DC motor – Steady-State Characteristics

From the user's viewpoint the extent to which speed falls when load is applied, and the variation in speed with applied voltage are usually the f i rst questions that need to be answered in order to assess the suitability of the motor for the job in hand. The information is usually conveyed in the form of the steady-state characteristics, which indicate how the motor behaves when any transient effects (caused for example by a sudden change in the load) have died away and conditions have once again become steady. Steady-state characteristics are usually much easier to predict than transient characteristics, and for the d.c. machine they can all be deduced from the simple equivalent circuit in Ill. 3.6.

Under steady-state conditions, the armature current I is constant and equation (3.7) simplifies to

V = E + IR or I + (V-E) / R (3.8)

This equation allows us to find the current if we know the applied voltage, the speed (from which we get E via equation (3.6)) and the armature resistance, and we can then obtain the torque from equation (3.5).Alternatively, we may begin with torque and speed, and work out what voltage will be needed.

We will derive the steady-state torque-speed characteristics for any given armature voltage V in Section 3.4.3, but first we begin by establishing the relationship between the no-load speed and the armature voltage, since this is the foundation on which the speed control philosophy is based.

No-load speed

By 'no-load 'we mean that the motor is running light, so that the only mechanical resistance is due to its own friction. In any sensible motor the frictional torque will be small, and only a small driving torque will therefore be needed to keep the motor running. Since motor torque is proportional to current (equation (3.5)), the no-load current will also be small. If we assume that the no-load current is in fact zero, the calculation of no-load speed becomes very simple. We note from equation (3.8) that zero current implies that the back e.m.f. is equal to the applied voltage, while equation (3.2) shows that the back e.m.f. is proportional to speed. Hence under true no-load (zero torque) conditions, we obtain

V = E = K_ E Phi n or n = V /K_ E Phi (3.9) where n is the speed. (We have used equation (3.8) for the e.m.f., rather than the simpler equation (3.4) because the latter only applies when the flux is at its full value, and in the present context it's important for us to see what happens when the flux is reduced.). At this stage we are concentrating on the steady-state running speeds, but we are bound to wonder how it's that the motor attains speed from rest. We will return to this when we look at transient behavior, so for the moment it's sufficient to recall that we came across an equation identical to equation (3.9) when we looked at the primitive linear motor in Section 1.We saw that if there was no resisting force opposing the motion, the speed would rise until the back e.m.f. equaled the supply voltage. The same result clearly applies to the frictionless and unloaded d.c. motor here.

We see from equation (3.9) that the no-load speed is directly proportional to armature voltage, and inversely proportional to field flux. For the moment we will continue to consider the case where the flux is constant, and demonstrate by means of an example that the approximations used in arriving at equation (3.9) are justified in practice. Later, we can use the same example to study the torque-speed characteristic.

Performance calculation - example

Consider a 500 V, 9.1 kW, 20 A, permanent-magnet motor with an armature resistance of 1 V . (These values tell us that the normal operating voltage is 500 V, the current when the motor is fully loaded is 20 A, and the mechanical output power under these full-load conditions is 9.1 kW.)When supplied at 500 V, the unloaded motor is found to run at 1040 rev/min., drawing a current of 0.8 A.

Whenever the motor is running at a steady speed, the torque it produces must be equal (and opposite) to the total opposing or load torque: if the motor torque was less than the load torque, it would decelerate, and if the motor torque was higher than the load torque it would accelerate. From equation (3.3), we see that the motor torque is determined by its current, so we can make the important statement that, in the steady state, the motor current will be determined by the mechanical load torque. When we make use of the equivalent circuit ( Ill. 3.6) under steady-state conditions, we will need to get used to the idea that the current is determined by the load torque - i.e. one of the principal inputs which will allow us to solve the circuit equations is the mechanical load torque, which isn't shown on the diagram. For those who are not used to electromechanical interactions this can be a source of difficulty.

Returning to our example, we note that because it's a real motor, it draws a small current (and therefore produces some torque) even when unloaded. The fact that it needs to produce torque, even though no load torque has been applied and it's not accelerating, is attributable to the inevitable friction in the cooling fan, bearings, and brushgear.

If we want to estimate the no-load speed at a different armature voltage (say 250 V),we would ignore the small no-load current and use equation (3.9), giving no-load speed at 250 V = (250 = 500) 1040 = 520 rev/min since equation (3.9) is based on the assumption that the no-load current is zero, this result is only approximate.

