For selection of reliable and efficient motor it is essential that the conditions of service are well known.
It is not sufficient to simply specify the output power in kW and the speed but it is also necessary to know the following additional particulars:
(i) Torque at the shaft during running, starting and at different loads.
(ii) Accelerating torque and braking torque.
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(iii) Switching frequency.
(iv) Efficiency of motor at different loads.
(v) Other working requirements.
In studying the behaviour of a motor selected for a particular driven unit, one of the first problems involved is to determine whether the speed-torque characteristic of the motor suits the requirements imposed by the speed-torque characteristic of the driven unit. Drive behaviour during the transient period of a start-up, braking, or speed change-over also depends upon how the speed-torque characteristics of the motor and the driven unit vary with speed.
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It is, therefore, imperative to study these characteristics in order to be able to select correctly the motor and obtain an economical drive.
1. Speed-Torque Characteristics of Machines or Mechanisms:
The speed-torque characteristic of a machine or mechanism given by the relation ω = f(TL) is defined as the relationship between the speed at which it is operated and the resisting or load torque it develops.
Different kinds of mechanisms and machines exhibit different speed-torque characteristics. However, several general conclusions may be drawn if we use the following empirical equation for the speed-torque characteristic of some driven unit of industrial equipment-
TL=T0 + (Trn-T0) (ω/ωn)x…(1.3)
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where TL is gross load (or resisting) torque developed by the unit at speed ω, to, T0 is the resisting torque developed by the unit due to friction in its moving parts, Trn is the resisting torque developed by the unit when it is driven at its nominal rated speed ωn and x is the exponential coefficient characterising the change in resting torque with the change in speed.
The above Eq. (1.3) permits the speed-torque characteristics of different kinds of machines and mechanisms to be roughly divided into the following categories:
i. Loads Requiring Constant Torque at All Speeds:
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Such a load presents to the motor a passive torque which is essentially independent of speed. It is characterized also by the requirement of an extra torque at very near zero speed. For this characteristic x = 0 and the load torque TL does not depend upon speed. The speed-torque characteristic for such loads is illustrated by vertical line in Fig. 1.4. Such loads are dry friction, cranes during hoisting, hoist winches, machine tool feed mechanism, piston pumps operating against a constant pressure head, and conveyors handling a constant weight of material per unit time. In power applications it is usually called the breakaway torque and in control systems, it is referred to as stiction (derived from sticking friction).
Since it changes sign with reversal of rotation, the dry friction torque characteristic is discontinuous, as illustrated in Fig. 1.4.
ii. Loads with Linear-Rising Characteristic:
Such speed- torque characteristics, illustrated by straight line II in Fig. 1.4, are exhibited by calendaring machines, eddy current brakes, separately excited dc generators supplying fixed ohmic resistance loads and fluid or viscous friction. In this case x = 1 and the load torque TL rises in direct proportion to the speed.
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iii. Loads with Non-Linear-Rising (Parabolic) Characteristic:
For such a characteristic x = 2 and the load torque TL is proportional to the square of the speed. Such a characteristic is illustrated by curve III in Fig. 1.4. A load with considerable windage, of which a fan is the extreme example, has a torque which varies almost as the square of the speed. Blowers, centrifugal pumps, propellers in ships or aeroplanes, water wheels, pipe friction, velocity head of pumps etc. also have the same type of speed-torque characteristics.
iv. Loads with Non-Linear Falling (Hyperbolic) Characteristic (or Constant Power Load):
For such a characteristic x = – 1 and the load torque TL is inversely proportional to the speed, while the power required to drive the given unit remains unchanged. Such a characteristic is illustrated by curve IV in Fig. 1.4. Certain types of lathes, boring machines, milling machines and other kinds of metal-cutting machine tools, steel-mill coilers fall under this category of loads.
The categories of loads listed above do not cover all the cases that may be met with in practice but give a good idea of the characteristics typical of a great many kinds of industrial equipment. In actual practice we may come across loads which are combination of these basic types of loads.
