To begin our studies on DC Electric Machines, let's introduce some definitions and establish some Basic principles of Electromagnetism.

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The study of Electromagnetism brings with it a very complex theory if we want to be exact. Therefore, in practice, it is desirable to simplify the problem by making some simplifications that lead to very satisfactory results and very close to the exact values.

In the course of this study we will come across several definitions and symbols to designate the variables. So, below is a list of the main ones.

The behavior of DC machines can be determined by applying four basic equations to the machine. The equations are the following:

eq. 103-01

eq. 103-02

eq. 103-03

eq. 103-04

Using these four equations as tools let's analyze the basic behavior of the DC machine.

As we saw in previous chapters, whenever a winding or coil rotates immersed in a magnetic field, there will be a magnetomotive force (fmm) induced (also known as the back electromotive force as it opposes the voltage applied), which we will call E_A. Considering the winding of the armature, which has a winding resistance designated by R_A and, being fed by a voltage DC, which we will call V_T, obeys the following equation when it is operating as engine.

This equation tells us that the magnetomotive force (fmm) plus the voltage drop across the armature winding resistance must be equal to the supply voltage V_T. From this equation, we can derive two more, or:

With this equation we can determine the magnetomotive force, knowing the other variables.

This equation allows us to calculate the armature current. These equations are easy understanding if we look at Figure 103-01, where we present the equivalent circuit of a DC motor in a configuration with independent excitation.

It should be noted that in DC machines, as we have the brushes in contact with the commutator, which lead to armature current, this causes an additional voltage drop in the armature circuit. This voltage drop, symbolized by V_B, must be added in eq. 103-05, as explained in eq. 103-08. Then, depending on the author, some problems will refer to voltage V_B. In general, if there is no specification in contrary, the voltage drop across the brushes is of the order of 1 V to 2 V when we have two brushes operating in series and we can consider it independent of the armature current.

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    2.1   Working Principle

Let's analyze how a DC machine starts working from the moment we apply a constant DC voltage, V_T, at the ends of the brushes that are in contact with the armature terminals. Considering the rotor (part of the machine where the armature winding is housed) at rest, when applying V_T, there is no induced voltage developed by the armature winding. In this case, the starting current of the motor will be simply I_A = V_T / R_A, because E_A= 0, as we can see from the eq. 103-07. For simplicity, we represent in Figure 103-02 a motor with a constant magnetic field provided by a permanent magnet and the rotor represented by a loop. As soon as there is an armature current, there will be a interaction between this current and the magnetic field, resulting in a force on the conductor, in the tangential direction, which will produce a torque of spin, starting the machine in its accelerated rotation movement. This emerging force is given by eq. 103-01.

With the machine in motion, the induced voltage E_A appears, defined by eq. 103-02. Note that in this instant, the induced electromotive force E_A and the supply voltage V_T coexist. The tension induced is directly proportional to the speed of rotation ω and acts opposing the external voltage V_T. At this moment, the armature current I_A is defined by eq. 103-07. The induced voltage grows to the limit of equaling the external voltage V_T. When E_A = V_T, the rotation speed of the machine stabilizes and we have I_A = 0. Then, the torque (also known as conjugate) cancels out. In this situation, we can say that the rotor floats under the action of the external voltage V_T. Bear in mind that any drop in machine speed will cause the voltage to induced E_A is less than the external voltage V_T and again the current appears I_A, generating with it a restoring torque, increasing the speed of the machine to the break-even point.

It is important to note that in this condition there are two operating conditions that we can analyze:

See Figure 103-03 for an exploded view of a DC motor showing its main parts, except the brushes.

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    2.2   Power and Losses in DC Machines

The power developed by a DC motor can be calculated from eq. 103-05, multiplying both sides by the armature current I_A. Doing this, we obtain the eq. 103-09.

