Rutherford was bewildered. He brilliantly succeeded in revealing the internal structure of the atom, however, in doing this, the scientist revealed the largest conflict in physical science. The gold foil experiment demonstrated that the atom is a tiny "planetary" system. However, the theory of electromagnetism predicted that such a system was categorically unstable - it would not last even "the blink of an eye". It was a paradoxical situation, and finding a way out of it seemed almost impossible. Nevertheless, one man - a young Danish physicist - succeeded.

Niels Bohr (1885–1962) came to England in 1911 after receiving his doctorate in Copenhagen and has since worked under first J. J. Thomson and then Rutherford. He understood that Rutherford's planetary model of the atom, supported by serious experimental data, was quite convincing. But at the same time, he understood that the laws of electromagnetism, which gave the world electric motors and dynamos, are no less convincing. Bohr's revolutionary resolution of the atomic paradox was both simple and bold. Bohr announced in 1913 that the laws of electromagnetism simply do not apply inside atoms. Electrons revolving around the nucleus do not emit electromagnetic waves and therefore do not fall in a spiral onto the nucleus. In short, the known laws of physics do not apply to the field of ultra-small objects.

The word "induction" in Russian means the processes of excitation, guidance, creation of something. In electrical engineering, this term has been used for more than two centuries.

After getting acquainted with the publications of 1821, describing the experiments of the Danish scientist Oersted on the deviations of a magnetic needle near a conductor with electric current, Michael Faraday set himself the task: convert magnetism to electricity.

After 10 years of research, he formulated the basic law of electromagnetic induction, explaining that inside any closed circuit, an electromotive force is induced. Its value is determined by the rate of change of the magnetic flux penetrating the circuit under consideration, but taken with a minus sign.

Transmission of electromagnetic waves over a distance

The first guess that dawned on the brain of a scientist was not crowned with practical success.

He placed two closed conductors side by side. Near one I installed a magnetic needle as an indicator of the passing current, and in the other wire I applied a pulse from a powerful galvanic source of that time: a volt column.

The researcher assumed that with a current pulse in the first circuit, the changing magnetic field in it would induce a current in the second conductor, which would deflect the magnetic needle. But, the result was negative - the indicator did not work. Or rather, he lacked sensitivity.

The scientist's brain foresaw the creation and transmission of electromagnetic waves over a distance, which are now used in radio broadcasting, television, wireless control, Wi-Fi technologies and similar devices. He was simply let down by the imperfect element base of the measuring devices of that time.

Power generation

After an unsuccessful experiment, Michael Faraday modified the conditions of the experiment.

For the experiment, Faraday used two coils with closed circuits. In the first circuit, he supplied an electric current from a source, and in the second he observed the appearance of an EMF. The current passing through the turns of winding No. 1 created a magnetic flux around the coil, penetrating winding No. 2 and forming an electromotive force in it.

During Faraday's experiment:

• turned on the pulse supply of voltage to the circuit with stationary coils;
• when the current was applied, he injected the upper one into the lower coil;
• permanently fixed winding No. 1 and introduced winding No. 2 into it;
• change the speed of movement of the coils relative to each other.

In all these cases, he observed the manifestation of the induction emf in the second coil. And only with the passage of direct current through the winding No. 1 and the fixed coils of guidance, there was no electromotive force.

The scientist determined that the EMF induced in the second coil depends on the speed at which the magnetic flux changes. It is proportional to its size.

The same pattern is fully manifested when a closed loop passes through. Under the action of the EMF, an electric current is formed in the wire.

The magnetic flux in the case under consideration changes in the circuit Sk created by a closed circuit.

In this way, the development created by Faraday made it possible to place a rotating conductive frame in a magnetic field.

It was then made from a large number of turns, fixed in rotation bearings. At the ends of the winding, slip rings and brushes sliding along them were mounted, and a load was connected through the leads on the case. The result was a modern alternator.

Its simpler design was created when the winding was fixed on a stationary case, and the magnetic system began to rotate. In this case, the method of generating currents at the expense was not violated in any way.

The principle of operation of electric motors

The law of electromagnetic induction, which Michael Faraday substantiated, made it possible to create various designs of electric motors. They have a similar device with generators: a movable rotor and a stator, which interact with each other due to rotating electromagnetic fields.

Electricity transformation

Michael Faraday determined the occurrence of an induced electromotive force and an induction current in a nearby winding when the magnetic field in the adjacent coil changes.

The current inside the nearby winding is induced by switching the switch circuit in coil 1 and is always present during the operation of the generator on winding 3.

On this property, called mutual induction, the operation of all modern transformer devices is based.

To improve the passage of the magnetic flux, they have insulated windings put on a common core, which has a minimum magnetic resistance. It is made from special grades of steel and formed in typesetting thin sheets in the form of sections of a certain shape, called a magnetic circuit.

Transformers transmit, due to mutual induction, the energy of an alternating electromagnetic field from one winding to another in such a way that a change occurs, a transformation of the voltage value at its input and output terminals.

The ratio of the number of turns in the windings determines transformation ratio, and the thickness of the wire, the design and volume of the core material - the amount of transmitted power, the operating current.

Work of inductors

The manifestation of electromagnetic induction is observed in the coil during a change in the magnitude of the current flowing in it. This process is called self-induction.

When the switch is turned on in the above diagram, the inductive current modifies the nature of the rectilinear increase in the operating current in the circuit, as well as during the trip.

When an alternating voltage, rather than a constant voltage, is applied to the conductor wound into a coil, the current value reduced by the inductive resistance flows through it. The energy of self-induction shifts the phase of the current with respect to the applied voltage.

This phenomenon is used in chokes, which are designed to reduce the high currents that occur under certain operating conditions of the equipment. Such devices, in particular, are used.