2. Load Torque-Time Characteristics:
Perhaps, the variation of load torque with time is of equal or greater importance in selection of motor. This variation in certain applications, can be periodic and repetitive, one cycle of variation being called a duty cycle.
Different types of loads from the point of view of load torque characteristics may be classified as follows:
(i) Continuous, constant loads such as paper making machines, centrifugal pumps or fans operating for a long time under the same conditions.
(ii) Continuous, variable loads such as hoisting winches, metal-cutting lathes, conveyors etc.
(iii) Pulsating loads such as reciprocating pumps and textile looms and, in general, all machines having crank shaft.
(iv) Impact load such as rolling mills, shearing machines, presses, forging hammers etc. Apparent, regular and repetitive load peaks or pulses occur in such loads.
(v) Short time loads such as motor-generator sets for charging of batteries; servo-motors used in remote control of drilling machine clamping rods.
(vi) Short time intermittent loads such as cranes and hoisting mechanisms, excavators, roll trains etc.
Certain machines (such as ball mills) do not strictly belong to any category mentioned above. If such loads (ball mills, stone crushers etc.) were characterized by frequent impacts of comparatively small peaks, it would be more appropriate to place them in the category of continuous variable loads rather than impact loads. Sometimes, it is quite cumbersome to make distinction between pulsating loads and impact loads, because both of them are periodic in nature.
One and same drive can be represented by a load torque which varies either with speed or with time. The most appropriate example is a fan load whose load torque TL is proportional to the square of speed, is also a continuous constant load.
3. Load Torques Varying With Shaft Displacement Angle:
In all machines having crankshafts, such as reciprocating pumps and compressors, frame-saws etc. load torque varies with the angular displacement of the shaft or rotor of the motor. For all such machines, the load torque TL can be resolved into two components—one of constant magnitude Tav and the other a variable TL‘ which changes periodically in magnitude depending on the angular position of the shaft. Such load torque characteristics, can, for simplicity, be represented by a Fourier series as a sum of oscillations of fundamental and harmonic frequencies, i.e.,
Where, θ = ωt, ω being the angular speed of the motor shaft driving the compressor.
During variations in speed, only small deviations from a fixed value of speed ωa occur, therefore, displacement can be represented by θ = (ωa + Δω)t. Thus the variable portion of the load torque may be given as-
The term rΔωt being very small in magnitude may be neglected. Thus, restricting to small deviations in angle from the equilibrium position, a load torque varying with angular displacement of shaft can be transformed to one which varies periodically w.r.t. time.
4. Load Torques Depending on Path or Position of Load during Motion:
In Art 1.9.1. load torques varying with speed were considered. However, load torques, that depend not only speed but also on the nature of the path traced out by the load during its motion, do exist both in hoisting mechanisms and transport systems. For example, the resistance to motion of a train moving up-gradient or taking a turn depends on the magnitude of the gradient or the radius of curvature of the track respectively.
Force due to the gradient is given as-
Fg = 1,000 W sin θ kg. …(1.6)
Where, W is the weight of train in tonne.
But in railway work gradient is expressed as rise in metres in a track distance of 100 m, and is denoted by ‘percentage gradient’ (G%)
i.e., G = Sin θ x 100
or sin θ = G/100
Substituting sin θ = G/100 in Eq. (1.6), we have-
Fg = 1,000 W × G/100 = 10 WG kg. …(1.7)
The tractive force required to overcome curvature resistance is given by the empirical formula given below-
Fc = 700,000W/R kg …(1.8)
Where, R is the radius of curvature in metres.
In hoisting mechanisms in which tail ropes or balancing ropes are not employed (Fig. 1.7), the load torque is not only due to the weight of the unloaded or the loaded cage but also due to that of the lifting ropes or cables, which depends on the position of two cages. When cage 1 is at the bottom most position and is to be lifted upwards, the entire weight of the rope is also to be moved up.
