The Inverting Amplifier | 反相放大器

📚 The Inverting Amplifier | 反相放大器

An inverting amplifier is one of the most common operational amplifier (op-amp) circuits studied in A-Level Physics. It produces an output voltage that is proportional to the input voltage but opposite in sign, and its gain is set entirely by two external resistors. This article explains the circuit, the key concept of the virtual earth, how the gain formula is derived, and important practical limits such as saturation and bandwidth.

反相放大器是 A-Level 物理中最常见的运算放大器电路之一。它产生的输出电压与输入电压成正比但符号相反，其增益完全由两个外部电阻决定。本文将解释电路结构、虚地这一关键概念、增益公式的推导，以及饱和与带宽等重要实际限制。

1. Operational Amplifier Basics | 运算放大器基础

An operational amplifier, or op-amp, is a high-gain differential voltage amplifier. It has two input terminals: the non-inverting input marked with a plus sign (+) and the inverting input marked with a minus sign (−). There is one output terminal. The op-amp amplifies the difference between the voltages applied to its two inputs.

运算放大器简称 op-amp，是一种高增益差分电压放大器。它有两个输入端：标有加号 (+) 的同相输入端和标有减号 (−) 的反相输入端，另外有一个输出端。运算放大器放大的是两个输入端之间的电压差。

In an open-loop configuration, the output voltage is given by:

在开环配置中，输出电压由下式给出：

V_out = A_ol × (V₊ − V₋)

where A_ol is the open-loop gain, V₊ is the non-inverting input voltage, and V₋ is the inverting input voltage. For an ideal op-amp, A_ol is assumed to be infinitely large.

其中 A_ol 是开环增益，V₊ 是同相输入电压，V₋ 是反相输入电压。对于理想运算放大器，A_ol 被认为是无穷大。

An ideal op-amp is usually described by the following ideal characteristics:

理想运算放大器通常用以下理想特性来描述：

Infinite open-loop voltage gain: A_ol → ∞
Infinite input impedance: Z_in → ∞, so no current flows into either input
Zero output impedance: Z_out = 0 Ω, so the output can drive any load without voltage drop
Infinite bandwidth and zero output offset

These ideal assumptions allow us to analyse op-amp circuits with simple rules. In reality, op-amps are close to ideal but not perfect, and later sections will discuss practical limitations.

这些理想假设使我们能够用简单规则分析运算放大器电路。现实中运算放大器接近理想但并非完美，后面的小节将讨论实际限制。

2. The Inverting Amplifier Circuit | 反相放大器电路

The inverting amplifier uses negative feedback to control the gain. An input resistor R_in is connected between the signal source and the inverting input (−). A feedback resistor R_f is connected between the output and the inverting input. The non-inverting input (+) is connected directly to ground, so V₊ = 0 V.

反相放大器利用负反馈来控制增益。输入电阻 R_in 连接在信号源与反相输入端 (−) 之间。反馈电阻 R_f 连接在输出端与反相输入端之间。同相输入端 (+) 直接接地，因此 V₊ = 0 V。

The input signal is applied to the free end of R_in, and the output is taken from the op-amp output terminal. Because the feedback path returns a fraction of the output to the inverting input, the circuit is said to have negative feedback. This stabilises the gain and makes it predictable.

输入信号加在 R_in 的自由端，输出从运算放大器输出端取出。由于反馈路径将输出的一部分送回反相输入端，因此该电路被称为具有负反馈。负反馈使增益稳定并且可以预测。

The circuit is called “inverting” because the output waveform is inverted relative to the input: a positive input produces a negative output, and vice versa.

该电路被称为“反相”是因为输出波形相对于输入是反相的：正输入产生负输出，反之亦然。

3. Virtual Earth Concept | 虚地概念

The virtual earth concept is central to understanding the inverting amplifier. Since the non-inverting input is grounded, V₊ = 0 V. If the op-amp is working in its linear region, the output voltage is finite. Because the open-loop gain A_ol is very large, the difference V₊ − V₋ must be very small for the output not to saturate.

虚地概念是理解反相放大器的核心。由于同相输入端接地，V₊ = 0 V。如果运算放大器工作在线性区，输出电压是有限的。由于开环增益 A_ol 非常大，为了使输出不饱和，差值 V₊ − V₋ 必须非常小。

As A_ol → ∞, we can approximate V₊ − V₋ ≈ 0, so V₋ ≈ V₊ = 0 V. The inverting input is therefore held at almost 0 V, even though it is not physically connected to ground. It is called a virtual earth or virtual ground.

