ELE847 Advanced Electromechanical Systems Course Notes 2008 Edition

Department of Electrical and Computer Engineering ELE847 Advanced Electromechanical Systems Course Notes 2008 Edition

ELE847 Advanced Electromechanical Systems Table of Contents 1. Course Outline.... 1 2. Lab Manual... 6 3. Problems (with Answers).... 22 4. Lecture Slides 27 Advanced Electromechanical Systems 1

1. Course Outline Course Description A course on modeling and simulation of electromechanical systems. The main topics include: modeling of dc motors, dc motor dynamic performance, reference frame theory, modeling of induction and synchronous machines, small signal (linearized) analysis, solid state converters, advanced motor speed control schemes, and simulation techniques. The modeling and simulation techniques developed in this course provide a useful tool for the analysis and design of electric machines, power electronics circuits and dc/ac motor drives. Prerequisite All required third year courses. Course Organization This course consists of three hours of lecture and one hour of laboratory per week. Course Material Text: "Analysis of Electric Machinery and Drive Systems" by P.C. Krause, O.Wasynczuk and S.D.Sudhoff, published by Wiley-IEEE Press, 2002. ISBN 0-471-14326-X Instructor Bin Wu, Ph.D., P.Eng., Professor Room ENG328, 245 Church Street, Toronto Department of Electrical and Computer Engineering Ryerson University (416) 979-5000 ext: 6484 Advanced Electromechanical Systems 2

Course Evaluation Theoretical component 70% Mid-term Examination 25% Final Examination 45% Laboratory component 30% 4 Labs including post-lab reports 20% 1 Project including a formal report 10% In order to achieve a passing grade, the student must achieve an average of at least 50% in both theoretical and laboratory components. Course Material 1. Course Outline 2. Lab Manual 3. Problems Download from http://www.ee.ryerson.ca/~bwu/courses.html 4. Lecture Slides 5. Selected Chapters from "Analysis of Electric Machinery and Drive Systems" by P.C. Krause, et al. Purchase at Alicos Copy Centre 66 Gerrard Street, E. Toronto, M5B 1G3 Tel: (416) 977-6868 Copied under license from access Advanced Electromechanical Systems 3

Lecture Topics 1 DC Motor Dynamic Performance and Speed Control (10hrs) 1.1 Introduction 1.2 DC Motor Dynamic Models and Transfer functions 1.3 Computer Simulation Techniques 1.4 Dynamic Performance of DC Motors 1.5 Thyristor (SCR) Rectifiers 1.6 DC Motor Speed Control and Simulation 2 Reference Frame Theory (3hrs) 2.1 Introduction 2.2 Equations of Transformation 2.3 Stationary and Arbitrary Reference Frames 2.4 Transformation Between Reference Frames 3 Theory of Induction Machines (8hrs) 3.1 Introduction 3.2 Modeling of Induction Machines 3.3 Commonly Used Reference Frame 3.4 Induction Motor Dynamic Performance 3.5 Steady State Operation 3.6 Induction Motor Small Signal Models 4 Induction Motor Speed Control (12hrs) 4.1 Introduction 4.2 Simulation of Voltage and Current Source Inverters 4.3 Pulse Width Modulation (PWM) Techniques 4.4 Induction Motor Control Schemes 4.5 Field Oriented Control and Simulation 5 Theory of Synchronous Machines (5hrs) 5.1 Introduction 5.2 Modeling of Synchronous Machines 5.3 Synchronous Machine Dynamic Performance 5.4 Steady State Operation 5.5 Small Signal Models 5.6 Speed Control of Synchronous Motors Advanced Electromechanical Systems 4

Laboratory Schedule Lab Class Topic Week # 1 DC Motor Dynamic Performance and Solid-state Rectifiers 2 & 3 2 DC Motor Speed Control 4 & 5 3 Induction Motor Dynamic Performance 6 & 7 4 Pulse-width-modulated (PWM) Inverters and Harmonic Analysis 8 & 9 Project Simulation of a High Performance Induction Motor Drive 10, 11 &12 Note - Each lab class is composed of two lab sessions. - Post-lab reports should be handed in one week after the second lab session. - A formal report should be prepared for the project. Advanced Electromechanical Systems 5