When both the cages are at the same height, the weight of the rope to be lifted becomes zero, since the weight of the ropes on both sides balance each other, being equal in length. When cage 1 is at a higher position than cage 2, a portion of the weight of the rope acts in such a way as to assist the upward motion of the cage 1. Ultimately when cage 1 reaches the top most position, the whole weight of the rope assists the upward movement.
The force resisting the upward movement of the load, Fr due to varying weight of the rope depending on the position of the load is given as-
Where, Wr is the total weight of the rope in kg, h is the desired maximum height to which the cage is to be moved upwards, in metres and x is the height of the cage at any arbitrary position from the bottom most position in metres.
For large values of h, the force Fr affects to a large extent, the performance of the drive used in hoisting mechanisms because in such a situation the weight of the rope may be considerably larger than that of the load to be lifted upwards. If we use tail ropes, as illustrated by means of dotted lines in Fig. 1.7, the weight of the connecting rope can be balanced and almost smooth movement of the cages can be had.
5. Speed-Torque Characteristics of Electric Motor:
A motor speed-torque characteristic is defined as the relationship between the speed at which it operates and the torque it develops, i.e., ω = f(T).
Practically all electric motors—shunt wound, series wound, compound dc motors, squirrel-cage and slip-ring induction motors, and ac commutator motors, have drooping speed- torque characteristics, i.e., their speed falls off as the load torque increases. However, the degree by which the speed changes with the change in torque differs for various types of motors, it being characterized by the so-called hardness of their speed-torque characteristics.
Electric motor speed-torque characteristics may be classified into three main groups:
1. Absolutely Hard (flat) Speed-Torque Characteristic:
A characteristic exhibiting no change in speed with change in load torque. Synchronous motors operate with such a characteristic (horizontal straight line I in Fig. 1.8).
2. Hard Speed-Torque Characteristic:
A characteristic showing a speed which drops only slightly with increase in torque. A hard characteristic is exhibited by a shunt wound dc motor, this also being true of induction motors over the operating region of the speed-torque characteristic (curve II in Fig. 1.8).
The speed-torque characteristic of an induction motor exhibits a “hardness” that differs according to what point along the characteristic is taken into consideration (Fig. 1.9). Between the points of the maximum torque in motor operation Tmax M and the maximum torque in generator operation Tmax G an induction machine will exhibit a rather hard characteristic.
3. Soft Speed Characteristic:
A characteristic showing considerable drop in speed with rise in torque. Series wound dc motor possesses such characteristic, especially along the low-torque portion of the characteristic (curve III in Fig. 1.8). For such motors, the degree of hardness of the characteristic varies all along the curve.
Compound wound dc motors, depending on the degree of hardness their speed-torque characteristics display, may be considered as hard- or soft-characteristic motors.
6. Joint Speed-Torque Characteristic of an Electric Motor:
The joint operation of an electric motor and the unit it drives, when the speed has a steady value, corresponds to a condition of balance between the driving torque of the motor and the resisting torque developed by the driven unit at a given speed. When the resisting or load torque developed at motor shaft by the driven unit undergoes some change, the speed and torque developed by the motor will change automatically so as to restore stable operation at a new value of speed and load torque.
In case of non-electric prime movers (water turbine, steam turbine or diesel/petrol engine) the balance between the resisting torque and the driving torque is achieved by using respective type of governor to control the inflow of energy to the prime mover by increasing or decreasing the flow of water, steam or fuel. In electric motors, the role of the automatic governor is performed by the emf of the motor. This ability of electric motors to maintain the balance of the drive system on changes in the resisting (load) torque developed by the driven unit is extremely valuable because this torque very often is unstable to a certain degree.
This can be illustrated with the help of Fig. 1.10 which illustrates the speed-torque characteristic (curve III) of a dc shunt motor and the two characteristics I and II of the production unit driven by the motor (for example, a conveyor).
