Control Lab1

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering

ECNG 2005

LABORA LA BORATORY TORY & PROJECT DESIGN DESIGN III III

Lab # 1: 1: DC Motor Stati Static c and Dynamic Dynami c Characteristics

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles

Contents 1.

General Information ........................................................ ................................................................................................................ ........................................................ 3

2.

Lab Learning Outcomes .................................................. ........................................................ 4

3.

Pre-Lab ................................................ ........................................................ .................................................................................... ............................ 4

4.

3.1.

Required Reading Resources ..................................................... .......................................................................................... ..................................... 4

3.2.

Other Resources ...................................................... .............................................................................................................. ........................................................ 5

3.3.

Pre-Lab Exercise..................................................... Exercise ............................................................................................................. ........................................................ 5

In-Lab .................................................. ........................................................ .................................................................................. .......................... 25 4.1.

In-Lab Procedure .................................................... .......................................................................................................... ...................................................... 25

List of Figures

Figure 3.1: DC Motor operation and construction........................................................................ 7 Figure 3.2: The Motor and Inertial Load Simplified Block Diagram............................................. 9 Figure 3.3: Magnetically Induced Force on a DC Motor Armature .............................................11 Figure 3.4: DC Motor Electric Circuit.......................................................................................... 12 Figure 3.5: Simplified Open Loop Block Diagram of the DC Motor........................................... 20 Figure 4.1: DCMCT Trainer Module and Schematic (Quanser) ..................................................26 Figure 4.2: Modeling Module of the QICii Software .................................................................. 28 Figure 4.3: Locating the Push-Button and the LEDs....................................................................32 Figure 4.4: Step Response Test Input and Output.........................................................................45 List of Tables Table 3.1: Open-Loop System Nomenclature ............................................................................... 8 Table 3.2: Modeling Pre-Laboratory Assignment Results .......................................................... 22 Table 4.1: QICii Modelling Module Nomenclature..................................................................... 29 Table 4.2: Default Parameters for the Modelling the Modelling Module............................................................ 32 Table 4.3: Motor Resistance Experimental Results...................................................................... 37 Table 4.4: Back-EMF Constant Experimental Results ................................................................ 41 Table 4.5: Module Parameters for the Step Response Test ......................................................... 46 Table 4.6: Results Summary Table............................................................................................... 52 Table 4.7: DCMCT Model Parameter Specifications................................................................... 54 Table 4.8: DCMCT Sensor Parameter Specifications................................................................ ..56 1


Contents 1.

General Information ........................................................ ................................................................................................................ ........................................................ 3

2.

Lab Learning Outcomes .................................................. ........................................................ 4

3.

Pre-Lab ................................................ ........................................................ .................................................................................... ............................ 4

4.

3.1.

Required Reading Resources ..................................................... .......................................................................................... ..................................... 4

3.2.

Other Resources ...................................................... .............................................................................................................. ........................................................ 5

3.3.

Pre-Lab Exercise..................................................... Exercise ............................................................................................................. ........................................................ 5

In-Lab .................................................. ........................................................ .................................................................................. .......................... 25 4.1.

In-Lab Procedure .................................................... .......................................................................................................... ...................................................... 25

List of Figures

Figure 3.1: DC Motor operation and construction........................................................................ 7 Figure 3.2: The Motor and Inertial Load Simplified Block Diagram............................................. 9 Figure 3.3: Magnetically Induced Force on a DC Motor Armature .............................................11 Figure 3.4: DC Motor Electric Circuit.......................................................................................... 12 Figure 3.5: Simplified Open Loop Block Diagram of the DC Motor........................................... 20 Figure 4.1: DCMCT Trainer Module and Schematic (Quanser) ..................................................26 Figure 4.2: Modeling Module of the QICii Software .................................................................. 28 Figure 4.3: Locating the Push-Button and the LEDs....................................................................32 Figure 4.4: Step Response Test Input and Output.........................................................................45 List of Tables Table 3.1: Open-Loop System Nomenclature ............................................................................... 8 Table 3.2: Modeling Pre-Laboratory Assignment Results .......................................................... 22 Table 4.1: QICii Modelling Module Nomenclature..................................................................... 29 Table 4.2: Default Parameters for the Modelling the Modelling Module............................................................ 32 Table 4.3: Motor Resistance Experimental Results...................................................................... 37 Table 4.4: Back-EMF Constant Experimental Results ................................................................ 41 Table 4.5: Module Parameters for the Step Response Test ......................................................... 46 Table 4.6: Results Summary Table............................................................................................... 52 Table 4.7: DCMCT Model Parameter Specifications................................................................... 54 Table 4.8: DCMCT Sensor Parameter Specifications................................................................ ..56 1