2. Lab Manual Lab 1 DC Motor Dynamic Performance and Solid State Rectifiers Part A DC Motor Dynamic Performance A1. Objectives - Build a Simulink model for a separately exited DC motor; and - Study dynamic performance of the motor. A2. Pre-lab Exercises 1) Copy file Dcm1.mdl from /home/courses/ele847/ into your current working directory. 2) Open Dcm1.mdl and study all blocks and subsystems contained in this model (refer to your lecture notes for comparison). 3) Start the simulation and study the waveforms of i a, T e and n using scope blocks. 4) Print the model including block diagrams of all subsystems. A3. Lab Procedures A3.1 Model Building Build a dynamic model for a separately excited DC motor. A suggested system block diagram is shown in Fig. 1. Refer to your lecture notes for details. The DC motor has the following nameplate data and parameters: 5hp, 1220rpm, 240V, 16.2A, r a = 0.6Ω, r f = 240 Ω, L AA = 0.012H, L AF = 1.8H, L FF = 120H, J = 1.0kg.m 2. The load torque T L is assumed to be zero. File: Dcm2.mdl Step Armature Voltage [V] Va Ia Load Current [A] Scope ia To Workspace 0 Const Load Torque [N.m] TL Te Wm Motor Torque [N.m] -K- Gain Scope1 n Motor Speed [rpm] Scope2 Te To Workspace1 n To Workspace2 Step1 Field Voltage [V] Vf if Field Current Scope3 DC Motor2 (Masked subsystem) Clock t To Workspace3 Fig. 1 Suggested system block diagram of a separately excited DC motor. Print the model including block diagrams of all subsystems. Advanced Electromechanical Systems 6

A3.2 Dynamic Performance During Starting 1) Assuming that a dc supply of 50V is applied to the armature winding at the same time when the field winding is switched to a 240V dc supply, find: - the maximum starting current I a,max, and - the ratio of the maximum starting current to the motor rated current. Is this starting current acceptable in practice? 2) Plot the transient waveforms of armature current i a, electromagnetic torque T e and motor speed n. 3) A dc supply of 240V is applied to the armature and field windings simultaneously. Find the value of an external resistance in the armature circuit such that the maximum starting current is limited to 30A. A3.3 Transients During Sudden Changes in Load Torque The dc motor is running steadily under no load conditions with a 240V dc voltage applied to both armature and field windings (Note: no external resistor is added to the armature circuit). Assume that the load torque is suddenly increased to its rated value. Plot the transient waveforms of armature current i a, electromagnetic torque T e and motor speed n. Find speed regulation under this operating condition ωr( no load ) ωr( rated ) (Speed Regulation = ). ω r( rated ) Part B Solid-state Rectifiers B1. Objectives - Build Simulink models for diode and thyristor rectifiers; and - Investigate rectifier characteristics. B2. Pre-lab Exercises 1) Single phase diode rectifier - Copy file Rectd1.mdl from /home/courses/ele847/ into your working directory. - Open Rectd1.mdl and study all blocks and subsystems contained in this model (refer to your lecture notes for details). - Run the Simulink model and study the waveforms of v a, v d1, v d2, v dc and i dc. 2) Single phase thyristor (SCR) rectifier - Copy file Rectt1.mdl from /home/courses/ele847/ into your working directory. - Open Rectt1.mdl and study all blocks and subsystems contained in this model (refer to your lecture notes for details). - Run the model and study the waveforms of v a, v d1, v d2, v dc and i dc. Advanced Electromechanical Systems 7

