In the previous chapter we saw that a magnetic field exerts a lateral force on electrons that move on a wire. And that force is transmitted to the wire, since electrons cannot leave it. In this way, we will study the behavior of a conducting wire that is conducting an electric current and is under the influence of a magnetic field.

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The observation of Ampère on the magnetic forces between conductive wires covered by electric current, it takes us to a more analytical study of this phenomenon. As a first step, we will find the force exerted by a uniform magnetic field on a straight conductive wire through which an electric current I flows. For clarity, let's look at two different cases: first, when the field and current have the same direction and sense; second, when the current and the field are orthogonal to each other.

In Figure 76-01 we see the representation of the two situations mentioned above. By the right hand rule, each charge of the electric current experiences a modulus force equal to q v B , directed to the left. Consequently, the entire length of the wire within the magnetic field experiences a left-oriented force that is simultaneously perpendicular to the direction of the field and the direction of the current.

If, in Figure 76-01 , we invert the direction of the magnetic field OR , the direction of the electric current, the force exerted on the wire will change direction and will be directed to the right.

Notice the difference between electrical force and magnetic force : the electrical force is always parallel to the electric field, while the magnetic force is always orthogonal to the magnetic field.

If we want to determine the modulus of the force, we must relate the current I in the conductive wire to the load q that travels through it. Let us suppose a segment of the conducting wire, of length L , through which a current I circulates. We define this current as the ratio between the amount of charge q that moves in this segment and the time Δt elapsed during the route by the segment. Obviously, the elapsed time is given by Δt = L / v. Using that relation and, knowing that q = I Δt, we can easily conclude that:

We can define the vector L^→ that has the orientation of v^→, with module L. So let's get the following relation: IL^→= qv^→. Substituting this relation in eq. 74-01 we find the force, in its vector form, that a conductive wire experiences when it is under the action of a magnetic field, that is:

We can write eq. 76-02 in scalar form, considering θ as the angle between the vectors L^→ and B^→, obeying the rule of right hand.

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In his experiments, Ampère made an arrangement with two parallel wires, a and b, being traversed by electric currents, which we will call I_a and I_b, as shown in Figure 76-03 . For our analysis, we will consider two situations: the currents flow in the same direction or, the currents flow in opposite directions.

Let us consider that the conducting wires are spaced at a distance d from each other. Also, consider a length L of them. Let's start by assuming that the two currents flow in the same direction, as shown in Figure 76-03 . Watch that the current I_a , which flows through wire a , produces a magnetic field in the position where the wire b is located, facing downwards, according to the right hand rule.

This magnetic field, B_a^→ generated by I_a , produces a force of attraction in the wire b represented by F_ab^→ in the figure above. As the two wires are in parallel, certainly the magnetic field generated by wire a will be constant over the entire length of wire b . With that, we can start from equation that allows you to calculate the magnetic field generated by a very long wire and, as we studied in chapter 74 , the eq. 74-17 provides:

If we work in the scalar form, we can use the eq. 76-03 , studied in the previous item, remembering that θ = 90°and therefore sen 90 ° = 1 . In this way, replacing by the respective values and considering the field produced by the wire a , the attraction will be given by:

If we work in vectorial form, we must give the axes adequate orientations in relation to Figure 76-03.

In Figure 76-04 we see how the axes were oriented. Note that the x axis is in the direction of the force. The y axis is in the direction of the wire length, while the z axis is in the direction of the magnetic field. For determining the force F_ab we will use the eq. 76-02 , studied in the previous item. Note that this equation requires that the cross product be carried out between the vector L and the vector B_a.

Let's rewrite the eq. 76-02 , adapting it to the case we are analyzing and making the nomenclature compatible with Figure 76-04 . Like this:

According to Figure 76-04 , it is important to note that the currents point at positive direction of the y axis . In turn, the magnetic field points in the direction negative of the z axis . So, placing the signs correctly on the product vector, we get:

Notice that we replace the vector B_a as the product among your module and the k versor. As the field is pointing in the negative direction of the z-axis , we use the - k versor. For the case of the vector L we also use the product between your module and the J versor. On the other hand, we know that the product vector between the verses j and k is equal to j x k = i. Based on this result, we can write the product that interests us, or j x (- k) = - i . So, we were able to determine the magnetic force that the wire a exerts on the wire b , in module, direction and sense, that is:

Note the negative sign in eq. 76-07 indicating that the force is pointing in the negative direction of the x-axis and it is a attraction force. It is worth mentioning that if we reverse the direction of one of the currents, the force will be pointing in the direction positive of the x-axis , indicating that there is a repulsion force between the conducting wires.