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles List of Equations Equation 4.1: k t = k m ………………………… …………………………………………………… ………………………………………….………10 ……………….………10 Equation 4.2: Gw, v = Equation 4.3: Gw, v = Equation 4.4: Gw,T =

b s + a

…………………………………………………………………..…..17

k

+1

s τ

…………………………………..………………………………….18

K Td τ Td s

Equation 5.1: w s ( s) =

+1

……………………………………………………………………18

KV m ( s ) s τ

+1

…………………………………………………………………..30

Equation 5.2: h=0.01s …………………………………………………………………………..30 …………………………………………………………………………..30 Equation 5.3: wm =

sθ m T f s + 1

Equation 5.4: Gw, v = Equation 5.5 K =

k s τ

+1

……………………………………………………………………..30

……………………………………………………………………..45

Δ y …………………………………………………….… ………………………… ………………………….……………………..46 …………………..46 Δu

Equation 5.6: Gw, v ( s) =

Equation 5.7: w s ( s) =

k m

⎛ k ⎞ ⎜ J eq + m ⎟( Lm s + Rm ) ⎜ Rm ⎠⎟ ⎝ 2

1 k m (τ s + 1)(τ e s + 1)

……………………………………………….51

……………………………………………………..51

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ECNG 2005 LABORATORY & PROJECT DESIGN III http://myelearning.sta.uwi.edu/ Semester II 2008 / 2009

1. GENERAL INFORMATION Lab #:1 Name of the Lab:

DC Motor Static and Dynamic Characteristics

Lab Weighting:

10%

Delivery mode:

 Lecture

Estimated to total 1 study hours :

 Online 

Lab

Venue for the Lab: Lab Dependencies

Recommended prior know knowledge 3 and skills :

2

The theoretical background to this lab is provided in ECNG2009 Theoretical content link: 1) Sample dynamic systems and their mathematical descriptions: DC motor powered servo systems; 2) Mechanical, thermal and flow systems and state-space representations Pre-Requisites Pre-Requisites – ECNG2009 ECNG2009 To undertake this lab, students should be able to: a. Utilize the key mathematical prerequisites for the course: complex numbers, polynomial functions, L’aplace Transforms b. Utilize L’aplace Transfer Functions as an effective alternative to differential equations for mathematically describing system dynamics c. Obtain the poles and zeros of LTI systems d. Determine the input/output response of Linear Time Invariant (LTI) systems using the L’aplace transfer function e. e. Discuss why linear model representations are favored for dynamic systems modelling f. Use block diagrams to represent linear systems

1

Estimate includes teaching time, study time, and student preparation time for classes and labs. Include any Co-requisites, Post-requisites, or Forbidden course /lab combinations with respective code (C/P/F). 3 Lecturers can state lab input requirements in terms of student behaviours. 2

3


Course Staff

Position/Role

E-mail 

Lucia Cabrera-Jones Andre Morris

Teaching Assistant Laboratory Technician

Phone

Office



2462

322

3193

Control Systems Lab

Office Hours

2. LAB LEARNING OUTCOMES Upon successful completion of the lab assignment, students will be able to:

Cognitive Level

1. Describe the operation of a DC motor using first principles 2. Apply first principles to develop a second order linear mathematical representation of an armature controlled DC motor that models the effect of armature voltage and load torque on motor speed and position. 3. Calculate a simplified first order model of the armature controlled DC motor 4. Utilize the model developed to estimate the static and dynamic characteristics of an armature controlled DC motor

3. PRE-LAB Due Date: Submission Procedure: Estimated time to completion:

Submit to TA

3.1. Required Reading Resources Katsuhiko Ogata, Prentice Hall 1997 . Modern Control Engineering Engineering (4td Ed) .

4


3.2. Other Resources Course Web site: http://www.eng.uwi.tt/depts/elec/staff/copeland/

3.3. Pre-Lab Exercise 3.4.1. Background

Traditionally, Servomechanisms are feedback systems used for controlling mechanical speed or position. Typical applications include conveyor belts, consumer equipment drives (e.g. video and audio tapes, CDs, DVDs, hard-disc drives), aileron control in air craft, crane lifts etc. Servomechanism systems are the most common application of control theory in the electrical engineering discipline. In fact, the term is now applied, not just to systems employing mechanical elements, but to more general feedback control systems including biological ones. A servomechanism (or servo) system is comprised of four major components 1. 2. 3. 4.

A motor (electric, pneumatic or hydraulic) – translates energy into motion Controller and control amplifier – provides control o f motion Velocity and position feedback sensors – provides measurement of motion Gearbox or belt/pulley system (optional) – facilitates matching of the motor and load characteristics

The servomotor must be matched to the intended application. Usually the most important specification pertains to the level of torque that can be developed over a given range of speed. For example, DC motors would be used when a large amount of torque must be developed at zero or low speeds (as in passenger electric trains). On the other hand, less expensive AC motors are favored for higher speed applications. In both cases, the gearbox or belt/pulley system helps to increase the torque and/or speed range of the motor. The control strategy used depends on the precision of control required. Low precision systems may use no controls at all (open loop). Feedback increases the level of precision.

5

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles This lab focuses on the modeling and control of DC motors in servomechanism systems. Students will develop a linear model(s) of the motor system that will be later used to derive a prototype control strategy. 3.4.2 DC Motors

A DC motor is used to convert electrical energy to mechanical energy usually in the form of rotation or linear motion. Rotational motors are by far the most common. Rotation is effected through the magnetic interaction of two systems: the armature and stator systems.

