AMAZON multi-meters discounts AMAZON oscilloscope discounts
1. Introduction
This Section explains how a motor should be rated for use on steady-state, intermittent or incremental duty. For steady-state applications where the motor output is defined in terms of a constant and continuous speed and torque, motor rating and selection from a range of motor specifications is normally straightforward. If the motor torque is required intermittently, selection is guided by specifications which give torque and speed limits for intermittent duty in general. Care is needed here in assessing the likely effects of the intermittent operation of the motor on the peak rise in motor temperature.
In incremental motion the requirements are not usually defined only in terms of motor output torque and speed. The other loading conditions normally include load inertia and angle of rotation, or load mass and distance covered, plus the load velocity profile over the period of movement. The required motor speed and torque profiles may be calculated for individual motors on a trial and error basis, and the experienced designer may often select the most suitable motor without too much iteration. To assist in the process, a selection method is given in this Section which indicates the most suitable motor for given incremental loading conditions.
Motor suppliers normally provide all the data which the user needs to assess the thermal performance of a motor, but the values of the constants should never be assumed to apply exactly to a particular motor. Calculations which give the final temperature for the most likely 'worst case' normally assume the stator resistance to be 10% higher, and the torque constant 10% lower than the nominal values given in the motor specification. The numerical examples in this Section show calculations of nominal torques and temperatures using nominal constants, and also give the results at the extreme tolerance of the constants.
2. Motor heating
Heat is generated in the brushless motor as a result of the fir loss in the stator winding and iron losses in the stator and rotor. In addition, some heat arises from the friction between the bearing seal and the rotor shaft. Rotor windage loss is normally very low. Iron losses are produced by induced eddy currents in the stator and rotor, and are partly a function of the PWM frequency of the stator winding current. Motors which exhibit rotor heating do not appear to have lower losses when fitted with laminated rather than solid hubs. This is because the eddy currents favor the magnets as the closest part of the rotor to the stator winding. Laminated hubs do of course have advantages as far as manufacturing costs are concerned.
Motor temperature rises as losses are generated, and the thermal resistance Rth quoted by the manufacturer gives a temperature rise in degrees centigrade per watt of total power loss. However, the final steady-state temperature differs at the various parts of the motor structure; for example, the magnets may become hotter than the stator core. The figure for Rth must therefore be high enough to allow any part of the motor to remain below the maximum temperature, normally 150 C.
Soac curves
The thermal characteristic of a brushless motor is usually given as the boundary of the Safe Operation Area for Continuous operation, which takes the form shown in FIG. 1. The curve shows the maximum continuous torque rating at all speeds from zero to maximum in the way already described for the brushed motor in Section 1.
FIG. 1 The Soac thermal boundary
The Soac curve is drawn from the results of practical tests. The motor is loaded and the current gradually adjusted until the above-ambient steady-state temperature at any part of the motor reaches l l0~ This means that at an ambient temperature of 40~ no part of the motor exceeds the maximum allowable figure of 150 degree The speed and current are measured, and the torque is calculated by multiplying the current by the torque constant of the motor. (KT is independent of motor speed.) At zero speed the rated stall torque is:
Ts = KTIs
where Is is the maximum continuous current at zero speed, when the heating is entirely the result of the i2R loss.
Speed-sensitive loss
If the motor could operate without generating a speed-sensitive loss, the Soac characteristic would rise vertically from T~. The total losses and motor temperature are constant and maximum at all points on the curve. The i2R loss at speed is therefore lower than that at zero speed by the amount of the speed- sensitive loss Psp- At all points on the curve the total losses are therefore giving
I2R + Psp -- t2sR ~ Psp - (I2s -/2)R'
Resistance R ~ is the stator resistance at the maximum motor temperature. Motor specifications normally give the 'cold' stator resistance R at 25 degree and so at 150degree the resistance is R' - R[1 + 0.00385(150 - 25)] - 1.48 R where the figure of 0.00385 is the temperature coefficient of resistance of copper for a temperature rise above 25degree At any speed, the speed-sensitive loss is therefore given by R
- Psp 1"48~-~T (T2s - '2
where Tsoac is the rated torque given by the Soac curve. The speed-sensitive loss may amount to 25-35% of the total power loss when the brushless motor runs on load at the center of the speed range. A restriction on the average operation period of the motor may be imposed at mid to high speeds, where a substantial speed-sensitive loss is possible.
