Calculating and projecting magnet DC

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Introduction

Electro technical device consisting of a current-carrying winding and the ferromagnetic core which is magnetized when current flows through the winding is an electromagnet. If the electromagnet is supplied by the direct current it is called DC electromagnet. Electromagnets are usually used for creation of a magnetic stream (in electric cars) and efforts (in drive mechanisms). A DC electromagnet system is widely used at commutating apparatuses and electrical switchgear. The system comprising: a fixed magnet yoke, and an armature guided in a displacement stroke in the passage by a guide device, a working air gap, movable and fixed contacts.

Depending on a way of creation of a magnetic stream and character of operating magnetizing force electromagnets are subdivided into 3 groups: a direct current Electromagnet neutral, a direct current Electromagnet polarized, an alternating current Electromagnet.The action of electromagnet depends both on size of a magnetic stream, and on the electric current direction in a working winding. An electromagnet distinguishes also on a number of other signs: on a way of inclusion of windings -- with parallel and consecutive windings; on a kind of work -- working in long, faltering and short-term modes; on speed of action -- high-speed and the slowed down action etc.

This course design is intended for getting skills on calculation of electromagnets. In this work next problems are solved: direct problem (the armature is in open stage; such parameters are calculated: permeances of the air-gaps, magnetic fluxes, magnetic potential drops in metal and, as a result, magnetic motive force), calculation of winding (calculated parameters are: diameter and cross-section of wire, resistance of wire, current, flowing in wire and precise value of magnetic motive force), inverse problem (armature is in closed stage; calculated parameters are: permeances of air-gaps, magnetic flux, inductance of metal and magnetic intensity), calculation of dynamic characteristics of electromagnet (found characteristics are: moving, pick up, actuation time and electromagnetic force for actuation of calculated electric apparatus).



1. Determination of magnetic motive force of electromagnet

2. Initial data for DC electromagnet design

The electromagnet consists of the joke, the core, windings, and the armature. The scheme of the electromagnet is shown on the figure (1.1) Initial data is given in the table 1.1.

Figure 1.1 - DC Electromagnet

Table 1.1 - Initial data for DC electromagnet design

Geometrical sizes, mm	Oper.air-gap, mm	Induction, B(д), T	Magnet circuit material	Winding power voltage U, V
a	b	l	h	d		д2	д		0.15	Structural steel of quality 20	36
3	20	60	30	8	5	0.1	3	0.1

1.2 Calculation of air-gap magnetic permeances

The first step is the construction of the equivalent circuit to a magnetic one without taking in account reluctance of iron subcircuits.

Figure 1.2 - Equivalent circuit of the electromagnet.

There are operating air gaps , parasitic air-gaps . The respective permeances mast be calculated and also total permeance of air gaps Л_дУ.

To define magnetic permeance of operating air gap Л_д one must use the method based on specific magnetic permeances:

;

where Л_T is the frontal permeance, H;

Л_B is the fringe permeance, H.

Frontal magnetic permeance is the permeance of magnetic flux straight from the armature to the joke. It may be defined by formula:

where м_o is magnetic permeability, м_o = 12,56•10^-7 H/m;

d is the diameter of the core, d = 25•10^-3 m;

д is the size of the operating air-gap, д = 10•10^-3 m.

Using formula (1.2) and substituting all values one may obtain:

Fringe permeance may be found by specific permeances:

where л_p and л_z are specific magnetic permeances that may be defined using curves of specific magnetic permeances by [2] :



where z is the width of flank magnetic flux, z = 6 mm.

So, using formula (1.3):

The value of fringe permeance will be

And then calculate by formula (1.1):

To define parasitic permeance Л_д1 one must use the formula for frontal permeance, because fringe permeance may be neglected because the air gap is more than 10 times smaller than the size of the pole:

where a is the height of the joke, a = 8 mm;

b is the width, b = 70 mm;

д₁ is the parasitic air gap

So by formula (1.4):



Than parasitic air gap permeance Л_д2 must be determined:

To define magnetic permeance Л_д3 of the air gap д₃ one must determine the value of the air gap, and then to use formula (1.1):

Figure 1.3 - Geometrical finding of air gap д₃

To determine the frontal permeance of the air-gap д₃ one must use the formula:



where:

д₃ is the operating air gap, д₃ = 0,02 m;

м_o is magnetic permeability, м_o = 12,56•10^-7 H/m.

Than one must determine the value of the flank permeance of this air-gap:

where л_p and л_z are specific magnetic permeances that may be defined using curves of specific magnetic permeances for square conductors by [2] :

Then, substituting obtained values in formula (1.5) one may obtain:



The sum of two obtained magnetic permeances for air-gap д₃ gives the total permeance for this gap:

And to determine the total magnetic permeance one must neglect the and use the formula for series and parallel connection:

.

