Control Lab1

8/18/2019 Control Lab1

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THE UNIVERSITY OF THE WEST INDIES

ST. AUGUSTINE, TRINIDAD & TOBAGO, WEST INDIES

FACULTY OF ENGINEERING

Department of Electrical & Computer Engineering

ECNG 2005

LABORATORY & PROJECT DESIGN III

Lab # 1: DC Motor Static and Dynamic

Characteristics


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Lab#1: Modeling the DC Motor from First Principles

1

Contents

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

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

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

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

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

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

4. 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............................................. 9Figure 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........................................... 20Figure 4.1: DCMCT Trainer Module and Schematic (Quanser) ..................................................26

Figure 4.2: Modeling Module of the QICii Software .................................................................. 28Figure 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 .......................................................... 22Table 4.1: QICii Modelling Module Nomenclature..................................................................... 29

Table 4.2: Default Parameters for the Modelling Module............................................................ 32

Table 4.3: Motor Resistance Experimental Results...................................................................... 37Table 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............................................................................................... 52Table 4.7: DCMCT Model Parameter Specifications................................................................... 54

Table 4.8: DCMCT Sensor Parameter Specifications................................................................ ..56


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List of Equations

Equation 4.1: …………………………………………………………………….………10mt k k =

Equation 4.2:a s

bG vw

+=, …………………………………………………………………..…..17

Equation 4.3:1

,+

= s

k G vw

τ …………………………………..………………………………….18

Equation 4.4:1

,+

= s

K G

Td

Td T w

τ ……………………………………………………………………18

Equation 5.1:1

)()(

+=

s

s KV sw m s

τ …………………………………………………………………..30

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

Equation 5.3:1+

= sT

sw

f

mm

θ ……………………………………………………………………..30

Equation 5.4:1, +

= s

k G

vw τ

……………………………………………………………………..45

Equation 5.5u

y K

Δ

Δ= …………………………………………………….……………………..46

Equation 5.6:

( )mmm

meq

mvw

R s L R

k J

k sG

+⎟⎟

⎠

⎞⎜⎜⎝

⎛ +

=2,

)( ……………………………………………….51

Equation 5.7: )1)(1(

1

)( ++=

s sk sw em s τ τ ……………………………………………………..51


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ECNG 2005LABORATORY & 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% Estimated total

study hours1:

Delivery mode: Lecture

Online

Lab

Venue for the Lab:

Lab Dependencies2 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 – ECNG2009

Recommended

prior knowledge

and skills3:

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 dynamicsystems modelling

f. Use block diagrams to represent linear systems

1 Estimate includes teaching time, study time, and student preparation time for classes and labs.2 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.

http://myelearning.sta.uwi.edu/http://myelearning.sta.uwi.edu/http://myelearning.sta.uwi.edu/


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Course Staff Position/Role E-mail

Phone

Office Office

Hours

Lucia Cabrera-Jones Teaching

Assistant

2462 322

Andre Morris Laboratory

Technician

3193 Control

Systems

Lab

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:

Submit to TA

Estimated time to

completion:

3.1. Required Reading Resources

Katsuhiko Ogata, Prentice Hall 1997. Modern Control Engineering (4td Ed) .


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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 mechanicalspeed 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. A motor (electric, pneumatic or hydraulic) – translates energy into motion2. Controller and control amplifier – provides control of motion3. Velocity and position feedback sensors – provides measurement of motion4. 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.

http://www.eng.uwi.tt/depts/elec/staff/copeland/http://www.eng.uwi.tt/depts/elec/staff/copeland/


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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 toderive 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 themotor. 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.


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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


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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 fieldcurrent. 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.


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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)

N S

N S

ebea

Ra La

Amplifier

Vm M

Jm, bm

JL, bL

Inertial and

viscous load

ωm

Motor

Gearbox

ωL

T

(b)

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

(b) expanded block diagram


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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-motive-force (back EMF) constant, k m i.e.,

k t = k m. 4.1

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


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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 therelationship, expressed in the Laplace domain characterizing, between the armature current

(ia) and voltages (ea, eb).