Observing carefully the eq. 103-09, we clearly see that we have three potencies involved, that is:

Being more rigorous in the analysis of the Joule losses, we can add the losses in the brushes and in the field winding. With this we can express the losses, represented by P_loss, as:

We have already studied it in chapter 76 in item 5.2 - AC electric power generator   (click here!)  the theory regarding induced voltage in the armature winding due to the magnetic field generated, in the case of the motor under analysis, by the winding machine field. For clarity, let's repeat the equation developed in that item.

This equation was developed thinking about a generator with a single turn and then generalized to N turns. In the case of a motor, we have a lot more variables to consider, as the armature winding is riding on a cylinder composed of laminated iron with a commutator and brushes interconnected to it. Furthermore, we have a certain number of poles forming the induction field. In this way, we can think of a constant, which we will call K, encompassing all the mechanical and geometric variables of an engine design. The value of this constant it depends of machine design. So, we can generalize that, the induced voltage E_A for a DC machine, can be expressed through the eq. 103-11, where in this equation we express the product B A sin φ as the magnetic flux Φ in the air gap, as the angle between the coil area and the magnetic induction field is 90°. Since sin 90° = 1, then

Note that the induced voltage depends on the magnetic flux provided by the field winding and the speed of rotation. of the machine. In this equation, ω is given in rad/s. If we want the rotation speed in rpm (designated by the letter n) and vice versa, we must make the transformation according to eq. 103-12.

The constant K is defined by eq. 103-13.

In the eq. 103-13, the variables are:

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    2.3   Torque in DC Machines

To determine the torque, we will refer to eq. 76-16, studied in chapter 76, in item 4.2.1 - Mathematical Details (click here!)   and let's go repeat it here for clarity.

Note that the equation that determines the induced voltage (eq. 103-10) and the equation that determines the torque (eq. 103-14) differ just because the induced voltage depends on the rotation speed (ω) of the machine, while the torque depends on the armature current (I_A).

Adapting this equation to the nomenclature used here, we have:

In chapter 76, when we studied work and power in item 4.1.3 ((click here!)   we saw the relationship between power (measured in watt), torque (measured in newton.meter) and angular velocity (measured in rad/s) through eq. 76-13. Let's repeat this equation for clarity.

At the beginning of this item, through eq- 103-09 we studied the mechanical power developed in the motor shaft, given by:

Now we can equate the equations eq. 103-16 and eq. 103-17 giving rise to eq. 103-18, or:

This equation tells us that the instantaneous electrical power associated with the velocity voltage is equal to the mechanical power instantaneous associated with the magnetic torque, with the direction of the magnetic flux determining whether the machine works as a motor or generator.

The flux in the air gap between the fields and the rotor is produced by the combination of magnetomotive forces (mmf) provided by all field windings. The magnetization characteristics that determine the flux of the fmm are provided by the magnetization curve which depends on the geometry and type of iron used in the construction of the rotor and stator of the machine.

Since the armature electromotive force is directly proportional to the product of magnetic flux and velocity, makes it more convenient to express the magnetization curve in terms of armature emf E_Ao at a certain constant speed, which we will designate as ω_Ao. In this way, we can combine the equations eq. 103-15 and the eq. 103-18 and find a relation that determines an fem E_A for a given stream at any other speed ω_A. Thus, we can write:

Based on eq. 103-19, we can find the final relationship to determine the induced armature voltage to any other desired speed. Thus, we get the eq. 103-20.

This relationship is linear and this is possible because over a reasonably wide range of excitation, the reluctance of iron is negligible compared to that of the air gap. In this region, the flux is linearly proportional to the total fmm of the windings of field.

If the rotational speed of the machine is given in rpm, then we can use eq. 103-21 to find the new value of E_A.

Still based on eq. 103-19, we can write another important equation that relates the speed of the machine, the induced voltage and magnetic flux. This equation is given by eq. 103-21abelow.

It is also possible to replace the machine speed given in rad/s for the speed given in rpm. Just replace ω with n, keeping the indexes.

