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 Linear and Non-Linear Modelling of the 2260-885 PM DC Motor February 24, 2003
 1.0 Introduction - Microscopy

1.1 Background

With the development of rare-earth permanent magnets, it is possible to achieve very high torque-to-volume PM DC motors at reasonable cost. Furthermore, the advances made in brush and commutator technology have made these parts virtually maintenance-free. The DC motor has become quite popular in high performance control systems .

For, analytical purposes, it is necessary to establish mathematical models for DC motors for control applications. The DC motor is basically an open-loop system. However, it has a built-in feedback loop caused by back emf. Back emf represents the feedback of a signal that is proportional to the negative speed of the motor. The back-emf constant Kb represents an added term to the resistance Ra and the viscous friction coefficient B. Therefore, the back-emf is equivalent to an electrical friction, which tends to improve the stability of the motor .

1.2 Scope

A block diagram representation and transfer function of a Maxon 2260-885 PM DC motor has been provided. A preliminary linear model was developed using the block model, motor specifications and physical properties of the disk (applied load). Six experiments were conducted to observe the behaviour of the real system.

From experiment one, the motor’s response to different input voltages and loads were observed. The coulomb friction was calculated by observing the motor deceleration after the e-stop has been engaged. The results of this experiment were compared to the results produced by the theoretical linear model as described above. From this experiment, new values for the viscous friction B, and effective inertia J could be determined and substituted into the now-refined linear model. Further adjustments to B, J as well as L yielded a more representative linear model that better matched experiment one results.

The linear model was further refined by considering non-linearity’s such as voltage and current saturation of physical components as well as the effect of coulomb friction. The results of the final model were compared to the other five experiments. A small analysis accompanied each experiment.

 2.0 Preliminary Linear Model

The linear model was modelled after the Block diagram of the PM DC servomotor : Where:

Constants:
KT = motor torque constant 0.1 Nm/A
KB = back emf constant 0.1 V/rad/s
RA = armature resistance 1.44 Ω
LA = armature inductance 0.00056 H
KA = amplifier constant 4
B = viscous friction UKNOWN
J = effective inertia UKNOWN

Inputs and Outputs:
VIN = input voltage user defined input
VM = applied motor voltage calculated internally
VB = back emf calculated internally
IA = armature current calculated internally
T = generated torque calculated internally
Ω = motor speed output
TL = disturbance UNKNOWN

Figure 1: Theoretical Linear Model

The second order, linear model is completed when constants B, J and disturbance TL are specified. TL is an external torque applied to the motor in the form of coulomb friction. Since TL is constant, it is non-linear and thus set to 0 for this linear model. To determine the constants B and J it is necessary to reduce the second order transfer function to a first order transfer function. In effect, L must be set to 0 to achieve results for B and J. However, for this lab, the first order simplification will only be made to determine B and J - once they are found,
L will be reconsidered.

 3.0 Theoretical Linear Model

3.1 Theoretical Viscous Friction

Viscous friction B was determined to be 1.781×10-4 using equation 1 [Appendix A]. Equation 1

Equation 1 was developed by rearranging the first order simplification of the system transfer function at steady state.

The effective inertia J, which includes the load, was calculated to J = 0.0011(kg∙m2) using equation 2 [Appendix A]. Equation 2

Where

JM = inertia of motor armature

N  = gear ratio 5.2:1

The motor inertia Jm, which does not include the load, is given in equation 3 . It is clear that the inertia of the system with the disk is much greater than without the disk by approximately a factor of 10. Equation 3

With J and B determined, the theoretical linear model is now completed (with and without disk). This model can be further refined by comparing its results to experimental results.

 4.0 Refined Linear Model

From experiment 1, several modifications to the theoretical model could be made. An experimental value of viscous friction B was determined using data from the 5V, with load test. The 5V test produces the highest steady state speed, thus the non-linear, constant coulomb friction which is present in every test, would have the smallest percent effect on the final speed. This would yield a more accurate value of viscous friction B.