0 1 2

Magnetic

field

strength,

Armature

Motor

Armature


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eb =km ωmea

Ra La

M

Ia

Tm

ωm

Td

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?


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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 DCmotor.

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. zerothickness 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:


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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 maximumvelocity: ωmax.

Solution:


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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 thecorresponding 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


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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 andmeasuring both corresponding steady-state current and speed (in radians persecond).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:

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

Hint:

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

can be considered as cero.

Solution:


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0 1 2

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

,v

bG

s aω

=+

[4.2]

Solution:


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0 1 2

3. Express and evaluate Gω,V(s) as a function of the parameters K and τ, defined such as:

1,

+=

s

k G v

τ ω

[4.3]

Solution:

0 1 2

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

defined below:

, ( ) 1

Td

T

Td

K

G sω τ = + [4.4]

Show that τ τ =Td

Solution:


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0 1 2

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

Solution:

0 1 2

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


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Vm

Tdm

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

Solution:

0 1 2

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

τe previously evaluated, justify the approximation.

Solution:


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0 1 2


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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 velocityωmax

Rad/s

2 Motor maximum current Imax A

2 Motor maximum torque Tmax Nm

3.5.2 Dynamic relationships

5 Motor torque constant k m Nm/A

5 Motor armature resistance R m Ω

6 Open-loop model parameter a kg.m/(Ws4)


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6 Open-loop model parameter b 1/(V.2

)

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)

Table 3.2- Modeling Pre-Laboratory Assignment Results


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4. IN-LAB

Allotted CompletionTime:

4 hours

Required lab

Equipment:

1. The Quanser DCMCT rig2. 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 control the motor using analog electronics or

a PC. The DCMCT consists of (Fig 4.1)

1. a potentiometer for precision speed and position sensing2. 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.


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1. Maxon DC Motor

2. Removable inertial Load3. Linear Power Amplifier4. High Resolution Optical

encoder5. Ball Bearing Servo

Potentiometer

6. Removable Belt to drivethe potentiometer

7. i. PC Interface Option:this is implemented by

using DtoA and AtoD

convertersii. Analog Controller

Option: to implement

controllers using analog

electronic circuits

8. Breadboard Option: toimplement controllerswith your own circuits

9. Embedded/PortableOption: The QIC installsin this socket to perform

embedded control in

place of PC-based

control

10. Serial Port (used byQICii)

11. PIC Reset Switch12. User Switch: Momentary

Action Pushbutton

Switch For Manual

Interaction

13. Inertial Load Storage Pin14. Jumper J6: to switch

between HIL and QICuse

15. 6—mm Power Jack16. Power Supply I leader: J417. Analog Signals I leader:

ii 1

Figure 4.1 : DCMCT Trainer Module and Schematic (Quanser)


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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 modelof 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.


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Figure 4.2:- Modeling module of the QICii software

2.5.1. M


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otor First Principles

ID # Label Parameter Description Unit

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

Generator

Type of Generator For The Input Voltage

Signal

5 Amplitude Generated Signal Amplitude Input Box V

6 Frequency Generated Signal Frequency Input Box Hz

7 Offset Generated Signal Offset Input Box V

8 Speed ωm Scope With Actual (in red) And Simulated

(in blue) Motor Speeds

rad/s

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

Signal

s

Table 4.1:- QICii Modelling Module Nomenclature


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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 andmotor 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:-

( )( )

1

m s

KV s s

sω

τ =

+ [5.1]

The implemented digital controller in the QIC runs at 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:-

1

mm

f

s

T s

θ ω =

+ [5.3]

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


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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 againon the QIC .The two LEDs should start flashing.

6. Press the User Switch, which is close to the flashing light. Automatically the removableinertial 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.


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Reset

-

Force on armature conductor, F

Figure 4.3:-Locating the push-button and the LEDs

Signal

Type

Amplitude

[V]

Frequency

[Hz]

Offset

[V]

K

[rad/(V.s)]

τ

[s]

T f

[ s]

Square

Wave

2.0 0.4 0.0 10.0 0.2 0.01

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.


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In case of disconnection:

1. Press the Reset button again on the QIC .The two LEDs should startflashing.

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 thesteps 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?


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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?


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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?