The design feature of the magnetic circuit at the inductor is the cut of the plates, which is created to further increase the magnetic resistance to the magnetic flux due to the formation of an air gap.

Chokes with a split and adjustable position of the magnetic circuit are used in many radio engineering and electrical devices. Quite often they can be found in the designs of welding transformers. They reduce the magnitude of the electric arc passed through the electrode to the optimum value.

Induction Furnaces

The phenomenon of electromagnetic induction manifests itself not only in wires and windings, but also inside any massive metal objects. The currents induced in them are called eddy currents. During the operation of transformers and chokes, they cause heating of the magnetic circuit and the entire structure.

To prevent this phenomenon, the cores are made of thin metal sheets and insulated between themselves with a layer of varnish that prevents the passage of induced currents.

In heating structures, eddy currents do not limit, but create the most favorable conditions for their passage. are widely used in industrial production to create high temperatures.

Electrical measuring devices

A large class of induction devices continues to operate in the energy sector. Electric meters with a rotating aluminum disk, similar to the design of power relays, resting systems of pointer meters operate on the basis of the principle of electromagnetic induction.

Gas magnetic generators

If, instead of a closed frame, a conductive gas, liquid or plasma is moved in the field of a magnet, then the charges of electricity under the action of magnetic field lines will deviate in strictly defined directions, forming an electric current. Its magnetic field on the mounted electrode contact plates induces an electromotive force. Under its action, an electric current is created in the connected circuit to the MHD generator.

This is how the law of electromagnetic induction manifests itself in MHD generators.

There are no such complex rotating parts as the rotor. This simplifies the design, allows you to significantly increase the temperature of the working environment, and, at the same time, the efficiency of power generation. MHD generators operate as backup or emergency sources capable of generating significant electricity flows in short periods of time.

Thus, the law of electromagnetic induction, justified by Michael Faraday at one time, continues to be relevant today.

The first law of electromagnetism describes the flow of an electric field:

where ε 0 is some constant (read epsilon zero). If there are no charges inside the surface, but there are charges outside it (even very close to it), then all the same average the normal component of E is zero, so there is no flow through the surface. To show the usefulness of this type of statement, we will prove that equation (1.6) coincides with Coulomb's law, if only we take into account that the field of an individual charge must be spherically symmetric. Draw a sphere around a point charge. Then the average normal component is exactly equal to the value of E at any point, because the field must be directed along the radius and have the same magnitude at all points on the sphere. Our rule then states that the field on the surface of the sphere times the area of ​​the sphere (i.e., the flux flowing out of the sphere) is proportional to the charge inside it. If you increase the radius of a sphere, then its area increases as the square of the radius. The product of the average normal component of the electric field and this area must still be equal to the internal charge, so the field must decrease as the square of the distance; thus the field of "inverse squares" is obtained.

If we take an arbitrary curve in space and measure the circulation of the electric field along this curve, then it turns out that in the general case it is not equal to zero (although this is the case in the Coulomb field). Instead, the second law holds for electricity, stating that

And, finally, the formulation of the laws of the electromagnetic field will be completed if we write two corresponding equations for the magnetic field B:

And for the surface S, bounded curve WITH:

The constant c 2 that appeared in equation (1.9) is the square of the speed of light. Its appearance is justified by the fact that magnetism is essentially a relativistic manifestation of electricity. And the constant ε 0 was set in order for the usual units of electric current strength to arise.

Equations (1.6) - (1.9), as well as equation (1.1) - these are all the laws of electrodynamics. As you remember, Newton's laws were very easy to write, but many complex consequences followed from them, so it took a long time to study them all. The laws of electromagnetism are incomparably more difficult to write, and we must expect the consequences of them to be much more complicated, and now we will have to understand them for a very long time.

We can illustrate some of the laws of electrodynamics with a series of simple experiments that can show us at least qualitatively the relationship between electric and magnetic fields. You get to know the first term in equation (1.1) by combing your hair, so we won't talk about it. The second term in equation (1.1) can be demonstrated by passing a current through a wire suspended over a magnetic bar, as shown in Fig. 1.6. When the current is turned on, the wire moves due to the fact that a force acts on it F = qvXB. When a current flows through the wire, the charges inside it move, that is, they have a speed v, and the magnetic field of the magnet acts on them, as a result of which the wire moves away.

When the wire is pushed to the left, the magnet itself can be expected to experience a push to the right. (Otherwise, this whole device could be mounted on a platform and get a reactive system in which momentum would not be conserved!) Although the force is too small to notice the movement of a magnetic wand, the movement of a more sensitive device, say a compass needle, is quite noticeable.

How does the current in the wire push the magnet? The current flowing through the wire creates its own magnetic field around it, which acts on the magnet. In accordance with the last term in equation (1.9), the current should lead to circulations vector B; in our case, field lines B are closed around the wire, as shown in fig. 1.7. It is this field B that is responsible for the force acting on the magnet.

Equation (1.9) tells us that for a given amount of current flowing through the wire, the circulation of the field B is the same for any curve surrounding the wire. Those curves (circles, for example) that lie far from the wire have a longer length, so the tangent component B must decrease. You can see that B should be expected to decrease linearly with distance from a long straight wire.

We said that the current flowing through the wire forms a magnetic field around it, and that if there is a magnetic field, then it acts with some force on the wire through which the current flows. So, one should think that if a magnetic field is created by a current flowing in one wire, then it will act with some force on the other wire, through which the current also flows. This can be shown by using two freely suspended wires (Fig. 1.8). When the direction of the currents is the same, the wires attract, and when the directions are opposite, they repel.