Using equation 4, the viscous friction was found to be -0.0001517(Ns/m) [Appendix A]. This value is theoretically impossible and is a result of a motor speed greater than 200rad/s. This phenomenon could have occurred if the amplifier produced a gain slightly larger than the specified factor of 4. Equation 4

Note:       Equation 4 is similar to equation 1, except the amplifier constant was included, with the presumption that the input voltage is the voltage supplied to the amplifier, not the motor voltage as in equation 1.

To obtain more meaningful results, a second 5V, with load test was conducted where the viscous friction was found to be 0.0000843 (Ns/m) [Appendix A]. The motor speed was found to be 197.6 rad/s, which was acceptable. To confirm this result, the viscous friction was calculated to be 0.0000701 (Ns/m) for the 5V, without load test. This also confirms that the steady state speed of the motor is independent of load.

An experimental value of J was determined using data from the 1V, with load and without load tests. This will yield two experimental values for inertia to coincide with the two theoretical values of inertia. The 1V tests produced the only profiles that clearly did not exhibit saturation in either loading case, which is desirable because saturation affects the time constant needed to compute J [Figure 2]. Figure  2: Experimental Data

The experimental motor inertia Jm, which does not include the load was calculated to be Jm = 0.000355 (Kgm2). This is about twice as much as the theoretical value. This can be attributed to the extra inertia provided by components external to the motor such as the gear box.

The effective inertia J, which includes the load was calculated to be J = 0.00122 (Kgm2) using equation 5 [Appendix A].This value is very close to the theoretical value. This suggests that even though the experimental motor inertia calculation was significantly different from the theoretical value, its overall contribution to the effective inertia is minimal compared to the dominant inertia of the disk. Equation 5

Where =         time constant - time for system to

reach 63% of steady state value

4.3 Refined Armature Inductance

The armature inductance LA was given as 0.00056H in the theoretical linear model. However, slight modifications can be made to this value to better approximate experimental data. However, the modification should only be slight; the actual inductance cannot be far from 0.00056H. The inductance is a second order component of the system, thus modifying it will adjust overshoot, if any and the shape of the model’s response in the transient phase.

Unfortunately, to achieve reasonable results, LA was required to be increased to 0.00275 H. This has especially improved the refined model’s response to non -loaded tests as will be demonstrated in Section 5.

 5.0 The Non – Linear Model

The non-linear model is an extension of the linear model with various non-linear additions. The viscous friction B and the effective inertia J determined in the refined linear model will be kept for this model. The non-linear additions will include the rated motor current – current saturation, the maximum voltage output of the DAQ – voltage saturation, disturbance torque to the motor caused by coulomb friction, and dead zone. Figure 3:  Non linear model

5.1 Voltage and Current Saturation

The amplifier voltage saturation is provided as ± 5V and the current saturation was provided as ± 3.2A. Their respective icons were added to the non-linear model as illustrated in Figure 3.

However, current saturation is permitted to be significantly higher at ± 6.4A for a maximum of 2 seconds. Current saturation could occur when the motor is required to accelerate a load, which occurs in the transient phase. At steady state, the motor will draw only minimal current to overcome coulomb and viscous friction; current saturation will no longer be a factor. Observing that the transient phase never exceeds 2 seconds [Figures 4-11], it could be safely assumed that current saturation for the linear model is ± 6.4A. This assumption greatly improves the performance of the non-linear model as will be demonstrated in Section 5.

5.2 Coulomb Friction

In addition to viscous friction, coulomb friction also influences motor operation. The total friction opposing motor movement is the sum of the constant coulomb friction, and viscous friction which is a function of motor velocity. The effect of coulomb friction can be simulated as an effective opposing disturbance torque.

Coulomb friction was calculated to be -0.02127 N [Appendix A]. This was achieved by observing the motor’s behaviour after the e-stop was engaged [Figure 13]. Without power, the only force slowing the motor was viscous and coulomb friction. As the motor angular velocity approached 0, the significance of coulomb friction increased. At zero velocity, viscous friction no longer applied and the angular deceleration α caused by coulomb friction was determined. The effective coulomb torque TL was determined by the product of the effective inertia J (with load) and angular deceleration α.