B3. Lab Procedures B3.1 Three phase diode rectifier 1) Build a Simulink model for a three phase diode rectifier using the circuit diagram discussed in the lecture class. The system block diagram should be similar to that given in Rectd1.mdl. 2) Run the Simulink model and plot the waveforms of v a, v d1, v d2, v dc and i dc assuming that the phase voltage of the three phase ac supply is 110V (60Hz). 3) Calculate the average value of the rectifier output voltage (v dc ) based on the simulated waveforms. B3. 2 Three phase thyristor rectifier 1) Build a Simulink model for a three phase thyristor rectifier using the circuit diagram discussed in the lecture class. The system block diagram should be similar to that given in Rectt1.mdl. It is assumed that the output current of the rectifier is continuous. Hint: Add Variable Transport Delay blocks to the three phase diode model you have built in Part B3.1. 2) Assume that the phase voltage of the ac supply is 110V (60Hz). Run the Simulink model and plot the waveforms of of v a, v d1, v d2, v dc and i dc with the delay angle of 30 and 90 degrees respectively. 3) Derive an expression which can be used to calculate the average value of the rectifier output voltage (v dc ). Verify this expression using simulated waveforms. General Instruction for Post-lab Reports (Lab 1 to 4) The post-lab report (typed) should include the following items: 1) Cover page (including course/lab title, your name, student ID, date) 2) Abstract (a paragraph of about 150 words) 3) Theory (one page, 1.5 line space) 4) Required simulation waveforms 5) Required calculation results and answers to the questions if any 6) Conclusions (200 300 words) 7) Appendix: Simulink models, including block diagrams of all subsystems. Advanced Electromechanical Systems 8

Lab 2 DC Motor Speed Control Objectives - To investigate characteristics of various dc motor speed control schemes. - To learn how to tune PI regulator parameters. Part A Open-loop Speed Control A1. System Block Diagram The block diagram of a dc motor speed control system is shown in Fig. 1. The specifications and requirements for masked subsystem blocks in the diagram are as follows: AC Supply: A three-phase balanced power supply with phase voltage of 120V (rms) and frequency of 60Hz. SCR Rectifier: A three-phase full-wave thyristor rectifier with continuous dc current. DC Motor2: A separately excited dc motor. The motor has the following nameplate data and parameters: 5hp, 1220rpm, 240V, 16.2A, r a = 0.6Ω, r f = 240 Ω, L AA = 0.012H, L AF = 1.8H, L FF = 10H, J = 0.1kg.m 2 and B m = 0. Note: The values of motor field self-inductance L FF and moment of inertia J are not the same as those used in Lab 1. Firing Circuit: A cos -1 function should be implemented to make the output voltage (V dc ) of the rectifier proportional to the input voltage (V α ) of the firing circuit. The maximum and minimum input voltages of the firing circuit should be 1.0 and -1.0V respectively. Therefore, a limiter should be used in the firing circuit to limit its input voltage. Dcdrv1.mdl 3 phase Va Vb Vc Vdc Vdc Va Ia Armature Current Ia AC Supply t_d acos fcn included SCR Rectifier Firing Circuit 29.2 Load (N.m) TL Te Wm Electromagnetic Torque (N.m) Speed (rad/s) -Krpm Gain Te n 0.5 V_alpha 240 Vf (V) Vf DC Motor2 if Field Current If Reference (0 ~ 1) Fig.1 A separately excited dc motor with open-loop speed control. Advanced Electromechanical Systems 9

A2. Lab Procedures A2.1 Build a dynamic model for the dc motor drive system shown in Fig.1. All the requirements given in Section A1 should be satisfied. A2.2 System Dynamic Performance 1) Assume that the field voltage is 240V, the input voltage to the firing circuit is 0.5V and the load torque is rated. It is also assumed that the voltages and load torque are applied to the drive system simultaneously. Suggested simulation parameters: start time = 0, stop time = 0.4sec and differential equation solver = ode4 (Runge Kutta) with a fixed step size of 0.0001sec. (Select Parameters from Simulation menu, choose Fixed-step from Solver Options, and then select ode4). 2) Run the model and determine: - the peak value of the starting (armature) current (A), electromagnetic torque (N.m) and rotor speed (rpm); - the maximum speed overshoot (%); and - the value of ripple current I a and ripple torque T e in steady state. Question: How to reduce the ripples? - the average value of the dc voltage V dc. A2.3 Plot the transient waveforms of armature current, electromagnetic torque and rotor speed. Part B DC Motor Drive with Current Feedback B1. System Block Diagram The block diagram of a current controlled dc motor drive system is shown in Fig. 2. The specifications and requirements for masked subsystem blocks in the diagram remain the same as those given in Section A1. The suggested parameters for the current PI regulator are: Time constant: 0.01sec. Gain: 0.02 Upper limiting level: 1.0 Lower limiting level: 0.0 Dcdrv2.mdl Va Vb 3 phase Vc AC Supply t_d acos fcn included SCR Rectifier Firing Circuit Vdc 29.2 Load (N.m) Va TL Ia Te Wm Mux Mux Torque (N.m) -Krpm Gain Ia Te n PI V_alpha Current PI (with limiters) 240 Vf (V) Vf DC Motor2 if Field Current If Sum2 32.4 Current Ref (A) Fig.2 DC motor speed control with current feedback. Advanced Electromechanical Systems 10