In short, electric currents, like magnets, create magnetic fields. But then what is a magnet? Since magnetic fields are created by moving charges, can it not turn out that the magnetic field created by a piece of iron is actually the result of the action of currents? Apparently, that's the way it is. In our experiments it is possible to replace the magnetic stick with a coil of wound wire, as shown in Fig. 1.9. When the current passes through the coil (as well as through a straight wire above it), exactly the same movement of the conductor is observed as before, when a magnet was in place of the coil. Everything looks as if a current circulates continuously inside a piece of iron. Indeed, the properties of magnets can be understood as a continuous current within the iron atoms. The force acting on the magnet in Fig. 1.7 is explained by the second term in equation (1.1).

Where do these currents come from? One source is the movement of electrons in atomic orbits. In iron this is not the case, but in some materials the origin of magnetism is precisely this. In addition to rotating around the nucleus of an atom, the electron also rotates around its own axis (something similar to the rotation of the Earth); it is from this rotation that a current arises, which creates a magnetic field of iron. (We said "something like the rotation of the Earth," because in fact, in quantum mechanics, the issue is so deep that it does not fit well into classical concepts.) In most substances, some electrons spin in one direction, and some in the other, so that magnetism disappears, and in iron (for a mysterious reason, which we will discuss later) many electrons rotate so that their axes point in the same direction and this is the source of magnetism.

Since the fields of magnets are generated by currents, there is no need to insert additional terms into equations (1.8) and (1.9) that take into account the existence of magnets. These equations are about all currents, including circular currents from rotating electrons, and the law turns out to be correct. It should also be noted that, according to equation (1.8), there are no magnetic charges similar to electric charges on the right side of equation (1.6). They have never been discovered.

The first term on the right side of equation (1.9) was discovered theoretically by Maxwell; he is very important. He says change electrical fields causes magnetic phenomena. In fact, without this term, the equation would lose its meaning, because without it the currents in open circuits would disappear. But in fact, such currents exist; the following example speaks of this. Imagine a capacitor made up of two flat plates. It is charged by current flowing into one of the plates and outflowing from the other, as shown in Fig. 1.10. Draw a curve around one of the wires With and pull on it a surface (surface S 1) that will cross the wire. In accordance with equation (1.9), the circulation of the field B along the curve With is given by the amount of current in the wire (multiplied by from 2). But what happens if we pull on the curve another surface S 2 in the form of a cup, the bottom of which is located between the plates of the capacitor and does not touch the wire? No current passes through such a surface, of course. But a simple change in the position and shape of an imaginary surface should not change the real magnetic field! The circulation of field B must remain the same. Indeed, the first term on the right-hand side of equation (1.9) is combined with the second term in such a way that for both surfaces S1 and S 2 the same effect occurs. For S 2 the circulation of the vector B is expressed in terms of the degree of change in the flow of the vector E from one plate to another. And it turns out that the change in E is connected with the current just so that equation (1.9) is satisfied. Maxwell saw the need for this and was the first to write the complete equation.

With the device shown in Fig. 1.6, another law of electromagnetism can be demonstrated. Disconnect the ends of the hanging wire from the battery and attach them to a galvanometer - a device that records the passage of current through the wire. Stands only in the field of a magnet swing wire, as the current will immediately flow through it. This is a new consequence of equation (1.1): the electrons in the wire will feel the action of the force F=qv X B. Their speed is now directed to the side, because they are deflected along with the wire. This v, together with the vertically directed field B of the magnet, results in a force acting on the electrons along wires, and the electrons are sent to the galvanometer.

Let us suppose, however, that we leave the wire alone and begin to move the magnet. We feel that there should be no difference, because the relative motion is the same, and indeed the current flows through the galvanometer. But how does a magnetic field act on charges at rest? In accordance with equation (1.1), an electric field should arise. A moving magnet must create an electric field. The question of how this happens is answered quantitatively by Eq. (1.7). This equation describes many practically very important phenomena occurring in electrical generators and transformers.

The most remarkable consequence of our equations is that, by combining equations (1.7) and (1.9), one can understand why electromagnetic phenomena propagate over long distances. The reason for this, roughly speaking, is something like this: suppose that somewhere there is a magnetic field that increases in magnitude, say, because a current is suddenly passed through the wire. Then it follows from equation (1.7) that the circulation of the electric field should occur. When the electric field begins to gradually increase for circulation to occur, then, according to equation (1.9), magnetic circulation must also occur. But the rise this the magnetic field will create a new circulation of the electric field, etc. In this way, the fields propagate through space, requiring neither charges nor currents anywhere but the source of the fields. It is in this way that we see each other! All this is hidden in the equations of the electromagnetic field.

Translation of an article fromhttp://www.coilgun.eclipse.co.uk/ by Roman.

Fundamentals of electromagnetism

In this section, we will look at general electromagnetic principles that are widely used in engineering. This is a very brief introduction to such a complex topic. You must find yourself a good book on magnetism and electromagnetism if you want to understand this section better. You can also find most of these concepts detailed in Fizzics Fizzle (http://library.thinkquest.org/16600/advanced/electricityandmagnetism.shtml).

electromagnetic fieldsandstrength

Before we consider a special case - coilgun -a, we need to briefly familiarize ourselves with the basics of electromagnetic fields and forces. Whenever there is a moving charge, there is a corresponding magnetic field associated with it. It can arise due to the current in the conductor, the rotation of an electron in its orbit, the plasma flow, etc. To facilitate the understanding of electromagnetism, we use the concept of the electromagnetic field and magnetic poles. Differential vector equations that describe this field have been developed James Clark Maxwell.

1. Measurement systems

Just to make life more difficult, there are three measurement systems that are popularly used. They're called Sommerfield, Kennely and Gaussian . Since each system has different elements (names) for many of the same things, it can be confusing. I will use Sommerfield The system shown below:

Quantity
Field (Tension)
magnetic flux		weber (W)
Induction		tesla(T)
Magnetization
Magnetization intensity
Moment

Table 1 Measurement system

2. LawBio- Savara

Using the Biot-Savart law, you can determine the magnetic field created by an elementary current .