It should be noted that the electromagnetic force on a conductor is directly proportional to the current and this means that the current is proportional to the load to which it is subjected. Reviewing the eq. 103-07, we verify that the current depends directly on the difference between V_T and E_A is inversely proportional to the armature resistance. So, for a given armature resistance, the greater the load, the greater must be the difference between V_T and E_A and, as a consequence, the machine will lose rotation, because smaller E_A, smaller must be ω. So, we conclude that the greater the resistance of the armature winding, for a given load, the machine will reduce its rotation speed and, conversely, the smaller the resistance of the armature winding, for a given load, the less variation in the rotation speed of the machine. From the above, we easily deduce that for a machine to maintain an almost constant speed, regardless of the load, the resistance must be very close to zero.

On the other hand, to complete this analysis, it is possible to make an evaluation when we vary the flow of the machine and see what happens with the speed of rotation when we apply load. Let's base ourselves on the eq. 103-15. If the flux density, Φ, is small, for the same armature current, we obtain a small torque. This means that the rotational speed of the machine will drop to a worrying level. If we want to increase the torque, we must increase the armature current. Although, we know that the armature current can vary within certain limits, because if we increase it too much, it can happen winding to be destroyed.

Due to all these factors mentioned above, it is always desirable to design the machine operating with the maximum flux density, as this guarantees an armature current adequate to the torque developed by the machine, without harming the armature winding.

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    2.4   Magnetization Curve in DC Machines

Therefore, by equation eq. 103-11, the voltage E_A is directly proportional to the flux and the rotational speed of the machine. Here is a question: what is the relationship between the voltage E_A and the field current (I_F) of the machine?

When we studied transformers, we saw that when we apply an electric current in a winding, a magnetomotive force. On DC machines the same thing happens. So, when we apply the current I_F in the field winding, we generate a magnetic field in the air gap of the machine and, as a consequence, we obtain a voltage induced E_A which was expressed by eq. 103-11, shown below.

This voltage induced produces a magnetic flux in the machine according to the magnetization curve. It is very practical to represent the magnetization curve in a graph E_A x I_F for a given speed ω_o, since we know that the magnetizing current is directly proportional to the magnetomotive force and the voltage E_A directly proportional to the magnetic flux. In Figure 103-04 we can see a magnetization curve.

In order to obtain the maximum possible power per kilogram of a machine, most motors and generators is designed to operate near the saturation point on the magnetization curve (i.e. knee of the curve). In Figure 103-04 we can see that until we reach the field current I_F1, (point A) we have one linearity in the curve. From that point, we have the knee with the consequent beginning of iron saturation. As Consequently, a large increment in field current is required to obtain a small increase in E_A when the operating point is close to full load. Note this detail in the figure above.

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3.   DC Machine as Motor

In general, DC motors are driven from a DC power source, unless otherwise specified. We will also assume that the DC motor supply voltage is constant, as this simplifies motor analysis and comparison between different types of DC motors.

In principle, there are five main types of general purpose DC motor. Are they:

Let's analyze each of them separately.

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    3.1   Motors with Independent Excitation

An independently excited DC motor is a motor whose field circuit is supplied from a isolated source of constant voltage, which we will call V_F. The equivalent circuit of an excitation motor independent is shown in Figure 103-05.

While the motor is at rest, without turning, there is no fmm induced in the armature. Then, as the resistance of the winding of armature is very small, as a consequence we will have a very large rushing current. Therefore, in order to prevent the armature current reaches excessive values, it is common to insert a variable resistance, called break-in rheostat or starting, in series with the armature winding. As soon as the machine starts to spin, developing the fmm, it is possible to deactivate gradually the rheostat. Normally, the value of the rheostat is calculated so that the starting current does not exceed twice the current engine rating. It should be noted that, at the moment of starting, if the flux is small, due to a small current in the winding of field, the motor torque, expressed by eq. 103-15, will also be small. If this conjugate fails to overcome the conjugate resistance of the machine, it will stand still, not generating the fmm in the armature and, consequently, the current in the armature will reach excessive values, which may destroy the armature winding.

