📚 The Operational Amplifier (Op-Amp) | 运算放大器
The operational amplifier, or op-amp, is a high-gain voltage amplifier with two inputs and one output. It is one of the most versatile building blocks in analogue electronics and appears in amplifiers, filters, comparators and many sensor circuits.
运算放大器（简称运放）是一种高增益电压放大器，具有两个输入端和一个输出端。它是模拟电子学中最通用的基本模块之一，广泛应用于放大器、滤波器、比较器和许多传感器电路中。
1. What is an Operational Amplifier? | 什么是运算放大器？
An op-amp is usually drawn as a triangular symbol with two input terminals and one output terminal. The input marked with a plus sign is the non-inverting input, while the input marked with a minus sign is the inverting input. Two power supply connections, often labelled +Vs and -Vs, provide the energy needed for the output voltage swing.
运放通常画成三角形符号，带有两个输入端和一个输出端。标有加号的输入端为同相输入端，标有减号的输入端为反相输入端。两个电源连接端通常标为 +Vs 和 -Vs，为输出电压摆幅提供能量。
The output voltage of an ideal op-amp is proportional to the difference between the voltages at the non-inverting and inverting inputs. The word ‘operational’ originally referred to the use of these amplifiers in analogue computers to perform mathematical operations such as addition, integration and differentiation.
理想运放的输出电压与同相输入端和反相输入端的电压之差成正比。’运算’一词最初指这些放大器在模拟计算机中用于执行加法、积分和微分等数学运算。
Vout = Aol(V+ − V−)
Aol represents the open-loop voltage gain, which is extremely large in an ideal op-amp. This equation is the starting point for understanding both linear amplifiers and comparators.
Aol 表示开环电压增益，在理想运放中该值极大。这个方程是理解线性放大器和比较器的出发点。
2. Ideal Op-Amp Characteristics | 理想运算放大器特性
For analysis, CIE assumes an ideal op-amp having the following properties: infinite open-loop voltage gain, infinite input impedance, zero output impedance, infinite bandwidth and zero offset voltage. No real op-amp meets these exactly, but the ideal model gives accurate predictions for many circuits using negative feedback.
在分析中，CIE 假设理想运放具有以下特性：无限开环电压增益、无限输入阻抗、零输出阻抗、无限带宽和零失调电压。实际运放不能完全达到这些指标，但在采用负反馈的许多电路中，理想模型能给出准确的预测。
- Infinite open-loop gain | 无限开环增益: Aol → so the input difference is forced to nearly zero in feedback circuits.
- Infinite input impedance | 无限输入阻抗: no current flows into either input terminal.
- Zero output impedance | 零输出阻抗: the output voltage remains constant when the load current changes.
- Infinite bandwidth | 无限带宽: the gain is independent of frequency.
These ideal assumptions simplify circuit calculations. In many CIE questions, you can use them unless the question specifically asks about practical limitations.
这些理想假设简化了电路计算。在许多 CIE 考题中，除非题目明确要求考虑实际局限，否则你都可以使用这些理想条件。
3. Open-Loop Behaviour and Saturation | 开环行为与饱和
When no feedback is connected, the op-amp operates in open-loop mode. Because the open-loop gain is extremely large, even a tiny difference between V+ and V− drives the output to its maximum or minimum possible voltage. This is called saturation.
当没有连接反馈时，运放工作在开环模式。由于开环增益非常大，即使 V+ 与 V− 之间只有极小的差值，也会使输出达到其最大或最小可能电压。这称为饱和。
The saturated output voltages are usually slightly less than the supply voltages. These limits are denoted +Vsat and -Vsat. For example, with ±15 V supplies, the saturation voltages may be around ±13 V.
饱和输出电压通常略小于电源电压。这些限值记为 +Vsat 和 -Vsat。例如，使用 ±15 V 电源时，饱和电压可能约为 ±13 V。
If V+ > V− then Vout = +Vsat; if V+ < V− then Vout = −Vsat
This open-loop behaviour is deliberately used in comparator circuits, where a rapid switch between two output levels is required. It is also the reason why negative feedback is needed for linear amplification.
这种开环行为被有意用于比较器电路，因为比较器需要在两个输出电平之间快速切换。这也是线性放大需要负反馈的原因。
4. Negative Feedback | 负反馈
Negative feedback means feeding a portion of the output signal back to the inverting input. This reduces the overall gain but greatly improves stability, bandwidth and linearity. Most amplifier circuits use negative feedback to give a predictable gain set by external resistors.
负反馈是指将输出信号的一部分送回到反相输入端。这会降低总增益，但显著改善稳定性、带宽和线性度。大多数放大器电路利用负反馈来获得由外部电阻决定的稳定增益。
With negative feedback and a large open-loop gain, the op-amp adjusts its output to make the voltage difference between the two inputs almost zero. This leads to two useful rules for ideal op-amp analysis:
在负反馈和大开环增益下，运放会调整输出，使两个输入端之间的电压差几乎为零。由此得到理想运放分析的两条有用规则：
- Virtual short rule | 虚短规则: V+ = V− because the output forces the difference toward zero.
- Zero input current rule | 零输入电流规则: I+ = I− = 0 because the input impedance is infinite.
These two rules allow you to analyse most ideal op-amp circuits quickly. The virtual short rule applies only when negative feedback is present and the op-amp is not saturated.
这两条规则使你能快速分析大多数理想运放电路。虚短规则仅在存在负反馈且运放未饱和时适用。
5. The Inverting Amplifier | 反相放大器
In the inverting amplifier, the input signal is applied through an input resistor Rin to the inverting input. A feedback resistor Rf connects the output back to the inverting input, while the non-inverting input is connected to 0 V. Because V+ =
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