The armature system is comprised of a winding on a soft iron core coupled to the shaft of the motor. The stator generates a fixed magnetic field that threads the armature system. This can be achieved by use of a permanent magnet (permanent magnet DC motor) or an electromagnet comprised of a coil (stator field winding) wound on magnetic material. Application of a voltage to the armature winding sets up a separate magnetic field which interacts with the field generated in the stator system resulting in motion of the armature and shaft. If the armature excitation were to be maintained, the shaft would rotate to a steady state position. Continuous rotation can be achieved by cleverly switching the armature excitation. Figure 3.1 provides a diagrammatic summary of the details discussed above.

6


Typical DC Motor

Armature system on shaft. Commutator segments are shown to the left of the shaft

Stator system

FIGURE 3.1 DC Motor operation and construction (a)The rotating magnet moves clockwise because like poles repel. (b) The rotating magnet is being attracted because the poles are unlike. (c) The rotating magnet is now shown as the armature coil. Its polarity is switched by the brushes and commutator segments to effect continuous motion.

Source: DC Motor Theory, by Thomas E. Kissell, Industrial Electronics, Second Edition, Prentice Hall PT

In general, DC motors are set up for ARMATURE CONTROL or FIELD CONTROL. For armature control, the field current is kept constant while the motor speed is varied by changing the armature current; since a constant field current implies a constant magnetic 7

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles field, permanent magnet DC motors are by nature armature controlled. In field control, the armature current is kept constant while the motor speed is varied by variation of the field current. Armature control is usually favoured because field control systems have an inherently under-damped speed characteristic. Field control systems, however, require less control power. 3.4.3 Pre-Laboratory Assignments:-Modeling the DC Motor from first principles .

Pre-laboratory Exercises must be completed before the laboratory exercise. The following nomenclature is used for the open-loop modeling of the DC motor.

Table 3.1:-Open-Loop System Nomenclature N.B:- The back emf constant k b = k m once we work in S.I units.

8

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles 3.5 Pre-Laboratory Assignments: First Principles

The motor, inertial load, power amplifier, encoder along with the signal conditioning required to obtain estimate velocity is modeled by the Motor and Inertial Load subsystem, as represented in Figure 3.2. The block has one input: the voltage to the motor Vm and one output: the angular velocity of the motor ωm. Additionally, a second input is also considered: the disturbance torque, Td, applied to the inertial load.

(a)

Ra

La

Amplifier

Vm

N

ea

JL, bL

S

M

eb N

S

ωm

T ωL

Jm, bm Motor

Gearbox

Inertial and viscous load

(b) Figure 3.2:- The motor and Inertial Load Subsystem: (a) simplified block diagram (b) expanded block diagram

9

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles In the following, the mathematical model for the Motor and Inertial Load subsystem is derived through first principles. Motor: First Principles 1. In S.I units motor torque constant k t is numerically equal to the back-electro-motiveforce (back EMF) constant , k m i.e., k t = k m.

Note:-This laboratory exercise uses SI units throughout. k m is used to represent both the torque constant and the back-electro-motive force constant.

Considering a single current-carrying conductor moving in a magnetic field, derive an expression for the torque generated by the motor as a function of current and an expression for the back EMF voltage produced as a function of the shaft speed. You may use Figure 3.3 in this regard. Show that both expressions are affected by the same constant, as implied in relation [4.1]. Explain.

Solution:

0

1

2

10

4.1


Magnetic field strength,

Armature

Armature Motor

Figure 3.3: Magnetically induced force on a DC motor armature

2. Figure 3.4 is a schematic of the armature circuit of a standard DC motor. Derive the relationship, expressed in the Laplace domain characterizing, between the armature current (ia) and voltages (ea, eb).

0

1

2

11

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles Ra

La Td Ia

ea

ωm

eb =km ω m

M

Tm Jeq

Figure 3.4:- DC Motor Electric Circuit

3. Using the previous result determine and evaluate the motor electrical time constant, τ e. Assume that the shaft is stationary. The parameters of the motor are listed in Appendix.1. System Parameters

Solution:

0

1

2

4. Assume τ e is negligible and simplify the motor electrical relationship previously determined. What is the simplified electrical equation?

12

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles Solution:

0

1

2

5. Determine the equivalent moment of inertia of the motor rotor and the load, assuming n=1 since the motor drives the load directly (there is not gear). Neglecting the friction in the system, derive from first dynamic principles the mechanical equation of motion of a DC motor. Solution:

0

1

2

6. Calculate the moment of inertia of the inertial load which is made of aluminum. Also, evaluate the motor total moment of inertia. Assume that the load is a perfect disc i.e. zero thickness and uniformly distributed mass. Resume the system parameter values J eq , K m and Rm.using the data sheet that is given in Table A.1 7. Solution:

13


0

1

2

3.5.1 Static Relations

Modeling by experimental tests on the process is a complement to first principles modeling. In this section we will illustrate this by static modeling. Determining the static relations between system variables is very useful even if it is often neglected in control systems studies. It is useful to start with a simple exploration of the system.

Answer the following questions . 1. Assuming no disturbance and zero friction, derive an expression for the motor maximum velocity: ωmax. Solution:

14

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles 0

1

2

2. Determine the motor maximum current, Imax, and maximum generated torque, Tmax.(Torque /Speed characteristics).Explain.

