А) Continuous running duty of electromagnet

In a preliminary calculation suppose usually, that the cooling surface of direct-current coil S_col are external S_ex and internal S_in its surface and ignore cooling of butt ends. Equality (fig. 1.1) is thus just:

S_ex = π∙ D_ex∙H = π∙ m ∙(1 + 2 n)∙ d ² _c, (1.5, a)

S_in = π ∙D_in∙H = π ∙m∙d ² _c, (1.5, b)

where D_ex = d_с + 2 А; D_in ≈ d_c, and accordingly height of Н and width A winding of coil is expressed through a qualificatory size d_c and dimensionless coefficients of proportion of m and n, i.е.

Н = m∙d_с, A = n∙d_с. (1.6)

In accordance with equation of coil heating in the continuous running duty we have:

Θ _per = R∙I ² / [ h∙ (S_ex + α∙S_in)], (1.7)

where h − coefficient of heat emission from the surface of coil, W /⁰∙ cm ²; its value can be chosen for the exceeding of temperature Θ _per under formula:

h = 9,3∙10^-4 ( 1 + 0,0059∙Θ _per

α − coefficient, taking into account influence of windung method of coil on its heat emission: self-supported, shrouded, winded on a pipe, winded on a core or wireframe coil (h_in = α∙h − coefficient of heat emission from the internal surface of coil, α = 0,9 − for the self-supported shrouded coil; α = 1,7 − for a coil, winded on a pipe; α = 2,7 − for a spool, winded on a core; α = 0 − for coils, having isolating framework of material, badly conducting warm, and coils of alternating current).

I − current, flowing in a coil, a; R − resistance of coil, refered toϑ _per = Θ _per + 35⁰ С.

As is generally known,

R = 10^-4ρ∙ w∙l_av / S_m = 10^-4ρ∙π∙ D_av∙w / S_m = ρ∙π∙(1 + n) d_c∙w / (10⁴∙ S_m) (1-8)

where ρ − specific resistance of wire metal of coil at ϑ _per, Ω∙ mm ²/ m. Specific resistance ρ of wire material it is necessary to refer to the permissible temperature of coil: ϑ _per = Θ _per + ϑ _env, here ρ = ρ₀(1 + α₀∙ϑ _per), where ρ₀ is specific resistance at 0⁰С; α₀ is a temperature coefficient of wire metal. For a copper wire α₀ = 0,00393 within the limits of heating of 10÷100⁰С, ρ₀ = 0,0162. For a leader copper a value ρ for most often meeting working temperatures is driven to the table.2:

Table. 2

ϑ per (0С)	ρ (Ω mm 2/ m	ϑ per (0С)	ρ (Ω mm 2/ m	ϑ per (0С)	ρ (Ω mm 2/ m
	0,01754		0,02170		0,02339
	0,01857		0,02236		0,02370
	0,01991		0,02300		0,02443

l_av, D_av − length and diameter of middle loop of coil, cm;

S_m – cross-section of wire metal of coil, см ²;

w − number of coil loops.

On determination of aperture occupation ratio of winding we will get:

f_ap = S_m∙w / HA = S_m∙w / m∙n∙d_c ²,(1.9)

from where we find the cross-section of wire metal

S_m = f_ap∙m∙n∙d_c² / w (1.10)

Putting (1.10) in (1.8), and then (1.5) and (1.8) in (1.7), we will define:

Θ _per = ρ(1 + n)∙ w ² I ² / [10⁴∙ f_ap ∙ m ²∙ n∙h∙ (1 + 2 n + α)∙ d_c ³]

From here it is easily to define necessary coils MMF:

w∙I = √ [10⁴∙ f_ap ∙ m ²∙ n∙ (1 + 2 n + α)∙ h∙ Θ _per ∙ d_c ³] / ρ(1 + n) (1.11)

Putting a value w∙I from (1.11) in (1.4), we find a basic formula, defined force of electromagnet:

