LAB REPORT: MOTOR SPEED CONTROL

ANDY BENNETT

1. Summary

This report details the use of voltage and current derived feedback in order to effect a simple motor speed controller. The primary illustration of this will be in showing that once the speed of the motor has been set, changes in the load applied to the motor will produce little change in the actual speed of rotation. The report explains how each type of feedback works and the factors affecting and controlling its operation. The results of the experiments conducted are presented in graphical form. The raw data and the necessary calculations to produce the graphs and equations cited can be found in the log book. This report focuses on providing a detailed expalanation of the application of feedback in static systems. This report does not cover the dynamic properties of the motor. Rather, it attempts to provide a full report on a part of the experiment as opposed to a more superficial report on the whole experiment.

2. Introduction

This report shows how a Power Op-Amp can be used with negative feedback to regulate, in one configuration; the voltage, and in another; the current that is supplied to the armature of a dc motor. This allows control over the speed at which the motor spins. It is important to be able to control the speed of the motor accurately and predictably in almost all applications in which a motor is used. Examples include automated machinary and electrial vehicles. As a basis for comparison, the motor’s characteristics are measured in the “open loop” state (Figure 1).

2.1. The Motor Assembly. The motor assembly (Figure 2) consists of a dc motor coupled, via a friction coupler, to a “brake” motor. The “brake” motor is the same as the driving motor. The “brake” motor has only a resistor connected across it in order to provide resistance to the driving motor. When the resistance is infinite (open circuit) there is no braking effect. When the resistance is zero (short circuit) there is the maximum braking effect.

Date: January 21, 2003.

Figure 1. The “open loop”, unloaded characteristics of the motor

Figure 2. The Motor Assembly

A tachogenerator is also connected to the driving motor. This generator produces a pulsed dc output, where the amplitude and the frequency of the output are proportional to the speed of the drive shaft.

3. The Motor

The motor has a permenant magnetic field, so the field flux, Φ, is assumed to remain constant at all times. Therefore, the only way of varying the speed at which the motor spins (given a constant load) is to vary the voltage across the armature of the motor. From the model of the motor given, ^{1} (and appliction of Kirchoff’s Voltage Law) the following equations are produced;

v_{a} = E + R_{a}i_{a}  (1)

Where v_{a} is the total voltage across the armature of the motor, E is the EMF produced across the motor itself and R_{a}i_{a} is the voltage across the armature resistance. The EMF across the motor is proportional to the speed at which it is spinning:

E = k_{e}θ̇  (2)

When a load is applied to the motor, the speed at which is rotates will decrease. From Equation 2 it can be seen that the EMF across the motor will also decrease. The voltage v_{a} has been set, so therefore, in order to be consistent with Equation 1, the voltage across the armature resistor must increase. Since the resistance is constant, this will result in an increase in the current flowing in the armature.

From the model given, the torque supplied by the drive shaft is

T = k_{e}i_{a}  (3)

By combining Equations 1, 2 and 3 and rearranging, Equation 4 is produced:

T = \frac{k_{e}v_{a}}{R_{a}} − \frac{k_{e}θ̇}{R_{a}}  (4)

From Equation 4 it can be seen that the torque in the drive shaft is inversly proportional to the speed at which it is spinning. i.e. a larger load requires a larger torque in order to keep that motor spinning at the previous speed. This in turn requires a larger armature current. By varying the load on the motor, and keeping the speed constant (at 1, 000rpm in this case) by changing the voltage across the armature, it is possible to determine the values of the motor constants k_{e} and R_{a}.

By using Figure 3.a and Ohm’s Law (V = IR), we can see that the armature resistance is 8.5Ω. The Y-intercept of this line is K_{e}Θ̇. This gives a value for K_{e} as 96.5mVs/c. It is also possible to calculate the values of these constants by holding the voltage across the armature constant and taking readings for Θ̇ and i_{a} with various loads applied to drive shaft (Figure 3.b). This gives R_{a} as 7.8Ω and K_{e} as 95mVs/c: The error between the values obtained through the different methods is small and insignificant.

1In the Autumn Term 2002 Lab Book.

Figure 3. Measurements to determine the motor constants

4. Voltage Derived Feedback Control

In order to be able to regulate the speed of the motor, it is necessary to know how fast it is spinning at a given time. From the analysis above, it can be seen that neither the voltage across or the current through the armature individually reveal how fast the motor is actually spinning. However, there is a tachogenerator attached to the drive shaft. (Figure 2) Since this is not driving anything itself: It is being driven, the ”“load” will be constant. Therefore, the voltage across it will be directly proportional to the speed at which it is being driven As long as this voltage is kept constant, the motor will be spinning at a constant speed. By feeding this signal back into the Op-Amp that is regulating the voltage supplied to the motor, when external influences, such as the loading of the drive shaft, affect the motor’s speed, the Op-Amp will increase the power available to the motor. An ideal Op-Amp is such that it has zero output resistance. This means that an arbitrarily large current can be drawn from the output without causing a drop in the voltage at that point in the circuit. Therefore, when the Op-Amp makes more power available, it will increase the voltage across the motor. The armature resitor will then draw a correspondingly larger amount of current, and the voltage will not be affected.