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We have already used the concept of magnetic force to describe the behavior of some type of conductor, By this conductor flow an electric current when immersed in a magnetic field. Now, let's study the concept of torque , also known as conjugate , that is produced when a force is applied to bodies that can rotate around some point.

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The torque, whose name comes from a Latin word meaning "to twist" , can be described, in a simple way, like spinning or twisting a force, F . When we apply a force to an object, with some tool, in order to make the object rotate, we are applying a torque.

The unit of torque in SI is newton. meter . Note, however, that in SI the work also has newton.meter as its unit , which is equivalent to 1 joule . But, torque can never be expressed in joule. Torque is represented by the letter Greek "tau", τ.

The torque can be considered positive or negative . It will be positive if the body rotates counterclockwise when applying force. And it will be negative if the body rotates clockwise when applying force.

In Figure76-05 we see three disks fixed in its center C , with freedom to rotate around its axis. Notice that in the drawing on the left the force is being applied exactly on the line that runs through the center of the disc. This force will not cause a rotation movement.

However, in the center design we see that the force is applied to the disk in such a way that its action line passes to the left of the rotation axis. This will make the disc have a tendency to rotate clockwise . This is possible because we create a torque , that is, the action of spinning the disc. Likewise, in the drawing on the right of Figure76-05 , the force is applied to the disk in such a way that its line of action passes to the right of the axis of rotation. In this case, the disc will have a tendency to rotate counterclockwise.

The torque or conjugate of an object is defined as the product of the applied force to the object times the shortest distance between the line of action of the force and the axis of rotation of the object. In Figure 76-06 we show how a force applied to the disk displaced to the right acts of the action line that passes through the axis of rotation. The torque generated will cause the disc to rotate counterclockwise . Note that the angle θ is defined as the angle between the line of the force action line and the straight line of the disc radius extension.

Thus, the distance d from the point of application of the force in relation to the axis of rotation of the disk is given by d = r sen (180° - θ) = r sen θ, using the properties of trigonometry. Considering r a vector that points from the axis of rotation to the point of application of force and F as the applied force, then the conjugate (or torque) can be expressed by

Let us remember that the vector product follows the right-handedness convention., that is, the right hand rule. So, in reference to Figure 79-06 , the counterclockwise torque points upwards, leaving the page, while the hourly torque points downwards, entering the page.

It should be noted that the component F sin θ in eq. 76-09 , represents the component of the force F in the direction perpendicular to the position vector r . In other words, torque can also be interpreted as the product of the distance from the point of application to the origin by the perpendicular component of the force. The torque produced by the force is due solely to to the perpendicular component of the force.

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There is a particular case of forces acting on a point. It consists of a pair of forces called F^→ and - F^→, equal and opposite, with lines of parallel action, as shown in Figure 76-07 . These forces are called binary and do not have the ability to produce translation, only rotation . Note that to determine the torque we must use the fact that r = r_PQ sin θ. With that, we fall into the eq. 76-08.

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The relation between torque and acceleration is given by the Newton's law of rotation which can be described by eq. 76-10

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In the rectilinear movement, work is defined as the product of an applied force that moves for a certain distance. In the rotation movement, work is defined as the product of a conjugate, J, applied by a certain angle θ. Considering the constant conjugate, the work can be described by

If the conjugate is not constant with the angle variation, then we must integrate it in relation to the angle.

On the other hand, power is the rate of production of work, or the increase in work per unit of time. By this definition, power is given by

In general, the unit of measurement for power is joule per second or, which is equivalent, watt . But HP = 746 watts or CV = 735 watts can also be used.

Based on eq. 76-12, and assuming a constant torque we can determine the power in a system with rotation movement given by

This equation (eq. 76-13) will be the correct relationship between power, torque and angular speed if the power is measured in watts , the conjugate in newton-meters and the angular speed in radians per second.

It should be noted that the eq. 76-13 is very important in the study of electrical machines, because it can describe the mechanical power on the axis of an engine or generator.

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In the previous items we studied what is meant by torque , acceleration in rotation , torque , etc ... and now we are going to apply these concepts to calculate torque or conjugate that a loop suffers when an electric current circulates through it, I , and is under the action of a uniform magnetic field, B. We present in Figure 76-08 a schematic representation of a system that will help us understand its functioning.

