0 | ~75 Ω |
| Bandwidth / 带宽 | ∞ | Gain-bandwidth product ~3 MHz |
| Input offset voltage / 输入失调电压 | 0 | ~3 mV |
Because the open-loop gain is effectively infinite, the voltage difference between the two input terminals, V+ − V−, must be essentially zero for any finite output voltage. This idea leads directly to the virtual earth concept in inverting circuits.
由于开环增益实际上是无穷大的，因此对于任何有限的输出电压，两个输入端之间的电压差 V+ − V− 必然基本上为零。这一思想直接导出了反相电路中的虚地概念。
3. The Inverting Amplifier Configuration | 反相放大器电路组态
The inverting amplifier uses an op-amp with two external resistors. A feedback resistor Rf is connected between the output and the inverting input (–). An input resistor Rin is placed in series with the signal source, feeding into the inverting input. The non-inverting input (+) is connected directly to 0 V (ground). The signal voltage Vin is applied to the free end of Rin.
反相放大器使用一个运放和两个外部电阻。反馈电阻 Rf 连接在输出端与反相输入端 (–) 之间。输入电阻 Rin 与信号源串联，接入反相输入端。同相输入端 (+) 直接连接到 0 V（接地）。信号电压 Vin 加在 Rin 的自由端。
This arrangement forces the output to be an amplified, inverted version of the input. Negative feedback through Rf stabilises the circuit and determines the closed-loop voltage gain.
这种安排使输出成为输入被放大并反相后的版本。通过 Rf 引入的负反馈使电路稳定，并决定了闭环电压增益。
4. Virtual Earth Concept | 虚地概念
With the non-inverting input grounded (V+ = 0) and the op-amp maintaining V− ≈ V+, the inverting input sits at approximately 0 V. This point is called a virtual earth (or virtual ground). It is not physically connected to ground, but it behaves as if it is at ground potential for small changes in signal voltage. No current flows into the op-amp inputs because of the extremely high input impedance.
由于同相输入端接地 (V+ = 0)，且运放保持 V− ≈ V+，因此反相输入端大约处于 0 V。这一点称为虚地。它并没有物理连接到地，但对于信号电压的微小变化，它的性能如同处于地电位。由于运放输入阻抗极高，没有电流流入运放输入端。
The virtual earth simplifies circuit analysis: the full input voltage appears across Rin, and the current through Rin must be equal to the current through Rf because no current enters the op-amp.
虚地简化了电路分析：全部输入电压都降在 Rin 上，而流过 Rin 的电流必定等于流过 Rf 的电流，因为没有电流进入运放。
5. Deriving the Voltage Gain | 电压增益的推导
We can find the closed-loop gain using Kirchhoff's current law (KCL) at the virtual earth node. Let the current through Rin be Iin and the current through Rf be If. Since no current enters the op-amp inverting input,
我们可以利用虚地节点处的基尔霍夫电流定律 (KCL) 求出闭环增益。设流过 Rin 的电流为 Iin，流过 Rf 的电流为 If。由于没有电流流入运放反相输入端，
Iin = If
Because the virtual earth keeps the inverting terminal at 0 V, the voltage across Rin is simply Vin − 0 = Vin, giving
由于虚地把
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