F ₀ = [4∙10⁴ ∙μ ₀ ∙ φ²∙ε²∙χ² ∙ f_ap∙ τ² ∙m ² ∙n∙ (1 + 2 n + α)∙ h∙ Θ _per ∙ d_c ⁵] / [ρ(1 + n) δ₀²] (1.12)

From the last formula we find the key size of electromagnet:

d_c = ⁵ √ [2∙10³∙ ρ(1 + n)∙ F ₀∙ δ₀²] / [φ²∙ε²∙χ² ∙ f_ap∙ τ² ∙m ² ∙n∙ (1 + 2 n + α)∙ h∙ Θ _per ] (1.13)

Entering denotation

C ₁ = [2∙10³∙ρ(1+ n)] / [φ²∙χ² ∙f_ap∙ τ² ∙m ² ∙n∙ (1+2 n +α)∙ h∙ Θ _per ],(1.14)

we will rewrite formulas (1.12) and (1.13) so:

F ₀ = ε² ∙d_c ⁵(1.15)

and d_c = ⁵ √ [ C ₁∙ F ₀∙ δ₀² / ε²], (1.16)

where F ₀ − force, kg, and δ₀ – air gap, cm − are set by the critical terms of design;

ε _eq, ε − are dimensionless coefficients, taking into account buckling of flux and define equivalent conductivity of basic gap or her derivative ε = ε (δ₀, d_c), ε _eq = ε∙δ₀, d_c;

С ₁− constant, determined by factors, included in a formula (1.14), namely: ρ, f_ap − constants, determined by the type of the chosen wire and method of its winding;

Θ _per, h, α − constants, determined by the terms of the permissible heating;

m, n, τ, φ − constants, determined by the optimally chosen sizes of coil and magnetic core;

2 ∙10³ − constant, determined by chosen system of units.

About the practically recommended values of indicated constants, determined the task solution, approaching optimal terms, it will be rendered below. Possibility of close estimation of buckling coefficient ε, which in case of basic gap, formed by a cylindrical core and flat armature, can be expressed by next dependence: ε² = 1 + [2,08 / (τ∙ d_c / δ₀)].

A value ε for most electromagnets of the examined form is within the limits of 1 ÷ 1,6, however depends considerably on sizes δ₀ and d_p = τ∙ d_c.

Thus, the coefficient ε or ε _eq included in a formula (1.16) is the function of gap δ₀ and diameter d_c, that complicates the determination of key size d_c from this equality.

Therefore for the calculation of d_c next methodology can be recommended:

We will transform a formula (1.16) so:

F ₀ / δ₀³ = (1/ С ₁)(d_c/ δ₀)⁵∙ε², или F ₀ / δ₀³ = χ⁵∙ε² / С ₁, (1.17)

where χ = d_c / δ₀ (1.18)

In the interval of the supposed change of δ₀ / d_с, or χ = d_c / δ₀, for example 0,1 ≤ (δ₀ / d_с) < 1 or 1< χ ≤ 10, at constant value С ₁ and τ; set by Fig. 1.2 values χ = 1, 2, 3.., it is easily to define by a formula (1.17) a value F ₀ / δ₀³ and, thus, to build the curve of functional dependence of F ₀ / δ₀³ = f (fig. 1.2 and table. 3).

Then at the reverse raising of task, i. е. at a set value F ₀ and δ₀, determine the size of F ₀ / δ₀³ and, using the got curve, find the relation χ = d_c /δ₀ and, so, for set value of δ₀ the sought value d_c.

At the change of values С ₁ and τ a family of analogical curves can be got.

By found value of key size d_с it is easily to find the cross-section of wire metal S_m and number of coil loops w.

Really, because on (1.8)

R = 10^-4∙ρ∙π∙(1 + n)∙ d_c∙w / S_m,, then w∙I = w∙U/R = 10⁴∙ U∙S_m / [π∙ρ∙(1 + n)∙ d_c ], from where S_m = [ρ∙π∙(1 + n)∙ d_c∙w∙I ] / 10⁴∙ U.