Solution:

0

1

2

3. During the in-laboratory session you will be experimentally estimating the motor resistance Rm. This can be done by applying constant voltages to the motor and measuring the corresponding current while holding the motor shaft stationary. Derive an expression that will allow you to solve for Rm under these conditions. Explain.

Solution:

0

1

2

15

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles 4. (Locked Rotor Test) During the in-laboratory session you will be experimentally estimating the motor torque constant k m. This can be done by applying constant voltages to the motor and measuring both corresponding steady-state current and speed (in radians per second).Assuming that the motor resistance is known, derive an expression that will allow you to solve for km. Explain

What is the effect of the inertia of the inertial load on the determination of the motor constant? Solution:

0

1

2

3.5.2 Dynamic Models: Open-Loop Transfer Functions

Answer the following:

Draw the block diagram and determine the transfer function, Gω,V(s), of the motor from voltage applied to the motor to motor speed. Explain Hint: 1.

The motor armature inductance Lm should be neglected. The friction of the motor is so small that can be considered as cero.

Solution:

16


0

2.

1

2

Express and evaluate Gω,v(s) as a function of the parameters a and b, defined such as: b Gω ,v = s + a

[4.2]

Solution:

17


3.

1

2

Express and evaluate Gω,V(s) as a function of the parameters K and τ, defined such as: k Gω , v = s + 1 τ

[4.3]

Solution:

0

4.

1

2

Determine and evaluate the transfer function, Gω,T(s), from disturbance torque applied to the inertial load to motor speed. Express Gω,T(s) as a function of the parameters K Td and τTd, as defined below:

G ω ,T ( s ) = Show that τ Td

K Td τ Td

+1

[4.4]

= τ

Solution:

18


0

1

2

Derive the motor open-loop block diagram clearly showing the effect of all major parameters above (see class notes). Solution: 5.

0

6.

1

2

Simplify the open-loop block diagram obtained so that it has the block structure depicted in Figure 3.5 HINT: Determine the composite transfer function for the block drawn in 7 above.

19

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles Td Vm

m

Figure 3.5:-Simplified Open Loop Block Diagram of the Dc Motor

Solution:

0

1

2

The transfer function Gω,V(s) previously derived is only an approximation since the inductance of the motor has been neglected. Considering the motor electrical time constant τe previously evaluated, justify the approximation. Solution: 7.

20


0

1

2

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THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles 3.6 Pre-Laboratory Results Summary Table

Table 3.2 below MUST be completed before you come to the in-laboratory session to perform the experiments .

Question

Description

Symbol

Value

Unit

3.5 Motor First Principles

3

Motor electrical time constant

τe

s

6

Disc load moment of inertia

J1

Kg.m 2

6

Total moment of inertia

Jeq

Kg.m 2

3.5.1 Static relationships

1

Motor maximum velocity

ωmax

Rad/s

2

Motor maximum current

Imax

A

2

Motor maximum torque

Tmax

Nm

Nm/A

3.5.2 Dynamic relationships

5

Motor torque constant

k m

5

Motor armature resistance

R m

Ω

6

Open-loop model parameter

a

kg.m/(Ws )

4

22


Open-loop model parameter

b

1/(V.

6

Open-loop steady-state gain K

K

rad/(V.s)

6

Open-loop time constant

τ

s

6

Open-loop torque disturbance gain

K τ d

rad/(N.m.s)

2

)

Table 3.2- Modeling Pre-Laboratory Assignment Results

23


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THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering

4. IN-LAB Allotted Completion Time: Required lab Equipment:

4 hours

1. The Quanser DCMCT rig 2. PC with serial port and operational JAVA engine which is needed to power the GUI used in this lab

4.1. In-Lab Procedure 4.1.1 Lab Specifics

The motor used in this lab is a Maxon 18-Watt permanent magnet DC motor. The motor is mounted on the DC Motor Control Trainer (DCMCT) rig manufactured by Quanser. This trainer rig allows the user/student to operate and c ontrol the motor using analog electronics or a PC. The DCMCT consists of (Fig 4.1) 1. a potentiometer for precision speed and position sensing 2. a digital shaft encoder for precision speed and position sensing. 3. a QIC Processor Core consisting of a 16F877 PIC microcontroller. This allows for control and monitoring of the DCMCT from a PC connected to the DCMCT serial port.

Various options can be selected by onboard jumpers. For this series of lab, we will use the QIC Processor Core under PC control to provide the excitation signals and monitor motor variables.

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles 1. 2. 3. 4.

Maxon DC Motor Removable inertial Load Linear Power Amplifier High Resolution Optical encoder 5. Ball Bearing Servo Potentiometer 6. Removable Belt to drive the potentiometer 7. i. PC Interface Option: this is implemented by using DtoA and AtoD converters ii. Analog Controller Option: to implement controllers using analog electronic circuits 8. Breadboard Option: to implement controllers with your own circuits 9. Embedded/Portable Option: The QIC installs in this socket to perform embedded control in place of PC-based control 10. Serial Port (used by QICii) 11. PIC Reset Switch 12. User Switch: Momentary Action Pushbutton Switch For Manual Interaction 13. Inertial Load Storage Pin 14. Jumper J6: to switch between HIL and QIC use 15. 6—mm Power Jack 16. Power Supply I leader: J4 17. Analog Signals I leader: ii 1

Figure 4.1 : DCMCT Trainer Module and Schematic (Quanser)

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THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles Students are required to refer to the analytical derivation of the motor model in the course text and notes. This will assist in the estimation of the system model in this exercise.