Confusion sometimes arises when the horizontal axis of the Soac plane is treated as a torque which is related solely to motor speed. This misinterpretation allows an erroneous 'torque loss' to be calculated by subtracting the torque at a point on the curve from the torque at the point of stall. It should be remembered that the horizontal axis gives the torque available at various speeds when the motor temperature is at its maximum limit. Differences between the squares of the rated torques shown along the axis are used when the speed-sensitive loss is calculated, as already described.
Heatsinks and blowers
The Soac curve is drawn from the results of rating tests in which the motor is attached to a heat-absorbing frame. When the motor is installed in its working environment, its front flange is usually bolted directly to the frame of the driven machinery, the frame providing a heatsink. If the frame has been thermally isolated from the motor mounting point in some way, a heatsink should be included between the motor flange and the frame.
There are two situations where the cost of a blower may be justified at the planning stage of an installation. One is where a combination of a very low motor inertia and a high torque is required. Low inertia limits the choice of motor to the small end of the range, where output torque ratings are also relatively low. Forced cooling must then be used to increase the rating. The other case where a blower is sometimes used is at the other end of the motor range. If the very largest machine available cannot provide the output required, its rating may be increased with a blower. Between these two extremes, blowers are not usually part of the best solution to a particular load specification and should be avoided if possible.
The cost of a blower and motor is normally of the same order as the cost of the next larger motor. Any saving in cost is clearly negligible in comparison to the cost of failure of the blower, followed by the motor.
3. Steady-state rating
The rating of a brushless motor in terms of its continuous, constant torque output may be assessed in the way described in Section 1 for the brushed machine. The maximum continuous stall current Is is normally given in the data sheet, and the Soac curve is always available. If the form factor of the current supplied to a trapezoidal motor or to a brushed DC motor is other than unity, the same considerations apply to both.
Example 1
A motor is to run continuously. Estimate the speed-sensitive loss using the Soac diagram of FIG. 2 (motor M06). Estimate the stall torques, and the maximum output torques at a speed of 4500 rpm, available from the trapezoidal and sinusoidal forms of the motor. The form factor of the current supplied to the trapezoidal motor is 1.1. The motor constants are
KT( trap) = 0.42 Nm/A
KT(sin) = 0.51 Nm/A
RLL = 1.29 ohm
Speed-sensitive loss
Referring to FIG. 2 (M06) and using the trapezoidal constants gives the estimate of the loss at 4500 rpm as
The figure of 37 W applies to both motors, remembering that the thermally effective resistance for the sinusoidal form is R = 1.5 RLL.
FIG. 2 Soac curves
Torques
Trapezoidal motor
The maximum continuous output torque at 4500 rpm is 3.2 Nm when the motor current has a form factor of unity. When the form factor is 1.1, the torque must be reduced by 1/1.1 so that the rms value of the input current remains the same. The maximum average torque available at 4500 rpm is then
T = 3.2 /1.i = 2.9 Nm
and the average, continuous stall torque becomes 3.7 Ts = 1.1 = 3.3Nm
Sinusoidal motor
If the input current is purely sinusoidal, the continuous torque at 4500 rpm is T~oac - 3.2 Nm.
maximum
4. Intermittent torque
Intermittent operations may be broadly divided into two types.
This section deals with the simple torque profile, where there are intervals between the application of a constant motor torque. Cases where the load moves incrementally over a specific velocity-time profile are left until Sub-Section 5.
Motor temperature
The main effects of intermittent operation on the temperature of the brushed motor were dealt with in Section 1. It was seen that the intermittent rating of the brushed motor is affected by the presence of a temperature ripple, which is most pronounced at the rotor winding. The same effects occur at the stator winding of the brushless motor. FIG. 3 shows an intermittent output torque, applied every t' seconds over a time of tp. The remainder of the diagram shows the associated power loss and the steady-state temperature of the winding after the motor has been running on the intermittent cycle for some time. The diagram is similar to that of Figure 1.15, and shows above-ambient winding temperatures of Omin, Oav and Opk. The average, above-ambient temperature is
Oav- RthPloss(av)
where Ploss(av) is the average value of the motor losses and Rth is the published value of the thermal resistance Rth in ~ A large ripple can obviously lead to overheating if the average loss is large enough by itself to raise the average winding temperature by 110 degree At an ambient temperature of 40 degree the maximum winding temperature of 150 degree would be exceeded by
Opk -- Oav.
