**2. Theoretical Analysis**

### *2.1. Principle of NHCB*

As shown in Figure 1, an NHCB mainly includes an MS, an SS and a metal-oxide varistor (MOV) [14]. The current limiting reactor is mainly used to limit the fault current. SS is mainly composed of FPEDs through series and parallel connection, and the most commonly used FPEDs are GTO, IGCT, IGBT, and IEGT. According to the literature [9], the interruption process of NHCB mainly includes vacuum arc commutation and the interruption process of SS. The interruption process of NHCB is shown in Figure 2. In this paper, we will focus on the vacuum arc commutation during NHCB interruption. *Im*, *Is*, and *Imov* represent the currents of the MS, SS, and MOV, respectively.

**Figure 1.** The schematic of a natural-commutate hybrid DC circuit breaker (NHCB).

**Figure 2.** Interruption process of NHCB.

### *2.2. Vacuum Arc Commutation Model*

*I*

The typical waveform of the vacuum arc commutation is shown in Figure 3 [18]. According to the research of the literature [16,17], the vacuum arc commutation model of NHCB is shown in Figure 4a. Then, according to the model shown in Figure 4a, the vacuum arc commutation should follow Formula (1). As shown in Formula (2), Formula (1) can be further rewritten as the relational expression about the current commutating speed d*Ic*/d*tc*. *Uarc* represents the vacuum arc voltage. *Use* represents the on-state voltage of the SS at the end of vacuum arc commutation. *Ls* represents the stray inductance of the SS. *In* represents the currents of NHCB. *Isi* represents the current of the SS at the initial moment of vacuum arc commutation. *Ise* represents the current of the SS at the end of vacuum arc commutation. *Ic* is called the commutation current, which represents the current commutated to the SS during the vacuum arc commutation. *Ice* is called the final commutation current, which represents the current commutated to the SS from the beginning to the end of the vacuum arc commutation. *tc* represents the time it takes for the commutation current to increase from zero to *Ic*. *Tc* represents the duration of vacuum arc commutation.

$$
\mathcal{U}\_{\rm arc} = L\_s \frac{\mathbf{d}I\_c}{\mathbf{d}t\_c} + \mathcal{U}\_s \ 0 \le I\_c \le I\_{c\nu} \tag{1}
$$

$$\frac{\mathrm{d}I\_{\mathrm{c}}}{\mathrm{d}t\_{\mathrm{c}}} = \frac{\mathrm{U}\_{\mathrm{arc}} - \mathrm{U}\_{\mathrm{s}}}{L\_{\mathrm{s}}} \; 0 \le I\_{\mathrm{c}} \le I\_{\mathrm{c}c}.\tag{2}$$

**Figure 3.** Example of typical vacuum arc commutation.

**Figure 4.** The vacuum arc commutation model of NHCB (**a**) before and (**b**) after simplification.

For FPEDs, such as IGBT, IGCT, and IEGT, the relationship between their on-state voltage *U*CE and collector current *I* is shown in Formula (3) [22]. As shown in Figure 5, the value of their on-state voltage *UCE* is basically linear with the collector current *I*. Therefore, as shown in Figure 5, the relationship between *UCE* and *I* can be linearly processed, and the relationship between *UCE* and *I* is obtained as shown in Formula (4). Because SSs are made of FPEDs through series and parallel connection, the relationship between *Us* and *Is* can also be obtained as shown in Formula (5). *U*CE0 represents the on-state voltage of the FPEDs when the collector current is zero. *r* represents the on-state resistance of the FPEDs. *U*0 represents the approximate on-state voltage of the FPEDs when the collector current is zero. *R* represents the approximate on-state resistance of the FPEDs. *Us*0 represents the approximate on-state voltage of the SS when the *Is* is zero:

$$\mathcal{U}\_{CE} = \mathcal{U}\_{CE0} + rI\_{\prime} \tag{3}$$

$$\mathcal{U}L\_{\text{CE}} = \mathcal{U}l\_0 + \mathcal{R}l\_\prime\tag{4}$$

$$
\mathcal{U}\_s = \mathcal{U}\_{s0} + \mathcal{R}\_s I\_s. \tag{5}
$$

**Figure 5.** Collector-emitter saturation voltage characteristics of FZ1200R17HP4\_B2 [23].

As shown in Figure 3, because the change in the vacuum arc voltage is relatively small during the vacuum arc commutation [24], *Uarc*-*avg* can be taken as the vacuum arc voltage value, and *Uarc*-*avg* is the average value of the maximum value *Uarc*-*max* and the minimum value *Uarc*-*min* of the vacuum arc voltage. According to Formula (5), the model shown in Figure 4a can be rewritten as shown in Figure 4b. Then, according to the model shown in Figure 4b, the vacuum arc commutation should follow Formula (6). It is easy to know that *Is* = *Isi* + *Ic*, and according to Formula (5), the system of equations shown in Formula (7) can be easily obtained. *Usi* represents the on-state voltage of the SS at the initial moment of vacuum arc commutation:

$$\frac{dI\_{\rm c}}{dt\_{\rm c}} = \frac{\mathcal{U}\_{\rm arc\,avg} - \mathcal{U}\_{\rm s}}{L\_{\rm s}} \; 0 \le I\_{\rm c} \le I\_{\rm cc} \tag{6}$$

$$\begin{cases} \ll lL\_{si} = lI\_{s0} + R\_s I\_{si} \\\ll lL\_s = lI\_{si} + R\_s I\_c \ 0 \le I\_c \le I\_{cc} \end{cases} \tag{7}$$

