## 2 Theory and Simulation of XS Anode
### Crosstalk Effect of the Inter-Electrode Capacitance  
The main factor that affects the resolution of the detector is electronic noise. Electronic noise is mainly generated by charge-sensitive amplifiers and signal input capacitors. For single-layer XS anodes, there will be capacitive coupling between strips, which interferes with adjacent strip signals and reduces strip signals, thereby reducing the signal-to-noise ratio and thus affecting the resolution of the XS anode.  
Total capacitive load on the amplifier input:  
\[C_{in}=C_{i}+C_{ie}+C_{a}+(1+A)\cdot C_{f}+C_{p} \tag{3}\]  
In the formula, \(C_{i}\) is the strip capacitance, \(C_{ie}\) is the capacitance between different strips, \(C_{a}\) is the amplifier input capacitance, \(C_{f}\) is the feedback capacitance, \(C_{p}\) is parasitic capacitance, and \(A\) is the gain of the open circuit charge sensitive amplifier. Generally speaking, the feedback capacitance is very small (only a few pf), and the capacitance \(C_{ie}\) between the strips will affect the collected signal. The greater the crosstalk, the less accurate the signal will be, thus reducing the resolution.  
\[N=N_{0}+N_{c}C_{in} \tag{4}\]  
Figure 3: Anode model. (**a**) XS anode equivalent model, (**b**) XS anode minimum unit.  
\(N_{0}\) is the preamplifier noise without capacitive load, and \(N_{c}\) is the average electronic noise. By calculating the inter-electrode capacitance, the electronic noise-limited spatial resolution of the XS anode detector can be estimated.  
\[Q_{m}=Q-Q_{i}=Q-2 VC_{i} \tag{5}\]  
\[Q=2 VC_{i}+C_{dyn}V \tag{6}\]  
\[V=\frac{Q}{2C_{I}+C_{dyn}} \tag{7}\]  
\[Q_{m}=Q-\frac{2 QC_{i}}{2C_{I}+C_{dyn}} \tag{8}\]  
\(C_{dyn}\) is considered to be the dynamic capacitance of the charge-sensitive amplifier. From Formula (8), it can be known that when \(C_{dyn}>>C_{i}\), the capacitance of the external electrode to ground can be approximated as \(C_{i}\). The charge \(Q_{m}\) measured by the amplifier is equal to the total charge \(Q\) minus the charge \(Q_{i}\) lost to the outside of the strips. It can, thus, be seen that a large dynamic capacitance will ensure minimum charge loss, yielding maximum signal-to-noise ratio, thereby maximizing the efficiency of the electronic device and improving resolution.  
For an ideal XS anode, the relationship between capacitance \(C\), charge \(Q\), and potential \(V\) is  
\[Q=C_{mutual}V,Q=\begin{bmatrix}Q_{1}\\ \vdots\\ Q_{n}\end{bmatrix},C_{mutual}=\begin{bmatrix}C_{11}&\cdots&C_{1n}\\ \vdots&\ddots&\vdots\\ C_{n1}&\cdots&C_{m}\end{bmatrix},\;V=\begin{bmatrix}V_{1}\\ \vdots\\ V_{n}\end{bmatrix} \tag{9}\]  
However, high efficiency signal transmission is also very important in practical applications. Fundamentally speaking, analyzing and solving the crosstalk problem generated during signal transmission is essentially a problem of solving [PERSON]'s equations under certain boundaries and excitation conditions. The expression for the charge on the strips can be written as the Maxwell capacitance matrix:  
\[Q^{\prime}=C_{maxwell}V,Q^{\prime}=\begin{bmatrix}Q^{\prime}_{1}\\ \vdots\\ Q^{\prime}_{n}\end{bmatrix},C_{maxwell}=\begin{bmatrix}\sum\limits_{i=1}^{n}C _{1i}&\cdots&-C_{1n}\\ \vdots&\ddots&\vdots\\ -C_{n1}&\cdots&\sum\limits_{i=1}^{n}C_{ni}\end{bmatrix} \tag{10}\]  
In the formula, \(C_{ii}\) is the capacitance of i horizontal strips and 32 vertical strips to the vacuum chamber, \(C_{ij}\) is the capacitance between band i horizontal strips and j vertical strips, and \(C_{ij}=C_{ji}\). Substituting Formula (10) into Formula (9), the relationship between ideal charge and actual charge can be obtained, and the charge can be corrected to eliminate crosstalk.  
\[Q^{\prime}=C_{maxwell}C^{-1}_{mutual}Q \tag{11}\]