FIG. 3 Effect of intermittent torque on the motor winding temperature
The rate of rise of 'the motor temperature' from ambient to steady state depends on the thermal time constant degree'th. For a typical motor, ~'th is of the order of 35 minutes. The published value is normally the overall time constant of the main mass, which for the brushless motor is taken to be the stator winding, stator iron and motor case. The temperatures of these three motor components do not vary at the same rate. The i2R heating energy passes from the winding and through the stator core before reaching the motor case of the motor, and so the winding must heat up before the stator core. The winding heats quickly, particularly at the start of the torque pulse, and also cools quickly at the start of the pulse gap. This means that the average time constants of the rising and falling waveforms of winding temperature in FIG. 3 are lower than the value of ~-th quoted for the motor as a whole [9]. The ripple on the winding temperature becomes more pronounced as t' is increased. The ripple also increases as tp is reduced, assuming the average losses are kept the same by increasing the torque. The extreme case would be if the motor produced four times its continuous rated torque over a pulse width of tp- t'/16. As a rule of thumb, the ripple in the steady-state winding temperature will normally be confined to a band of
+IO~ when Tth > lOOt'
...where Tth > 25 minutes and the maximum average temperature rise is 100~ When 7"th < 100t', the ripple may still fall within the •176 band at the higher end of the tp/t' range. Such cases should be considered individually.
Calculations are relatively easy when the period of the intermittent operation is less than 1% of the thermal time constant of the motor. The maximum ripple is +10~
The designed maximum rise in the winding temperature is 110~ and so the maximum average losses are those which lead to an average winding temperature of 110- 10, or 100 degree above ambient. The losses are found by adding the iZR loss to the speed-sensitive loss as estimated from the Soac diagram.
Example 2
The sinusoidal motor of Example 1 is to provide the intermittent torque shown in FIG. 4 at a speed of 2500 rpm.
What is the maximum height Tmax of the torque pulse,
assuming that the peak temperature rise of the winding is limited to 110 o C?
The motor constants are as follows:
RIlL = 1.29 g2 Kr = 0.51 Nm/A Tth -- 35 minutes Rth -- 0.75~
The current at Tmax is I = Tma__.__.~x __ Tma...._..~x m.
KT 0.51
The effective stator resistance for the purpose of calculation of the i2R loss in the sinusoidal motor is 1.5 RLL. The effective stator resistance at the average, steady-state temperature is
R' = 1.5 RLL[1 + 0.00385( Oss - 25)]
The ratio of the thermal time constant of the motor to the period of the intermittent operation is 35 x 60/18, or 117.
We may assume that the maximum ripple will be within +10~ and that the average temperature of the winding can be allowed to rise by 110- 10, or 100~ In the extreme case, the average winding temperature would then be 140~ at an ambient of 40~ The maximum, average, effective stator resistance is therefore
R' = 1.5 • 1.2911 + 0.00385(140 - 25)] = 2.79 f~
The i2R loss in watts in the stator winding is or I2R '-- (Tmax/0.51) 2 • 2.79 I2R '- 10.73 T2ax
Following the approach used in Example 1 gives the speed- sensitive loss at 2500 rpm as
Psp ,~ 20W The total peak and average losses are Ploss(pk) = (10.73 T2ax --1- 20)
and tp
Ploss(av) = ~Ploss(pk) = (1.19T2max + 2)
where t' - 9tp. The average temperature rise of the winding is Oav -- RthPloss(av)
or giving
100 = 0.75(1.19TEax + 2)~ Tmax -- 10.5 Nm
FIG. 4 Motor torque for Example 2
Tolerances
The above figures are based on the nominal values of torque constant and winding resistance. In an extreme case, where KT is below and R above the nominal value by 10%, the maximum torque would be
Tmax - 9.0 Nm Motor tests
The motor in the above example was tested on the intermittent duty cycle shown in FIG. 5 under the following conditions
T = 10.5Nm
t' = 9tp
Tth -- 78t'
O0 = 29~
The ratio t'/tp was equal to 9 for the both the test and Example 5.2. The test was, however, made pessimistic by reducing the ratio 7"th/t t from the figure of 117 used for the example to 78, well below the recommended minimum of 100. When the temperature of the test motor had reached steady-state conditions, the variation in the winding temperature was recorded. The results are shown in Table 5.1, together with the nominal figures predicted in Example 5.2. At the ambient temperature of 29~ the average rise of the test motor was 97 degree with a ripple of + 4 degree Correcting the figures to allow for the maximum ambient temperature of 40 degree gives a rise for the motor of 100 + 5 degree The test motor had nominal values of resistance and torque constant, and so its average rise in temperature may be expected to be in close agreement with the value used in the example.