Dead zone was included to simulate experimental behaviour when a sinusoid was used for the input. The motor tended to stick around input values of zero – the motor did not exhibit a pure sinusoidal response but stopped around zero until a sufficient voltage forced it to move [Figure 16].

 6.0 Comparisons of Models to Experiments

6.1 Theoretical Linear Model

The theoretical linear model served to provide a skeleton from which better models could be constructed. The linear model adequately predicts the steady state response for the 1V, 2.5V, -2.5V, and 5V inputs as illustrated in Figures 4-11. However, the transient response was not satisfactorily predicted.

5.2 Refined Linear Model

The refined linear model significantly improved the transient response for non-loaded tests as illustrated in Figures 4,6,8,10. This was achieved by adjusting the armature inductance iteratively until a favourable match was found with experimental data. Unfortunately, this resulted in a significantly higher than expected inductance value. Despite this, the model’s response has improved significantly so it was decided to keep this modification. Modifying the armature inductance had only a minor impact on the transient response of loaded tests; in these tests the transient response is governed primarily by non-linear factors.

Slight modifications were made to the viscous friction which accounts for the slight improvement in the steady state response on the refined model over the theoretical model.

The major contribution of the refined model is its superiority in modelling the transient state of non-loaded tests. The theoretical and refined models behave more comparably when the motor is subjected to a load. The transient response of the refined model when loaded can be improved by considering non-linearities.

6.3 Non Linear Model

The non-linear model significantly improves the transient response of loaded tests as illustrated in Figures.5,7,9,11. The most significant resulted from current saturation.

When the motor is loaded, it draws current to accelerate the load. The motor can only draw so much current, which limits the torque that the motor can provide (the maximum torque is determined by the product of the motor torque constant Kt and the maximum current). Depending on the load (total effective inertia) the motor would only be able to increase acceleration until the motor reaches it maximum torque (draws maximum current). Different loads would allow the motor to accelerate at different rates as illustrated in Figure 1. In particular, note the steeper slope of non-loaded tests compared to the loaded tests.

The current saturation limits the rate of acceleration by capping the amount of allowable current. Thus in the refined linear model, there is no limit to how much the motor can accelerate when loaded as illustrated by the significantly steeper lines of its responses. The inclusion of current saturation forces the motor to respond more slowly and linearly (constant acceleration) when fully torqued. This dramatically improves the non-linear model’s transient response in loaded tests compared to the refined linear model.

The non-linear model incorporates the correct steady state response as determined by the theoretical model, the accurate transient response to loaded tests as determined by the refined model and the accurate transient response to non-loaded tests as determined by its own non-linear components. This has resulted is a fairly accurate model that can predict the motor’s entire response to different inputs when loaded or not. The qualitative merits of the three models have been presented in Table 1.

The performance of the non-linear model has been compared with experimental results stemming from experiments 2-6 in Appendix 3. The results has shown that the non-linear model responds well to different inputs: step and sinusoidal to produce the appropriate response. Moreover, the non-linear model has successfully duplicated results fro the e-stop and controlled stop tests, suggesting that the coulomb friction coefficient was correct. For the different controllers, the non-linear model produced result similar to the experimental results. Figure  4: 1 volt no load  Figure  6:  2.5 volts no load Figure 7:  2.5 volts with load Figure 8:  negative 2.5 volts no load Figure 9:   negative 2.5 volts with load Figure 10:  5 volts no load Figure 11:  5 volts with load

 Qualitative comparison of 3 models No Load Rating Load Rating 1V ±2.5V 5V 1V ±2.5V 5.0V Theoretical Steady State 3 1 1 Good 2 1 1 Good Transient 5 5 4 Poor 2 4 5 Poor Refined Steady State 2 1 1 Good 1 1 1 Good Transient 2 2 2 Satisfactory 1 3 5 Poor Non Linear Steady State 2 1 1 Good 1 1 1 Good Transient 2 2 1 Satisfactory 1 1 1 Good Grade (1 best - 5 very poor)

Table 1:  Qualitative comparison of 3 models

 7.0 Experimental Results

7.1 Experiment 1 Figure 12:  Comparison of motor response with and without load

The objective of this experiment is to investigate the open loop response of the motor velocity to various step inputs.   Figure 12 compares the response of the motor with and without load.  It is apparent that the system is non-linear and time invariant.  It is non-liner because the input (voltage) to output (velocity) ratio is not the same as shown in table 1.

