📚 Inverting Amplifier Circuit Analysis and Gain Calculation | 反相放大器电路分析与增益计算
The inverting amplifier is one of the most fundamental operational amplifier (op-amp) configurations in analog electronics. It produces an output voltage that is inverted in polarity and scaled by the ratio of two resistors. This article provides a complete CIE A-Level Physics treatment of the circuit, including its ideal assumptions, virtual ground concept, gain derivation, input and output impedance, bandwidth limitations, non-ideal effects, and a worked calculation example.
反相放大器是模拟电子学中最基本的运算放大器（运放）电路之一。它产生一个极性相反、并由两个电阻比值缩放比例的输出电压。本文针对 CIE A-Level 物理课程，系统讲解该电路，包括理想假设、虚地概念、增益推导、输入与输出阻抗、带宽限制、非理想效应以及一个计算实例。
1. What Is an Inverting Amplifier? | 什么是反相放大器？
An inverting amplifier uses negative feedback to set a precise voltage gain. The input signal is applied through an input resistor to the inverting input of an op-amp, while the non-inverting input is connected to ground. The output is fed back to the inverting input through a feedback resistor.
反相放大器利用负反馈来设定精确的电压增益。输入信号通过一个输入电阻加到运放的反相输入端，同相输入端接地。输出端通过一个反馈电阻接回反相输入端。
Because the non-inverting input is grounded, the amplifier output is inverted relative to the input. The gain magnitude depends only on the external resistor values, not on the op-amp itself, as long as the op-amp is ideal and negative feedback is present.
由于同相输入端接地，放大器输出相对于输入是反相的。只要运放是理想的且存在负反馈，增益大小仅取决于外部电阻值，而与运放本身无关。
2. Ideal Operational Amplifier Assumptions | 理想运算放大器假设
In A-Level analysis, the op-amp is treated as an ideal device. The following assumptions are used to simplify circuit calculations:
在 A-Level 的分析中，运放被视为理想器件。以下假设用于简化电路计算：
- Infinite open-loop voltage gain: A very small voltage difference between the inputs produces a large output.
- Infinite input impedance: No current flows into either input terminal.
- Zero output impedance: The output terminal can supply any current without voltage drop.
当这些假设成立时，运放可以视为一个电压控制的电压源，其输入电流为零，输出电阻为零，开环增益为无穷大。
These assumptions lead to two key rules: the voltage between the input terminals is zero (virtual short), and the input currents are zero. In an inverting amplifier, since the non-inverting terminal is grounded, the inverting terminal is forced to the same potential — approximately 0 V.
这些假设导出两条关键规则：两个输入端之间的电压为零（虚短），且输入电流为零。在反相放大器中，因为同相端接地，反相端被强制到相同的电位——近似为 0 V。
3. Circuit Topology of the Inverting Amplifier | 反相放大器的电路拓扑
The inverting amplifier circuit has three essential components: an op-amp, an input resistor Rin, and a feedback resistor Rf. The input signal Vin is connected to the inverting input through Rin. The non-inverting input is tied directly to ground.
反相放大器电路包含三个基本部分：一个运放、一个输入电阻 Rin 和一个反馈电阻 Rf。输入信号 Vin 通过 Rin 连接到反相输入端，同相输入端直接接地。
The output Vout is fed back to the inverting input through Rf. This negative feedback stabilises the gain and reduces distortion. The closed-loop gain is set entirely by the resistance ratio Rf/Rin.
输出 Vout 通过 Rf 反馈到反相输入端。这种负反馈使增益稳定并减小失真。闭环增益完全由电阻比值 Rf/Rin 决定。
4. The Virtual Ground Concept | 虚地概念
In the inverting amplifier, the non-inverting input is grounded, so V+ = 0 V. Because of the virtual short rule, the inverting input also sits at approximately 0 V. This is called a virtual ground: the node is not physically connected to ground, but its voltage is held at ground potential by the feedback action.
在反相放大器中，同相输入端接地，因此 V+ = 0 V。由于虚短规则，反相输入端也近似处于 0 V。这被称为虚地：该节点并非物理上连接到地，而是通过反馈作用被保持在接地电位。
The virtual ground is crucial because it allows us to calculate the current through Rin simply as I = Vin / Rin. Since no current enters the op-amp input, this same current must flow through Rf.
虚地概念至关重要，因为它使我们能够简单地计算通过 Rin 的电流为 I = Vin / Rin。由于没有电流进入运放输入端，这个电流必然流过 Rf。
5. Derivation of the Closed-Loop Gain | 闭环增益推导
Using the virtual ground, the voltage across Rin is Vin − 0, so the input current is:
利用虚地，Rin 上的电压为 Vin − 0，所以输入电流为：
I = Vin / Rin
Since the op-amp input draws no current, the same current flows through Rf. The voltage at the output side of Rf is Vout, while the left side is at virtual ground. Therefore the voltage drop across Rf is 0 − Vout, and we have:
由于运放输入不取电流，同样的电流流过 Rf。Rf 输出端电压为 Vout，而左端处于虚地。因此 Rf 上的电压降为 0 − Vout，于是有：
0 − Vout = I × Rf
Substituting I = Vin / Rin gives:
将 I = Vin / Rin 代入得：
Vout = − (Rf / Rin) × Vin
Therefore the closed-loop voltage gain is:
因此闭环电压增益为：
Av = Vout / Vin = − Rf / Rin
The negative sign indicates a 180° phase inversion. The magnitude of the gain is simply the ratio of the feedback resistor to the input resistor.