Comments

Positive

Direction

Negative

Direction


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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 displayis 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.


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Sample:

i

V m(i)

[V]

Offset in Measured

Current: I bias [A]

0 0

Sample:

i

V m(i)

[V]

Measured Current:

I meas(i) [A]

Corrected for Bias:

I m(i) [A]

Resistance:

Rm(i) [Ω]

1 -5

2 -4

3 -3

4 -2

5 -1

6 1

7 2

8 3

9 4

10 5

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.


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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.


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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.


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Sample:

i

V m(i)

[V]

I bias

[A]

Ravg

[Ω]

0 0

Sample:

i

V m(i)

[V]

Measured Speed:

ωm(i) [rad/s]

I meas(i)

[A]

I m(i)

[A]

k m(i)

[V.s/rad]

1 -5

2 -4

3 -3

4 -2

5 -1

6 1

7 2

8 3


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9 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

with the specified value and discuss your results ( mavg k )


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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?


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0 1 2

4.9 Estimate the Measurement Noise

The measurement noise can be determined experimentally as follows:


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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 andcompare 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 tothe disturbance frequency? Justify your answer. Do you also observe any repeatablefluctuations 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?


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0 1 2

4.10 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:

1,

+=

s

K G v

τ ω

[5.4]

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

Figure 4.4 Step response test input and output


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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:

y K

u

Δ=

Δ [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

Signal

Type

Amplitude

[V]

Frequency

[Hz]

Offset

[V]

K

[rad/(V.s)]

τ

[s]

Square

Wave

2 0.4 3 0 0.0

Table 4.5 Module Parameters for the step response test


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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.


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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


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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:


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• 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,

and quantization in the encoder 2π/4096 = 1.5 10-3

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.


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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 dynamics.

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:-

( ), 2

( ) mv

meq m m

m

k G s

k J L s R

ω =

⎛ ⎞+ +⎜ ⎟⎝ ⎠

R

[5.6]

which can also be expressed as:-

( )( ),1

( )1 1

v

m e

G sk s s

ω

τ τ =

+ + [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


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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 positionsignal 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

Value

In-Lab

Result

Unit

5.2. Static Relations

1. Motor Maximum Velocity ωmax rad/s

1. Positive Coulomb Friction

Voltage

Vfp N/A V

1. Negative Coulomb Friction

Voltage

Vfn N/A V

2. Motor Armature Resistance R m Ω

3. Motor Torque Constant k m N.m/A

4. Open-Loop Steady-State Gain K rad/(V.s)


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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


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Appendix 1: System Parameters

Symbol Description Unit Value

Motor

mk Motor Torque Constant Nm/A 0.0502

m R Motor Armature Resistance Ω 10.6

m L Motor Armature Inductance mH 0.82

Motor maximum continuous torque Nm 0.033

Motor power rating W 18

m J Moment Of Inertia Of Motor Rotor Kg.m2 1.16E(-6)

mτ Motor Mechanical -Time Constant s 0.005

i M Inertial Load Disc Mass kg 0.068

ir Inertial Load Disc Radius M 0.0248

Linear Amplifier

Vmax Linear Amplifier Maximum Output Voltage V 15

Linear Amplifier Maximum Output Current A 1.5

Linear Amplifier Maximum Output Power W 22

Linear Amplifier Maximum Dissipated Power

with heat sink R load =4 Ω

W 8

Linear Amplifier Gain V/V 3

Table 4.7: DCMCT Model Parameter Specifications


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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

Encoder

Line Count 1024 Lines/rev

Resolution (in quadrature) 0.0879o/count

Type TTL

Encoder signals A, B, Index

Potentiometer

Calibration at POT RCA jack 39 o/V

Calibration at QIC A/D input 78o/V

Resistance 10 K Ω

Bias voltage ±4.7 V

Electrical range 350o

Tachmoeter

Calibration at TACH RCA jack 667 RPM/V

Calibration at QIC A/D input 1333 RPM/V

Table 4.8: DCMCT sensor parameter specifications


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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


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A signed plagiarism declaration form must be submitted with your assignment.

End of Lab#1: Modeling the DC Motor from First Principles

Control Lab1

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