Figure 2.1

Ex.. 2.1

where H field component at a distance r , created by current i , current in the elementary section of the conductor of length l . u unit vector directed radially from l .

We can determine the magnetic field created by the combination of several elementary currents using this law. Consider an infinitely long conductor carrying a current i . We can use Biot-Savart's law to obtain a basic solution for the field at any distance from the conductor. I will not give the derivation of this solution here, any book on electromagnetism will show this in detail. Basic solution:

Ex.. 2.2

Figure 2.2

The field with respect to the current-carrying conductor is cyclic and concentric.

(The direction of the magnetic lines (vectors H, B) is determined by the gimlet (corkscrew) rule. If the translational movement of the gimlet corresponds to the direction of the current in the conductor, then the rotational direction of the handle will indicate the direction of the vectors.)

Another case that has an analytical solution is the axial field of a coil with current. So far, we can get an analytical solution for the axial field, but this cannot be done for the field as a whole. To find the field at some arbitrary point, we need to solve complex integral equations, which is best done using digital methods.

3. Ampère's law

This is an alternative method for determining the magnetic field, using a group of current-carrying conductors. The law can be written as:

Ex. 3.1

where N current-carrying conductor number i and llinear vector. The integration should form a closed line around the current-carrying conductor. Considering an infinite current-carrying conductor, we can again apply Ampère's law as shown below:

Figure 3.1

We know that the field is cyclic and concentric around a current-carrying conductor, soHcan be integrated around the ring (around the conductor with current) at a distance r , which gives us:

Ex . 3.2

The integration is very simple and shows how Ampère's law can be applied to get a fast solution in some cases (configurations). Knowledge of the field structure is necessary before this law can be applied.

(Field (strength) in the center of the circular field (coil with current))

4. Solenoid field

As the charge moves in the coil, it creates a magnetic field, the direction of which can be determined using the right hand rule (take your right hand, bend your fingers in the direction of the current, bend your thumb, the direction indicated by the thumb points to the magnetic north of your coil) . The convention for magnetic flux says that magnetic flux starts at the north pole and ends at the south. ( The convention for the direction of flux has the flux emerging from a north pole and terminating on a south pole ). The field and flux lines are closed turns around the coil. Remember that these lines don't really exist, they just connect points of equal value. It's a bit like contours on a map, where the lines show points of the same height. The height of the ground changes continuously between these contours. Similarly, the field and magnetic flux are continuous (the change is not necessarily smooth - a discrete change in permeability causes a sharp change in the value of the field, a bit like rocks on a map).

Figure 4.1

If the solenoid is long and thin, then the field inside the solenoid can be considered almost uniform.

5. Ferromagnetic materials

Perhaps the most well-known ferromagnetic material is iron, but there are other elements such as cobalt and nickel, as well as numerous alloys such as silicon steel. Each material has a special property that makes it suitable for its application. So what do we mean by ferromagnetic material? It's simple, a ferromagnetic material is attracted to a magnet. While this is true, it's hardly a useful definition, and it doesn't tell us why attraction occurs. A detailed theory of the magnetism of materials is a very complex topic involving quantum mechanics, so we will stick to a simple conceptual description. As you know, the flow of charges creates a magnetic field, so when we detect the movement of a charge, we must expect an associated magnetic field. In ferromagnetic materials, the orbits of electrons are distributed in such an order that a small magnetic field is created. Then this means that the material consists of many tiny current-carrying coils, which have their own magnetic fields. Usually, coils oriented in the same direction are combined into small groups called domains. The domains point in an arbitrary direction in the material, so there is no overall magnetic field in the material (the resulting field is zero). However, if we apply an external field to the ferromagnetic material from a coil or a permanent magnet, the coils with currents turn in the direction with this field.(However if we apply an external field to the ferromagnetic material from a coil or permanent magnet, the current loops try and align with this field - the domians which are most aligned with the field "grow" at the expense of the less well aligned domains ). When this happens, the result will be magnetization and attraction between the material and the magnet/coil.

6. Magneticinductionandpermeability

Receiving a magnetic field has an associated magnetic flux density, also known as magnetic induction. InductionB connected to the field through the permeability of the medium through which the field propagates.

Ex. 6.1

where 0 is the permeability in vacuum and r relative permeability. Induction measured in teslas (T).

(The intensity of the magnetic field depends on the medium in which it occurs. Comparing the magnetic field in a wire located in a given medium and in vacuum, it was found that, depending on the properties of the medium (material), the field is stronger than in vacuum (paramagnetic materials or media ), or, conversely, weaker (diamagnetic materials and media).The magnetic properties of the medium are characterized by the absolute magnetic permeability μ a.

The absolute magnetic permeability of vacuum is called the magnetic constant μ 0 . The absolute magnetic permeability of various substances (media) is compared with the magnetic constant (magnetic permeability of a vacuum). The ratio of the absolute magnetic permeability of a substance to the magnetic constant is called magnetic permeability (or relative magnetic permeability), so that

Relative magnetic permeability is an abstract number. For diamagnetic substances μ r < 1, например для меди μ r= 0.999995. For paramagnetic substances μ r> 1, e.g. for air μ r= 1.0000031. In technical calculations, the relative magnetic permeability of diamagnetic and paramagnetic substances is assumed to be 1.

For ferromagnetic materials, which play an extremely important role in electrical engineering, the magnetic permeability has different values ​​depending on the properties of the material, the magnitude of the magnetic field, temperature and reach values tens of thousands.)

7. Magnetization

The magnetization of a material is a measure of its magnetic ‘strength’. The magnetization may be inherent in the material, such as a permanent magnet, or it may be caused by an external magnetic field source, such as a solenoid. The magnetic induction in a material can be expressed as the sum of the magnetization vectorsM and magnetic fieldH .