To define specific leakage permeance it's necessary to use the formula of determination of leakage permeance for cylinder parallel to a plane:

where n = h/r=6,4;

r is the radius of the core, r = 12,5 mm.

Then by formula (1.6):

It is seen, that b<4, 70< 4•80, mm. The corresponding coefficient varies k_a=0,85…0,92. The chosen one is k_a =0,92:



The bulging ratio for the operating air-gap у_вп is the ratio of total permeance to the frontal permeance:

1.3 Calculation of magnetic flux and electromagnetic force

The magnetic flux through the operating air-gap Ц_д is product of induction by cross-section and by bulging ratio:

(1.7)

Where B_д is induction in air-gap, T;

S_T is the pole face area,m², it can be defined as:

So by formula (1.7):

The derivative of operating air-gap permeance is determined as:



The sign « - « can be neglected, as it shows only direction of the force.

Electromagnetic force is defined by the energetic formula:

where (Iw)_д is MMF in the working air gap, which is defined according to Ohm's law:

With respect to formula (1.8):

.

The value of the electromagnetic force is respectively high, taking into account values of working air gaps.

1.4 Calculation of magnetic circuit

To calculate the magnetic circuit with accuracy it had been divided the magnetic circuit into three sub circuits. Calculations are performed with the help of leakage koefficients, and it is shown on figure 1.4:



Figure 1.4 - Magnetic circuit divided into three sub circuits

The magnetic flux in certain sub circuit can be calculated by means of leakage coefficient:

(1.8)

Where у_x - leakage factor in certain sub circuit.

For the clapper-type electromagnet leakage factor is equal to[2]:

Where x is the middle distance from the working air-gap to the sub circuit.

And for example for second parasitic air-gap sub circuit the leakage factor will be:



And by formula (1.8):

For other sub circuits values of leakage factors and fluxes are given in table (1.2).

The magnet induction for an iron sub circuits should be determined as ratio of flux to cross-section:

where S_i is the cross-section of the respective iron sub circuit, m².

In case of sub circuit 1c it is equal to:

The next step will be calculating of the dimensions of every sub circuit:

For sub circuits 1c and 1a: ;

For sub circuits 2c and 2a: ;

For sub circuits 3c and 3a: ;

For the base x will be equal to the length, and for the armature x is equal to the product

Then, one must determine all values of leakage factor for air-gap sub circuits using necessary dimension and put all obtained values into the table 1.2.

After that each value of magnetic flux for every sub circuit must be calculated.

And then, by formula (1.9) one must calculate magnetic induction for each sub circuit, for example for sub circuit 1c:

where S is the cross-section of each sub circuit, which must be determined as follows:

For sub circuits 1c, 2c, and 3c: .

For sub circuits 1a, 2a, 3a, also base and armature: .

The magnetic intensity H_i is determined by the magnetization curve [5]. Its value for all sub circuits is given in table (1.2).

The magnetic potential fall for each sub circuit is equal to product of magnetic intensity to its length:

(1.10)

where l_i is the length of certain sub circuit, calculated in the table (1.2).

As it had been divided the electromagnet onto 3 equal zones then:



And for armature and base:

And by formula (1.10):

For all other sub circuits the calculated data is given in table 1.2.

The sum of all magnetic potential falls is:

The required value of MMF of electromagnet winding is determined as the total magnetic potential drops across iron sub circuits as well as both operating and parasitic air-gaps:



And taking into consideration safety factor (k_s=1,3-1,5) MMF of winding will be equal to:

Table 1.2 - Calculated data

	д	д1	д2	д3	arm	1c	2c	3c	1a	2a	3a	base
x, m	0	0	32	0	0	10	30	50	10	30	50	60
у	1	1	1.03	1	1	1.005	1.01	1.02	1.005	1.01	1.02	1
?, Wb	2.83	2.83	2.91	2.83	2.83	2.84	2.86	2.89	2.84	2.86	2.89	2.9
S, m2	-	-	-	-	9.4	5.02	5.02	5.02	6	6	6	9.4
B, T	-	-	-	-	0.3	0.56	0.57	0.58	0.47	0.476	0.48	0.3
H, A/m	-	-	-	-	154	160	162	164	157	157	158	154
l, m	-	-	-	-	74	20	20	20	20	20	20	74
Umi	-	-	-	-	11.3	3.2	3.24	3.28	3.14	3.14	3.16	11.3



2. Electromagnet winding design

2.1 Calculation of winding parameters

Diameter of a winding turn can be found by empirical formula:

where с₀ is specific resistance of the wire, for copper its value is equal to с₀=1.62;

б is the temperature coefficient of the wire, ;

is permissible temperature у_perm=105 C?;

l_w.m. is the average length of a winding, and it's found as it is shown below:

And by formula (2.1):

Using [2], it had been chosen the diameter of a wire and coefficient of filling of a wire _.