负号表示相位反转 180°。增益的大小就是反馈电阻与输入电阻的比值。
6. Input Impedance and Output Impedance | 输入阻抗与输出阻抗
From the signal source’s point of view, the inverting input is a virtual ground. Therefore the input impedance of the inverting amplifier is approximately Rin:
从信号源的角度看，反相输入端是虚地。因此反相放大器的输入阻抗近似为 Rin：
Zin ≈ Rin
This is a key difference from the non-inverting amplifier, which has a very high input impedance. To present a large input impedance, Rin must be large, but that also reduces the current and may affect noise performance.
这是与非反相放大器的一个关键区别，非反相放大器具有非常高的输入阻抗。为了获得大的输入阻抗，Rin 必须很大，但这也减小了电流并可能影响噪声性能。
The output impedance of the ideal op-amp is zero, so the closed-loop output impedance is also essentially zero. This means the amplifier can drive a load without significant voltage loss.
理想运放的输出阻抗为零，因此闭环输出阻抗也基本上为零。这意味着放大器可以驱动负载而不会产生显著的电压损失。
7. Bandwidth and Gain–Bandwidth Product | 带宽与增益带宽积
A real op-amp has a finite open-loop gain that decreases with frequency. The gain–bandwidth product (GBW) is a constant: the product of the closed-loop gain magnitude and the closed-loop bandwidth equals the unity-gain bandwidth of the op-amp.
实际运放的开环增益是有限的，并且随频率升高而下降。增益带宽积（GBW）是一个常数：闭环增益大小与闭环带宽的乘积等于运放的单位增益带宽。
|Av| × fBW = GBW
For an inverting amplifier with a gain of −10 and an op-amp with GBW = 1 MHz, the usable bandwidth is:
对于一个增益为 −10 的反相放大器，若运放的 GBW = 1 MHz，则可用带宽为：
fBW = 1 MHz / 10 = 100 kHz
Higher gains reduce bandwidth, while lower gains increase it. This trade-off is essential when designing amplifiers for specific frequency ranges.
增益越高，带宽越窄；增益越低，带宽越宽。在设计特定频率范围的放大器时，这一权衡至关重要。
8. Non-Ideal Effects and Practical Limitations | 非理想效应与实用限制
Real op-amps are not perfect. Several non-ideal effects can affect the inverting amplifier:
实际运放并不完美。几种非理想效应会影响反相放大器：
- Input bias current: Small currents flowing into the inputs create an offset voltage across Rin and Rf, shifting the output.
- Input offset voltage: The small voltage difference required to force the output to zero appears as an error at the output.
- Finite open-loop gain: The virtual ground is not exact; the gain becomes slightly lower than the ideal value.
- Output voltage saturation: The output cannot exceed the supply rails.
输入偏置电流：流入输入端的微小电流会在 Rin 和 Rf 上产生失调电压，使输出偏移。
输入失调电压：迫使输出为零所需的微小输入电压差会作为误差出现在输出端。
有限开环增益：虚地并不完全精确，实际增益略低于理想值。
输出电压饱和：输出不能超过电源轨电压。
In A-Level questions, these effects are usually neglected. However, you should know the ideal assumptions and recognise when a non-ideal op-amp would produce a different result.
在 A-Level 试题中，这些效应通常被忽略。然而，你应该了解理想假设，并认识到非理想运放可能产生不同的结果。
9. Worked Example and Calculations | 例题与计算
Example: An inverting amplifier has Rin = 10 kΩ and Rf = 50 kΩ. The input voltage is Vin = 1 V. Assume the op-amp is ideal.
例题：反相放大器中，Rin = 10 kΩ，Rf = 50 kΩ，输入电压 Vin = 1 V。假设运放是理想的。
(a) Calculate the closed-loop gain.
（a）计算闭环增益。
Av = − Rf / Rin = − 50 / 10 = −5
(b) Determine the output voltage.
（b）求输出电压。
Vout = Av × Vin = −5 × 1 = −5 V
(c) What is the input impedance seen by the signal source?
（c）信号源看到的输入阻抗是多少？
Zin = Rin = 10 kΩ
(d) If the op-amp has a gain–bandwidth product of 1 MHz, what is the closed-loop bandwidth?
（d）若运放增益带宽积为 1 MHz，闭环带宽是多少？
fBW = GBW / |Av| = 1 MHz / 5 = 200 kHz
10. Common Applications and Summary | 常见应用与总结
The inverting amplifier is used in many signal-processing circuits. It can scale a signal, combine multiple inputs, or convert current to voltage. A summing amplifier is formed by adding several input resistors to the inverting node:
反相放大器在许多信号处理电路中使用。它可以缩放信号、组合多个输入或实现电流电压转换。求和放大器是通过在反相节点上增加多个输入电阻而构成的：
Vout = − Rf ( V1/R1 + V2/R2 + V3/R3 )
A transimpedance amplifier uses a feedback resistor to convert an input current into an output voltage, with the virtual ground providing a low-impedance node for the current source.
跨阻放大器利用反馈电阻将输入电流转换为输出电压，虚地为电流源提供了低阻抗节点。
In summary, the inverting amplifier is defined by the gain Vout = −(Rf/Rin)Vin, has an input impedance of Rin, and a bandwidth determined by the gain–bandwidth product. Understanding its analysis is an essential skill for CIE A-Level Physics and practical electronics.
总之，反相放大器的定义是 Vout = −(Rf/Rin)Vin，其输入阻抗为 Rin，带宽由增益带宽积决定。掌握其分析是 CIE A-Level 物理和实际电子学中一项基本技能。
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