It has already been pointed out that the steady-state armature current depends on the small difference between the back e.m.f.Eand the applied voltageV. In a converter- fed drive it is vital that the current is kept within safe bounds, otherwise the thyristors or transistors (which have very limited overcurrent capacity) will be destroyed, and it follows from equation(3.8)that in order to prevent the current from exceeding its rated value we cannot afford to letVandEdiffer by more thanIR, whereIis the rated current.
It would be unacceptable, for example, to attempt to bring all but the smallest of d.c. motors up to speed simply by switching on rated voltage. In the example studied earlier, rated voltage is 500 V, and the armature resistance is 1U. At standstill the back e.m.f. is zero, and hence the initial current would be 500/1ẳ500 A, or 25 times rated current! This would destroy the thyristors in the supply converter (and/
or blow the fuses). Clearly the initial voltage we must apply is much less than 500 V;
and if we want to limit the current to rated value (20 A in the example) the voltage needed will be 201, i.e. only 20 volts. As the speed picks up, the back e.m.f. rises,
and to maintain the full current,Vmust also be ramped up so that the difference betweenVandEremains constant at 20 V. Of course, the motor will not accelerate nearly so rapidly when the current is kept in check as it would if we had switched on full voltage, and allowed the current to do as it pleased. But this is the price we must pay in order to protect the converter.
Similar current-surge difficulties occur if the load on the motor is suddenly increased, because this will result in the motor slowing down, with a consequent fall inE. In a sense we welcome the fall inE because this is what brings about the increase in current needed to supply the extra load, but of course we only want the current to rise to its rated value: beyond that point we must be ready to reduceV, to prevent an excessive current.
The solution to the problem of overcurrents lies in providing closed-loop current-limiting as an integral feature of the motor/drive package. The motor current is sensed, and the voltageVis automatically adjusted so that rated current is not exceeded continuously, although typically it is allowed to reach 1.5 times rated current for 60 seconds. We will discuss the current control loop in Chapter 4.
5.1 Dynamic behavior and time-constants
The use of the terms‘surge’and‘sudden’in the discussion above will doubtless have created the impression that changes in the motor current or speed can take place instantaneously, whereas in fact afinite time is always necessary to effect changes in either. (If the current changes, then so does the stored energy in the armature inductance; and if speed changes, so does the rotary kinetic energy stored in the inertia. For either of these changes to take place in zero time it would be necessary for there to be a pulse of infinite power, which is clearly impossible.)
The theoretical treatment of the transient dynamics of the d.c. machine is easier than for any other type of electric motor but is nevertheless beyond our scope.
However, it is worth summarizing the principal features of the dynamic behavior, and highlighting the fact that all the transient changes that occur are determined by only two time-constants. Thefirst (and most important from the user’s viewpoint) is the electromechanical time-constant, which governs the way the speed settles to a new level following a disturbance such as a change in armature voltage or load torque. The second is the electrical (or armature) time-constant, which is usually much shorter and governs the rate of change of armature current immediately following a change in armature voltage.
When the motor is running, there are two‘inputs’that we can change suddenly, namely the applied voltage and the load torque. When either of these is changed, the motor enters a transient period before settling to its new steady state. It turns out that if we ignore the armature inductance (i.e. we take the armature time-constant to be zero), the transient period is characterized byfirst-order exponential responses in the speed and current. This assumption is valid for all but the very largest motors.
We obtained a similar result when we looked at the primitive linear motor in Chapter 1 (see Figure 1.16).
For example, if we suddenly increased the armature voltage of a frictionless and unloaded motor from V1 to V2, its speed and current would vary as shown in Figure 3.11.
There is an immediate increase in the current (because we have ignored the inductance), reflecting the fact that the applied voltage is suddenly more than the back e.m.f.; the increased current produces more torque and hence the motor accelerates; the rising speed is accompanied by an increase in back e.m.f., so the current begins to fall; and the process continues until a new steady speed is reached corresponding to the new voltage. In this particular case the steady-state current is zero because we have assumed that there is no friction or load torque, but the shape of the dynamic response would be the same if there had been an initial load, or if we had suddenly changed the load.
The expression describing the current as a function of time (t) is:
i ẳ
V2V1 R
et=s (3.11)
The expression for the change in speed is similar, the time dependence again featuring the exponential transient term et/s. The significance of the time-constant (s) is shown in Figure 3.11. If the initial gradient of the current–time graph is projected it intersects thefinal value after one time-constant. In theory, it takes an infinite time for the response to settle, but in practice the transient is usually regarded as over after about four orfive time-constants. We note that the transient response is very satisfactory: as soon as the voltage is increased the current imme- diately increases to provide more torque and begin the acceleration, but the accelerating torque is reduced progressively to ensure that the new target speed is approached smoothly. Happily, because the system is first-order, there is no suggestion of an oscillatory response with overshoots.
Figure 3.11 Response of d.c. motor to step increase in armature voltage.
Analysis yields the relationship between the time-constant and the motor/
system parameters as
s ẳ RJ
k2 (3.12)
whereRis the armature resistance,Jis the total rotary inertia of motor plus load, and kis the motor constant (equations(3.3) and (3.4)). The appropriateness of the term
‘electromechanical time-constant’should be clear from equation(3.12), becauses depends on the electrical parameters (Randk) and the mechanical parameterJ.The fact that if the inertia was doubled, the time-constant would double and transients would take twice as long is perhaps to be expected, but the influence of the motor parametersRandkis probably not so obvious.
The electrical or armature time-constant is defined in the usual way for a series L,Rcircuit, i.e.
sa ẳ L
R (3.13)
If we were to hold the rotor of a d.c. motor stationary and apply a step voltageVto the armature, the current would climb exponentially to afinal value ofV/Rwith a time-constantsa.
If we always applied pure d.c. voltage to the motor we would probably wantsa
to be as short as possible, so that there was no delay in the build-up of current when the voltage is changed.
But given that most motors are fed with voltage waveforms which are far from smooth (see Chapter 2), we are actually rather pleased tofind that because of the inductance and associated time-constant, the current waveform (and hence the torque) is smoother than the voltage waveform. So the unavoidable presence of armature inductance turns out (in most cases) to be a blessing in disguise.
So far we have looked at the two time-constants as if they were unrelated in the influence they have on the current. We began with the electromechanical time- constant, assuming that the armature time-constant was zero, and saw that the dominant influence on the current during the transient was the motional e.m.f. We then examined the current when the rotor was stationary (so that the motional e.m.f. is zero), and saw that the growth or decay of current is governed by the armature inductance, manifested via the armature time-constant.
In reality, both time-constants influence the current simultaneously, and the picture is more complicated than we have implied, as the system is in fact a second- order one. However, the good news is that for most motors, and most purposes, we can take advantage of the fact that the armature time-constant is much shorter than the electromechanical time-constant. This allows us to approximate the behavior by decoupling the relatively fast‘electrical transients’in the armature circuit from the much slower‘electromechanical transients’which are apparent to the user. From
the latter’s point of view, only the electromechanical transient is likely to be of interest.