In order to avoid the inconvenience mentioned above, it is recommended that the magnetic flux, at the time of the start, has the maximum value possible, which is obtained by completely excluding the rheostat from the field winding.

From the facts exposed above, we conclude that the independent excitation motor is not suitable for services that require starting the same with maximum load.

On the other hand, when the motor runs no load, the useful torque applied to the motor shaft is zero. In this condition, the only The torque that the motor must overcome is the torque due to frictional resistance and the braking effect caused by iron losses. In general, this torque reaches a maximum of 5% of the useful torque value of the motor.

If the useful torque is small, the value of the armature current will also be, as can be concluded from eq. 103-15. The small current will cause a small voltage drop across the armature and the value of fmm approaches the value of V_T. In this case, the machine speed will be maximum, its value being defined by eq. 103-22.

The maximum speed value of the machine is obtained when it works at no load, with the field winding rheostat fully inserted in the field circuit, as in this condition the value of the magnetic flux is minimal.

In the case of the machine with load it absorbs an armature current proportional to the resistive torque, according to a eq. 103-23.

Therefore, increasing the torque on the motor shaft increases the armature current. The increase in current causes a increase in voltage drops and a decrease in fmm and the number of machine revolutions. So the number of machine revolutions can be obtained from the eq. 103-24.

The factor represented by R_A I_A, even when the machine runs at full load, is a small fraction of the supply voltage. Therefore, the maximum decrease in speed is relatively small. Furthermore, there is another factor that contributes to keeping the speed practically constant with the variation of the load. Note that with increasing load there will be a decreasing the numerator of the fraction in the above equation. This is due to the fact that an increase in armature current causes an increase in armature reaction and, consequently, a demagnetizing effect. The decrease in the magnetic flux, caused by the increase in the reaction of armature, makes the denominator of eq. 103-24 decreases, partially offsetting the decrease in the numerator. In these conditions, the speed value of the machine remains sensibly constant.

So, we conclude that large variations in the torque value are accompanied by small variations in the machine speed. This is one of the characteristics of the DC motor in the independent excitation configuration: practically constant speed when we vary the load value.

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    3.2   Motors with shunt excitation

A Shunt DC motor has a response very similar to an independently excited DC motor. Let's analyze how a shunt DC motor behaves when the load on its shaft increases. if the load is increased, in which case the load torque will exceed the induced torque in the machine and the motor will start to lose speed. When this happens, the generated internal voltage E_A decreases and, consequently, the armature current of the motor I_A increases, as eq. 103-23. As the current increases, the induced torque increases until it is equal to the load torque, at a lower mechanical speed of rotation, depending on the eq. 103-24.

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    3.2.1   Output Characteristics

The output characteristic of a shunt dc motor can be obtained from from the induced voltage and torque equations plus Kirchhoff's voltage law (LKT). As we have already studied, the LKT equation for a DC motor in shunt is given by eq. 103-05, repeated below.

Remembering that the induced voltage E_A is given by eq. 103-11, shown below for clarity.

So, substituting eq. 103-11 in eq. 103-05 and, taking into account the eq. 103-23, let's go get the eq.103-25 below.

From eq.103-25 we can determine the speed of rotation of the machine after some algebraic work, or

Note that in eq. 103-26, we have that V_T is a constant, as it is the motor supply voltage. Also R_A and the product K  Φ are constants that depend on the machine design. So this equation is the equation of a straight line with a negative slope, which represents the relationship between the speed of rotation ω_A of the machine as a function of the torque generated by the motor. It is important to bear in mind that for the motor speed to vary linearly with the conjugate, the other terms of this expression must be constant when the charge vary. If V_T is not constant, then voltage variations will affect the shape of the torque versus speed curve.