In this lab you will be asked to derive the theoretical open-loop model of the system and to assess its performance limitations. The DCMCT system is designed in such a way that a good model can be derived from first principles. The physical parameters can all be determined by simple experiments. Using QICii and the QET you will apply inputs to the process and observe its outputs thus allowing you to estimate system parameters using static and dynamic measurements. The model is to be validated by comparing the measured step response with that obtained from a simulation of the derived model.

Our model will not consider the effect of nonlinearities, although these affect the system primarily via amplifier and motor saturation. Other effects such as higher order dynamics and measurement noise are also ignored.

4.2 Module Description

In this section you would be using the QICii Modeling module to determine the open loop model of the DC motor. The user interface for the module should be similar to the one shown in Figure 4.2. Table 4.1 lists the main elements comprising the QICii Modeling module user interface. Every element is uniquely identified through an ID number and located in Figure 4.2.

27


Figure 4.2:- Modeling module of the QICii software

2.5.1. M

28

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles otor First Principles ID #

Label

Parameter Description

1

Speed

ωm

Motor Output Speed Numeric Display

rad/s

2

Current

Im

Motor Armature Current Numeric Display

A

3

Voltage

Vm

Motor Input Voltage Numeric Display

V

4

Signal

Type of Generator For The Input Voltage

Generator

Signal

5

Amplitude

Generated Signal Amplitude Input Box

V

6

Frequency

Generated Signal Frequency Input Box

Hz

7

Offset

8

Speed

ωm

Scope With Actual (in red) And Simulated rad/s (in blue) Motor Speeds

9

Voltage

Vm

Scope With Applied Motor Voltage (red)

V

10

K

K

Motor Model Steady-State Gain Input Box

rad/(V.s)

11

τ

τ

Motor Model Time Constant Input Box

s

12

Tf

Tf

Time Constant of Filter for Measured s Signal

Generated Signal Offset Input Box

Unit

V

Table 4.1:- QICii Modelling Module Nomenclature

29

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles The Modeling module program runs the process in open-loop using the motor voltage which is given by the signal generator. Two PLOT windows show the time histories of motor seed and motor voltage.

QICii runs a simulation of the system in parallel with the hardware. The output of the simulation can be used for model fitting and validation. The input of the simulation is equal to the motor voltage and the output of the simulation is displayed (blue trace) in the same window as the actual motor speed (red trace). The simulation model parameters K and τ can be adjusted from the front panel. The simulated motor speed, ωs, is obtained from the simulated transfer function and actual motor voltage using the assumed transfer function model:-

ω s ( s ) =

KVm ( s ) τ s + 1

[5.1]

The implemented digital controller in the QIC runs a t a sample rate of 100 Hz, i.e.,

h=0.01s

[5.2]

Note that the actual speed is obtained by filtering the position signal using the following filter:ω m

=

sθ m T f s + 1

[5.3]

where θm is the position of the motor shaft measured by the encoder.

30

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles 4.3 Module Startup

In order to power up the DCMCT you will first need to launch the QICii software as follows: a. Go Start-Programs-Quanser-QICii-QIcii . b. A command window will appear before the QICii screen is launched

Once this screen appears follow these instructions below in order to start running the lab.

1. Make sure the drop down menu on the left at the top of the QICii screen is set to Modelling and the port is set to COM1. 2. Press the Download program button on top of the QICii window

3. Click on the Write (F4) button of the PIC downloader popup window

4. Push the Reset button on the QIC, Figure 5.2 to start the down load.

5. Once the download is complete, close the pop-up window. Press the Reset button again on the QIC .The two LEDs should start flashing.

6. Press the User Switch, which is close to the flashing light. Automatically the removable inertial load will start to spin.

7. Press the Connect/Disconnect button to Connect and hence, display the trace.

The default module parameters loaded after download are given in Table 4.2.

31


Reset -

Force on armature conductor, F

Signal

Figure 4.3:-Locating the push-button and the LEDs Amplitude Frequency Offset K τ

T f

Type

[V]

[Hz]

[V]

[rad/(V.s)] [s]

[ s]

Square

2.0

0.4

0.0

10.0

0.01

0.2

Wave

Table 4.2:- Default Parameters For The Modelling Module 4.3.1 Using the QICii interface

Following are some useful TIPS to facilitate your laboratory experience:

Entering data: The lab requires that you change the input voltage several times. Thi scan be done on the PC keyboard or using the green up/down arrows on your QICii interface.

NB: If you are using your PC keyboard, make sure to press the ENTER key after entering the new value.

32


In case of disconnection: 1. Press the Reset button again on the QIC .The two LEDs should start flashing. 2. Press the User Switch, which is close to the flashing light. 3. Press the Connect/Disconnect button to Connect and hence, display the trace.