B2. Lab Procedures B2.1 Build a dynamic model for the dc motor drive system shown in Fig.2. All the requirements given in Section B1 should be satisfied. B2.2 System Dynamic Performance 1) Assume that the field voltage is 240V, load torque is rated and current reference is 32.4A (twice the rated current). It is also assumed that the field voltage, the load torque and the current reference are applied to the drive system simultaneously. Suggested simulation parameters: start time = 0, stop time = 1sec and solver type = ode4 (Runge Kutta) with a fixed step size of 0.0002 sec. 2) Run the model and plot the waveforms of starting current, electromagnetic torque and rotor speed; 3) Based on the simulation results, answer the following questions: - During the motor starting, the rotor speed increases linearly with time. Why? Use equations to assist explanation if necessary. - The armature current and motor torque do not have similar waveforms at the very beginning of the starting process. Why? - The armature current is kept constant during starting. Why? Is this a desirable feature? - If the moment of inertia is doubled, is the starting time doubled too? Please verify. Part C DC Motor Drive with Current and Speed Feedbacks C1. System Block Diagram The block diagram of a dc motor drive system is shown in Fig. 3. The specifications and requirements for masked subsystem blocks in the diagram remain the same as those given in Section B1. The parameters for the speed PI regulator are tentatively set at: Time constant: 0.3sec Gain: 1.2 Upper limiting level: 32.4 (A) Lower limiting level: 0.0 C2. Lab Procedures C2.1 Build a dynamic model for the drive system shown in Fig.3. All the requirements given in Section C1 should be satisfied. C2.2 Dynamic Performance 1) Assume that the field voltage is 240V, the load torque is 10N.m and the speed reference is 63.9 rad/s. The field voltage, load torque and speed reference are applied to the drive system simultaneously. Suggested simulation parameters: start time = 0, stop time = 1sec, and solver type = ode4 (Runge Kutta) with a fixed step size of 0.0002sec. Advanced Electromechanical Systems 11

Dcdrv3.mdl Va 3 phase Vb Vc Vdc Va Ia Mux Mux Ia AC Supply t_d acos fcn included SCR Rectifier Firing Circuit 10 Load (N.m) TL Te Wm -K- Gain rpm Te n V_alpha PI Current PI (with limiters) 240 Vf (V) Vf if DC Motor2 (Separately Excited) If Sum1 Ia Ia (Ref) PI Speed PI (with limiters) Note: Blocks with a drop shadow represent masked subsystems. Sum2 Wm (rad/s) 63.9 Speed Ref (rad/s) Fig.3 DC motor speed control with current and speed feedbacks. 2) Run the model and plot the waveforms of armature current and rotor speed; 3) Based on simulation results, calculate the speed overshoot (%). 4) Determine the speed PI regulator parameters such that the speed overshoot is approximately 5% and the speed settling time as short as possible. C2.3 Print the Simulink model including all subsystems. Post-lab Report Refer to Lab 1 for general instruction on post-lab report. Advanced Electromechanical Systems 12

Lab 3 Three-phase Induction Motor Dynamic Performance Objectives - To build a Simulink model for three phase induction motors; and - To investigate induction motor dynamic performance. Part A Induction Motor Dynamic Model The block diagram of a three-phase induction motor supplied by a three-phase power supply is shown in Fig. 1. The specifications and requirements for masked subsystem blocks in the diagram are as follows: AC Supply A three-phase power supply with phase voltage of 127V (rms) and frequency of 60Hz. File: Lab3.mdl Mux Ia Mux Vqs & Vds To Workspace 3 phase Va Vb Vc 3-phase To 2-phase Vqs Vds iqs ids 2-phase To 3-phase Ia Te To Workspace1 AC Supply 3-2 Transform 0 Tl Te 2-3 Transform Load Torque Te 0 W Wrm 60/(2*pi) Clock t To Workspace3 Stator Frame IM_dq_Arbi Induction motor model in the arbitrary frame Gain n n To Workspace2 Fig. 1 System block diagram. 3-phase to 2-phase Transformation: Both 3-phase variables and 2-phase variables are in the stationary frame. Use the transformation equations derived in the lecture or Equation 3.3-4 on Page 111 of textbook. Note: the angle θ between the stationary and arbitrary frames should be set to zero. Advanced Electromechanical Systems 13