Another internal motor effect that can also affect the shape of the torque versus speed curve is armature reaction. If one engine reacts armadure, so flux weakening effects will reduce your flux when the load increases. As the eq. 103-26 shows, for any load, the effect of a flux reduction is the increase in motor speed relative to the speed at which the motor would turn if there were no armature reaction.

Of course, if a motor has windings of compensation, there will be no flow weakening problems in the machine, which will be constant. If there are compensating windings in a shunt DC motor, then so that its flow is constant regardless of the load, and if the velocity and motor armature current are known for any load value, then its speed may be calculated for any other load value, provided that the armature current for that load is known or can be determined.

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    3.2.2   Nonlinear Analysis of a DC Motor Shunt

The flux Φ and consequently the generated internal voltage E_A of a DC machine is a non-linear function of its magnetomotive force. So anything that altering the magnetomotive force of a machine will produce a non-linear effect on the generated internal voltage of the machine. Since it is not possible to calculate analytically the changes of E_A, we must use the magnetization curve of the machine to determine accurately its E_A, for a given magnetomotive force. The two contributions principal to the magnetomotive force of the machine come from its field current and your armature reaction, if present.

Since the magnetization curve is a straight graph of E_A versus I_F for a given speed ω_A, the effect of change in machine field current can be determined directly from its magnetization curve.

If a machine exhibits armature reaction, its flux will be reduced every load increase. In a shunt dc motor, the total magnetomotive force is the field circuit magnetomotive force minus the original magnetomotive force of the armature reaction (AR).

Since magnetization curves are expressed as plots of E_A versus current I_F, it is customary to define an equivalent field current, which produces the same output voltage as the combination of all the magnetomotive forces in the machine. If we locate the equivalent field current on the magnetization curve, then we can determine the resulting voltage E_A. The equivalent field current of a shunt DC motor is given by eq. 103-28.

There is another effect that must be considered when using nonlinear analysis to determine the generated internal voltage of a DC motor. The magnetization curves of a machine are plotted for a particular speed, usually the rated speed. How can we determine the effects of a given current of field if the motor is running at a speed other than nominal?

For a given effective I_F field current, the flux in a machine is fixed. In this way, the internal voltage generated is related to the speed, being expressed in revolutions per minute, through the eq. 103-21, shown below.

Where, in this equation, E_Ao and n_Ao represent the voltage and speed reference values, respectively. If the reference conditions are known from the curve of magnetization and the actual voltage E_A is obtained from Kirchhoff's voltage law, then it will be possible to determine the actual velocity n_A from eq. 103-21. As an example of a problem which analyzes an armature reaction DC motor, see   Problem 103-5

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    3.2.3   Speed Control of a DC Motor in Shunt

There are two common methods possible to control your self the speed of a shunt DC motor. Commonly used methods have already been seen in the simple linear machine of Chapter 101. So the methods are:

Let's study each of them separately.

To understand what happens when the field resistor of a DC motor is changed, assume that the field resistor increase the value and observe the response. If the field resistance R_F increases, then the field current I_F will decrease, and when this happens, the flux will also decrease. A decrease in flow causes an instantaneous drop in internal generated voltage E_A which leads to a large current increase of armature in the machine, as shown in eq. 103-7, repeated below.

The induced torque in a motor is given by eq. 103-15. Since the flow of this machine decreases when the current I_A increases, how will the variation occur of the induced conjugate? The easiest way to answer this question is through an example. In [26] we have an example that will answer this question. Figure 103-06 shows a shunt DC motor with an armature resistance equal to 0.25 Ω. It is currently operating on a supply voltage of V_T = 250 V and an internal generated voltage E_A = 245 V.

Therefore, by eq. 103-07 the armature current is I_A = 20 A. What will happen to this engine if there is a 1% decrease in flow? If the flow decreases by 1% then E_A should decrease also in 1%, according to eq. 103-11. Therefore, the new value of E_A will be: