B) Recursive short-time mode

Operation of electromagnet in the recursive short-time mode is determined by the alternate swithing on of its coil on a turn-on time t_on and subsequent after this shutoff on a time of pause t_p.

As it is generally known, such mode is characterized by relative duty ratio ( DR_%, which is ratio in percent of t_on toward duration of cycle t_cl = t_on + t_p.

DR_% = (t_on / t_cl)∙100

It is thus assumed that during switching-on a current I = const flows through a coil, in a shutoff period − I = 0.

Thus, at determination of electromagnetic force, created by an electromagnet during turn-on time it is necessary to include a full rated value of MMF w∙I in a calculation of force, and so, induction.

Thermal calculation of coil at this mode it is possible to make coming from assumption, that coil is heated by the equivalent warming current I_ht < I, flowing through it long time, i.е.

Θ_per = R∙I²_ht / h∙S_cl

As it is generally known, I_ht is determined by a formula: I / I_ht = p_cr, where р_сr – overload factor by current, equal for the recursive short-time mode:

р_сr = √[(1− e^-ton/T) / (1 – e^-tcl/T)],

where Т − time constant of coil heating, s; е − natural logarithmic base.

If time of cycle t_cl is considerably less time constant of coil heating Т, that is correct for most electric devices coils (t_on << T), then a factor р_cr can be expressed so:

р_cr = √(100/DR_%) (1.23)

Simple transformations, similar to used for the continuous running duty, enable to define a full MMF of coils for a recursive short-time mode:

w∙I = р_cr√[10⁴∙ f_ap∙m²∙n∙(1 + 2n + α)∙h∙Θ_per∙d_c³/ρ(1 + n)] (1.24)

Thus, at the recursive short-time flowing of current I in the coil a permissible by heating MMF can be increased in р_cr time as compared to the recursive short-time.

Necessary for creation of force F₀ an induction В₀ is determined, as well as before, by (1.3) :

В₀ = χ∙μ₀∙φ∙(wI) / δ₀ = [χ∙μ₀∙φ∙ р_cr / δ₀]∙√ [10⁴∙f_ap∙m²∙n∙(1 + 2n + α)∙h∙Θ_per∙d_c³/ρ(1 + n)] (1.24)

and, so, a key size of electromagnet core is defined in this case by a formula

d_с = ⁵√(C₁F₀∙δ²₀ / р²_cr∙ε²) (1.25) From comparison of (1.16) and (1.25) it is clear, that if the current I flows through a coil in the recursive short-time duty, then the electromagnet core can be chosen with the diminished size in 1/ р_cr^2/5 times as compared to the sizes of electromagnet of the same type, operating in the continuous running duty at the same current in a coil.

For electromagnets, operating in the recursive short-time duty, like formulas (1.19), (1.20) and (1.21) it is possible to get:

MMF of coils:

w∙I = (4,5∙10³∙р_cr∙d_c / φ∙χ∙τ)∙√d_c/C₁ (1.26)

2) Cross-section of wire :

S_m = [1,41∙ρ∙(1 + n)∙р_cr∙d²_c / φ∙χ∙τ∙U]∙ √d_c/C₁ (1.27)

number of coil loops :

w = (U/ р_cr)∙√[10³∙f_ap∙n / ρ∙(1 + n)∙(1 + 2n + α)∙h∙Θ_per∙d_c] or

w = [φ∙χ∙τ∙f_ap∙n∙m∙U/1.41∙ρ∙(1+n)∙р_cr]∙√(C₁/d_c) = C₂∙U/ р_cr∙√(C₁/d_c)