*4.1. Error Analysis*

In practical applications, sample resolution and voltage ripple cause sampling errors, which affect the calculated result of insulation resistance. With the *v*<sup>11</sup> of the *M*<sup>1</sup> phase as an example, ∆*v*11\_1, ∆*v*11\_2, and ∆*v*11\_3 are the sampling errors of *v*11\_1, *v*11\_2, and *v*11\_3, respectively. *E*1*<sup>R</sup>* is the actual value of *E*1. *E*1*<sup>C</sup>* is the measured value of *E*1. Considering the sampling error, the expression of *E*1*<sup>R</sup>* and *E*1*<sup>C</sup>* according to Equation (6) is shown as

$$\begin{cases} E\_{1R} = \frac{v\_{11\\_2} - v\_{11\\_3}}{v\_{11\\_1} - v\_{11\\_2}} \\\ E\_{1C} = \frac{(v\_{11\\_2} + \Delta v\_{11\\_2}) - (v\_{11\\_3} + \Delta v\_{11\\_3})}{(v\_{11\\_1} + \Delta v\_{11\\_1}) - (v\_{11\\_2} + \Delta v\_{11\\_2})} \end{cases} \tag{10}$$

∆*E*<sup>1</sup> is the error of *E*<sup>1</sup> and is defined as follows:

$$
\Delta E\_1 = E\_{1\overline{C}} - E\_{1R} \tag{11}
$$

By substituting Equation (11) into Equation (10), ∆*E*<sup>1</sup> can be rewritten as

$$
\Delta E\_1 = \frac{(\Delta v\_{11\\_2} - \Delta v\_{11\\_3}) - E\_{1R}(\Delta v\_{11\\_1} - \Delta v\_{11\\_2})}{(v\_{11\\_1} - v\_{11\\_2}) + (\Delta v\_{11\\_1} - \Delta v\_{11\\_2})} \tag{12}
$$

where ∆*v*11\_1, ∆*v*11\_2, and ∆*v*11\_3 are uncontrollable components and *E*1*<sup>R</sup>* is a fixed value. (*v*11\_1–*v*11\_2) is inversely proportional to ∆*E*1. Similarly, *v*12, *v*21, and *v*<sup>22</sup> can result in the same conclusion. In the application, the larger ∆*t*11, ∆*t*12, ∆*t*21, and ∆*t*<sup>22</sup> are, the smaller the result error is. The larger the difference between the *R<sup>a</sup>* and *R<sup>b</sup>* is, the smaller the result error is. The higher the *v*dc is, the smaller the result error is.