当 A_ol → ∞ 时，我们可以近似认为 V₊ − V₋ ≈ 0，因此 V₋ ≈ V₊ = 0 V。于是反相输入端几乎保持在 0 V，尽管它并未实际接地。这被称为虚地或虚拟接地。

This means that the inverting input behaves as a fixed 0 V point for signal currents, but no current actually flows into the op-amp input itself because of the infinite input impedance.

这意味着对信号电流而言，反相输入端表现为一个固定的 0 V 点，但实际上由于输入阻抗无穷大，没有电流流入运算放大器输入端本身。

4. Deriving the Gain Equation | 增益公式推导

To derive the voltage gain, apply Kirchhoff’s current law at the inverting input node. Since the op-amp input impedance is infinite, no current enters the inverting input. Therefore the current through R_in must equal the current through R_f.

为了推导电压增益，在反相输入节点应用基尔霍夫电流定律。由于运算放大器输入阻抗无穷大，没有电流进入反相输入端。因此流过 R_in 的电流必定等于流过 R_f 的电流。

The input current is given by:

输入电流由下式给出：

I_in = V_in / R_in

Since the inverting input is at virtual earth (0 V), the output voltage is effectively applied across R_f, but in the opposite direction. The feedback current is:

由于反相输入端处于虚地 (0 V)，输出电压实际上加在 R_f 两端，但方向相反。反馈电流为：

I_f = (0 − V_out) / R_f = −V_out / R_f

Equating the two currents gives:

令两个电流相等，得到：

V_in / R_in = −V_out / R_f

Rearranging gives the closed-loop voltage gain:

整理后得到闭环电压增益：

A_v = V_out / V_in = −R_f / R_in

This is the key formula for the inverting amplifier. The gain depends only on the ratio of the two resistors and is independent of the op-amp’s own open-loop gain, provided the op-amp is ideal and not saturated.

这是反相放大器的关键公式。增益仅取决于两个电阻的比值，并且与运算放大器自身的开环增益无关，前提是运算放大器为理想且未饱和。

5. Sign and Magnitude of Gain | 增益的符号与大小

The minus sign in the gain equation indicates that the output is inverted relative to the input. A positive input voltage produces a negative output voltage, and a negative input produces a positive output. This corresponds to a 180° phase shift for sinusoidal signals.

增益公式中的负号表示输出相对于输入是反相的。正输入电压产生负输出电压，负输入产生正输出。对于正弦信号，这相当于 180° 的相位偏移。

The magnitude of the gain is determined by the ratio R_f / R_in. If R_f > R_in, the magnitude of the gain is greater than 1 and the circuit amplifies the input. If R_f < R_in, the magnitude is less than 1 and the circuit attenuates the input. If R_f = R_in, the gain is exactly −1, producing a unity-gain inverter.

增益的大小由比值 R_f / R_in 决定。如果 R_f > R_in，增益的幅度大于 1，电路对输入进行放大。如果 R_f < R_in，幅度小于 1，电路对输入进行衰减。如果 R_f = R_in，增益恰好为 −1，构成单位增益反相器。

For example, if R_in = 10 kΩ and R_f = 50 kΩ, then A_v = −50 kΩ / 10 kΩ = −5. An input of +0.2 V would give an output of −1.0 V.

例如，如果 R_in = 10 kΩ，R_f = 50 kΩ，则 A_v = −50 kΩ / 10 kΩ = −5。若输入为 +0.2 V，输出将为 −1.0 V。

6. Input and Output Resistance | 输入与输出电阻

The input resistance of the inverting amplifier is approximately equal to R_in. This is because the inverting terminal is at virtual earth, so the signal source sees R_in connected between the input terminal and ground. The input resistance is therefore finite and usually lower than that of a non-inverting amplifier.

反相放大器的输入电阻约等于 R_in。这是因为反相端处于虚地，信号源看到的是连接在输入端与地之间的 R_in。因此输入电阻是有限的，通常低于同相放大器的输入电阻。

The output resistance of the inverting amplifier is very low because of negative feedback. An ideal op-amp has zero output resistance, and with feedback the closed-loop output resistance is reduced further. This allows the amplifier to drive connected loads without a significant drop in output voltage.

由于负反馈，反相放大器的输出电阻非常低。理想运算放大器具有零输出电阻，加入反馈后闭环输出电阻进一步降低。这使得放大器能够驱动连接的负载而不会造成明显的输出电压下降。

In exam answers, it is useful to state that the input resistance is approximately R_in and the output resistance is approximately zero under ideal conditions.

在考试答案中，可以说明在理想条件下输入电阻约为 R_in，输出电阻约等于零。

7. Bandwidth and Frequency Response | 带宽与频率响应

An ideal op-amp has infinite bandwidth, meaning it can amplify all frequencies equally. Real op-amps, however, have a finite gain-bandwidth product, often abbreviated as GBW. This means the maximum possible gain decreases as frequency increases.

