📚 The Non-Inverting Amplifier | 同相放大器
In the study of operational amplifiers (op-amps), the non-inverting amplifier is one of the most fundamental configurations. It provides voltage amplification while preserving the phase (sign) of the input signal. Understanding its behaviour, gain derivation and practical limitations is essential for A-Level Physics and electronics.
在运算放大器（运放）的学习中，同相放大器是最基本的配置之一。它能够放大电压，同时保持输入信号的相位（符号）不变。理解其行为、增益推导和实际限制对于A-Level物理和电子学至关重要。
1. Introduction to Operational Amplifiers | 运算放大器简介
An operational amplifier is a high-gain differential voltage amplifier. It has two inputs: the inverting input (–) and the non-inverting input (+), and a single output. In open-loop mode, its voltage gain is extremely high (typically >10⁵). To make it useful for linear amplification, negative feedback is applied.
运算放大器是一种高增益差分电压放大器。它有两个输入端：反相输入端（–）和同相输入端（+），以及一个输出端。在开环模式下，其电压增益极高（通常大于10⁵）。为了使其能够用于线性放大，需要施加负反馈。
2. The Non-Inverting Amplifier Circuit | 同相放大器电路
The non-inverting amplifier configuration is shown below. The input signal Vin is applied directly to the non-inverting (+) input. A feedback network consisting of two resistors, R1 and Rf, connects the output back to the inverting (–) input. One end of R1 is tied to ground (0 V), while Rf is connected between the output and the inverting terminal. This arrangement produces a closed-loop gain greater than or equal to unity.
同相放大器电路结构如下：输入信号Vin直接加在同相输入端（+）。由两个电阻R1和Rf组成的反馈网络将输出端连接回反相输入端（–）。R1的一端接地（0 V），Rf连接在输出端和反相输入端之间。这种结构产生大于或等于1的闭环增益。
3. Ideal Op-Amp Assumptions | 理想运放假设
For circuit analysis, we assume an ideal op-amp with the following properties: infinite open-loop voltage gain AOL, infinite input impedance, zero output impedance, infinite bandwidth, infinite slew rate, and zero input offset voltage. These assumptions simplify the derivation of the closed-loop gain and other characteristics.
在电路分析中，我们假设运放是理想的，具有以下特性：无穷大开环电压增益AOL、无穷大输入阻抗、零输出阻抗、无穷大带宽、无穷大转换速率以及零输入失调电压。这些假设简化了闭环增益和其他特性的推导。
4. Virtual Short Concept | 虚拟短路概念
Because of negative feedback combined with the enormous open-loop gain, the voltage difference between the two input terminals, V+ – V–, is forced to be almost zero. Hence V+ ≈ V–. Since the non-inverting input receives Vin, we have V– = Vin. This is the 'virtual short' – a short circuit for voltage but not for current, as the input impedance is extremely high.
由于负反馈与极大的开环增益相结合，两个输入端之间的电压差V+ – V–被强制几乎为零。因此V+ ≈ V–。因为同相输入端接收Vin，所以V– = Vin。这就是“虚拟短路”——对电压而言如同短路，但对电流并非短路，因为输入阻抗极高。
5. Deriving the Closed-Loop Gain | 推导闭环增益
Since the inverting terminal draws negligible current, the resistors R1 and Rf form a simple potential divider from Vout to ground. The voltage at the inverting terminal is the voltage across R1: V– = [R1 / (R1 + Rf)] × Vout.
由于反相输入端吸取的电流可忽略不计，电阻R1和Rf形成从Vout到地的简单分压器。反相输入端的电压即为R1两端的电压：V– = [R1 / (R1 + Rf)] × Vout。
Using the virtual short condition V– = Vin, we obtain Vin = [R1/(R1+Rf)] Vout. Rearranging gives the closed-loop voltage gain AV = Vout/Vin = (R1+Rf)/R1 = 1 + Rf/R1. This is the fundamental gain equation for the non-inverting amplifier.
利用虚拟短路条件V– = Vin，得到Vin = [R1/(R1+Rf)] Vout。整理后得到闭环电压增益AV = Vout/Vin = (R1+Rf)/R1 = 1 + Rf/R1。这就是同相放大器的基本增益方程。
AV = 1 + (Rf / R1)
6. Input and Output Impedance | 输入输出阻抗
The input impedance of the non-inverting amplifier is ideally infinite because the signal is fed directly into the op-amp's high-impedance non-inverting terminal. In practice, it is limited by the op-amp's internal input resistance but remains very high (typically megaohms). The output impedance is very low (ideally zero) because negative feedback reduces the effective output resistance by a factor of (1 + AOLβ), where β = R1/(R1+Rf).