FIG. 5 Torque cycle for motor test
Table 1 Steady-state temperature ripples
Thermal models
In Example 2 and in the motor test, the demand was known in terms of an intermittent torque. In practice there may be other intermittent effects which are more difficult to analyse.
An extreme example would be where pulses of the maximum allowable motor current occur when the motor is at standstill. Thermal models are useful in such cases, where, for example, the thermal resistance of the motor is split into the winding-to-case and case-to-ambient values, and the thermal capacity into the winding and case capacities [9].
5. Incremental motion
The aim of this section is to assist in the selection of a motor for applications where the load is to be driven incrementally over a specified load velocity profile, and where the load and motor inertias should if possible be matched. Calculations are based on the i2R loss, plus an allowance for the speed-sensitive loss.
It was shown in Section 4.5 that the trapezoidal load velocity profile is most efficient when the periods of acceleration, deceleration and constant speed are all equal. Such a profile may not be realizable in practice, and the three periods may differ from each other. In this section we take an approximate account of the profile shown in FIG. 16 and repeated here in FIG. 6, where the load accelerates and decelerates over the proportions pl and p2 of the motion time tp. The total period of the increment is t'.
FIG. 6 Trapezoidal profile of load velocity ( Pl -t- P2 < 1 )
Motor selection for the rotating load
The principles of optimization were introduced in Section 4 on the basis of the motor-load dynamics. The thermal ratings of the motors used in the examples were assumed to be high enough to cope with demands of the given loads. This section describes a method of motor selection based on the thermal rating needed for a particular load.
Stator heating energy
In Section 4.5, the energy produced in the form of heat by the stator iER loss was found for the case of a load consisting of a rotating mass of inertia JL, and an opposing torque TL. When the load rotates through 0p in time tp, the energy is cp
... where J- (Jm + JL/G 2) and G is equal to unity when the load is driven directly from the motor. For the purpose of optimization, we took account of any trapezoidal profile by using a general expression for the profile constant Cp. The constant takes its minimum value of 13.5 when the trapezoidal profile of load velocity is equally distributed. In this section, however, we are interested in an approximate method of motor selection and so the exact value of the profile constant is not important. To allow for some asymmetry we will use a value of 15 in the initial selection of the motor. When TL = 0, the stator heating energy is therefore
15 RJ 2 G 202 p K,2/3 Tp
Selection criterion
For matched motor and load inertias, and zero load torque, the square of the optimum reducer ratio has been shown to be and so
_ JL Jm
:G~- Jm+G---~o GZ=4JrnJL
For the matched inertia case, we may therefore substitute 4JmJL for J2G2 in the last expression above for e. At the maximum allowable motor temperature of 150~ the steady-state i2R power loss for the matched inertia case becomes
_ , ,.. 60R'Jm JL 02 3 t eloss(max) e/ t KElp t
where R'= 1.48R. For most incremental motion applications, rth>> t' and any heating variations will be filtered out. The maximum allowable rise in motor temperature is 110 degree and so
Ploss(max) < llO Rth
The mechanical time constant of the motor is RJm "I'm -- KTKE
For the trapezoidal motor, KE -- KT. Combining the last three equations above gives the limiting condition as Rth'rm < 1.2 t3t' JLO2p
Sinusoidal and trapezoidal motors have much the same mechanical time constant, thermal resistance and rating. The last expression should therefore apply to both forms. So far we have dealt with only the i2R loss for the inertial part of the load. When the load includes an opposing torque TL, it can be shown that the i2R loss increases by the factor
61 = 0.511 + v/(1 + A1)] where A1 -- OpJL
Approximately 50 degree more may be added to cover the speed- sensitive loss, although this figure is very pessimistic at the lower end of the speed range. When we take account of all the i2R and speed-sensitive losses, the motor selection criterion becomes
RthTm < Rating coefficient
0.8 t'
Table 2 lists the values of motor constants given in typical specification sheets and also the values of the rating coefficient Rth'rm ~ ms/W, for a range of sinusoidal motors which increase in size down the table. KT is given as the total torque constant for the three phases. Four different voltage specifications are available for each motor shown on the table, but the choice of voltage does not significantly affect the rating. The speed limits are set by dividing the no-load voltage per 1000 rpm gradient by the peak supply voltage for the particular motor chosen for each line of the table.
Table 2 Constants for various sinusoidal motors
The values of RthTm are directly affected by the value of Tin and do not fall smoothly as the size of the motor increases.