 Input (Voltage) Output (Velocity) Ratio 1 36.4 0.0275 2.5 98.5 0.0254 5 198.5 0.0252 -2.5 -98.5 0.0254

Table 2:  Input to output ratio

From the figure 12 it is also apparent that the response of the motor with the load reaches steady state slower than that without load.  This is because of the increase in inertial load.

It is also clear that slopes exhibited at the transient state for the 5 and 2.5 volts (with or without load) are the same, indicating the same acceleration, but differs when compared to the response produced by 1 volt input (with or without load).  This phenomenon is due to current saturation.  From the block diagram of the motor (figure 1) it is clear that current is proportional to voltage, thus an increase in voltage will result in an increase in current which in turn result in an increase in acceleration. However the current is limited at a certain threshold to prevent damage to the motor.  This limit is controlled by the amplifier and is know as current saturation.  Thus an increase in voltage above a certain threshold won’t result in an increase in current, limiting the angular acceleration.  This is clear with the 2.5 and 5 volts since the acceleration is the same.  It is also apparent that there is a small lag when the voltage is first applied, which represent the coulomb friction needed to be overcome by the motor in order to gain velocity. Figure 13:  Comparison of stopping curves using different techniques

Figure 13 compares the response of the motor when the emergency stop is activated at steady state to that when an equal and opposite voltage is applied to the input at steady state.  When the emergency stop is engaged it effectively disconnects the motor from the amplifier, preventing the flow of armature current.  Thus there are no opposite forces produced by the motor. The only force acting on the disc is the coulomb friction which is a constant, which causes a constant deceleration.  The second curve on figure 13 is the response when an equal and opposite voltage is applied to the motor at steady state.  In this case the net voltage (0 volts) produces a back electromagnetic field which produces an opposing torque which acts on the disc to bring it to a stop faster.  The affect of friction is negligible compared to the opposing torque, in this case.   The discontinuity in the curve is caused by the backlash of the disk.

7.2 Experiment 2 Figure 14:  Motor response to a ramp input Figure 15:  Motor response to ramp input

The objective of this experiment is to investigate the linearity of the motor velocity as a function of applied motor voltage.  Figure 15 shows the motor response to a voltage input of 0.5 volts per second.  The response shows no visible transient state since the voltage is incremented in very small steps which allows the motor to reach its corresponding steady state respond quickly with no apparent lag.  This makes the response almost linear.  The voltage saturates at 5 volts thus the velocity is constant after 5 volts.  In figure 4 an opposite ramp input was applied at 4 seconds, this caused the combined voltage input to remain constant after 4 seconds.  The resulting velocity response reacted by maintaining a steady state.

7.3 Experiment 3 Figure 16:  Motor response to a 1 volt 0.1 Hz  input Figure 17:  Frequency response of motor

This experiment was conducted to investigate the frequency response of the motor.   A one volt amplitude was applied at different frequencies and the response was recorder.   Figure 16 shows the motor response to a 1 volt 0.1 Hz input.  It is clear that the voltage and the response frequency is the same.  It can also be noted that there exists a slight lag when the voltage changes sign.  This is due to the time need by the motor to overcome the coulomb friction.

Figure 17 is a bode diagram of all the frequency tests conducted in this experiment.  It is apparent that the system is stable up to 2 H, and reaches its resonant frequency around 3.5 Hz.  Beyond 5 Hz the stability and the response of the system degrades.

From figure 12 in experiment 1 it is evident that it takes approximately 0.5 seconds to reach steady state when a one volt input is applied to the motor without the load. This value corresponds to a frequency of 2 Hz.  Thus when the frequency is increase beyond 2 Hz, the period of the voltage input will be smaller than 0.5 seconds.  This will prevent the motor from reaching its steady state velocity, which corresponds to the degradation of the system after 3.5 Hz as seen in figure 5.  As the frequency approaches 100 Hz the response of the system becomes negligible since it does not have enough time to overcome the coulomb friction.

