C) Short-time duty

At this mode the electromagnet coil can endure considerably a greater current load, than at continuous running duty. It enables to decrease its sizes, and so, key size of electromagnet core.

At determination in the preliminary calculation of coil heating, switched on during the small interval of time (turn-on time t_on), about a few

seconds, it is possible to consider that all heat, radiated in a coil, is expended on heating of its active material (for example, copper), i.е. to ignore heat emission in an external environment and heating of insulants, included in its construction.

Equation of coil heating in this case will be:

R∙I²∙t_on = с∙G∙ Θ_per, (1.29)

where c − specific heat capacity of active material of wire, J/g∙⁰C;

G − weight of active material of wire, g;

I − current of this short-time duty, A.

Weight of active wire material can be determined in such way:

where γ_m is a specific gravity, g/сm³; l_av is a length of middle loop of coil, cm; S_m – cross-section of wire metal, сm²; w − number of coil loops.

Substitution of value G from (1.30) and R from (1.8) in equation (1.29) gives: (I / S_m)² = 10⁴∙c∙γ_m∙ Θ_per / ρ∙t_on.

A current density j, [A/cm²] in the coil section equals to:

j = I / S_m = √[10⁴∙c∙γ_m∙Θ_per / ρ∙t_on] (1.31)

At load duration in one second (t_on = 1s, onesecond current), a current density is determined by a formula:

j = √[c∙γ_m∙Θ_per / 10^-4ρ] and, so, j = j₁ / √t_on.

For coils of a copper wire, if to accept: γ_m = 8,9 g/сm³, c = 0,39 J/g∙⁰C, the permissible current density at the onesecond load practically can be accepted in accordance to a table. 1.2, where the values of temperatures ϑ_m.per and corresponding to it values of j₁ are brought accepted by ГОСТ.

Table 4

Class of isolation	Y	A	Compounded coils
ϑm.per (0C)	90	105	120
Θm.per = ϑper − 35 (0C)	55	70	85
j1 (A/сm2)	10∙103	10,8∙103	11,7∙103

As follows from a table. 4, at a preliminary calculation it is possible on the average to accept j₁ = 11000 A/сm² = 110 A/mm², or with some reserve on heating j₁ = 100 A/mm²: j = 10⁴ / √t_on.

On the other hand, because by a formula (1.10)

S_m = f_ap∙m∙n∙d_c² / w, then j = I / S_m = w∙I / f_ap∙n∙m∙d_c², from where

w∙I = j∙ f_ap∙n∙m∙d_c² (1.33) and, so, by (1.3)

В₀ = χ∙μ₀∙φ∙(wI) / δ₀ = χ∙μ₀∙φ∙j∙f_ap∙n∙m∙d_c² / δ₀ (1.34)

Substituted a value S₀ (1.2) and В₀ (1.34), we will define the size of electromagnetic force by (1.1) :

F₀ = 4∙ χ²∙μ₀∙φ²∙j²∙f_ap²∙n²∙m²∙d_c⁶∙ε²∙τ² / δ₀² (1.35)

from where with a glance of (1.31) it is possible to find a key size of electromagnet core for short-time duty with the set turn-on time:

d_c = ⁶√[2∙10³∙ρ∙t_on∙δ₀²∙F₀ / (χ²∙φ²∙c∙γ_m∙f_ap²∙n²∙m²∙ε²∙τ²∙Θ_per)] or, if to take on a close value of j =10⁴ / √t_on,

d_c = ⁶√[0,2∙t_on∙δ₀²∙F₀ / (χ²∙φ²∙f_ap²∙n²∙m²∙ε²∙τ²)] (1.36)

In general case we have:

d_c = ³√[(C₃∙ δ₀/ε)√ F₀∙ t_on] (1.37)

where C₃ = √[2∙10³∙ρ / χ²∙φ²∙c∙γ_m∙f_ap²∙n²∙m²∙τ²∙Θ_per] = √{[C₁∙h∙(1 + 2n + α) / [n∙(1 + n)∙f_ap∙c∙γ_m]} (1.38)

at the close value of j = 10⁴ / √t_on

С₃ = 0,14/ (χ∙φ∙f_ap∙n∙m∙τ) (1.38,а)

Because ε, in turn, depends on d_с, then, as well as before, for determination of d_с can be recommended the following methodology.

We will transform a formula (1.37) so: (d_с / δ₀)² = С₃∙(√ F₀∙ t_on) / (δ₀²∙ε). From here

(√F₀) / δ₀² = χ³∙ε / (С₃√t_on) (1-39)

Set by values χ, it is possible to define (√F₀) / δ₀² and then, as well as before, to build graphic dependence (√F₀) / δ₀² in a function of χ.

At solution of reverse task by the known values of F₀ and δ₀² determine (√F₀) / δ₀² and find χ = d_с / δ₀ by a chart. By value χ and given δ₀ find the key size of electromagnet core d_c = χ∙δ₀,

and so, coil sizes and cross-section of wire metal:

S_m = π∙ρ∙(1 + n)∙j∙f_ap∙n∙m∙d_c³ / 10⁴∙U, (1 -40)

or at the close value of j = 10⁴ / √t_on

S_m = π∙ρ∙(1 + n)∙f_ap∙n∙m∙d_c³ / U∙√t_on. . (1.40,а)

Because w∙S_m = НА∙f_ap, then a number of coil loops of electromagnet, operating in the short-time duty, is equal to: w = 10⁴∙U / π∙ρ∙(1 + n)∙j∙d_c or at the close value of j

w = U∙√t_on / π∙ρ∙(1 + n)∙d_c.

We will designate C₄ = 10² / [π∙(1 + n)∙√(ρ∙c∙γ_m∙Θ_per)], or at a close value j: C₄ = 1 / [π∙ρ∙(1 + n)].

Then the number of electromagnet coil loops, operating in the short-time duty, can be expressed so:

w = (U∙C₄∙√t_on) / d_c (1.41)

Induction in a working air-gap δ₀ can be defined by found value of d_с, χ and ε from a formula (1.1) and (1.2), or before determination of d_с approximately by a formula:

B₀ = [4,8∙10^-5 / (τ∙³√ С₃)]∙³√[F₀ / (δ₀∙√t_on)].