IM_dq_Arbi Induction motor d-q model in the arbitrary reference frame. This subsystem must be masked and the motor parameters must be specified in its dialog box. The inputs of the subsystem are d-q axis voltages (v qs, v ds ), load torque (T L ) and the speed of the arbitrary reference frame (ω) while the outputs are d-q axis current (i qs, i ds ), electromagnetic torque (T e ) and the rotor mechanical speed ω rm. Build the induction motor d-q model using the equivalent circuit given in Fig. 4.5-1 (Page 151). The zero-axis equivalent circuit can be neglected since this is a 3-phase balance system. The torque-speed relationship is described by Eq. 4.3-8 and the electromagnetic torque generated by the motor can be calculated according to Eq. 4.6-4. 2-phase to 3-phase Transformation Both 2-phase variables and 3-phase variables are in the stationary frame. Use the transformation equations derived in the lecture or Equation 3.3-6 on Page 111. Note: The angle θ between the stationary and arbitrary frames should be zero. Part B Free Acceleration Characteristics The induction motor under investigation is rated at 3hp, 220V, 8.4A and 1710rpm. The parameters of the motor are given in Table 4.10-1 (Page 165). B1. Motor free acceleration with rated stator voltage 1) The motor is started under no load conditions with the rated stator voltage. Suggested simulation parameters: start time = 0, stop time = 0.5sec and differential equation solver = ode4 (Runge Kutta) with a fixed step size of 0.0002sec. (Select Parameters from Simulation menu, choose Fixed-step from Solver Options, and then select ode4). 2) Run the model and determine: - the maximum peak value of the stator current and electromagnetic torque during free acceleration; - the average starting torquet e, start ; and - the motor starting time t st. The definition for T e, start and t st is given in Fig. 2. 3) Plot the waveforms of T e versus n, T e versus t, i a versus t, n versus t. Advanced Electromechanical Systems 14

n n ss (0.05)n ss t st (a) Motor speed T e T e1 T e, start t T e2 (b) Electromagnetic torque Fig. 2 Motor speed and torque waveforms during free acceleration. B2. Motor free acceleration with a reduced voltage 1) The motor is started under no load conditions with 50% rated stator voltage. 2) Run the model and determine: - the maximum peak value of the stator current and electromagnetic torque during free acceleration; - the average starting torque; and - the motor starting time. 3) Plot the waveforms of T e versus t, i a versus t, n versus t. 4) Compare the simulation results obtained from B1 and B2, and make your conclusions. Post-lab Report Refer to Lab 1 for general instruction on post-lab report. Advanced Electromechanical Systems 15

Lab 4 Three-phase Voltage Source Inverter and PWM Techniques Objectives - To build Simulink models for three phase voltage source inverters; and - To investigate PWM inverter performance. Part A Three Phase Voltage Source Inverter A1. Model Building The block diagram of a three phase voltage source inverter with a three phase RL load is shown in Fig. 1. The specifications for masked subsystem blocks in the diagram are as follows. Square Wave Generator This block generates three square wave signals for the inverter. These signals should have the same amplitude (1.0V) with a duty cycle of 50%. The phase shift between any two signals is 120 degrees. This is a masked subsystem. The frequency of the square waves should be passed to the subsystem through its dialog box. Lab4a.mdl Van G1 Van Ia Gating Square Wave Gating Generator G3 G5 VSI Vbn Vcn RL Load Ib Ic Mux Mux1 Load Current 250 Three Phase VSI RL Load Vdc Fig. 1 System block diagram. Three Phase Voltage Source Inverter Use the algorithm discussed in the lecture class to build the model for this inverter. Advanced Electromechanical Systems 16

