# Error Correction Method of TIADC System Based on Parameter Estimation of Identification Model
## 2 Research on the Calibration Method
### Determination and Analysis of the Model Structure  
Figure 3 shows the flow chart for the TIADC system to establish an identification model, including experimental design, data processing, system analysis, determination of model structure, parameter estimation, and model verification. First, through the analysis of the TIADC system, a model structure is established, and a reasonable experiment is designed according to the prior knowledge and motion law of the system to ensure that the identification can achieve the desired effect. The data used for identification need to be processed in advance to improve the accuracy of identification and the availability of the identification model. After establishing the structure of the model, the values of some parameters in the model are unknown, so it is necessary to estimate the unknown parameters according to the input and output data. The model obtained by parameter estimation must be verified by the model to test its effectiveness until the performance of the identified model is close to that of the actual system. In the TIADC system, if a channel mismatch is not considered, the ADC modules of each channel can be reasonably equivalent to the first-order RC circuit. After experimental  
Figure 3: Process of establishing the TIADC system identification model. Figure 2: Working Mechanism of TIADC Error Correction System Based on Parameter Estimation of Identification Model. verification, the analog filter in the conditioning module can be realized using the second-order RLC circuit. The equivalent model of a single sub-channel in the TIADC system in an ideal state is shown in Figure 4.  
First, according to [PERSON]'s law, the differential equation of the equivalent model circuit of a single sub-channel in a TIADC system under an ideal state is as follows:  
\[\left\{ \begin{array}{l}LC_{1}\frac{d^{2}U_{1}}{dt^{2}}+R_{1}C_{1}\frac{dU_{1 }}{dt}+U_{o1}=U_{i}\\ R_{2}C_{2}\frac{dU_{1}}{dt}+U_{o}=U_{o1}\end{array} \right. \tag{1}\]  
Among them, \(U_{i}\) represents the voltage of the analog input signal, \(U_{o1}\) represents the output voltage of the analog input signal after filtering circuit, \(U_{o}\) represents the output voltage of \(U_{o1}\) after ADC sampling and keeping the front end. After Laplace transform of the equivalent model circuit differential equation, it can be transformed into the following form:  
\[\left\{ \begin{array}{l}\left(LC_{1}s^{2}+R_{1}C_{1}s+1\right)U_{o1}(s)=U_{i }(s)\\ \left(R_{2}C_{2}s+1\right)U_{o}(s)=U_{o1}(s)\end{array} \right. \tag{2}\]  
From Equation (2), the transfer function equations of the equivalent model of a single sub-channel in the TIADC system under an ideal state can be obtained as follows:  
\[\left\{ \begin{array}{l}G_{1}(s)=\frac{U_{d1}(s)}{U_{i}(s)}=\frac{1}{LC_{1}s ^{2}+R_{1}C_{1}s+1}\\ G_{2}(s)=\frac{U_{o}(s)}{U_{d1}(s)}=\frac{1}{R_{2}C_{2}s+1}\end{array} \right. \tag{3}\]  
Considering the actual factors, the channel mismatch errors, such as DC bias \(\Delta o\), static gain \(\Delta g\) and aperture delay \(\Delta t\) of each channel are introduced into the ADC equivalent model, as shown in Figure 5.  
Let \(\omega_{n}=1/\sqrt{LC_{1}}\), \(\xi=\frac{R}{2}\sqrt{C_{1}/L}\), \(\omega_{c}=1/R_{2}C_{2}\), where \(\omega_{n}\) denotes the undamped oscillation frequency of the system, \(\xi\) denotes the damping ratio of the system, \(\omega_{c}\) denotes the cutoff frequency of the system, and \(e^{\Delta t}\) denotes the time delay processes of the system. The values of \(\omega_{n}\), \(\xi\), \(\omega_{c}\) are related to the actual hardware circuit. Then, the transfer function of a single sub-channel in the TIADC system identification model can be expressed as  
\[G(s)=\frac{\omega_{n}^{2}}{s^{2}+2\xi\omega_{n}s+\omega_{n}^{2}}\cdot\frac{(1+ \Delta g)e^{\beta\Delta t}}{s/\omega_{c}+1}+\Delta o \tag{4}\]  
Figure 4: Equivalent model of a single sub-channel in a TIADC system under an ideal state.