If we insist on being more precise, we must first calculate the original value of the back e.m.f., using equation (3.8), which gives E = 500 (0.8 1) = 499.2V As expected, the back e.m.f. is almost equal to the applied voltage. The corresponding speed is 1040 rev/min, so the e.m.f. constant must be 499.2/1040 or 480 V/1000 rev/min. To calculate the no-load speed for V = 250 V, we first need to know the current. We are not told anything about how the friction torque varies with speed so all we can do is to assume that the friction torque is constant, in which case the no-load current will be 0.8 A regardless of speed. With this assumption, the back e.m.f. will be given by

E = 250 - (0 . 8 x1) = 249.2V

and hence the speed will be given by no-load speed at 250 V = 249.2 /480 x 1000 = 519.2 rev/min

Ill. 3.7 No-load speed of d.c. motor as a function of armature voltage.

The difference between the approximate and true no-load speeds is very small and is unlikely to be significant. Hence we can safely use equation (3.9) to predict the no-load speed at any armature voltage and obtain the set of no-load speeds shown in Ill. 3.7.This diagram illustrates the very simple linear relationship between the speed of an unloaded d.c. motor and the armature voltage.

Behavior when loaded

Having seen that the no-load speed of the motor is directly proportional to the armature voltage, we need to explore how the speed will vary when we change the load on the shaft.

The usual way we quantify 'load 'is to specify the torque needed to drive the load at a particular speed. Some loads, such as a simple drum type hoist with a constant weight on the hook, require the same torque regardless of speed, but for most loads the torque needed varies with the speed. For a fan, for example, the torque needed varies roughly with the square of the speed. If we know the torque/speed characteristic of the load, and the torque/speed characteristic of the motor, we can find the steady-state speed simply by finding the intersection of the two curves in the torque-speed plane. An example (not specific to a d.c. motor) is shown in Ill. 3.8.

Ill. 3.8 Steady-state torque -speed curves for motor and load showing location (X) of steady-state operating condition At point X the torque produced by the motor is exactly equal to the torque needed to keep the load turning, so the motor and load are in equilibrium and the speed remains steady. At all lower speeds, the motor torque will be higher than the load torque, so the net torque will be positive, leading to an acceleration of the motor. As the speed rises towards X, the acceleration reduces until the speed stabilizes at X.

Conversely, at speeds above X the motor’s driving torque is less than the braking torque exerted by the load, so the net torque is negative and the system will decelerate until it reaches equilibrium at X .This example is one which is inherently stable, so that if the speed is disturbed for some reason from the point X, it will always return there when the disturbance is removed.

Turning now to the derivation of the torque/speed characteristics of the d.c. motor, we can profitably use the previous example to illustrate matters. We can obtain the full-load speed for V = 500 V by first calculating the back e.m.f. at full load (i.e. when the current is 20 A).From equation (3.8) we obtain E = 500 (20 1) = 480V .

We have already seen that the e.m.f. constant is 480 V/1000 rev/min, so the full load speed is clearly 1000 rev/min. From no-load to full-load, the speed falls linearly, giving the torque -speed curve for V = 500 V shown in Ill. 3.9.Note that from no-load to full-load, the speed falls from 1040 rev/min to 1000 rev/min, a drop of only 4%.Over the same range the back e.m.f. falls from very nearly 500 V to 480 V, which of course also represents a drop of 4%.

We can check the power balance using the same approach as in Section 1.7 of Section 1. At full load the electrical input power is given by VI, i.e.500 V 20 A = 10 kW. The power loss in the armature resistance is I^ 2 R = 400 1 = 400 W. The power converted from electrical to mechanical form is given by EI ,i.e.480 V x 20 A = 9600 W .

We can see from the no-load data that the power required to overcome friction and iron losses (eddy currents and hysteresis, mainly in the rotor) at no-load is approximately 500 V 0 : 8 A = 400 W , so this leaves about 9.2 kW. The rated output power of 9.1 kW indicates that 100 W of additional losses (which we won't attempt to explore here) can be expected under full-load conditions.

Two important observations follow from these calculations. Firstly, the speed drop with load is very small. This is very desirable for most applications, since all we have to do to maintain almost constant speed is to set the appropriate armature voltage and keep it constant. Secondly, a delicate balance between V and E is revealed. The current is in fact proportional to the difference between V and E (equation (3.8)), so that quite small changes in either V or E give rise to disproportionately large changes in the current. In the example, a 4% reduction in E causes the current to rise to its rated value. Hence to avoid excessive currents (which cannot be tolerated in a thyristor supply, for example), the difference between V and E must be limited. This point will be taken up again when transient performance is explored.