Ex. 7.1

(Electrons in atoms, moving along closed orbits or elementary circuits around the nucleus of an atom, form elementary currents or magnetic dipoles. The magnetic dipole can be characterized by the vector - magnetic moment dipole or elementary electric current m , the value of which is equal to the product of the elementary current i and elementary platform S , Fig.8e.0.1, limited by an elementary structure.

Rice. 8d.0.1

Vectorm directed perpendicular to the site S ; , its direction is determined by the gimlet rule. A vector quantity equal to the geometric sum of the magnetic moments of all elementary molecular currents in the body under consideration (substance volume) is magnetic moment of the body

Vector quantity determined by the ratio of the magnetic moment M ’ to volumeV , called average body magnetization or medium magnetization intensity

If the ferromagnet is not in an external magnetic field, then the magnetic moments of individual domains are directed in a very different way, so that the total magnetic moment of the body turns out to be equal to zero, i.e. ferromagnet is not magnetized. The introduction of a ferromagnet into an external magnetic field causes: 1-turn of the magnetic domains in the direction of the external field - the process of orientation; 2-an increase in the size of those domains whose moments are close to the direction of the field, and a decrease in domains with oppositely directed magnetic moments - the process of displacement of the domain boundaries. As a result, the ferromagnet is magnetized. If, with an increase in the external magnetic field, all spontaneously magnetized sections are oriented in the direction of the external field and the growth of domains stops, then the state of the limiting magnetization of the ferromagnet, called magnetic saturation.

At a field strength H, magnetic induction in a non-ferromagnetic medium (μ r= 1) would be equal to B 0 =μ 0 H. In a ferromagnetic medium, this induction is added to the induction of an additional magnetic field Bd= μ 0 M.Resulting magnetic induction in a ferromagnetic material B= B 0 + Bd=μ 0 ( H+ M).)

8. Magnetomotive force (mfs)

This is an analogue of the electromotive force (EMF) and is used in magnetic circuits to determine the magnetic flux density in different directions of the circuit. MDS measured in ampere - turns or simply in amperes . The magnetic circuit is equivalent to resistance and is called magnetic resistance, which is defined as

Ex . 8.1

where lchain path length, permeability andAcross-sectional area.

Let's take a look at a simple magnetic circuit:

Rice . 8.1

The torus has an average radius r and cross-sectional area A . MDS is generated by a coil with N coils in which current flows i . The calculation of magnetic resistance is complicated by nonlinearities in the permeability of the material.

Ex . 8.2

If the magnetic resistance is determined, then we can calculate the magnetic flux that is present in the circuit.

9. Demagnetizing fields

If a piece of ferromagnetic material, in the form of a bar, is magnetized, then poles will appear at its ends. These poles generate an internal field that tries to demagnetize the material - it acts in the opposite direction of the field that creates the magnetization. As a result, the internal field will be much smaller than the external one. The shape of the material matters a lot to the demagnetizing field, a long thin rod (large length/diameter ratio) has a small demagnetizing field compared to, say, a wide shape like a sphere. In the development perspective coilgun this means that a projectile with a small length/diameter ratio requires a stronger external field to achieve a certain state of magnetization. Take a look on the chart below. It shows the resulting internal field along the axis of two projectiles - one 20 mm long and 10 mm in diameter and the other 10 mm long and 20 mm in diameter. For the same external field, we see a big difference in internal fields, the shorter projectile has a peak of about 40% of the long projectile's peak. This is a very successful result, showing the difference between different forms of projectiles.

Rice . 9.1

It should be noted that poles are formed only where there is a continuous permeability of the material. On a closed magnetic path, like a torus, poles do not arise, and there is no demagnetizing field.

10. Force acting on a charged particle

So, how do we calculate the force acting on a conductor with current? Let's start by looking at the force acting on a charge moving in a magnetic field. ( I "ll adopt the general approach in 3 dimensions ).

Ex . 10.1

This force is determined by the intersection of the velocity vectorsvand magnetic inductionB, and it is proportional to the magnitude of the charge. Consider the charge q = -1.6x 10 -19 K, moving at a speed of 500m/s in a magnetic field with an induction of 0.1 T l as shown below.

Rice . 10.1. The effect of a force on a moving charge

The force experienced by the charge can be simply calculated as shown below:

Speed ​​vector 500i m/s and induction 0.1 k T , so we have:

Obviously, if nothing resists this force, the particle willdeviate (it will have to describe a circle in the plane x-y for the case above). There are many interesting special cases that can be obtained with free charges and magnetic fields - you have only read about one of them.

11. Force acting on a conductor with current

Now let's refer what we learned to the force acting on a conductor with current. There is two different ways to get the ratio .

We can describe the conditional current as a measure of charge change

Ex . 11.1

Now we can differentiate the force equation given above to get

Ex. 11.2

We combine these equations, we get

Ex. 11.3

d l is a vector showing the direction of the conditional current. The expression can be used to analyze a physical organization such as a DC motor. If a the conductor is straight, then this can be simplified to

Ex. 11.4

The direction of the force always creates a right angle to the magnetic flux and the direction of the current. When is the simplified form used?, the direction of the force is determined by the right hand rule.

12. Induced voltage, Faraday's law, Lenz's law

The last thing we need to consider is the induced voltage. This is simply an extended analysis of the effect of a force on a charged particle. If we take a conductor (something with a mobile charge) and give it some speed V , relative to the magnetic field, a force will act on the free charges, which pushes them to one of the ends of the conductor. In a metal bar there will be a charge separation where the electrons will be collected at one of the ends of the bar. Picture below shows the general idea.