Characteristic I corresponds to the no-load condition of the conveyor unit, while characteristic II corresponds to a greater level of load torque developed by the conveyor when it handles the required flow of material. Initially, at the time when the conveyor operates at no load, the motor torque T = T, and the motor operates at speed ω1. As soon as the conveyor starts to maintain a flow of material, the increase in load on the motor acts to brake the motor and reduce its speed. This makes the motor to develop a smaller emf.
Consequently the armature current increases and the motor begins to develop a larger driving torque. The motor torque grows until a point of balance is reached at which the torque developed by the motor = resisting torque of the driven unit, i.e., T = T2 (where the speed is ω2). This new point also constitutes the one that is common both to the speed-torque characteristic II of the conveyor and the speed-torque characteristic III of the motor.
In studying the operation of the motor and the unit it drives, it is sometimes convenient to make use of the so- called joint speed-torque characteristic of the electric drive, a curve representing the algebraic sum of the speed-torque characteristic of the driven unit and that of the drive motor.
Speed-torque characteristics of a fan and the drive motor and the joint speed-torque characteristic of a motor fan unit are represented by curves I, II and III respectively in Fig. 1.11.
When the unit attains a steady speed ωs, the motor operates with the torque T = load torque, TL. The torque indicated on the joint characteristic will be zero under such a condition. Operation of the unit at the steady speed ωs will be seen to be stable in this case because any increase in speed leads to a negative change (drop) in torque, while any drop in speed results in a positive change (increase) in torque.
Curve III is, therefore, an example of a joint speed-torque characteristic of a drive that will be able to operate with stability. If the joint characteristic had the form of curve IV, operation would not be stable because slight increase in speed gives rise to acceleration as the motor torque exceeds the load torque. On the other hand slight decrease in speed gives rise to retardation because the motor torque becomes less than the load torque.
The conditions of drive operation in the steady state discussed above constitute those required for static stability of the drive, and are only applicable when speed and torque vary slowly. During periods of transient (rapid) change involving the dynamic stability, the conditions of drive stability will be different.
Usually, when an electric drive is designed for a specific drive, its speed-torque characteristic is known beforehand. The problem of attaining stable operation in steady-state condition at known speeds and load torques of the driven unit therefore consists in selecting a motor whose speed- torque characteristic will be compatible with that of the driven unit.
This may be achieved by first selecting a suitable type of motor and then changing accordingly the electrical parameters of its circuits. Sometimes, in order to provide the required speed-torque characteristics it becomes imperative to set up special power and control circuits for involved switching of the drive motor and the control apparatus.
7. Dynamics of Motor-Load Combination:
In translational motion, an active or driving force Fd is counterbalanced by a resisting force Fr developed by the driven machine and by an inertia force m dv/dt arising from the change in speed. When the body involved has a mass m expressed in kg and a velocity v expressed in m/s, the inertia force, like other forces, will be expressed in newton’s (kg-m/s2).
The equation for force equilibrium in translation of a body may accordingly, be written in the following form-
The equation for torque equilibrium in translation of a body may, accordingly, be written in the following form-
The above Eq. (1.11) shows that the torque TM developed by the motor is counterbalanced by a resisting or load torque TL exerted at its shaft and by an inertia or dynamic torque J (dω/dt). In above Eqs. (1.10) and (1.11) it is assumed that the mass m of the bodies involved and the polar moment of inertia J of the drive remain constant, an assumption which holds good for a large number of industrial machines and mechanisms. In certain drives, it becomes necessary to deal with a variable polar moment of inertia as in the case of crank drives.
From analysis of Eq. (1.11), it is possible to determine the different states at which an electric drive causing rotational motor can remain:
1. When TM > TL, dω/dt > 0, i.e., the drive will undergo acceleration, in particular, picking up speed to attain rated speed.