A representative family of torque -speed characteristics for the motor discussed above is shown in Ill. 3.9.As already explained, the no load speeds are directly proportional to the applied voltage, while the slope of each curve is the same, being determined by the armature resistance: the smaller the resistance the less the speed falls with load.

These operating characteristics are very attractive because the speed can be set simply by applying the correct voltage.

The upper region of each characteristic in Ill. 3.9 is shown dotted because in this region the armature current is above its rated value, and the motor cannot therefore be operated continuously without overheating. Motors can and do operate for short periods above rated current, and the fact that the d.c. machine can continue to provide torque in proportion to current well into the overload region makes it particularly well-suited to applications requiring the occasional boost of excess torque.

Ill. 3.9 Family of steady-state torque -speed curves for a range of armature voltages A cooling problem might be expected when motors are run continuously at full current (i.e. full torque) even at very low speed, where the natural ventilation is poor. This operating condition is considered quite normal in converter-fed motor drive systems, and motors are accordingly fitted with a small air-blower motor as standard.

This book is about motors, which convert electrical power into mechanical power. But, in common with all electrical machines, the d.c. motor is inherently capable of operating as a generator, converting mechanical power into electrical power. and although the overwhelming majority of motors will spend most of their working lives in motoring mode, there are applications such as rolling mills where frequent reversal is called for, and others where rapid braking is required. In the former, the motor is controlled so that it returns the stored kinetic energy to the supply system each time the rolls have to be reversed, while in the latter case the energy may also be returned to the supply, or dumped as heat in a resistor.

These transient modes of operation may better be described as 'regeneration 'since they only involve recovery of energy originally provided by the motor.

Continuous generation is of course possible using a d.c. machine provided we have a source of mechanical power, such as an internal combustion (IC) engine. In the example discussed above we saw that when connected to a 500V supply, the unloaded machine ran at 1040 rev/min, at which point the back e.m.f. was very nearly 500 V and only a tiny positive current was flowing. As we applied mechanical load to the shaft the steady-state speed fell, thereby reducing the back e.m.f. and increasing the armature current until the motor toque was equal to the opposing load torque and equilibrium is attained. We saw that the smaller the armature resistance, the less the drop in speed with load.

Conversely, if instead of applying an opposing (load) torque, we use the IC engine to supply torque in the opposite direction, i.e. trying to increase the speed of the motor, the increase in speed will cause the motional e.m.f. to be greater than the supply voltage (500 V).

This means that the current will flow from the d.c. machine to the supply, resulting in a reversal of power flow. Stable generating conditions will be achieved when the motor torque (current) is equal and opposite to the torque provided by the IC engine. In the example, the full-load current is 20A, so in order to drive this current through its own resistance and overcome the supply voltage, the e.m.f. must be given by

The corresponding speed can be calculated by reference to the no-load e.m.f. (499.2 V at 1040 rev/min) from which the steady generating speed is given by

N gen 1040 = 520 499 : 2 i : e : N gen = 1083 rev = min :

On the torque-speed plot ( Ill. 3.9) this condition lies on the down ward projection of the 500 V characteristic at a current of 20A. We note that the full range of operation, from full-load motoring to full load generating is accomplished with only a modest change in speed from 1000 to 1083 rev/min.

It is worth emphasizing that in order to make the unloaded motor move into the generating mode, all that we had to do was to start supplying mechanical power to the motor shaft. No physical changes had to be made to the motor to make it into a generator -the hardware is equally at home functioning as a motor or as a generator -which is why it's best referred to as a 'machine'. (How nice it would be if the IC engine could do the same; whenever we slowed down, we could watch the rising gauge as its kinetic energy is converted back into hydrocarbon fuel in the tank!) To complete this section we will derive the analytical expression for the steady-state speed as a function of the two variables that we can control, i.e. the applied voltage ( V ),and the load torque ( T L).Under steady-state conditions the armature current is constant and we can therefore ignore the armature inductance term in equation (3.7);

and because there is no acceleration, the motor torque is equal to the load torque. Hence by eliminating the current I between equations (3.5) and (3.7) and substituting for E from equation (3.6) the speed is given by

v = V k R k 2 T L ( 3 : 10 )

This equation represents a straight line in the speed/torque plane, as we saw with our previous worked example. The first term shows that the no-load speed is directly proportional to the armature voltage, while the second term gives the drop in speed for a given load torque. The gradient or slope of the torque -speed curve is R = k 2, showing again that the smaller the armature resistance, the smaller the drop in speed when load is applied.