Figure 4. Circuits for voltage and current derived feedback

The negative feedback circuit in Figure 4.a produces a motor with the characteristics shown in Figure 5. Figure 5 was produced by setting V_{ref} so that the unloaded motor was spinning at 500rpm, then applying various loads to the motor and measuring Θ̇, i_{a} and v_{a}

V_{ref} was constant throughout the test, so the speed, Θ̇ should have remained constant. By comparison of Figures 1 and 5 it can be seen that the feedback has “flattened” the graph somewhat, therefore improving the motor’s behaviour with respect to loading. i.e. a greater load can be applied to the controlled motor before the speed drops, compared to the “open loop” motor. Since the current in the armature, i_{a} is a measure of the load on the motor, and the speed Θ̇ is a measure of how much the load is affecting the motor, it is possible to express the stability of the system in terms of rpm / A. For the original “open loop” system (Equation 5):

S_{open} = \frac{ΔΘ}{ΔI} = −815rpm/A  (5)

… and the voltage controlled system (Equation 6):

S_{voltage} = \frac{ΔΘ}{ΔI} = −89rpm/A  (6)

For the ideal motor, the graph would be flat, so S would appear to be undefined. However, from Equations 5 & 6 and from logic it can be seen that S_{ideal} = 0. The voltage controlled system does not have this ideal characteristic, but it improves on the “open loop” system more than ten fold.

Figure 5. Motor characteristics with voltage and current derived feedback

Figure 5 also shows that as the current, i_{a}, increases, so does the voltage across the motor, v_{a}. Equation 7 shows why this is so:

v_{a} = k_{e}Θ̇ + R_{a}i_{a}  (7)

K_{e} and R_{a} are both constants. We want to keep Θ̇ constant. Therefore, since i_{a} increases with load and, if the other terms in Equation 7 are constant, v_{a}αi_{a}, so must v_{a}.

The feedback ensures that V_{ref} controls the speed of the motor, rather that the actual voltage applied across the armature. This means that there exists a unique motor speed for every value of V_{ref}. In the “open loop” investigation, it was found that the motor does not start to turn until there is at least 1/4V across it. This is because the energy supplied by that voltage and the corresponding armature current is absorbed by the motor in overcoming its internal friction. Given that the input to the system now controls the motor’s actual speed rather than the motor itself, a well defined interface can be provided between other circuits. This potentially allows any motor to be substituted without affecting the rest of the machine and / or circuitry in which it is used. This is of benefit because every motor behaves in a slightly different way.

5. Current Derived Feedback Control

Given that it is the current flowing in the armature that dictates the torque suppied by the drive shaft, rather than the voltage across the motor, it may be possible to improve the characteristics of the system further. Therefore, instead of deriving the output of the Op-Amp from the voltage across the motor, it was proposed that the system be controlled based on the torque that the motor was supplying. - A certian load requires a certain torque to drive it at a certain speed. Therefore it is proposed that the speed of rotation and the torque in the drive shaft form the controlling parameters instead of the speed of rotation and the voltage applied. As stated previously, Equation 3 shows that the torque supplied is directly proportional to the current flowing in the armature. This value was therefore fed back into the Op-Amp (Figure 4.b). This also has the advantage that, by selecting appropriate component values, the maximum current available to the motor can be limited. This prevents the Op-Amp from increasing the power available to the Op-Amp “ad infinitum” if a particularly large load was to be applied or the motor became jammed. Limiting the power available will ensure that the motor does not burn itself out by attempting to deal with more power than it is rated for. By setting the speed and varying the load applied to the drive shaft, the graph in Figure 5 was obtained. The motor was also completely stalled in order to check that the curent (⇒ the power) supplied to the motor did not exceed the specified value. Figure 5 reveals that for all, reasonable, loads the motor continues to spin at the specified speed, almost exactly meeting the characteristics that are being sought. However, when the motor is forcibly stopped, the Op-Amp will stop providing power before the current being drawn by the motor exceeds the predefined maximum. The ideal characteristics would require that the speed only drops away once the maximum current was being drawn. i.e. When the maximum current was being drawn, further loading would cause the motor to behave in a similar way to the “open loop” system. That is, as the load gradually increased, the speed of the drive shaft would gradually decrease. This would produce a “square wave like” feature in the graph.

- points would exist at the maximum current for all values of speed less than the requested one.

This method of feedback control is considered superior the the voltage derived method because it provides extensive control over the system and also produces almost exactly the desired characteristics: The system has well defined, stable, characteristics. The system also benefits from the advantages of the voltage controlled system: It allows a well defined interface to be developed between itself and other ciruit components. Additionally, limits can be imposed that ensure that it is always possible to predict the region in which the system will be operating: It is known that the current will never exceed a certain amount.

6. Conclusions

• The “open loop” system operates over a wide range and is unstable. - Its characteristics can easily be improved. The position of the system changes a lot as the load is varied.

• The voltage controlled system is based on the speed of the motor and makes more marked improvements over the “open loop” system: The system was measured to be more that ten times more stable. - Changes in the system are less pronounced as the load is varied. This voltage controlled system also offers the advantage that a well defined interface can be definied so that the speed can be accurately be controlled by other modules, independent of the actual motor used.

• The voltage controlled system is based on the torque applied to the load and almost completely satisfies the sought criteria. The current controlled system offers the advantages of the voltage controlled system and also the fact that limits can be enforced so that the system is always operating with in a “safe” region.

It has been shown that feedback can improve the characteristics of a system by an extremely large factor. There is however a price, albeit small, to pay for these advantages. The range over which a heavily controlled system can operate will be significantly less than the range over which the “open loop” system will operate. For example, the controlled system will not be able to reach high speeds because a significant amount of power will be required to drive large loads at this speed. This power is either unavailable, or the motor would be unable to deal with it without overheating or otherwise becoming impaired. The controlled systems therefore, operate over a much narrower range of speeds but they can drive a larger number, and greater range, of different loads at these particular speeds.