4.4 STATIC RELATIONS

Initial Experimental Tests Objectives

1. Determine the maximum velocity and compare with calculations. 2. Determine the Coulomb friction.

4.5 Experimental Procedure

A procedure of this type is very useful to make sure that a system functions properly. Follow the steps described below. 1.

Step 1.Run the system open-loop by changing the voltage of the motor. The motor voltage is set by the signal generator. With zero signal amplitude, increase the signal offset gradually from 1 to 5 with increments of 1, to generate a constant voltage. Observe the steady-state speed, current, and velocity? Record the values obtained. Include in your results a snapshot of the change in the steady state speed showing the transition of one speed to another. What happens to the variables as the offset increases?

33

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles Offset (V)

Current (A)

Velocity (rad/seg)

Comments

0

1

2

Step 2.

Although the motor maximum input voltage is 15 V, the Offset numeric input is limited to 5 V.

Determine the maximum velocity and compare with calculations made in the Pre-Lab section 3.5.1 Q1?

34


0

1

2

Step 3. Keeping the amplitude at zero Change the value of the offset (starting at zero) on the motor and increase it gradually in steps of 0.06 until the motor starts to move. Determine the voltage when this occurs. Repeat the procedure at least 3 times. Repeat the test with negative voltages Record the values obtained. Explain why the voltages obtained may vary?

Positive Direction Negative Direction

Comments

35


0

1

2

4.6 Estimate the Motor Resistance

Some of the parameters of the mathematical model of the system can be determined by measuring how the steady-state velocity and current changes with the applied voltage. To experimentally estimate the motor resistance, follow the steps described below:

Step 1.Set the generated signal amplitude to zero. If the signal offset is different from zero then the motor will spin in one direction, since a constant voltage is applied. You can change the applied voltage by entering the desired value in the Offset numeric control of the Signal Properties box. You can also read the actual motor current from the digital display. The value is in Amperes. Fill the following table (i.e. Table 1.5). For each measurement hold the motor shaft stationary by grasping the inertial load to stall the motor. Note that for zero Volts (Offset zero) you will measure a current, I bias, that is possibly non-zero. This is an offset in the measurement which you need to subtract from subsequent measurements in order to obtain the right current. Note also that the current value shown in the digital display is filtered and you must wait for the value to settle before noting it down . The readings of the measured currents I meas must be made to 3 d.p . The recorded value must be taken an average of 3 times when reading each sample.

36

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles Sample:

V m(i)

Offset in Measured

i

[V]

Current: I bias [A]

0

0

Sample:

V m(i)

Measured Current:

Corrected for Bias: Resistance:

i

[V]

I meas(i) [A]

I m(i) [A]

1

-5

2

-4

3

-3

4

-2

5

-1

6

1

7

2

8

3

9

4

10

5

Rm(i) [Ω]

Average Resistance: Ravg [Ω] Table 4.3:- Motor Resistance Experimental Results

Step 2: From Table 4.3, above; explain the procedure you used to estimate the resistance Rm and R avg.

37


0

1

2

Step 3.The system parameters are given in Table 4.7. Compare the estimated value for Rm (i.e. Ravg) with the specified value and discuss your results .

38


0

1

2

4.7 Estimate the Motor Torque Constant

Follow the steps described below to experimentally estimate the motor back-EMF constant:

Step 1.With the motor free to spin , apply the same procedure as above and fill the following table (i.e. Table 4.4). You can read a value for the motor angular speed from the digital display. Wait a few seconds after you enter a new voltage value as the displayed speed values are low-pass filtered . The angular speed value is in radians per seconds. The current measurement may have an offset which you will need to account for. The speed measurement will have a very small offset which will need to be compensated for. Calculate the motor back-EMF constant for each measurement iteration and then calculate an average for the 10 measurements. You should use the value of Ravg that you estimated in the previous section.

39

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles Sample:

V m(i) I bias

Ravg

i

[V]

[Ω]

0

0

Sample:

V m(i) Measured Speed:

I meas(i)

I m(i)

k m(i)

i

[V]

[A]

[A]

[V.s/rad]

1

-5

2

-4

3

-3

4

-2

5

-1

6

1

7

2

8

3

[A]

ωm

(i) [rad/s]

40


4

10

5

Average Back EMF-Constant: k m_avg [V.s/rad] Table 4.4:- Back-EMF Constant Experimental Results

Step 2.Explain the procedure you used to estimate k m , k m avg.

0

1

2

Step 3.The system parameters are given in Table A.1. Compare the estimated value for km

(k mavg )

with the specified value and discuss your results

41


0

1

2

4.8 Obtain the Motor Transfer Function

From the above estimates, obtain a numerical expression for the motor open-loop transfer function Gω,V. What are the estimated open-loop steady-state gain and time constant ? How does this compare with the open-loop transfer function you obtained in Section 4.2.3 Dynamic Models: Open-Loop Transfer Functions, Question 5?

42


0

1

2

4.9 Estimate the Measurement Noise

The measurement noise can be determined experimentally as follows: 43

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles 1.