Base speed and field weakening

Returning to our consideration of motor operating characteristics, when the field fl ux is at its full value the speed corresponding to full armature voltage and full current (i.e. the rated full-load condition)is known as base speed (see Ill. 3.10).

The motor can operate at any speed up to base speed, and at any torque (current) up to the rated value by appropriate choice of armature voltage. This full fl ux region of operation is indicated by the shaded area O abc in Ill. 3.10, and is often referred to as the 'constant torque ' region of the torque-speed plane. In this context 'constant torque ' signifies that at any speed below base speed the motor is capable of producing its full rated torque. Note that the term constant torque does not mean that the motor will produce constant torque, but rather it signifies that the motor can produce constant torque if required: as we have already seen, it's the mechanical load we apply to the shaft that determines the steady-state torque produced by the motor.

When the current is at maximum (i.e. along the line a,b in Ill. 3.10), the torque is at its maximum (rated) value. Since mechanical power is given by torque times speed, the power output along a,b is proportional to the speed, and the maximum power thus corresponds to the point b in Ill. 3.10.At point b, both the voltage and current have their full rated values.

To run faster than base speed the field fl ux must be reduced, as indicated by equation (3.9).Operation with reduced fl ux is known as field weakening ',and we have already discussed this perhaps surprising mode in connection with the primitive linear motor in Section 1.E.g., by halving the fl ux (and keeping the armature voltage at its full value),the no-load speed is doubled (point d in Ill. 3.10).The increase in speed is however obtained at the expense of available torque, which is proportional to fl ux times current (see equation (3.1)).The current is limited to rated value, so if the fl ux is halved, the speed will double but the maximum torque which can be developed is only half the rated value (point e in Ill. 3.10).Note that at the point e both the armature voltage and the armature current again have their full rated values, so the power is at maximum, as it was at point b .The power is constant along the curve through b and e ,and for this reason the shaded area to the right of the line bc is referred to as the 'constant power 'region. Obviously, field weakening is only satisfactory for applications which don't demand full torque at high speeds, such as electric traction.

The maximum allowable speed under weak field conditions must be limited (to avoid excessive sparking at the commutator), and is usually indicated on the motor rating plate. A marking of 1200/1750 rev/min, for example, would indicate a base speed of 1200 rev/min, and a maximum speed with field weakening of 1750 rev/min. The field weakening range varies widely depending on the motor design, but maximum speed rarely exceeds three or four times the base speed.

To sum up, the speed is controlled as follows:

Below base speed, the fl ux is maximum, and the speed is set by the armature voltage. Full torque is available at any speed.

Above base speed, the armature voltage is at (or close to) maximum and the fl ux is reduced in order to raise the speed. The maximum torque available reduces in proportion to the flux.

To judge the suitability of a motor for a particular application we need to compare the torque -speed characteristic of the prospective load with the operating diagram for the motor: if the load torque calls for operation outside the shaded areas of Ill. 3.10, a larger motor is clearly called for.

Full field ('constant torque ') region Weak field ('constant power')region

0 a b c e d

Base speed

Speed Torque

Ill. 3.10 Region of continuous operation in the torque -speed plane

Finally, we should note that according to equation (3.9), the no-load speed will become in finite if the flux is reduced to zero. This seems unlikely; after all, we have seen that the field is essential for the motor to operate, so it seems unreasonable to imagine that if we removed the field altogether, the speed would rise to infinity. In fact, the explanation lies in the assumption that 'no-load 'for a real motor means zero torque.

If we could make a motor which had no friction torque whatsoever, the speed would indeed continue to rise as we reduced the field flux towards zero. But as we reduced the flux, the torque per ampere of armature current would become smaller and smaller, so in a real machine with friction, there will come a point where the torque being produced by the motor is equal to the friction torque, and the speed will therefore be limited. Nevertheless, it's quite dangerous to open circuit the field winding, especially in a large unloaded motor. There may be sufficient 'residual 'magnetism left in the poles to produce significant accelerating torque to lead to a run-away situation. Usually, field and armature circuits are interlocked so that if the field is lost, the armature circuit's switched off automatically.

Armature reaction

In addition to deliberate field-weakening, as discussed above, the flux in a d.c. machine can be weakened by an effect known as 'armature reaction '.As its name implies, armature reaction relates to the influence that the armature MMF has on the flux in the machine: in small machines it's negligible, but in large machines the unwelcome field weakening caused by armature reaction is sufficient to warrant extra design features to combat it. A full discussion would be well beyond the needs of most users, but a brief explanation is included for the sake of completeness.