2. When TM < TL, dω/dt < 0, i.e. the drive will undergo deceleration and, particularly, coming to rest. Deceleration will evidently also occur at negative values of motor torque. A motor develops a negative torque when it passes over to braking operation.
3. When TM = TL, dω/dt = 0, i.e., the drive will run at a steady-state speed.
The above statements, namely, that when TM > TL the drive accelerates and that when TM < TL the drive decelerates, are valid only when load or restraining torque TL happens to be a passive torque. The reverse may occur with active torque loads. For example, if a motor is switched on for hoisting up a winch, while it is coming down on its own weight, until the direction of rotation changes, deceleration of the drive takes place when TM > TL. In case TM < TL in the above situation when the motor has been switched on for moving the winch up, the load will continue to come down and the motor will accelerate instead of decelerating.
The inertia or dynamic torque J (dω/dt) appears only during transient conditions, i.e., when the speed of the drive varies. During acceleration of the drive, the inertia torque opposes the drive motion, but during braking it maintains the motion of the drive. The inertia torque, both in magnitude and in sign, is determined as the algebraic sum of the motor torque and the resisting and load torque.
In view of the above, the signs for TM and TL in Eq. (1.11) corresponding to motoring operation of the driving machine and to passive load torque (or an active braking torque) respectively. In the general form, the torque equation can be written as-
Selection of sign to be placed before each of the torques in above Eq. (1.12) depends upon the operating condition and on the nature of the resisting or load torques. The equation of motion for a drive makes it possible to determine the dependence of torque, current, speed and path upon the time of operation under transient conditions. All the torques in the equation for motion have to be referred to some given element in the system. Most frequently, both the load torque and the dynamic torque are referred to the motor shaft.
Example:
A motor is coupled to a load having the following characteristics:
i. Motor: Tm = 15 – 0.5ωm
ii. Load: Tl = 0.5ω2m
Find out the stable operating point for this combination.
Solution:
The stable operation will be obtained when-
Tm = Tl
or 15 – 0.5ωm = 0.5ωm2
or ωm2 + ωm – 30 = 0
or ωm = 5 or -6
Discarding minus figure, we have-
ωm = 5 and T = 12.5
So stable operating point is (12.5, 5) Ans.
Referred Load Torques and Moments of Inertia:
A motor generally drives an industrial machine through some transmission system whose individual parts operate at different speeds. In carrying out practical calculations, a necessity arises to refer the torques and masses of individual parts to some convenient element, say, a certain shaft.
Load torques may be referred from one shaft to another on the basis of the power balance of the system. In this case account is taken of the power losses in the intermediate links of the transmission by introducing the respective efficiency values.
Let the speed of the motor shaft be ωM and the speed of shaft of the given industrial machine ωL.
On the basis of equality of power flow, we have-
or load torque referred to the motor shaft,
where TL is the load torque, ηT is the efficiency of transmission and i is speed transmission ratio and is equal to ωM/ωL.
When there are several stages in transmission between the drive motor and the driven machine, as illustrated schematically in Fig. 1.12, with the transmission ratios i1, i2,…, in and the respective transmission efficiencies ηT1, ηT2 …, ηTn , the load torque referred to the motor shaft is given as-
Moments of inertia are referred to a given shaft on the basis that the total amount of kinetic energy stored in the moving parts and referred to the given shaft remains unchanged. With the rotating parts having polar moments of inertia of JM, J1, J2, …, Jn and angular velocities of ωM, ω1, ω2, … ωn (Fig. 1.12), their dynamic action may be replaced by that of a single polar moment of inertia referred, say, to the motor shaft, and we may write the following equations-
Example:
A motor drives a rotational load through a reduction gear of teeth ratio a = 0.1 and efficiency as 90%. The load has a moment of inertia of 10 kg-m2 and a torque of 50 N-m. The motor has inertia of 0.4 kg-m2 and runs at a constant speed of 1,400 rpm. Determine the equivalent inertia of the motor and load combination referred to the motor side and the power developed by the motor.