Selection may be based on the combination of the rating coefficient with the higher or lower value of time constant as required. In the following examples it is assumed that the motor is to be chosen solely on the basis of its rating coefficient. Following an initial selection, the rms torque required from the prospective motor is calculated for both the nominal and extreme values of KT and R. The point of selection may then need to be adjusted to the next larger or the next smaller motor.
Example 4
A load is to be rotated incrementally over the velocity profile shown in FIG. 7. The loading constants are:
Op = 27r rad t ~ = 0.20 s
FIG. 7 Motor torque and velocity for Example 4
Motor selection
The load torque factor in the denominator of the rating criterion above is:
61 -- 0.511 + x/(1 + A1)] where A1 -- OpJL
Inserting the numerical values gives 61 - 1.4. An initial choice is made from Table 2 at the maximum value of Rth'rm which satisfies
0.8 x 0.12 3 x 0.2 RthTm <
0.0082 x (27r) 2 x 1.4
or RthTm < 0.61 x 10 -3
The smallest motor in Table 2 with a value of RthT"m below
0.61 is M09.
Reducer ratio
Having identified a motor, the next step is to calculate the optimum ratio of a reducer. FIG. 7 shows the motor speed profile as the motor accelerates, runs at constant speed and then decelerates. Using the method described in Section 4, the optimum ratio of a reducer between M09 and the load is found to be G~ = 4.4.
Motor speed
The constant speed of the load is:
~')C =
0p (Section 4.5)
tp[1 - 0.5(pl + P2)] 21r = 65.1 rad/s
0l [,
The constant motor speed in FIG. 7 is therefore w' = G~xwc = 286 rad/s = 2735 rpm Speed-sensitive loss
A pessimistic estimate of the speed-sensitive loss is obtained by taking ~' to be the motor speed throughout the incremental time tp. FIG. 2 shows the Soac curve for motor M09.
Using the method followed in Example 1 gives the loss at
2735 rpm as Psp "~ 70 W
[...]
FIG. 8 Motor torque and velocity for Example 5
[...]
5. Precautions
The temperature of 150 degree is the absolute maximum for any part of a brushless servomotor. Overheating of the motor or its environment is an obvious risk if the safety margin used in selecting the motor is too small. On the other hand, an overcautious approach can result in an unnecessarily large and expensive motor. The final choice is often influenced by several factors, and no figure can be given for a safety margin which suits all circumstances.
Throughout this Section we have allowed for 10% variation of the stator resistance and torque constant, between individual motors of the same type. We have catered for this variation by allowing a margin of 17% between the required rms torque and the rated torque of the motor. It is in fact possible for the torque constant to fall by 10% as the temperature rises, mainly due to the fall in the magnetic intensity of the permanent magnets. The same allowance for the stator resistance is, however, very generous. The normal variation is much smaller, being dependent, for example, on very small differences between the nominal and actual diameter of the wire used in the manufacture of the winding.
The conclusion is that the margin of 17% between the required and rated torque should be adequate, assuming that there is no other reason why the motor should be derated.
In practice there may be several thermal effects to be taken into account, apart from the effect of the tolerance allowed in the values of the motor constants. For example, an excessive amount of heat transmitted directly from a hot motor may cause distortion of the frame of a precision mechanism such as a ball screw. It is also important that the overall design of any system should allow for sufficient space between the hot surface of the motor and any heat-sensitive equipment, or allow for the accommodation of a larger motor which can run at a lower temperature. Care is also required in the way the motor is fitted to the frame of the installation. As already mentioned in Sub-Section 2, the thermal resistance of a motor is likely to be higher than the published value when it is thermally isolated from the flame. Thermal isolation can be deliberate, or it can be the accidental result of the insertion of electrical insulation material at all points of metal to metal contact with the flame. The maximum rms torque available in either case will be lower than the figure given by the Soac curve, which is plotted experimentally when the motor is bolted directly to a typical flame. If possible, a heatsink should be fitted whenever the flame has to be electrically or thermally isolated from the motor.
Assuming that the motor has been selected correctly on the basis of its ability to supply the required torque, and that all other factors have been taken into account, any overheating which does occur is normally the result of accidental misuse.
Computer controlled duty cycles are very easy to change, and a motor which has been correctly rated for a particular torque profile is likely to overheat if the profile is made more demanding. It is, of course, sometimes difficult to judge the range of the future demands of an application. Where space permits, the fitting of a blower will help an existing motor to cope with a moderate increase in demand.