Determine the measurement noise for speed control by running the motor with a constant voltage and observing the fluctuations in the velocity. Use two values of constant voltage and compare the differences.

Hint: In order to view the noise on the actual speed (red trace) use the magnifier

key,

which expands the Y axis.

2. Does the noise level depend on the velocity? Observe as the speed increases what happens to the disturbance frequency? Justify your answer. Do you also observe any repeatable fluctuations in your velocity signal? Suggest one probable source of these fluctuations?

Hint:

Can the fluctuations in your velocity signal be related to the motor position?

44


4.10

1

2

Dynamic Models: Experimental Determination Of System Dynamics

A linear model of a system can also be determined purely experimentally. The idea is simply to observe how a system reacts to different inputs and change the structure and parameters of a reference model until a reasonable fit is of the model and actual responses is obtained. The inputs can be chosen in many different ways and there is a large variety of methods.

4.10.1 The step response test

The step response test is carried out by applying a constant input to bring the system, which must be stable, to a suitable steady state point. The input is then changed rapidly to a new level and the output is recorded. A simple model of the form:

Gω ,v =

K s τ

+1

[5.4]

can now be easily fitted to the data (see Figure 5.3).

Figure 4.4 Step response test input and output

45


Assume that the input changes with Δu and that the corresponding changes in the steady state output are Δ y. An estimate of the steady-state gain is then given by:

K =

Δ y Δu

[5.5]

The quantity τ is approximately given by the time the output has reached 63% of its total change. 4.10.2 Experimental Procedure

Please read appendix which describes how to use the QICii plots to take measurements of the acquired data, to start and stop the plots, and to measure point coordinates on the plots Step 1. Apply a series of step inputs to the open-loop system by setting the QICii module parameters as described in Table 4.5

τ

Signal

Amplitude

Frequency

Offset

K

Type

[V]

[Hz]

[V]

[rad/(V.s)]

[s]

Square

2

0.4

3

0

0.0

Wave

Table 4.5 Module Parameters for the step response test

46


Step 2. The open-loop controller now applies a constant-amplitude voltage square wave to the motor. Step voltages are applied to the motor from the signal generator with a period that is so long that the system well reaches steady-state at each step. The motor should run at the corresponding constant speeds. Determine the parameters K and τ of the model defined in [5.4] and compare them with the model obtained by first principles in Section 3.5.2, Question 3.Explain

0

1

2

Step 3. The fact that K = 0 means that the model output is zero. Activate the model by changing the simulation parameters K and τ. to the values you previously estimated from the step response test. Do you obtain a good fit between the estimated and the actual responses? Explain and print your screen result.

47


0

1

2

Step 4 Compare with the results of first principles modeling in Section 4.2.3 Dynamic Models: Open-Loop Transfer Functions, Question 6. Is your model valid. Explain. Print the screen showing the comparative results.

0

1

2

48

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles 4.11

Concluding Remarks

4.11.1 Load Disturbances and Measurement Noise

There are typically two types of disturbances in a control system. Load disturbances that drive the system away from its desired behaviour and measurement noise that corrupts the information obtained from the sensors.

Since this motor does not do any useful work there are no real load disturbances in this case. A load disturbance can be simulated by gently touching the inertial load with your finger. Load disturbances can also be simulated by injecting an extra voltage on the motor. The major noise source for position control is due to the quantization of the angle measurements due to the encoder. 4.11.2 Automating the Tests

The experimental tests you have done can easily be automated. Measurement of motor resistance Rm and motor constant k m can be done as follows:

•

Resistance measurement: Keep the wheel fixed with a clamp. Sweep the voltage slowly for a full cycle, measure the current, display curve, and present the linear fit and a measure of deviation from linearity.

•

Current constant: Free wheel. Sweep the voltage slowly for a full cycle, measure the speed, display curve, and present the linear fit and a measure of deviation from linearity.

The system parameter estimation procedures can also be automated by replacing manual search by an optimization algorithm. Automated test procedures of this type are essential to ensure quality in mass manufacturing. 4.11.3 Nonlinearities

Many aspects of control can be dealt with using linear models. There are however some nonlinear aspects that always have to be taken into account. The major nonlinearities are:

49


• • •

Saturation of the motor amplifier. Friction in the motor. Quantization of the encoder. It is very important to keep in mind that all physical variables are limited. The amplifier that drives the motor has a 15V power supply which restricts the voltage from the amplifier to Vmax = 15 V. A consequence is that the current through the motor is also limited.

The limitation in signal ranges implies that the motor transfer functions Gω,V and Gω,T do not describe the system well for large signals.

The other main nonlinearities are due to Coulomb friction, approximately equivalent to 0.2-0.5V, -3 and quantization in the encoder 2π/4096 = 1.5 10 rad. 4.11.4 Unmodeled Dynamics

When determining physical parameters it is customary to assign a precision to the values. There are uncertainties due to variations in component values, temperature variation of the armature resistance. It is therefore natural to give some measure of accuracy to the transfer functions Gω,V and Gω,T. One way to do this is to give the accuracy of parameters such as K and τ. This does unfortunately not capture all relevant issues because the actual transfer function may be much more complicated than the simple first order system given by Equation [5.4], the system may even be nonlinear. This effect which is called unmodeled dynamics can be specified in many different ways. An estimate of the unmodeled dynamics is an essential aspect of modeling for control. It is equivalent to an error analysis in traditional measurements.