RL Load This is a three phase balanced RL load. The parameters of the load resistance and inductance should be passed to the subsystem through its dialog box. A2. Lab Procedure A2.1 Run the model and plot the waveforms of G 1, v an and i a under the following operating conditions: V dc = 250V, R load = 2Ω, L load = 0.01H and the output frequency of the inverter is 60Hz. A note on simulation parameters. You can either use fixed-step or variable-step differential equation solver. If you use a fixed-step differential equation solver with a large time step, you may not be able to obtain accurate results or the results may even be wrong. This is mainly due to the switching operation of the inverter and small time constants that the drive system may contain. If you choose Runge Kutta (ode4) method, you may try to use a step size of 10 µ s or smaller. A2.2 Plot the harmonic spectrum of the waveforms of v an and i a. Frequency range for the plot: 0 to 2kHz. Note: This is a common task for electrical engineers working in the area of power electronics and motor drives. Part B PWM Controlled Inverter B1. Model Building The system block diagram is shown in Fig. 2. The specifications and requirements for masked subsystem blocks in the diagram are as follows: Three Phase Sine Wave Generator This block generates a three phase balanced sine wave whose frequency and amplitude are controlled by the block input variables Freq and M d, where Freq is the reference frequency and M d is the modulation index. The maximum value of the sine wave amplitude is 1V. Advanced Electromechanical Systems 17

Lab4b.mdl 20 Van Freq Sine Wave Sine G1 G1 Van Ia 0.8 Md 3-phase SW Generator Carrier G3 G5 G3 G5 VSI Vbn RL Load Ib Mux Load Current Carrier Wave PWM Generator Vcn Ic Mux 1-phase CW Generator Three Phase VSI RL Load 250 Vdc Fig. 2 PWM Controlled Voltage Source Inverter Single Phase Carrier Wave Generator This is a masked subsystem where the frequency of the carrier (a triangular wave) is passed to the block through its dialog box. The carrier is not synchronized with the sine waves. Other Blocks The specifications for the other blocks are given in Part A. B2. Lab Procedure B2.1 Run the model and plot the waveforms of v an and i a under the following operating conditions: Assume that V dc = 250V, R load = 2Ω and L load = 0.01H. 1) Freq = 20Hz, M d = 0.8, and the frequency of the carrier wave is 240Hz; 2) Freq = 20Hz, M d = 0.8, and the frequency of the carrier wave is 1080Hz; 3) Freq = 20Hz, M d = 0.4, and the frequency of the carrier wave is 1080Hz; and 4) Freq = 60Hz, M d = 0.4, and the frequency of the carrier wave is 1080Hz. Compare the simulation results and make your conclusions. B2.2 Plot the harmonic spectrum of the waveforms of v an and i a in Part B2-4). Frequency range for the plot: dc to 2kHz. Compare the harmonic spectrum with that in Part A2.2, and then make conclusions. Post-lab Report Refer to Lab 1 for general instruction on post-lab report. Advanced Electromechanical Systems 18

Project Simulation of Induction Motor Drives Objectives To investigate characteristics of two induction motor speed control systems. Part A Induction Motor Speed Control Using a Six-step Voltage Source Inverter A.1 Model Building Build the Simulink model according to the block diagram shown in Fig. 1. You may use some of the models you built in the previous lab sessions. The parameters of the induction motor speed control system are as follows. Induction motor Nameplate data: 3φ, 3hp, 220V, 8.4A (rated) and 1710rpm. Use the motor parameters given in Table 4.10-1, P165, textbook. DC link resistor This resistor represents the power loss of the SCR rectifier and dc link bus. The equivalent resistance is 0.5Ω. Low pass filter This is a second order lower pass filter with dc gain k = 1 and quality factor Q = 1. The corner frequency of the filter should be the same as the reference frequency of the drive. The LP filter is used to extract the fundamental component from the six-step inverter output voltage V an. Power Supply 60Hz, 127V per phase. Load torque 12.5N.m (rated torque) Other constants K v = 0.95/60 and V comp = 0. Proja.mdl Mux Vdc Vdc Van Vqs iqs 3 phase Va Vb DC Link G1 3-phase To 2-phase Vds ids 2-phase To 3-phase Ia AC Supply Vc Vdo Filter G3 VSI Vbn 12.5 Tl Te 2-3 Transform V_alpha Sum t_d acos fcn included SCR Rectifier Firing Circuit Subsystem G5 3-phase Square Wave Generator Three Phase VSI Vcn Load Torque [N.m] 0 Stator Frame W IM_dq_Arbi Wrm Induction motor model in the arbitrary frame 60/(2*pi) Gain Te n Kv -K- 0.05 Vcomp LP Filter (2nd Order) Van1 60 Ref [Hz] Fig. 1 An Induction Motor Speed Control System Using a Six-step Inverter. Advanced Electromechanical Systems 19