理想运算放大器具有无穷大的带宽，也就是说它可以同等地放大所有频率。然而实际运算放大器的增益带宽积是有限的，通常缩写为 GBW。这意味着随着频率升高，最大可能的增益会下降。

For an inverting amplifier, the closed-loop bandwidth can be estimated by:

对于反相放大器，闭环带宽可以估算为：

Bandwidth ≈ GBW / |A_cl|

where A_cl is the closed-loop gain. Therefore a larger gain leads to a smaller bandwidth. If the gain is −10 and the GBW is 1 MHz, the bandwidth is about 100 kHz.

其中 A_cl 是闭环增益。因此增益越大，带宽越小。如果增益为 −10，GBW 为 1 MHz，则带宽约为 100 kHz。

At A-Level, you do not normally need to calculate bandwidth in detail, but you should understand that the ideal infinite-bandwidth assumption is only valid for moderate frequencies.

在 A-Level 阶段，通常不需要详细计算带宽，但应了解理想无穷带宽的假设只适用于中等频率范围。

8. Saturation and Output Limits | 饱和与输出限制

The output voltage of a real op-amp cannot exceed its supply voltages. If the op-amp is powered by ±15 V rails, the output can only swing between approximately +13 V and −13 V, depending on the device. When the required output voltage reaches these limits, the op-amp is said to saturate.

实际运算放大器的输出电压不能超过其电源电压。如果运算放大器由 ±15 V 电源供电，输出只能在约 +13 V 到 −13 V 之间摆动，具体取决于器件。当所需的输出电压达到这些极限时，运算放大器就称为饱和。

For an inverting amplifier, saturation occurs when:

对于反相放大器，当以下条件满足时发生饱和：

|V_out| = |−R_f / R_in × V_in| ≥ V_sat

where V_sat is the saturation voltage, typically slightly less than the supply voltage. Beyond this point the output cannot follow the linear gain equation and the waveform becomes clipped.

其中 V_sat 是饱和电压，通常略低于电源电压。超过这一点后，输出无法继续遵循线性增益公式，波形将被削波。

For example, if R_f / R_in = 10 and V_sat = 13 V, then any input voltage larger than ±1.3 V will cause saturation. This limits the useful input range of the amplifier.

例如，如果 R_f / R_in = 10，且 V_sat = 13 V，那么任何大于 ±1.3 V 的输入电压都会引起饱和。这限制了放大器有用的输入范围。

9. Summing Amplifier Extension | 加法放大器扩展

The inverting amplifier can be extended to sum several input voltages. If multiple input resistors R₁, R₂, R₃ … are connected to the virtual earth point at the inverting input, each input contributes a current V_n / R_n. Because no current enters the op-amp, the feedback current is the sum of all input currents.

反相放大器可以扩展为对多个输入电压求和。如果多个输入电阻 R₁、R₂、R₃ … 连接到反相输入端的虚地点，每个输入都会贡献一个电流 V_n / R_n。由于没有电流进入运算放大器，反馈电流等于所有输入电流之和。

This gives the output:

这样得到的输出为：

V_out = −R_f × (V₁/R₁ + V₂/R₂ + V₃/R₃ + …)

If all input resistors are equal to R_f, the circuit becomes a summing inverter with unity weighting. This is useful in audio mixers and analogue computers. The summing amplifier is not always required by every exam board, but it is a natural extension of the inverting configuration and reinforces the virtual earth principle.

如果所有输入电阻都等于 R_f，该电路就成为一个具有单位权重的求和反相器。这在音频混音器和模拟计算机中很有用。加法放大器不一定是每个考试局都要求的内容，但它是反相配置的自然扩展，有助于巩固虚地原理。

10. Worked Example | 例题解析

A student builds an inverting amplifier with R_in = 10 kΩ and R_f = 40 kΩ. The op-amp supply voltages are ±12 V, and the saturation voltage is measured as ±10 V.

一位学生搭建了一个反相放大器，其中 R_in = 10 kΩ，R_f = 40 kΩ。运算放大器的电源电压为 ±12 V，测得的饱和电压为 ±10 V。

(a) Calculate the voltage gain.

(a) 计算电压增益。

A_v = −R_f / R_in = −40 kΩ / 10 kΩ = −4

(b) Find the output voltage when the input is +0.8 V.

(b) 当输入为 +0.8 V 时，求输出电压。

V_out = A_v × V_in = −4 × 0.8 V = −3.2 V

V_in(max) = V_sat / |A_v| = 10 V / 4 = 2.5 V

Therefore the input must stay within the range −2.5 V to +2.5 V for linear operation.

因此，输入必须保持在 −2

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The Inverting Amplifier | 反相放大器