To have an indication of the accuracy of a model it is necessary both to have an estimate of the accuracy of its parameters and also an assessment of dynamics that has been neglected.

One obvious factor is that the controller and the computation of the velocity is implemented in a computer. The encoder gives values of the angle that are quantized with a resolution of 2π/4096 = 1.5 10-3 rad. Since the controller is implemented on a computer there are also dynamic effects.

50

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles A crude approximation is to assume that there is an extra time delay corresponding to half a sampling period.

The system has no sensor for velocity. The velocity is instead obtained by taking filtered differences of the position. A common rule of thumb is to approximate the effect of the computer by adding a delay of half a sampling interval or 0.005 s. Since the velocity is computed by taking differences of the angles between two sampling intervals there is an additional delay in the velocity signal of one sampling interval. Because of the extra sampling period required to compute velocity from the encoder position, the time delay will be approximately one and a half sampling interval. The signal is also filtered which introduces additional d ynamics.

The inductance of the rotor has already been mentioned previously. The model we have obtained is an approximation because we have neglected the inductance in the motor rotor. A more accurate transfer function from voltage to motor speed is thus:-

Gω ,v ( s ) =

k m

⎛ k m2 ⎞ ⎜ J eq + ⎟ ( Lm s + Rm ) R m ⎠ ⎝

[5.6]

which can also be expressed as:-

Gω ,v ( s ) =

1 km (τ s + 1)(τ e s + 1)

[5.7]

Full details can be found in the class notes.

Introducing the numerical values we find τ = 0.0929 s and τe = 0.0000774 s, which means

51

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles that the electrical time constant is much smaller than the time delay. The major contribution to the unmodeled dynamics is thus due to the effects of sampling. It is 0.005 s for the position signal and 0.015 s for the velocity signal (i.e. speed control). 4.11.5 In-Laboratory Results Summary Table

Table 4.6 should be completed using Table 3.2, which contains data from the pre-laboratory assignments, as well as experimental results obtained during the in-laboratory session.

Question

Section Description

Symbol Pre-Lab In-Lab Value

Unit

Result

5.2. Static Relations

1.

Motor Maximum Velocity

rad/s

1.

Positive Voltage

Coulomb

Friction Vfp

N/A

V

1.

Negative Voltage

Coulomb

Friction Vfn

N/A

V

2.

Motor Armature Resistance

R m

3.

Motor Torque Constant

k m

N.m/A

4.

Open-Loop Steady-State Gain

K

rad/(V.s)

ωmax

Ω

52


4.

Open-Loop Time Constant

τ

s

5.3.1 Dynamic Models: The Step response

2

Open-Loop Steady-State Gain

K

rad/(V.s)

2

Open-Loop Time Constant

τ

s

Table 4.6: Results Summary Table

53

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles Appendix 1: System Parameters Symbol Description

Unit

Value

Motor

k m

Motor Torque Constant

Nm/A

0.0502

Rm

Motor Armature Resistance

Ω

10.6

Lm

Motor Armature Inductance

mH

0.82

Motor maximum continuous torque

Nm

0.033

Motor power rating

W

18

J m

Moment Of Inertia Of Motor Rotor

Kg.m

τ m

Motor Mechanical -Time Constant

s

0.005

M i

Inertial Load Disc Mass

kg

0.068

r i

Inertial Load Disc Radius

M

0.0248

Linear Amplifier Maximum Output Voltage

V

15

Linear Amplifier Maximum Output Current

A

1.5

Linear Amplifier Maximum Output Power

W

22

2

1.16E(-6)

Linear Amplifier

Vmax

Linear Amplifier Maximum Dissipated Power W with heat sink R load =4 Ω

8

Linear Amplifier Gain

3

V/V

Table 4.7: DCMCT Model Parameter Specifications

54

THE UNIVERSITY OF THE WEST INDIES ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES FACULTY OF ENGINEERING Department of Electrical & Computer Engineering Lab#1: Modeling the DC Motor from First Principles Appendix 2: Sensor Parameters Description Current sense

Value

Unit

Current Calibration at ±10% at QIC A/D input

1.112

A/V

Current sensor resistor

0.1

Ω

Line Count

1024

Lines/rev

Resolution (in quadrature)

0.0879

o

Type

TTL

Encoder signals

A, B, Index

Encoder

/count

Potentiometer

Calibration at POT RCA jack

39

o

Calibration at QIC A/D input

78

o

Resistance

10

K Ω

Bias voltage

±4.7

V

Electrical range

350

o

Calibration at TACH RCA jack

667

RPM/V

Calibration at QIC A/D input

1333

RPM/V

/V /V

Tachmoeter

Table 4.8: DCMCT sensor parameter specifications

55


NB Analog sensor calibration constants for the QIC A/D converters are twice those for the RCA output jacks. This is because the RCA outputs are in the ±5V range while the QIC A/D inputs are in the 0-5V range.

Appendix: 3 How to use the QICii plots

56


57

Control Lab1

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