After solving Formulas (6) and (7), the relationship expression of *tc* about *Ic* is obtained as shown in Formula (8). It is easy to know that when *Ic* = *Ice*, *tc* = *Tc*, then according to Formula (8), the relationship between *Tc* and *Ice* can be obtained as shown in Formula (9). From Formula (9), we can see that *Ls*, *Rs*, *Ice*, *Uarc*-*avg*, and *Usi* are the main parameters that affect *Tc*. Meanwhile, according to Formula (6), if d*Ic*/d*tc* ≤ 0, the current will stop commuting to the SS branch, so d*Ic*/d*tc* > 0 should always be ensured during the vacuum arc commutation. By solving d*Ic*/d*tc* > 0, Formula (10) can be obtained:

$$t\_c = -\frac{L\_s}{R\_s} \ln \left( \frac{\mathcal{U}\_{\text{arc-avg}} - \mathcal{U}\_{si} - R\_s I\_c}{\mathcal{U}\_{\text{arc-avg}} - \mathcal{U}\_{si}} \right) \\ 0 \le I\_c \le I\_{c\varepsilon\prime} \tag{8}$$

$$T\_c = -\frac{L\_s}{R\_s} \ln \left( \frac{lL\_{\text{arc-avg}} - lL\_{si} - R\_s I\_{cx}}{lL\_{\text{arc-avg}} - lI\_{si}} \right) \tag{9}$$

$$
\mathcal{U}\_{\text{arc-avg}} - \mathcal{U}\_{\text{si}} - \mathcal{R}\_s I\_{\text{cc}} > 0. \tag{10}
$$

### *2.3. Influence of Main Parameters on Vacuum Arc Commutation Characteristics*

In this section, the effects of parameters, such as *Ls*, *Rs*, *Ice*, *Uarc*-*avg*, and *Usi*, on *Tc* are further studied. Firstly, it is easy to know from Formula (9), under the condition that other parameters remain unchanged, *Tc* has a linear relationship with *Ls*.

Assuming that the parameters *Ls* and *Rs* remain unchanged and *Ls*/*Rs* = b, let *k* = (*Uarc*-*avg* − *Usi*)/*RsIce*, then as shown in Formula (11), Formula (9) can be further rewritten as a function about *k*, and it is easy to know from Formula (10) that *k* > 1:

$$T\_{\mathbb{C}} = -\mathbf{b} \cdot \ln\left(1 - \frac{1}{k}\right) k > 1. \tag{11}$$

It is easy to know that changing the parameters *Uarc*-*avg*, *Usi*, and *Ice* is equivalent to changing the parameter *k*, and the relationship curve between *Tc* and *k* is plotted according to Formula (11), as shown in Figure 6. It can be seen from Figure 6, *Tc* decreases with increasing *k*, but the rate of decrease becomes slower and slower and even tends to be constant. *k* is the ratio of the parameters *Uarc*-*avg* − *Usi* and *RsIce*.

Assuming that the parameters *Ls*, *Ice*, *Uarc*-*avg* and *Usi* remain unchanged and *Ls* = a, the variation of *Tc* with *Rs* is explored. Let the value of *Rs* at the initial time be a fixed value *Rs*0. Correspondingly, *k* = (*Uarc*-*avg* − *Usi*)/*Rs*0*Ice* = *k*0; let the changed *Rs* = *xRs*0, and correspondingly, *k* = (*Uarc*-*avg* − *Usi*)/*xRs*0*Ice* = *k*0/*<sup>x</sup>*. Then, as shown in Formula (12), Formula (9) can be further rewritten as a function about *x*, and it is easy to know from Formula (10) that *x* < *k*0:

$$T\_{\mathcal{E}} = -\frac{\mathbf{a}}{\mathbf{x}} \ln \left( 1 - \frac{\mathbf{x}}{k\_0} \right) \mathbf{x} < k\_0. \tag{12}$$

When *k*0 takes the values of 4 and 8, respectively, as shown in Figure 7, the variation curves of *Tc* and *k* are plotted with *x* according to Formula (12). It can be seen from Figure 7, *Tc* increases with increasing *x*, and the rate of increase becomes faster and faster; *k* decreases with increasing *x*, but the rate of decrease becomes slower and slower. The change curve of *Tc* with *k* is plotted in Figure 7 as shown in Figure 8. As it can be seen from Figure 8, *Tc* decreases with increasing *k*, but the rate of decrease becomes slower and slower and even tends to be constant. *k*0 is the ratio of the parameters *Uarc*-*avg* − *Usi* and *Rs*0*Ice*, and *Rs*0 is the initial value of *Rs*.

**Figure 6.** The relationship curve between *tc* and *k*.

**Figure 7.** The variation curves of *Tc* and *k* with *x*.

**Figure 8.** The relationship curve between *Tc* and *k* in Figure 7.

It can be seen from Figures 6 and 8 that when *k* is small, *Tc* changes greatly with the increase of *k*, and when *k* is large, *Tc* changes little or even tends to be constant with the increase of *k*; therefore, the larger the value of *k*, the more e ffective it is to reduce the value of *Tc*.