7.4 Experiment 4 Figure 18:  Closed-loop motor velocity response using proportional controller

The purpose of this experiment is to investigate the closed-loop motor velocity response using a proportional controller.  A closed loop system is more accurate and adaptive than an open loop system because it compares the output signal to the reference input and makes changes to compensate.

In this case the angular velocity produced by the motor is fed back into the input where it is subtracted from the desired velocity to produce an error signal.  This error signal is then fed in to a controller which multiplies the error by a gain (Kp), which indicates the factor by which the system should respond.    The higher the gain the faster the system can respond, but it will result in more fluctuation (or steady state error), as seen in figure 18.  This type of control will be ideal in a situation where the motor receives a sudden load change and has to react quickly.

7.5 Experiment 5 Figure 19:  Motor position response using different proportional gains

The objective of this experiment is to study the motor response using an integral proportional controller.  This was accomplished by varying the proportional gain, which is presented in figure 19. It is apparent that as the proportional gain is increased the system fluctuates more when reaching steady state.  Also the amplitude of these fluctuations is higher with increasing gain values, and the system reaches steady state more slowly.  However, when compared to a simple proportional control system as seen in experiment 4, the steady state is more stable.   Thus a higher proportional gain is not desirable.  In this experiment a proportional gain that gives the best response is the 0.5 as it approaches steady state the fastest and with minimal overshoot when compared to the other two proportional gains.

7.6 Experiment 6 Figure 20:  Motor position response using derivative control

The purpose of experiment six was to investigate the effect of a PID control on a closed loop.  A PID controller provides proportional, integral, and derivative compensation to an existing system.  The derivative values of the PID control block in Simulink were varied and the results are shown in figure 20.  It is apparent from the figure that for a Kd value of 0.2 there exist an overshoot, but the system responds faster, and the reaches the steady state in the same time as the one with Kd value of 0.2.  The response for the curve with Kd value of 0.2 is in-between the two extremes, but has no overshoot.  The response for the curve with a Kd value of 0.4 reaches steady state in the slowest time, but has no overshoot.  The response with Kd value of 0.1, 0.2 and 0.4 correspond to a under-damped, critically damped and an over-damped response, respectively. For the Using a derivative value of 0.2 the response is critically damped,  Thus the best response is the critically damped response at a Kd value of 0.2.

 8.0 Conclusions

The best model of the Maxon 2260-885 PM DC motor is the non linear model as developed in this report. The non linear model is capable of predicting the motor’s steady state and transient response at different inputs and loads quite reasonably. This was determined by analytically determining the linear components of viscous coefficient, effective inertia, and armature inductance as well as the non linear components of current saturation, voltage saturation and coulomb friction.

Experiment 1 was used to observe the behaviour of the motor which permitted the development of the refined linear model. Experiment 2 was conducted to investigate the linearity of the motor velocity as a function of time; as the voltage increases slowly, the motor speed will increase linearly. Their relationship is linear. Experiment 3 was conducted to investigate the frequency response of the motor; the higher the frequency, the less the motor rotates. Experiment 4 to 6 was conducted to investigate different closed-loop motor behaviours.  It was determined that the PID controller produced the best results.

 Appendix B:  Comparison of Linear Models with Experimental Data Figure 21:  Experiment 3 comparison of a sine wave input Figure  22:  Experiment 3 comparison of frequency response

 Appendix C:  Comparison of Non-Linear Model to Experimental Data Figure  23:  Experiment 1 comparison emergency stop at steady state Figure 24:  Experiment 1 comparison of controlled stop at steady state Figure  25:  Experiment 2 ramp input comparison (a) Figure 26:  Experiment 2 ramp input comparison (b) Figure  27:  Experiment 3 frequency response comparison Figure 28:  Experiment 4 proportional controller comparison

 8.0 References

1.      K. C. Benjamin, Automatic Control Systems, 7th Edition, John Wiley & Sons, Inc., New York, 1995.

2.      Professor J. Huissoon, ME 360 Lab Manual, University of Waterloo, Waterloo, Winter 2003.