Table. 3

x	x 2	x 3	x5	ε 2 ε eq 2	x 5ε2 x 5εeq2	F 0 / δ03

On the other hand, MMF of coil by (1.11) with a glance (1.14) can be expressed so: w∙I = (4,5∙10³ / φ∙χ∙τ)∙ d_c∙√ (d_c/C ₁)

So, the cross-section of wire is determined by formula

S_m = 1,41∙ρ∙(1 + n)∙ d_c ²∙ √ (d_c/C ₁ / (φ∙χ∙τ∙ U), [ сm ²], (1.20)

and number of coil loops − by a formula:

w = f_ap∙HA / S_m = φ∙χ∙τ∙ U∙f_ap∙m∙n ∙√(C ₁/ d_c) / [1,41∙ρ∙(1 + n)] or

w = C ₂ ∙U∙ √ (C ₁/ d_c) (1.21)

where it is accepted: C ₂ = φ∙χ∙τ∙ f_ap∙m∙n / [1,41∙ρ∙(1+ n)] (1.22)

On occasion it is comfortably to use dependence:

w = U ∙√{10³∙ f_ap∙n / [ρ(1+ n)(1+2 n +α)∙ h ∙Θ _per∙d_c ]}

The formulas got higher upon calculation of key size of electromagnet and winding data of coil are correct in the wide range of induction, however at its values, exceeded a limit, corresponding to steel saturation, the got dependences lost a fisical sense.

In accordance with it at the calculation of electromagnet core it is necessary to specify the value of induction, which a key size d_с is got at.

A value of induction in steel, and so, its less value in a working air-gap, can be defined by the values of d_с, ε and given initial terms of design (F ₀ and δ₀), found in a preliminary calculation from a formula (1.1):

B ₀ = 0,56∙10^-4∙√ F ₀ / τ∙ε∙ d_c.

As specified, this value is specified in the subsequent project calculation of induction and under existent experience of design and production of electromagnets must not excel the values of order

(0,6÷0,8)∙10^-4, Wb / cm ².

For the exception of the repeated calculations the value of induction in a basic working gap can be got approximately, before the calculation of key size of core, by formula:

B ₀ ≈ 4,8∙10^-5∙[¹⁰√(F ₀³ / δ₀⁴)] / τ∙(⁵√ C ₁)

Nonconformance of the got value of induction with permissible indicates about unsuccessful choice of factors of preliminary calculation or accepted type of electromagnet for the set terms of its operation, required force and motion of armature.

Change of the got value of induction, as be obvious from the formula given above, at maintenance of the set terms of design (F ₀ and δ₀) can be made due to clarification of coefficients of preliminary calculation in the real limits of their change.

So, for example, at the chosen value of п only due to the change of relation of coil height to the width of its window β = m / n

in limits from 1 to 10, it is possible to change induction В ₀ in 1÷2,5 time. A value of induction can be corrected due to clarification and other coefficients of preliminary calculation (τ, n, Θ _per and other).

In electromagnets with small critical air-gaps and insignificant leakage flux the induction in a gap can some exceed the legitimate values indicated higher.

In some designs of electromagnets, at which the initial terms of planning require creation of small critical forces and relatively large critical air-gaps, for maintenance real structural ratio and optimal terms on heating the induction in a air-gap it is needed to take considerably below than recommended.

In electromagnets with considerable critical forces and small air-gaps at maintenance of desirable design correlations and legitimate values of induction it is necessary agree to incomplete use of coil by heating.

b) Recursive short-time mode

Operation of electromagnet in the recursive short-time modeis 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∙I ² _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∙m ² ∙n∙ (1 + 2 n + α)∙ h∙ Θ _per ∙ d_c ³/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 F ₀ an induction В ₀ is determined, as well as before, by (1.3):

В ₀ = χ∙μ₀∙φ∙(wI) / δ₀ = [χ∙μ₀∙φ∙ р_cr / δ₀]∙√ [10⁴∙ f_ap∙m ² ∙n∙ (1 + 2 n + α)∙ h∙ Θ _per ∙ d_c ³/ρ(1 + n)] (1.24)