A.2 Simulation tasks 1) Run the Simulink model and determine the value of the dc link capacitor such that the dc link voltage ripple is limited to 10% when the inverter operates at 20Hz with a rated V/f and the motor operates with rated torque. Plot the steady state waveforms of v do and v dc (e.g., 2 cycles of the supply frequency). 2) Start the drive system until a steady state operation is reached. Complete the following table. Inverter Output Frequency [Hz] 60 45 30 15 5 Vˆ an (Fundamental, peak, steady state) V an (Fundamental, rms) V / f Ratio (Volts,rms/Hz) Steady State Speed n (rpm) Synchronous Speed n s (rpm) Slip Speed (n s - n) 3) To compensate the voltage drop on stator winding resistance at low frequencies, let K v = 0.9/60 and V comp = 0.05. Run the drive system and complete the following table. Inverter Output Frequency [Hz] 60 45 30 15 5 Vˆ an (Fundamental, peak, steady state) V an (Fundamental, rms) V / f Ratio (Volts,rms/Hz) Steady State Speed n (rpm) Synchronous Speed n s (rpm) Slip Speed (n s - n) 4) Start the drive system at 60Hz under the operating conditions given in 2) until a steady state operation is reached. Plot the transient waveforms of motor speed n, stator current I a, motor torque T e, diode rectifier output voltage V do and inverter input voltage V dc. Part B Induction Motor Speed Control using a PWM Inverter B.1 Model Building Build the Simulink model shown in Figure 2. The system parameters remain the same as those given in Part A. B.2 Simulation tasks Repeat the simulation tasks specified in A.2-2. Advanced Electromechanical Systems 20

Fig2. An Induction Motor Drive Using a PWM Inverter. Report The formal report should include the following items: 1) Cover page (including project title, your name, student ID, date) 2) Abstract (a paragraph of about 200 words) 3) Theory (two full pages, 1.5 line space) 4) All required waveforms, tables and calculations 5) Comment on the size of the dc link capacitors used in both systems 6) Compare the simulation results obtained in Part A.2-2 and A.2-3 by answering the following questions: - Is the V/f ratio constant? You may draw V versus f curves for comparison. - Is the slip speed constant? Why? 7) Compare the simulation results obtained in Part A.2-2 and B.2 8) Comments on harmonic issues of the two systems 9) Conclusions (300 400 words) 10) Appendix: Simulink model in Part B including block diagrams of all subsystems. Advanced Electromechanical Systems 21

3. Problems Topic 1 DC Motor Dynamic Performance and Speed Control 1.1 Consider the dynamic equivalent circuit of a shunt dc motor given in Fig. 2.4-2 (page 79, textbook). Drive a block diagram for this motor. It is assumed that the armature voltage and load torque are input variables while the armature current, rotor speed and electromagnetic torque are output variables. Show these variables on the diagram. 1.2 Repeat Problem 1.1 for a series dc motor using the equivalent circuit given in Fig. 2.4-6 (page 83, textbook). Answer: Discussed in the lecture class. 1.3 Formulate the following transfer functions for a shunt dc motor under the assumption that the field current I f is constant: I a (s) (a) assuming that armature voltage is zero. This transfer function can be used to study T L (s) the dynamic response of armature current due to changes in load torque. K v I a(s) J τ a a Answer: = r T L(s) 2 1 S + ( 1/ τ a + Bm /J ) S + ( Bm /J +1/ τ m) τ a J where ra τ m = 2 K v I a (s) (b) assuming that the load torque is zero. This transfer function can be used to study the dynamic V a (s) response of armature current due to changes in armature voltage. J S + Bm I a(s) a J τ a Answer: = r V a(s) 2 1 S + ( 1 / τ a + Bm /J ) S + ( Bm /J +1/ τ m) τ a T e (s) (c) assuming that the load torque is zero. V (s) a Answer: T e(s) = V a(s) S 2 + ( 1 / + /J ) S + ( /J +1/ ) τ a K v B m J S + Bm ra J τ a 1 1.4 Derive an expression for motor speed ω r (s) in terms of armature voltage V a (s) and load torque T L (s) for a separately excited motor with a constant field current. Refer to Page 97 of textbook for answers. 1.5 Sketch to scale the dc side waveforms ( v d1, v d 2 and vdc = vd1 vd 2 ) of a single phase full-wave thyristor rectifier with a delay angle of 30 and 90 degrees respectively. 1.6 Repeat Problem 1.5 for a three phase full-wave thyristor rectifier. 1.7 Using standard Simulink blocks, derive a Simulink model for a single phase full-wave thyristor rectifier. τ a B m τ m Advanced Electromechanical Systems 22