Rice. 12.1 Induced voltage during transverse movement of the conductive bar

Any relative motion between the conductor and the induction of the magnetic field will result in an induced voltage generated by the movement of the charges. However, if the conductor moves parallel to the magnetic flux (along the axis Z in the figure above), then no voltage will be induced.

We can consider another situation where an open planar surface is pierced by a magnetic current. If we place a closed loop there C , then any change in magnetic flux associated with C will generate tension around C.

Rice . 12.2 Magnetic flux associated with the circuit

Now if we imagine the conductor as a closed loop in place C , then a change in the magnetic flux will induce a voltage in this conductor, which will move the current in a circle in this coil. The direction of the current can be determined by applying Lenz's law, which, in simple terms, shows that the result of the action is directed opposite to the action itself. In this case, the induced voltage will drive a current which will prevent the magnetic flux from changing - if the magnetic flux decreases then the current will try to keep the magnetic flux unchanged (counterclockwise), if the magnetic flux increases then the current will prevent this increase (clockwise ) (the direction is determined by the gimlet rule) . Faraday's law establishes the relationship between induced voltage, change in magnetic flux, and time:

Eqn 12.1

Minus takes into account Lenz's law.

13. Inductance

Inductance can be described as the ratio of the associated magnetic flux to the current that this magnetic flux creates. For example, consider a coil of wire with a cross-sectional area A in which flows I.

Rice. 13.1

The inductance itself can be defined as

Eqn 13.1

If there is more than one turn then the expression becomes

Eqn 13.2

where N- number of turns.

It is important to understand that inductance is only a constant if the coil is surrounded by air. When a ferromagnetic material appears as part of a magnetic circuit, then there is a non-linear behavior of the system, which gives a variable inductance.

14. transformationelectromechanical energy

The principles of electromechanical energy conversion apply to all electrical machines and coilgun not an exception. Before consideration coilgun let's imagine a simple linear electrical 'motor' consisting of a stator field and an armature placed in this field. This is shown in fig. 14.1. Note that in this simplified analysis, the voltage source and armature current do not have an inductance associated with them. This means that only the induced voltage in the system is a consequence of the movement of the armature with respect to the magnetic induction.

Rice. 14.1. Primitive linear motor

When voltage is applied to the ends of an armature, the current will be determined according to its resistance. This current will experience a force ( I x B ), causing the anchor to accelerate. Now, using the previously discussed section ( 12 Induced voltage, Faraday's law, Lenz's law ), we have shown the fact that a voltage is induced in a conductor moving in a magnetic field. This induced voltage acts opposite to the applied voltage (according to Lenz's law). Rice. 14.2 shows an equivalent circuit in which electrical energy is converted into thermal energy P T , and mechanical energy P M .

Rice . 14.2. Motor equivalent circuit

Now we need to consider how the mechanical energy of the armature relates to the electrical energy transmitted to it. Since the armature is located at right angles to the field of magnetic induction, the force is determined by the simplified expression 1 1.4

Ex . 14.1

since instantaneous mechanical energy is a product of force and speed, we have

Ex . 14.2

where v- anchor speed. If we apply Kirchhoff's law to a closed circuit, we get the following expressions for the current I.

Ex . 14.3

Now, the induced voltage can be expressed as a function of the armature speed

Ex . 14.4

Substituting vyp . 14.4 in 1 4.3 we get

Ex . 14.5

and substituting vyp.14.5 in 14.2 we get

Ex . 14.6

Now let's consider the thermal energy released in the anchor. It is determined by vyp. 14.7

Ex . 14.7

And finally, we can express the energy supplied to the anchor as

Ex . 14.8

Note also that mechanical energy (vyp.14.2) is the equivalent of current I multiplied by the induced voltage (vyr.14.4).

We can plot these curves to see how the energy delivered to the anchor is combined with the speed range.(We can plot these curves to show how the power supplied to the armature is distributed over a range of speeds).For this analysis to be relevant to coilgun , we will give our variables values ​​that match the accelerator coilgun . Let's start with the current density in the wire, from which we will determine the values ​​of the remaining parameters. The maximum current density during testing was 90 A /mm 2 , so if we choose the length and diameter of the wire as

l = 10 m

D = 1.5x10 -3 m

then the resistance of the wire and the current will be

R = 0.1

I = 160 A

Now we have values ​​for resistance and current, we can determine the voltage

V=16V

All these parameters are necessary to build the static characteristics of the motor.

Rice. 14.3 Performance curves for frictionless motor model

We can make this model a little more realistic by adding a friction force of, say, 2N, so that the reduction in mechanical energy is proportional to the armature speed. The value of this friction is deliberately taken more in order to make the effect of this more obvious. The new set of curves is shown in Figure 14.4.

Rice . 14.4. Performance curves with constant friction

The presence of friction slightly changes the energy curves so that the maximum armature speed is slightly less than in the case of zero friction. The most noticeable difference is the change in the efficiency curve, which now peaks and then drops off sharply when the anchor reaches " no-load speed. This shape of the efficiency curve is typical for a permanent magnet DC motor.

Also noteworthy is how force, and hence acceleration, depends on speed. If we substitute ex.14.5 into ex.14.1 we get an expression for F in terms of speed v.

Ex . 14.9

Having built this dependence, we will get the following graph

Rice. 14.5. The dependence of the force acting on the anchor on the speed

It is clear that the armature starts with the maximum accelerating force, which begins to decrease as soon as the armature begins to move. Although these characteristics give instantaneous values ​​of the actual parameters for a certain speed, they should be useful in order to see how the motor behaves over time, i.e. dynamically.

The dynamic response of a motor can be determined by solving a differential equation that describes its behavior. Rice. 14.6 shows a diagram of the effect of forces on the anchor, from which you can determine the resulting force described by the differential equation.