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

d_с = ⁵ √ (C ₁ F ₀ ∙ δ²₀ / р ² _cr∙ε ²) (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/C ₁ (1.26)

2) Cross-section of wire:

S_m = [1,41∙ρ∙(1 + n)∙ р_cr∙d ² _c / φ∙χ∙τ∙ U ]∙ √ d_c/C ₁ (1.27)

number of coil loops:

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

w = [φ∙χ∙τ∙ f_ap∙n∙m∙U /1.41∙ρ∙(1+ n)∙ р_cr ]∙√(C ₁/ d_c) = C ₂ ∙U/ р_cr ∙√(C ₁ /d_c)

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∙I ²∙ t_on = с∙G ∙ Θ _per, (1.29)

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

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 / сm ³; l_av is a length of middle loop of coil, cm; S_m – cross-section of wire metal, сm ²; 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 / cm ²] 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 = 1 s, onesecond current), a current density is determined by a formula:

j = √[ c ∙γ _m ∙Θ _per / 10^-4ρ] and, so, j = j ₁ / √ t_on.

For coils of a copper wire, if to accept: γ _m = 8,9 g / сm ³, c = 0,39 J / g∙ ⁰ C, 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 j ₁ are brought accepted by ГОСТ.

Table 4

Class of isolation	Y	A	Compounded coils
ϑ m.per (0 C)
Θ m.per = ϑ per − 35 (0 C)
j 1 (A / сm 2)	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 j ₁ = 11000 A / сm ² = 110 A / mm ², or with some reserve on heating j ₁ = 100 A / mm ²: j = 10⁴ / √t_on.

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

S_m = f_ap∙m∙n∙d_c ² / w, then j = I / S_m = w∙I / f_ap∙n∙m∙d_c ², from where

w∙I = j∙ f_ap∙n∙m∙d_c ² (1.33) and, so, by (1.3)

В ₀ = χ∙μ₀∙φ∙(wI) / δ₀ = χ∙μ₀∙φ∙ j∙f_ap∙n∙m∙d_c ² / δ₀ (1.34)

Substituted a value S ₀ (1.2) and В ₀ (1.34), we will define the size of electromagnetic force by (1.1):

F ₀ = 4∙ χ²∙μ₀∙φ²∙ j ²∙ f_ap ² ∙n ² ∙m ² ∙d_c ⁶∙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∙ F ₀ / (χ²∙φ²∙ c∙γ_m∙f_ap ²∙ n ² ∙m ² ∙ 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∙ F ₀ / (χ²∙φ²∙ f_ap ²∙ n ² ∙m ² ∙ ε²∙τ²)] (1.36)

In general case we have:

d_c = ³√[(C ₃ ∙ δ₀/ε)√ F ₀∙ t_on ] (1.37)

where C ₃ = √[2∙10³∙ρ / χ²∙φ²∙ c∙γ_m∙f_ap ²∙ n ² ∙m ²∙τ²∙Θ_per] = √{[ C ₁∙ h∙ (1 + 2 n + α) / [ n ∙(1 + n)∙ f_ap∙c∙γ_m ]} (1.38)

at the close value of j = 10⁴ / √ t_on

С ₃ = 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 = С ₃∙(√ F ₀∙ t_on) / (δ₀²∙ε). From here

(√ F ₀) / δ₀² = χ³∙ε / (С ₃√ t_on) (1-39)

Set by values χ, it is possible to define (√ F ₀) / δ₀² and then, as well as before, to build graphic dependence (√ F ₀) / δ₀² in a function of χ.

At solution of reverse task by the known values of F ₀ and δ₀² determine (√ F ₀) / 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_c ³ / 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_c ³ / 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 C ₄ = 10² / [π∙(1 + n) ∙√ (ρ∙ c∙γ_m ∙Θ_per)], or at a close value j: C ₄ = 1 / [π∙ρ∙(1 + n)].

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

w = (U∙C ₄ ∙√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:

B ₀ = [4,8∙10^-5 / (τ∙³√ С ₃)]∙³√[ F ₀ / (δ₀∙√ t_on)].