An arc chain model is established as shown in Figure 2.

For any current element, the force analysis is shown in the following Figure 3.

F f i + F d i + F z i = 0 (1)

where, F f i , F d i , F z i are the wind force, magnetic field force and air resistance respectively for the ith arc element.

The magnetic field force equation:

F d i = B i I i l i (2)

Wind force equation:

F f i = ρ v f 2 C s S i 2 (3)

F z i = C R ρ v i 2 r i l i (4)

where, B i is the magnetic induction intensity of the ith arc element; l i is the arc length of the ith arc element; v f is the wind speed; v i is the velocity of the ith arc element; S i is the area of the side cross section of the ith arc element; C s is the shape factor; C R is a constant; r i is the radius of the ith arc element.

The joint system of Equations (1)-(4) yields:

v i = F d i + F f i C R ρ r i l i (5)

The arc resistivity expression is:

ρ = m e v e i n e e 2 = m e e 2 ln Λ 32 π ε 0 2 T e 3 / 2 (6)

m e : electron mass, taken as 9.1 × 10⁻³¹ kg.

e : electron charged amount is 1.76 × 10⁻¹⁹ C.

ε 0 : vacuum medium constant 8.85 × 10⁻¹² F/m.

T e : electron temperature.

T e = n e V = I S V (7)

where, n e is the electron density, V is the volume of the arc column, and S is the cross-sectional area of the arc column.

ln Λ ≈ ln ( T 3 / 2 e 2 n 1 / 2 ) (8)

InΛ: Coulomb logarithm, where n is the number density of particles in the arc and T is the arc temperature. In general, InΛ take the value range of 10 - 20.

The moving state of the current element at a certain instant under the wind load is shown in Figure 4 below.

Then the arc resistance is:

R = R 1 + R 2 + R 3 + R 4 + R 5 (9)

As shown above, let the arc resistance R₁, R₂, R₃, R₄, R₅ corresponding to the arc length of l 1 , l 2 , l 3 , l 4 , l 5 , respectively.

Figure 5 shows the overlapping cross section of the first arc element and the second arc element.

h = r 1 sin β (10)

d = r 1 cos β (11)

where r 1 is the radius of the first current element; β is the angle between O_1A and O₁O₂.

S Δ O 1 A B = 1 2 d h = 1 2 r 1 2 sin β cos β (12)

S O 1 A C = r 1 2 β (13)

Therefore, the overlapping area S is:

S = 4 × ( S O 1 A C − S Δ O 1 A B ) = 4 × ( r 1 2 β − 1 2 r 1 2 sin β cos β ) (14)

d ( t ) = 1 2 v 2 t (15)

where, v 2 is the velocity of motion of the second arc element and t is the time.

sin β = h r 1 = r 1 2 − d ( t ) 2 r 1 (16)

cos β = d r 1 = v 2 t 2 r 1 (17)

Then the expression for the overlapping area S can be converted to:

S = r 1 2 ( 4 × arccos v 2 t 2 r 1 − v 2 t r 1 2 − d ( t ) 2 r 1 2 ) (18)

D = 0.26 I (19)

r 1 = 0.13 I (20)

where D is the diameter of the arc element.

According to the resistance formula, the simplification gives:

R 1 = ρ l 1 s 1 = l 1 m e e 2 I n Λ 0.5408 π I π ε 0 2 T e 3 / 2 (21)

R 2 = ρ l 2 s 2 = m e e 2 I n Λ 32 π ε 0 2 T e 3 / 2 B (22)

R 3 = ρ l 3 s 3 = l 3 m e e 2 I n Λ 0.5408 π I π ε 0 2 T e 3 / 2 (23)

R 4 = ρ l 4 s 4 = l 4 m e e 2 I n Λ 0.5408 π I π ε 0 2 T e 3 / 2 (24)

R 5 = ρ l 5 s 5 = l 5 m e e 2 I n Λ 0.5408 π I π ε 0 2 T e 3 / 2 (25)

R = R 1 + R 2 + R 3 + R 4 + R 5 = m e e 2 I n Λ 32 π ε 0 2 T e 3 / 2 ( l 1 + l 3 + l 4 + l 5 0.0169 π I + B ) (26)

Formula analysis:

Arc resistance and electron temperature is inversely proportional to the 3/2 times. As the arc temperature decreases, the arc resistance increases; finally, the arc tends to be extinguished.

Arc resistance gradually grows to infinity, the arc gradually extinguished. In the arc current after the zero number of ms, all meet the condition dR/dt > 0, the arc is judged to be extinguished.

Through the above calculation method can be obtained under different parameters of the arc burning time, as shown in Table 1.