Rice. 14.6 Diagram of the effect of forces on the anchor

F m and F d are the magnetic and opposing forces, respectively. Since stress is a constant value, we can use Eq. 14.1 and the resulting force Fa , acting on the anchor, will be

. 14.11

If we write acceleration and speed as derivatives of displacement x with respect to time and rearrange the expression , we get differential equation for motion anchors

vyr. 14.12

This is a non-homogeneous second-order differential equation with constant coefficients and can be solved by defining an additional function and a partial integral. The method of solving a straight line (all programs of mathematical universities consider differential equations), so I will simply give the result. One note - this particular solution uses initial conditions:

vyr. 14.14

We need to assign a value to the friction force, magnetic induction and armature mass. Let's choose friction. I will use the 2H value to illustrate how it changes the dynamic performance of the motor. Determining the value of induction that will produce the same accelerating force in the model as it does in the test coil for a given current density requires that we consider the radial component of the magnetic flux density distribution generated by the magnetized projectilecoilgun(this radial component creates an axial force). To do this, it is necessary to integrate the expression obtained by multiplying the current densityDetermining the volume integral of the radial magnetic flux density usingFEMM

The projectile becomes magnetized when we define for itB- Hcurve andhcvalues ​​inFEMMmaterial properties dialog. Valueswerechosenforstrictcompliancewithmagnetizediron. FEMMgives a value of 6.74x10 -7 Tm 3 for the volume integral of the magnetic flux densityB coil, so usingF= /4 we getB model = 3.0 x10 -2 Tl. This value of magnetic flux density may seem very small, considering the magnetic flux density inside the projectile, which is somewhere around 1.2Tl, however, we must understand that the magnetic flux unfolds in a much larger volume around the projectile with only a portion of the magnetic flux shown in the radial component. Now you understand that, according to our model,coilgun- This "insideout"(turned inside out) and "backtofront", in other words,coilgunthe immovable copper surrounds the magnetized part, which is moving. This does not create any problems. So the essence of the system is the connected linear force acting on the stator and the armature, so we can fix the copper part and allow the stator field to create movement. The stator field generator is our shell, let's assign a mass of 12g to it.

We can now plot displacement and velocity as a function of time, as shown in Fig. 14.8

Rice. 14.8. Dynamic behavior of a linear motor

We can also combine the expressions for velocity and displacement to obtain a function of velocity from displacement, as shown in Fig. 14.9.

Rice. 14.9. Characteristic of the dependence of speed on displacement

It is important to note here that a relatively long accelerator is needed for the anchor to begin to reach its maximum speed. This isIt hasmeaningforbuildingmaximum effectivepracticalaccelerator.

If we increase the curves, we can determine what speed will be reached at a distance equal to the length of the active material in the spool of the accelerator gun (78 mm).

Rice. 14.10. Increased speed versus displacement curve

These are remarkably close to those of the actual 3-stage accelerator, however this is just a coincidence as there are several significant differences between this model and the actualcoilgun. For example, incoilgunforce is a function of speed and displacement coordinates, and in the presented model, force is only a function of speed.

Rice. 14.11 - dependence of the total efficiency of the motor as a projectile accelerator.

Rice. 14.11. Cumulative Efficiency as a Function of Displacement Without Friction Loss

Rice. 14.11. Cumulative Efficiency as a Function of Displacement Considering Constant Friction Losses

The cumulative efficiency shows a fundamental feature of this type of electrical machine - energy is acquired by the armature when it accelerates first and up to ‘no- load’ speed is exactly half of the total energy delivered to the car. In other words, the maximum possible efficiency of an ideal (frictionless) accelerator would be 50%. If there is friction, then the cumulative efficiency will show the maximum efficient point that occurs due to the operation of the machine against friction.

Finally, let's look at the impactBon the dynamic characteristics of speed-displacement, as shown in Figures 14.10 and 14.11.

Rice. 14.11. InfluenceBon gradient speed-displacement

Rice. 14.12. Area of ​​small displacement where increasing induction gives more speed

This set of curves shows an interesting feature of this model, in which a large inductance of the field in the initial stage gives a greater speed at a particular point, but as the speed increases, the curves corresponding to the lower inductance catch up with this curve. This explains the following: You decide that a higher induction will give a greater initial acceleration, however, in accordance with the fact that a larger induced voltage will be induced, the acceleration will decrease more sharply, allowing the curve for the lower induction to catch up with this curve.

So what have we learned from this model? I think the important thing to understand is that, starting from a standstill, the efficiency of such a motor is very low, especially if the motor is short. The instantaneous efficiency increases once the projectile picks up speed due to the induced voltage reducing the current. This increases the efficiency because the energy loss in the resistance (obviously heat loss) decreases and the mechanical energy increases (see figs 14.3, 14.4), however, since the acceleration also falls, a progressively larger displacement is obtained, so the best efficiency curve will be used.(In short, a linear motor subjected to a step voltage "forcing function" is going to be quite an inefficient machine unless it is very long.)

This primitive motor model is useful in that it shows a case of typical weak efficiencycoilgun, namely the low level driving induced voltage. The model is simplified and does not take into account the non-linear and inductive elements of the practical circuit, therefore, in order to enrich the model, we need to include these elements in our electrical model circuit. In the next section, you will learn the basic differential equations for a single-stagecoilgun. In the analysis, we will try to obtain an equation that could be solved analytically (with the help of several simplifications). If this fails, I will use Runge Kutta's numerical integration algorithm.

The thermistor heat balance equation has the form

I2 R =ξ (Qп – Qс ) S,

where ξ - heat transfer coefficient, depending on the speed of the medium; Qp and Qc - respectively, the temperature of the thermistor; (converter) and environment;

S is the surface area of ​​the thermistor.

