## 2 Low-Offset Operational Amplifier
### Auto-Zeroing  
Auto-zeroing technology is a sample and hold technique that can dynamically suppress the offset voltage, noise, and temperature drift in circuits. Automatic zeroing technology stores the sampled DC offset and noise in the sampling capacitor through sampling, and then subtracts it from the signal carrying DC offset and noise, ultimately achieving the goal of suppressing DC offset and noise. The basic principle of automatic zeroing is shown in Figure 4, which is controlled by a digital clock to sample switches S\({}_{1}\)-S\({}_{6}\), and capacitors \(C_{1}\) and \(C_{2}\) are built around an operational amplifier g\({}_{\rm m}\).  
Figure 4: Auto-zeroing diagram.  
Figure 3: Chopper amplifier with SC notch filter.  
Clocks \(\varphi_{1}\) and \(\varphi_{2}\) are complementary clocks, and the method of eliminating misalignment is usually carried out in two stages. The first stage is detection and storage. When the first clock phase \(\varphi_{1}\) is at a high level, switches S\({}_{1}\) and S\({}_{2}\) are open, switches S\({}_{3}\)-S\({}_{6}\) are closed, and g\({}_{\rm m}\) is connected in a unity gain configuration. The offset voltage of the amplifier is stored in capacitor \(C_{1}\), as shown in (1).  
\[V_{A}(f)=\frac{A}{1+A}V_{OS} \tag{1}\]  
The second stage is the amplification stage, in which the next clock phase \(\varphi_{2}\) is at a high level. The amplifier g\({}_{\rm m}\) amplifies the input signal, which is then amplified by capacitors \(C_{1}\) and \(C_{2}\) before being further amplified by the amplifier. As shown in (2), the offset voltage \(V_{\rm os}\) cancels out the voltage stored in capacitor \(C_{1}\). At this time, the input offset voltage of the amplifier is shown in (3), and the output only changes with the input differential voltage, not affected by the input offset voltage, thereby eliminating the offset voltage. Similarly, low-frequency flicker noise components are also stored on capacitors \(C_{1}\) and \(C_{2}\), and therefore will be canceled out. However, the correlation between higher-frequency flicker noise components is small, so it cannot be effectively canceled out.  
\[(V_{in}+\frac{A}{1+A}V_{OS}-V_{OS})A=-V_{out} \tag{2}\]  
\[V_{OS\_in}=-\frac{A}{1+A}V_{OS} \tag{3}\]  
Additionally, according to the sampling theorem, the sampling process can induce high-frequency thermal noise outside the desired bandwidth to mix with in-band noise. This interaction increases the noise power spectral density (PSD) within the band, resulting in a higher in-band PSD compared to the original noise floor. The noise suppression effect of automatic zeroing technology is illustrated in Figure 5. As shown, under automatic zeroing conditions, low-frequency noise is primarily flicker noise, while in the absence of automatic zeroing, low-frequency noise is predominantly white noise. However, noise aliasing leads to an increase in the spectral density of baseband noise.  
Automatic zeroing technology offers a larger bandwidth and smaller ripple in the output voltage but comes with increased current consumption and a higher susceptibility to signal superposition, which results in elevated low-frequency noise. In contrast, chopping technology is less prone to signal superposition, but it has a narrower bandwidth, higher energy consumption, and significant output ripple at the chopping frequency. However, chopping technology generally requires lower power consumption. A comparison between the two technologies is presented in Table 1.  
Figure 5: Noise spectrum before and after auto-zeroing.