The way armature reaction occurs can best be appreciated by looking at Ill. 3.1 and noting that the MMF of the armature conductors acts along the axis defined by the brushes, i.e. the armature MMF acts in quadrature to the main flux axis which lies along the stator poles. The reluctance in the quadrature direction is high because of the large air spaces that the flux has to cross, so despite the fact that the rotor MMF at full current can be very large, the quadrature flux is relatively small; and because it's perpendicular to the main flux, the average value of the latter wouldn't be expected to be affected by the quadrature flux, even though part of the path of the reaction flux is shared with the main flux as it passes (horizontally in Ill. 3.1) through the main pole-pieces.

A similar matter was addressed in relation to the primitive machine in Section 1.There it was explained that it was not necessary to take account of the flux produced by the conductor itself when calculating the electromagnetic force on it. and if it were not for the nonlinear phenomenon of magnetic saturation, the armature reaction flux would have no effect on the average value of the main flux in the machine shown in Ill. 3.1:the flux density on one edge of the pole-pieces would be increased by the presence of the reaction flux, but decreased by the same amount on the other edge, leaving the average of the main flux unchanged. However if the iron in the main magnetic circuit's already some way into saturation, the presence of the rotor MMF will cause less of an increase on one edge than it causes by way of decrease on the other, and there will be a net reduction in main flux.

We know that reducing the flux leads to an increase in speed, so we can now see that in a machine with pronounced armature reaction, when the load on the shaft is increased and the armature current increases to produce more torque, the field is simultaneously reduced and the motor speeds up. Though this behavior isn't a true case of instability, it's not generally regarded as desirable! Large motors often carry additional windings fitted into slots in the pole-faces and connected in series with the armature. These 'compensating 'windings produce an MMF in opposition to the armature MMF, thereby reducing or eliminating the armature reaction effect.

Maximum output power

We have seen that if the mechanical load on the shaft of the motor increases, the speed falls and the armature current automatically in creases until equilibrium of torque is reached and the speed again becomes steady. If the armature voltage is at its maximum (rated) value, and we increase the mechanical load until the current reaches its rated value, we are clearly at full-load, i.e. we are operating at the full speed (determined by voltage) and the full torque (determined by cur rent).The maximum current is set at the design stage, and reflects the tolerable level of heating of the armature conductors.

Clearly if we increase the load on the shaft still more, the current will exceed the safe value, and the motor will begin to overheat. But the question which this prompts is 'if it were not for the problem of over heating, could the motor deliver more and more power output, or is there a limit'? We can see straightaway that there will be a maximum by looking at the torque-speed curves in Ill. 3.9.The mechanical output power is the product of torque and speed, and we see that the power will be zero when either the load torque is zero (i.e. the motor is running light)or the speed is zero (i.e. the motor is stationary).There must be maximum between these two zeroes, and it's easy to show that the peak mechanical power occurs when the speed is half of the no-load speed. However, this operating condition is only practicable in very small motors: in the majority of motors, the supply would simply not be able to provide the very high current required.

Turning to the question of what determines the theoretical maximum power, we can apply the maximum power transfer theorem (from circuit theory) to the equivalent circuit in Ill. 3.6.The inductance can be ignored because we assume d.c. conditions. If we regard the armature resistance R as if it were the resistance of the source V , the theorem tells us that in order to transfer maximum power to the load (represented by the motional e.m.f. on the right-hand side of Ill. 3.6)we must make the load 'look like 'a resistance equal to the source resistance, R .This condition is obtained when the applied voltage V divides equally so that half of it's dropped across R and the other half is equal to the e.m.f. E .

(We note that the condition E = V = 2 corresponds to the motor running at half the no-load speed, as stated above.)At maximum power, the current is V /2 R ,and the mechanical output power ( EI )is given by V 2 /4 R .

The expression for the maximum output power is delightfully simple. We might have expected the maximum power to depend on other motor parameters, but in fact it's determined solely by the armature voltage and the armature resistance. E.g., we can say immediately that a 12 V motor with an armature resistance of 1 V cannot possibly produce more than 36 W of mechanical output power.

We should of course observe that under maximum power conditions the overall efficiency is only 50% (because an equal power is burned o ff as heat in the armature resistance);and emphasize again that only very small motors can ever be operated continuously in this condition. For the vast majority of motors, it's of academic interest only because the current ( V /2 R ) will be far too high for the supply.