If the thermistor has the shape of a cylinder and is located across the flow so that the angle between the axis of the cylinder and the flow velocity vector is 90°, then the heat transfer coefficients for gases and liquids are determined by the formulas

cλ

cλ

Vd n

cλ

ξg =

ξl =

where V and υ are, respectively, the speed and thermal conductivity of the medium, d is the diameter of the thermistor;

c and n are coefficients depending on the Reynolds number Re = Vd/υ ;

P r = υ d - Prandtl number, depending on the kinematic viscosity and

thermal conductivity of the medium.

Such a converter (thermistor) is usually included in a bridge measuring circuit. Using the above expressions, the speed V can be measured.

5.2. Use of the laws of electromagnetism in measuring technology

On the basis of the phenomenon of electric repulsion of charged bodies, an electroscope device is arranged - a device for detecting electric charges. An electroscope consists of a metal rod, to which

a thin aluminum or paper sheet is hung. The core is reinforced with an ebonite or amber stopper inside a glass jar, which protects the leaf from air movement.

An electrometer is an electroscope with a metal case. If you connect the case of this device to the ground, and then touch its rod with some charged body, then part of the charge will transfer to the rod and the leaves of the electrometer will diverge at a certain angle. Such a device measures the potential difference between the conductor and the ground.

An oscilloscope is a device designed to observe, record and measure the parameters of the signal under study, as a rule, voltage, which depends on time. Light beam oscilloscopes use the electromechanical deflection of a light beam under the action of the voltage under investigation.

Cathode-beam oscilloscopes (CBE) are built on the basis of cathode-ray tubes. The deflection of the electron beam is carried out directly by an electrical signal.

The main unit of the ELO is a cathode ray tube (CRT), which is a glass evacuated bulb (Fig. 10), inside which there is an oxide cathode 1 with a heater 2, a modulator 3, anodes 4 and a system of deflecting plates 5 and 6. Part of the CRT, including into itself a cathode, modulator and anodes, is called an electron gun.

Rice. 10 Cathode ray tube

If a voltage is applied to the deflecting plates, the electron beam will deflect, as shown in Fig. eleven.

The investigated voltage Uy is usually applied to the vertically deflecting plates, and the developing voltage (in this case, a linearly changing periodic one with a period Tr) is applied to the horizontally deflecting plates.

Rice. 11. Getting an image on the CRT screen

Devices of the magnetoelectric system (ammeters, voltmeters and ohmmeters) are suitable for use in DC circuits, and when using detectors, also for AC purposes. The principle of operation of the measuring mechanism magnetoelectric The system uses the effect of the interaction of the field of a permanent magnet with a coil (frame) through which current flows. On fig. 12 shows a typical design (moving coil).

Rice. 12. Typical moving coil design Permanent magnet 1, core with pole pieces 2 and

the fixed core 3 constitute the magnetic system of the mechanism. In the gap between the pole pieces and the core, a strong uniform radial magnetic field is created, in which there is a movable rectangular coil (frame) 4, wound with a copper or aluminum wire on a frame. The coil is fixed between axle shafts 5 and 6. Coil springs 7 and 8 are designed to create a counteracting moment and, at the same time, to supply the measured current.

The frame is rigidly connected to the arrow 9. To balance the moving part, there are movable weights on the antennae 10.

Conversion equation:

α = I(ВnS / W),

where B is the magnetic induction in the gap;

α - angle of rotation of the moving part; S is the area of ​​the frame;

n is the number of coil turns;

W is the specific counteracting moment. 51

Devices of electromagnetic, electrodynamic, ferrodynamic and electrostatic systems are widely used as typical electromechanical ammeters, voltmeters, wattmeters and frequency meters.

The principle of operation of electrodynamic devices is based on the interaction of the magnetic fields of two coils through which current flows.

The device of such a measuring mechanism is shown in fig. thirteen.

Rice. 13. Electromechanical converter of the electrodynamic system

Inside the fixed coil 1, a movable coil 2 can rotate, the current to which is supplied through the springs.

The rotation of the coil is carried out by a torque caused by the interaction of the magnetic fields of coils 1 and 2. The counteracting moment is created by special springs (not shown in Fig. 13).

The transformation equation of this mechanism is:

α = W 1 ∂ ∂ M α I 1 I 2 ,

where W is the specific counteracting moment;

α - angle of rotation of the moving part; M is the mutual inductance of the coils.

This mechanism can be used to measure the constants

and alternating currents, voltages and power.

Ferrodynamic measuring mechanisms are essentially

are a kind of electrodynamic devices, from which they differ only in design, since the coil has a magnetically soft core (magnetic circuit), between the strips of which a moving coil is placed. The presence of the core significantly increases the magnetic field of the fixed coil, and hence the sensitivity.

In electrostatic devices the principle of interaction of electrically charged conductors is carried out.

One of the common designs of a detailed measuring mechanism is shown in fig. fourteen.

Fig.14. Electrostatic converter Movable aluminum plate 1 fixed together with arrow

on axis 3, can move, interacting with two electrically connected fixed plates 2. The input terminals (not shown), to which the measured voltage is applied, are connected to the movable and fixed plates.

Under the action of electrostatic forces, the movable plate is drawn into the space between the fixed plates. Motion

stops when the counteracting moment of the twisted plate becomes equal to the torque.

The transformation equation for such a mechanism has the form

α = 2 1 W ∂ d C α U 2 ,

where U is the measured voltage;

W is the specific counteracting moment; C is the capacitance between the plates.

Similar converters are used to develop voltmeters of direct and alternating currents.

The principle of operation of electromagnetic system devices is based on the interaction of a magnetic field created by a current in a fixed coil with a movable ferromagnetic core. One of the most common designs is shown in Fig. fifteen.

Rice. 15. Electromagnetic system converter:

I - coil, 2 - core, 3 - helical spring that creates a counteracting moment, 4 - air damper

Under the influence of a magnetic field, the core is drawn inward