*4.4. Correction of Sampled Value*

The actual sampled voltage value shows a certain deviation from the expected value. The red sampled point in Figure 6 shows that the exponential function curve cannot be formed. The result calculated by the sampled value must be a large error. Therefore, to obtain satisfactory results, the sampled values must be corrected with a loop iterative correction method. *<sup>v</sup>*ˆ11\_1(*i*),*v*ˆ11\_2(*i*),*v*ˆ11\_3(*i*),*v*ˆ12\_1(*i*),*v*ˆ12\_2(*i*),*v*ˆ12\_3(*i*),*v*ˆ21\_1(*i*),*v*ˆ21\_2(*i*),*v*ˆ21\_3(*i*),*v*ˆ22\_1(*i*),*v*ˆ22\_2(*i*), and <sup>e</sup>*v*22\_3(*i*) are set as the *i*th correction voltage values. After the 12 voltages of *v*11, *v*12, *v*21, and *v*<sup>22</sup> are sampled completely, the counter is set as *i* = 0. The 12 voltage values are substituted into the following equation, and the sampled value is used as the initial correction value.

$$\begin{cases} \begin{aligned} \hat{v}\_{11\\_1}(0) = v\_{11\\_1} \\ \hat{v}\_{11\\_2}(0) = v\_{11\\_2} \\ \hat{v}\_{11\\_3}(0) = v\_{11\\_3} \end{aligned} \end{cases} \begin{aligned} \begin{aligned} \hat{v}\_{12\\_1}(0) = v\_{12\\_1} \\ \hat{v}\_{12\\_2}(0) = v\_{12\\_2} \\ \hat{v}\_{12\\_3}(0) = v\_{12\\_3} \end{aligned} \end{cases} \begin{aligned} \hat{v}\_{21\\_1}(0) = v\_{21\\_1} \\ \hat{v}\_{21\\_2}(0) = v\_{21\\_2} \\ \hat{v}\_{21\\_3}(0) = v\_{21\\_3} \end{aligned} \end{cases} \begin{aligned} \hat{v}\_{22\\_1}(0) = v\_{22\\_1} \\ \hat{v}\_{22\\_2}(0) = v\_{22\\_2} \\ \hat{v}\_{22\\_3}(0) = v\_{22\\_3} \end{aligned} \tag{16}$$

**Figure 6.** Actual sampling point.

The three sampled voltages of each group cannot form an exponential curve of *E*ˆ <sup>1</sup>(*k*) due to the measurement error and ripple. To form the desired exponential curve, each voltage value can be estimated by the two other voltage values. <sup>e</sup>*v*11\_1(*i*),e*v*11\_2(*i*),e*v*11\_3(*i*),e*v*12\_1(*i*),e*v*12\_2(*i*),e*v*12\_3(*i*),e*v*21\_1(*i*),e*v*21\_2(*i*),e*v*21\_3(*i*),e*v*22\_1(*i*),e*v*22\_2(*i*), and <sup>e</sup>*v*22\_3(*i*) are set as the estimated voltage values. The estimated method can be derived from the following equation according to Equation (6).

$$\begin{cases} \widetilde{v}\_{11,1}(i) = [\mathfrak{b}\_{11,2}(i) - \mathfrak{d}\_{11,3}(i)]\var{E}\_{1}(k)^{-1} + \mathfrak{d}\_{11,2}(i) \\ \widetilde{v}\_{11,2}(i) = [\mathfrak{b}\_{11,1}(i)\var{E}\_{1}(k) + \mathfrak{d}\_{11,3}(i)]\var{//(1 + \hat{E}\_{1}(k))} \\ \widetilde{v}\_{11,3}(i) = \mathfrak{d}\_{11,2}(i) - [\mathfrak{d}\_{11,1}(i) - \mathfrak{d}\_{11,2}(i)]\var{E}\_{1}(k) \\ \widetilde{v}\_{21,1}(i) = [\mathfrak{b}\_{21,2}(i) - \mathfrak{d}\_{21,3}(i)]\var{E}\_{2}(k)^{-1} + \mathfrak{d}\_{21,2}(i) \\ \widetilde{v}\_{21,2}(i) = [\mathfrak{b}\_{21,1}(i)\var{E}\_{2}(k) + \mathfrak{d}\_{21,3}(i)]\var{//(1 + \hat{E}\_{2}(k))} \\ \widetilde{v}\_{21,2}(i) = [\mathfrak{b}\_{21,1}(i)\var{E}\_{2}(k) + \mathfrak{d}\_{21,3}(i)]\var{//(1 + \hat{E}\_{2}(k))} \\ \widetilde{v}\_{21,3}(i) = \mathfrak{d}\_{21,2}(i) - [\mathfrak{b}\_{21,1}(i) - \mathfrak{d}\_{21,3}(i)]\var{E}\_{2}(k) \end{cases} \tag{17}$$

