## 3 Principle of the Weak Signal Detection Circuit  
The lock-in amplifier principle uses the frequency difference between the output signal and the noise signal to correlate the target signal, thereby achieving detection of weak signals [20]. The eddy current detection coil's output detection signal \(S(t)\) consists of the effective signal \(E(t)\) and the noise signal \(n(t)\):  
\[E(t)=V_{s}sin(\omega_{s}t+\phi_{s}) \tag{3}\]  
Figure 2: Single-turn current loop. where \(V_{s}\) is the effective signal amplitude, \(\omega_{s}\) is the effective signal frequency, \(\phi_{s}\) is the effective signal phase angle; therefore the output signal is Equation (4) and the reference signal is Equation (5):  
\[S(t)=E(t)+n(t) \tag{4}\]  
\[R(t)=V_{r}sin(\omega_{r}t+\phi_{r}) \tag{5}\]  
where \(V_{r}\) is the amplitude of the reference signal, \(\omega_{r}\) is the frequency of the reference signal, \(\phi_{r}\) is the phase angle of the reference signal, and the output signal obtained by performing a multiplication operation between \(S(t)\) and \(R(t)\) is denoted \(V_{psd}(t)\):  
\[ \begin{array}{l}V_{psd}(t)=\frac{V_{s}V_{r}}{2}cos[(\omega_{s}-\omega_{r})t +(\phi_{s}-\phi_{r})]-\frac{V_{s}V_{r}}{2}cos[(\omega_{s}+\omega_{r})t+(\phi_ {s}+\phi_{r})]\\ +V_{r}n(t)sin(\omega_{r}t+\phi_{r})\end{array}  \tag{6}\]  
The first two items are valid signals, the latter is a signal with noise effects, the frequency of the valid signal \(\omega_{s}\) is equal to the frequency of the excitation signal \(\omega\), and the frequency of the reference signal \(\omega_{r}\) is set to \(\omega\); thus, Equation (6) can be simplified to Equation (7):  
\[V_{psd}(t)=\frac{V_{s}V_{r}}{2}cos(\phi_{s}-\phi_{r})-\frac{V_{s}V_{r}}{2}cos (2\omega t+\phi_{s}+\phi_{r})+V_{r}n(t)sin(\omega t+\phi_{r}) \tag{7}\]  
From Equation (7), it can be obtained that the signal after phase-sensitive detection is divided into three terms: the first term is the DC component, and the second and third terms are the high-frequency AC components. The output signal \(V_{out}\) is obtained after low-pass filtering of \(V_{psd}(t)\):  
\[V_{out}=\frac{V_{s}V_{r}}{2}cos(\phi_{s}-\phi_{r}) \tag{8}\]  
From Equation (8), it can be seen that the magnitude of \(V_{out}\) is related to the amplitude and phase of \(E(t)\), which are proportional to the amplitude and cosine of the phase difference between the effective signal and the reference signal. To mitigate large phase differences caused by discrepancies in output signal amplitudes, we employ Equation (9) for the reference signal \(R^{\prime}(t)\) and perform aforementioned processing on both the reference and output signals, resulting in \(V^{\prime}_{out}\) depicted in Equation (10):  
\[R^{\prime}(t)=V_{r}sin(\omega_{r}t+\phi_{r}+\frac{\pi}{2}) \tag{9}\]  
\[V^{\prime}_{out}=\frac{V_{s}V_{r}}{2}sin(\phi_{s}-\phi_{r}) \tag{10}\]  
where \(V_{out}\) is the in-phase component and \(V^{\prime}_{out}\) is called the quadrature component. Therefore, the amplitude Equation (11) and the phase difference Equation (12) are obtained. \[V_{S}=\frac{2\sqrt{(V^{2}_{out}+V^{\prime 2}_{out})}}{V_{r}} \tag{11}\]  
\[\Delta\phi=\phi_{s}-\phi_{r}=tan^{-1}(\frac{V^{\prime}_{out}}{V_{out}}) \tag{12}\]  
A lock-in amplifier consists of a target signal channel, a reference signal channel, and a correlator channel. From Equation (8) it can be seen that, in a single lock-in amplifier, the final output signal can be controlled by adjusting the phase of the reference signal and thus the value of the phase difference [21]. In studying the principle of lock-in amplification, it is noted that a single lock-in amplifier requires precise adjustment of the phase. In order to realize efficient processing of the weak signals from planar coil sensors, a circuit structure for the dual lock-in amplifier was adopted in this paper. As shown in Figure 3, two phase shift detectors (PSDs) were introduced, each of which used a reference signal with a phase difference of 90\({}^{\circ}\) to multiply with the target signal.