To define the cross-section of a winding it had been used the formula:



The number of the winding turns is determined by empirical formula:

Where j_perm is permissible current density, which in this case may be accepted in the range 2-4 A/мм² So the number of windings is:

Assuming wire height equal to height of magnetic circuit «window», its width h', m can be expressed as ratio of area of a wire to its height:

where S is the area of winding, m².

It can be found as ratio of product of number of turns by cross-section of a winding to coefficient of filling of a wire:

where is the crosssection of the winding;

is the filling coefficient of the wire.



And by formula (2.2):

2.2 Determination of the precised values of a pure resistance and MMF of a winding

The precised value of pure resistance R can be found as product of specific resistance of copper by average length of winding by number of turns divided by cross-section of a winding:

The precised values of MMF can be found as product of current by number of turns:

where I is the current in one winding. And it's found as ratio of voltage to resistance:

So by formula (2.3):



The precised value of the MMF has been calculated.

3. Determination of electromagnetic force

electromagnet calculation equivalent

The permeances can be calculated by formula (1.1), but since here the value of closed gap is too little, the flank permeance must be neglected and so for the working air-gap permeance will be equal to:

Then for parasitic air-gaps:

doesn't change so it's value is:

The magnetic resistances of air-gaps are inversely proportional to their permeances. Here are its respective values:



The total magnetic resistivity is calculated by the formula:

where are respectively magnetic resistances of all air-gaps, performed in the design.

The initial calculation of magnetic flux through the operating air-gap without taking into account magnetic potential fall across iron sub circuits is defined as ratio of MMF to total magnetic resistivity:

The magnetic induction through the magnetic circuit , without taking into account the magnetic leakage flux, can be found by the empirical formula:



First and furthers approximations of magnetic flux through the operating air-gap taking into account magnetic potential fall across iron sub circuit is defined as difference between MMF and magnetic potential falls divided on total magnetic resistivity:

where H is the magnetic intensity, determined by the graph of magnetic curves [5]:

;

is the length of magnetic flux in the armature and rectangle part of the core:

and is the length of magnetic flux in the cylindrical part of core:

So by (3.1) the second approximation with taking into account values of H, determined by the graph of the magnetic curves [5], will be look like:



Table 3.1 - Results of the approximation.

	0	1	2	…	7
B, T			19.6	…	25
Ф, Wb				…

After several approximations it had been defined the value of the magnetic flux:

The electromagnetic force based can be found by Maxwell's formula:



4. Determination of the actuation time of an electromagnet

The inductance of the winding for the initial value of the operating air-gap is the product of number of turns squared by total permeance for open armature:

The pick-up time of an electromagnet assuming safety factor in range 1,3-1,5 can be found by empirical formula:

where R is the active resistance of the winding.

The mass of movable parts and moving time of the electromagnet is product of density of armature by its volume:

where с_s is density of material,

V is the volume of armature:



Then by (4.1):

Moving time is found by next empirical formula:

where s is path of armature, s=д=0.003 m;

(P_e - P_p)_m - is the average value of acting and counteracting forces difference.

Assuming that the average magnitude of counteracting force comprises 70 % from the average value of electromagnetic force:

So by (4.2):

The actuation (response) time of the electromagnet is the sum of pick-up time and moving time: The electromagnet is a high-speed one.



Conclusions

During the course design it had been calculated all necessary parameters such as magnetic permeances, resistances, magnetic fluxes, induction. It was found the equivalent scheme of the circuit. Also it had been determined the time characteristics of the electromagnet: actuation time. The electromagnet is a high-speed one. These values fully depend on the features of the movable part - armature and the value of MMF, that constantly depend on current and voltage.

The widest and important using of the DC electromagnet is performed at commutating apparatuses and electrical switchgear, electric cars and the devices entering into systems of industrial automatics, in equipment of regulation, protection of electro technical installations. As a part of various mechanisms electromagnets are used as a drive for realization of necessary forward moving (turn) of working parts of cars or for creation of keeping force. As an example such the electromagnet of load-lifting cars, the electromagnet of couplings and brakes, electromagnets are applied in various actuators, contactors, switches, electric devices, etc. Depending on appointment the electromagnet may weigh from shares of grams to hundreds of tones, consuming electric capacity from shares of Watt to tens MWt.



References

1. Лоптев - Справочник по высокоточным электрическим аппаратам

2. Буткевич Г.В., Дегтярь В.Г., Сливинская А.Г. Задачник по электрическим аппаратам. - М.: Высшая школа, 1987. - 232 с.

3. Правила устройства электроустановок. - М.: Энергия, 2002. - 638 с.

4. Таев И.С. Электрические аппараты: Общая теория. - М.: Энергия, 1977. - 272 с.

5. Сахаров В.П. - Проектрирование электрических аппаратов. Общие вопросы проектирования: Учебное пособие для студентов электротехнических вузов. - М.: Энергия, 1971. - 560 с.

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