Note that a permanent magnet system is being used to produce a uniform magnetic field. An electrical current I is circulating in the counterclockwise direction. The loop, with a rectangular shape, whose largest side is represented by a and the smaller side by b , has as its area the product of the dimensions of its sides, that is, A = a . b

In the middle of the smaller side, there are two supports that hold the loop in place. Applying the right hand rule, we notice that the resulting force, F , on the side of the loop near the north pole of the magnet, it points upwards, as shown in the figure. And for the turns close to the south pole of the magnet, the - F force, points downwards. With these two forces, we have as a result a torque of forces being applied to the loop and, as a result, we obtain a torque or conjugate that will make the loop turn clockwise. This is a qualitative analysis that allows us to understand how phenomena are processed electromagnetic elements involved in that device. Now, let's do a quantitative analysis of the system.

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To begin, let's explain why no forces appear on the smaller sides of the loop. We should note that the smaller sides of the loop, as shown in Figure 76-08 , are parallel to the lines of force in the magnetic field. So, the angle between the vector L^→ and the vector B^→ or do 0°, or 180° and, in these two cases, the sine of these angles is equal to zero and, reporting to eq. 76-02 , we conclude that the resulting force is null . When the loop turns from an angle θ, forces will appear on the smaller sides of the loop, but they will be of the same module and opposite sense. As the two forces act on the same line (the rotation axis of the loop), they cancel each other and do not there is no contribution to the torque.

On the other hand, on the larger sides of the loop two forces of the same module and direction appear, but in opposite simplesmente. Furthermore, the two forces do not act on the same line, causing the binary to appear (studied in 4.1.1). This torque makes on the turn, actuate a torque that will make the turn turn clockwise. We know that the force modulus on the a side of the loop is given by F = I a B. To calculate the torque, we must find the distance from the side of the loop to the axis of rotation of the same. In the position where the loop is in Figure 76-08 , the distance is simply b / 2 . But we must take into account that the loop turns and, therefore, we must consider the contribution of this side as a function of the angle, θ, when rotating. Therefore, the contribution should be written as (b/2) sin θ. That way we can write the torque relative to the force F , as

This calculation considered only the F force. However, we must add the contribution of force - F . In module, the two values are equal. Therefore, adding b / 2 + b / 2 = b and into the equation at the end, the product will appear on both sides of the loop ( a. b ), which is exactly the area of the loop. So we can write the total torque on the loop as

Also, note that this calculation was made considering that the loop was constructed with a single loop of wire. Of course, for practical purposes we always use more than a loop to form what we call a coil . So when we use N turns, just multiply the value of eq. 76-15 by the number of turns of the coil, or simply

The product N I A is called the magnetic moment of the coil and it is always perpendicular to the plane formed by the coil area. We represent the magnetic moment by μ and its unit of measure is the amp square meter . So, eq.76-16 can be written as

Note that, again, this equation can be written in the form of a cross product, as we can see in the eq.76-18.

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In some practical applications it is interesting to have the possibility to control the field magnetic of a magnet. For example, making the field range from zero to a certain value. Using permanent magnets this is an arduous task. However, based on learning with electromagnetism, the idea arises of building the so-called electromagnet . What is an electromagnet?

In Figure 76-09 we see the figure of an electromagnet, which as you can see are some turns wrapped around a ferromagnetic material. This material, as it has a permeability 2000 to 6000 times the magnetic permeability of the air, allows us to create a field quite intense magnetic and with its controlled magnitude, via the value of the electric current, I , which passes through the solenoid. In this way, it is possible to switch on and off, through a key, the magnetic field.

Naturally, there is a need for a source of energy to power the electromagnet, as shown in the figure above. And so, with this technique it is possible to build an immense variety of electrical equipment. As an example, we can mention the relays, which are very used in automobiles to control the most varied functions, such as connecting on and off headlights, windshield wipers, direction arrow, signage, air conditioning, etc ...

In Figure 76-10 , on the left, we show the internal scheme of a relay. Note that the terminals B1 and B2 are the terminals that give access to the solenoid winding that makes up the relay. The terminal identified by the number 3 is the terminal where the mobile contact is connected, highlighted in red in the figure. Note that by not applying any control voltage to the relay, via terminals B1 and B2, the fixed contact 1 is connected to the mobile terminal 3, via fixed armor. When applying a control voltage to terminals B1 and B2, the solenoid will be energized and the moving contact will be attracted to the solenoid. As a result, we will have the mobile contact breaking the connection with the fixed contact 1 and being connected to the fixed contact 2 . To return to the previous situation, simply remove the voltage from the terminals B1 and B2 and the force of attraction of the solenoid will cease. With that, the reset spring remakes the connection between the 1 and 3.