The estimated value comparison rule is shown as

(  *<sup>v</sup>*ˆ11\_1(*i*) <sup>−</sup> <sup>e</sup>*v*11\_1(*i*) <sup>&</sup>lt; D, *<sup>v</sup>*ˆ11\_2(*i*) <sup>−</sup> <sup>e</sup>*v*11\_2(*i*) <sup>&</sup>lt; D, *<sup>v</sup>*ˆ11\_3(*i*) <sup>−</sup> <sup>e</sup>*v*11\_3(*i*) <sup>&</sup>lt; <sup>D</sup> *<sup>v</sup>*ˆ12\_1(*i*) <sup>−</sup> <sup>e</sup>*v*12\_1(*i*) <sup>&</sup>lt; D, *<sup>v</sup>*ˆ12\_2(*i*) <sup>−</sup> <sup>e</sup>*v*12\_2(*i*) <sup>&</sup>lt; D, *<sup>v</sup>*ˆ12\_3(*i*) <sup>−</sup> <sup>e</sup>*v*12\_3(*i*) <sup>&</sup>lt; <sup>D</sup> (  *<sup>v</sup>*ˆ21\_1(*i*) <sup>−</sup> <sup>e</sup>*v*21\_1(*i*) <sup>&</sup>lt; D, *<sup>v</sup>*ˆ21\_2(*i*) <sup>−</sup> <sup>e</sup>*v*21\_2(*i*) <sup>&</sup>lt; D, *<sup>v</sup>*ˆ21\_3(*i*) <sup>−</sup> <sup>e</sup>*v*21\_3(*i*) <sup>&</sup>lt; <sup>D</sup> *<sup>v</sup>*ˆ22\_1(*i*) <sup>−</sup> <sup>e</sup>*v*22\_1(*i*) <sup>&</sup>lt; D, *<sup>v</sup>*ˆ22\_2(*i*) <sup>−</sup> <sup>e</sup>*v*22\_2(*i*) <sup>&</sup>lt; D, *<sup>v</sup>*ˆ22\_3(*i*) <sup>−</sup> <sup>e</sup>*v*22\_3(*i*) <sup>&</sup>lt; <sup>D</sup> (18)

When Equation (18) is satisfied, the difference between the estimated value and the correction value is small, and the *i*th correction value is applied as the final correction value. Otherwise, the counter *i* is increased by 1, and further correction is be conducted as follows:

$$\begin{cases} \begin{aligned} & \left[\hat{\boldsymbol{\nu}}\_{11,1}(i+1)=\overline{\boldsymbol{\nu}}\_{11,1}(i)+\mathbb{B}[\hat{\boldsymbol{\nu}}\_{11,1}(i)-\overline{\boldsymbol{\nu}}\_{11,1}(i)] \\ & \hat{\boldsymbol{\nu}}\_{11,2}(i+1)=\overline{\boldsymbol{\nu}}\_{11,2}(i)+\mathbb{B}[\hat{\boldsymbol{\nu}}\_{11,2}(i)-\overline{\boldsymbol{\nu}}\_{11,2}(i)] \\ & \boldsymbol{\nu}\_{11,3}(i+1)=\overline{\boldsymbol{\nu}}\_{11,3}(i)+\mathbb{B}[\boldsymbol{\nu}\_{11,3}(i)-\overline{\boldsymbol{\nu}}\_{11,3}(i)] \\ & \boldsymbol{\nu}\_{11,4}(i+1)=\overline{\boldsymbol{\nu}}\_{21,1}(i)+\mathbb{B}[\boldsymbol{\nu}\_{21,1}(i)-\overline{\boldsymbol{\nu}}\_{21,3}(i)] \\ & \boldsymbol{\nu}\_{21,4}(i+1)=\overline{\boldsymbol{\nu}}\_{21,1}(i)+\mathbb{B}[\boldsymbol{\nu}\_{21,1}(i)-\overline{\boldsymbol{\nu}}\_{21,1}(i)] \\ & \boldsymbol{\nu}\_{21,2}(i+1)=\overline{\boldsymbol{\nu}}\_{21,2}(i)+\mathbb{B}[\boldsymbol{\nu}\_{21,2}(i)-\overline{\boldsymbol{\nu}}\_{21,2}(i)] \\ & \boldsymbol{\nu}\_{21,3}(i+1)=\overline{\boldsymbol{\nu}}\_{21,3}(i)+\mathbb{B}[\boldsymbol{\nu}\_{21,3}(i)-\overline{\boldsymbol{\nu}}\_{21,3}(i)] \end{aligned} \tag{19}$$

where B is the correction factor that satisfies 0 < B < 1. Then, the results of Equation (19) are substituted into Equation (17). This method cycles back and forth until the difference between the estimated and correction values satisfies Equation (18). The cycle is then stopped, and the final correction value is outputted.

The overall software flow chart of the method is shown in Figure 7.

**Figure 7.** Software flow chart of the proposed method.