同相放大器的输入阻抗在理想情况下为无穷大，因为信号直接馈入运放的高阻抗同相输入端。实践中，它受到运放内部输入电阻的限制，但仍然非常高（通常为兆欧级）。输出阻抗非常低（理想情况下为零），因为负反馈将有效输出电阻降低了(1 + AOLβ)倍，其中β = R1/(R1+Rf)。
7. Bandwidth and Gain–Bandwidth Product | 带宽与增益带宽积
Real op-amps have a finite gain– a non-inverting amplifier, the closed-loop bandwidth BW is related to GBP by BW = GBP / AV. Therefore, a higher voltage gain results in a proportionally lower bandwidth. This trade-off is important when selecting resistor values for high-frequency applications.
实际运放具有有限的增益带宽积（GBP）。对于同相放大器，闭环带宽BW与GBP的关系为 BW = GBP / AV。因此，电压增益越高，带宽按比例减小。这一权衡在为高频应用选择电阻值时非常重要。
8. Voltage Follower – A Special Case | 电压跟随器 – 特例
When Rf = 0 (short circuit) and R1 is removed (open circuit), the gain equation reduces to AV ollower or buffer. It provides unity gain with extremely high input impedance and very low output impedance, making it ideal for impedance matching between a high-impedance source and a low-impedance load.
当Rf = 0（短路）且R1被移除（开路）时，增益方程简化为AV = 1。这种电路称为电压跟随器或缓冲器。它提供单位增益，同时具有极高的输入阻抗和非常低的输出阻抗，非常适合高阻抗源与低阻抗负载之间的阻抗匹配。
AV = 1
9. Effect of Negative Feedback | 负反馈的影响
Negative feedback in the non-inverting amplifier stabilises the voltage gain against variations in op-amp parameters, reduces harmonic distortion, extends the bandwidth, and modifies input and output impedances. The feedback topology used is series–shunt (voltage–series), where the feedback voltage is applied in series with the input and sampled from the output voltage.
同相放大器中的负反馈稳定了电压增益，使其不受运放参数变化的影响，降低了谐波失真，扩展了带宽，并改变了输入和输出阻抗。所采用的反馈拓扑为串联–并联（电压–串联）类型，反馈电压与输入串联，并从输出电压取样。
10. Practical Limitations | 实际限制
While the ideal model is useful, real op-amps exhibit several non-ideal behaviours. The output voltage cannot exceed the supply rails – this is called saturation and limits the maximum output swing. The slew rate limits how fast the output voltage can change, causing distortion at high frequencies. Input offset voltage and bias currents produce small errors at the output, and finite open-loop gain and finite common-mode rejection ratio (CMRR) cause deviations from the ideal gain equation.
尽管理想模型很有用，但实际运放表现出许多非理想行为。输出电压不能超过电源轨——这被称为饱和，限制了最大输出摆幅。转换速率限制了输出电压的变化速度，在高频时会造成失真。输入失调电压和偏置电流会在输出端产生微小误差，而有限的开环增益和有限的共模抑制比（CMRR）则会造成与理想增益方程的偏差。
11. Comparison with Inverting Amplifier | 与反相放大器比较
It is instructive to contrast the non-inverting amplifier with the inverting configuration. The table below summarises key differences:
将同相放大器与反相配置进行比较很有意义。下表总结了关键差异：
| Property | Non-Inverting | Inverting |
|---|---|---|
| Voltage gain | 1 + Rf/R1 | – Rf/Rin |
| Input impedance | Very high (ideal ∞) | Equal to Rin (moderate) |
| Phase relationship | 0° (in phase) | 180° (inverted) |
| Minimum gain | 1 (unity) | Any value (can be <1) |
The non-inverting amplifier is preferred when a high input impedance is required, whereas the inverting amplifier can provide gain less than one and has predictable input resistance.
当需要高输入阻抗时，优先选择同相放大器；而反相放大器可以提供小于1的增益，并且具有可预测的输入电阻。
12. Applications of the Non-Inverting Amplifier | 同相放大器的应用
Non-inverting amplifiers are widely used in signal conditioning circuits. Typical applications include: pre-amplifiers for sensors and transducers (e.g. thermocouples, strain gauges), active filters (Sallen–Key topology), audio line drivers, and buffer stages in analogue-to-digital conversion. The voltage follower variant is commonly used to prevent loading effects when measuring voltages in high-impedance circuits.
同相放大器广泛用于信号调理电路。典型应用包括：传感器和换能器前置放大器（如热电偶、应变片）、有源滤波器（Sallen–Key 拓扑）、音频线路驱动器，以及模数转换中的缓冲级。电压跟随器变体常用于在高阻抗电路中测量电压时防止负载效应。
Published by TutorHao | Physics Revision Series | aleveler.com
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