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Electricity generators are very common in our daily lives. There are those that are located in hydroelectric, thermoelectric, nuclear power plants, etc ... They are high-power generators whose purpose is to supply energy to large cities. Usually these generators are of the type of alternating current. Its use is also very common in motor vehicles to supply energy to the vehicle. Of course, motor vehicles use direct current , based on a battery. So, to be able to use this type of generator in vehicles we must rectify the alternating voltage and turn it into direct current. How to do this was studied in chapters 62, 63 . In the technical literature, when it comes to using AC generators in motor vehicles, they are called alternators .

Another type of generator is that which provides direct current at its output. This type of generator has been widely used in the past and has been called dynamo . We will study them separately.

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The basic principle of an energy generator is based on the fact that a conductive wire moving under the influence of a magnetic field, develops a voltage induced at its ends as has already been studied. In Figure 76-11 we present a basic scheme of a generator based on this principle. Here the wire takes a rectangular geometric shape, with sides represented by a and b , its area being given by A = a . b.

For the generator to work, it must rotate under the action of an external force, for example, a propeller coupled to the shaft rotating under the action of the wind, an explosion engine or any other device. The loop is connected to the external environment through a device formed by a pair of brushes and a commutator. The wires are connected to the brushes that will conduct the energy for the load, it is not shown in the figure.

Note that the function of the switch is to maintain the same polarity as the voltage induced in the load. In Figure 76-12 we see the waveform at the generator output. Output voltage is zero when the loop is in the horizontal .

In addition, when the loop is horizontal, the brushes short-circuit the commutator, shorting the ends of the loop and ensuring that the output voltage is zero. As soon as the loop turns a small angle, the short breaks and the tension at the outlet begins to grow to its maximum when the turn is in the vertical. After reaching the maximum, with the continuous turn of the loop, the tension begins to decrease until reaching its null point, that is, when the loop is, again, horizontal. And so, the cycle is repeated.

It is clear that the waveform at the generator output follows the trigonometric shape, represented by the sine function. See the similarity of this waveform with the output of a full wave rectifier, studied in chapter 63. Therefore, the commutator has the function of rectifying the alternating voltage generated by the loop in full wave and producing a direct current voltage or, commonly known as DC , at the output . In the past this type of generator was known as dynamo and it was very common to use it in aero generators specially designed for use in rural areas, where there was no supply of electricity by electric companies. The automotive vehicles of that time also used dynamos as a source of energy for their operation.

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Another type of generator that can be built is the type that generates alternating current at its output. For this we must make a small change to the commutator.

In Figure 76-13 we have a diagram showing the necessary modification to transform a DC generator to a AC generator. Note the duplication of the commutator. Each of them has its own brush in charge of carrying the energy produced by the generator to the load. In fact, on AC machines it is common to adopt the name of slip rings , instead of commutator.

In the technical field, it is usually called collector . Naturally, one end of the loop is connected to the collector 1 , while the other end is connected to the collector 2.

The Figure 76-14 shows the waveform at the output of the AC generator. This waveform is perfectly described by the trigonometric function sine . And this function characterizes the alternating current , as it has a positive peak in the first half of the cycle and a negative peak in the other half of the cycle. In this way, we get a waveform that alternates periodically.

We can determine the value of the voltage induced in the loop considering that the magnetic flux to which the loop is subjected is given by

Note that we use the fact that θ = ω t and remembering that ω = 2 π f, on what ωis the angular frequency with which the loop turns. Here we will take into account that the generator has not only a loop, but N turns and using the Faraday's law , which is described by eq. 75-13 , we can write that

Then, by calculating the derivative of the function cos ωt, we get

Remembering that the product AB is equal to the magnetic flux, we can say that

"The voltage generated in the coil is a sinusoid whose amplitude is equal to the product of the magnetic flux present inside the machine due to the rotation speed of the machine and the number of turns that form the coil."

This is also true for real AC machines. In general, the voltage on any real machine will depend on three factors:

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As stated at the end of item 4.2 (Torque on a Conductive Loop) and repeating here

Well, there are some types of alternators where we keep the winding fixed and do the magnetic field is rotating, so that there is variation in the magnetic flux of the machine. Thereby, we guarantee an induced voltage in the coil. One of these types of alternators are those used in auto-vehicles. In Figure 76-15 we show a simplified model of this type of alternator.

The first important thing that we can notice about this alternator is that it has three separate windings, mechanically, at an angle of 120°. In the center, rotating, there is a permanent magnet for didactic issues. Therefore, rotating the magnet produces a variable flow in the windings and, consequently, an induced voltage appears in these coils. Now, we must connect in a way windings to obtain a voltage at the generator output.

In Figure 76-16 we see how the windings are connected inside an alternator for motor vehicles. We replace the windings with voltage sources for clarity. Note that the windings are connected in a Y or star configuration. This configuration is studied in chapter 81.

To achieve this configuration we must unite the wires of the windings represented by u, v and w (Figure 76-15). This union gives rise to the neutral terminal, indicated in Figure 76-16 by the letter N . And the wires x, y and w , which are where we have the induced voltages are connected directly to the diodes. Like a motor vehicle requires a battery as a power source, and it works with direct current, so there is a need to rectify the alternating current leaving the alternator. This is the purpose of the three diodes, connected one per phase, as shown in the figure above. Thus, at the terminals alternator output we obtain a direct current, indicated in the figure as V_cc.

In Figure 76-17, at the top, we can see the contribution of the three phases to the formation of alternator output voltage. In green, the x terminal (see Figure 76-15) with phase 0°. In blue, the y terminal (see Figure 76-15) with the phase -120° and in red, the z terminal (see Figure 76-15) with the + 120° phase. This formation the voltage in the generator is that which appears before the diodes. At the bottom of Figure 76-17 , we see the voltage at the generator output after being rectified by the three diodes. Note that when one of the tensions is downward, the next phase is upward.

This means that the voltage at the generator output does not fall below a minimum value, indicated in figure by the orange stripe. In this way, we obtain at the output a voltage with positive polarity , where the diodes eliminated the negative part of the alternator phase voltage. So, this tension positive at the alternator outlet is connected to the positive pole of the vehicle battery. The negative pole battery is connected to the vehicle chassis, where the alternator cable indicated is also connected with the letter N . This whole group forms the system that provides the necessary electrical energy to the perfect functioning of the vehicle.

On the other hand, we must understand that a vehicle battery only provides power when we go start the combustion engine. Once the combustion engine is operating, whoever supplies power to the vehicle is the alternator, with the battery in a "floating charge" regime, that is, it receives a small current (on the order of 1 amp) from the alternator so that its electrical charge is maintained. For this to happen, the alternator voltage has a value above the battery value at the output. For a 12 volt system, the voltage at the alternator output should be between 13.80 volts and 14.20 volts , depending on the manufacturer's specifications.

When a vehicle is in motion, there is a very large variation in the speed of engine rotation combustion, to which the alternator is mechanically interconnected via a belt. This generates a problem for the vehicle's electrical system, since based on eq. 76-21 , we verify that the alternator output voltage varies linearly with the angular velocity, ω. As we cannot change the number of turns of the alternator, nor the area of the loop, as are construction characteristics of it, so we only have the option to change the value of the field magnetic system. And how to do that? Let's remember what was studied in chapter 74 , in item 4.2 - Magnetic field produced by a solenoid , more specifically, reviewing to eq. 74-23 , repeated here for further understanding.

Note that this equation makes clear the linear relationship between the magnetic field value and the current flowing through the solenoid. Therefore, we conclude that to vary the magnetic field it is it is sufficient to vary the current flowing through it. So, we must use in solenoid to act as an electromagnet.

The Figure 76-18 shows how this is done on a real alternator. A rotor is used made of ferromagnetic material, with a suitable shape. Note the position of the winding inside the rotor. This winding together with the shape of the rotor alternately generates the necessary north and south poles to vary the magnetic flux in the fixed turns that form the alternator phases and are housed in the stator.

To supply electrical energy to the rotor solenoid, there are two collectors, that rotate with the rotor and a pair of brushes are attached to which they are connected to the power wires. Brushes are two contacts made with a type of hardened carbon for greater durability. And the collectors are made of an alloy of metals, called bronze . Therefore two conductive materials are used: carbon and bronze.

The Figure 76-19 shows an actual alternator stator. Notice how the coils are distributed along the circumference of the stator. Many coils in series form a phase that is 120° mechanically out of step with the other two. There are alternators that have only three output wires in the stator. When this is the case, there is a need to use six diodes to rectify the signal. Thus we managed to restore the alternator's neutral wire.