1.8 Repeat Problem 1.7 for a three phase full-wave thyristor rectifier. 1.9 Derive an expression which can be used to calculate the average dc output voltage of a three phase fullwave thyristor rectifier. Topic 2 Reference Frame Theory 2.1 Consider three-phase currents i as = cos t, i bs = t/2 and i cs = -sin t in the stationary reference frame. Find the values of iqs and ids in the arbitrary reference frame when the angle θ between the two reference frames is π/4 at t = π/3 sec (refer to Pages113-114, textbook). 2.2 Assume that i β and i α are variables in the stationary reference frame and i q and i d are variables in the arbitrary frame which rotates in space at an arbitrary speed of ω as shown in Fig. 2.1 below. Verify the following equations which can be used to transform the variables in the stationary frame to the arbitrary frame. i q = i β cosθ i α sinθ ; and i d = i β sinθ + i α cosθ (Two-phase to two-phase transformation). ω q-axis θ β axis (Stationary Frame) α axis (Stationary Frame) ω d-axis Fig. 2.1 Transformation between two reference frames 2.3 Derive equations which can be used to transform two-phase variables (i q and i d ) in the arbitrary reference frame to two-phase variables (i β and i α ) in the stationary frame. 2.4 Derive a coefficient matrix which can be used to transform abc variables in the stationary reference frame to qdo variables also in the stationary reference frame, assuming that the q-axis is coincident with the a-axis. 2.5 Derive an equation which can be used to transform abc variables in the stationary reference frame to qdo variables in the arbitrary reference frame which rotates in space at a speed of ω. 2.6 Derive an equation which can be used to transform qdo variables in the arbitrary reference frame to abc variables in the stationary reference frame. 2.7 Derive arbitrary-frame (q-d) equivalent circuits for a three-phase balanced capacitor bank. (Hint: Refer to Pages 119-120, textbook). 2.8 Derive arbitrary-frame (q-d) equivalent circuits for a three-phase RL circuit. It is assumed that 1) the RL circuit is three-phase balanced; 2) the resistors and inductors are connected in series; and 3) no mutual inductances exist between any two phases. (Hint: Refer to Pages 120-122, textbook). Advanced Electromechanical Systems 23

Topic 3 Theory of Induction Machines 3.1 A simplified version of induction motor dq model in the arbitrary reference frame is shown in Fig. 3.1, where L L is the total leakage inductance of the stator and rotor windings. It is also called the Γ equivalent circuit of the induction motor. a) Express dq-axis flux linkages in terms of motor inductances and currents in a matrix form. b) Express dq-axis currents in terms of motor inductances and flux linkages. Based on the derived equations, draw a block diagram using standard Simulink blocks. r s ωλ ds L L (ω-ω r )λ dr r r v qs pλ qs L m pλ qr v qr q-axis r s ωλ qs L L (ω-ω r )λ qr r r v ds pλ ds L m pλ dr v dr d-axis Fig. 3.1 Induction motor dq model in the arbitrary reference frame, where the stator and rotor winding leakage inductances are lumped together (the Γ equivalent circuit). 3.2 Derive equations for the calculation of the dq voltages specified in Fig. 3.1. Assume that this is a wound rotor induction motor where the rotor winding is open. Following the same procedure discussed in the lecture class, derive a Simulink model for the induction motor. r s ω s λ ds L ls L lr ( ω s -ω r )λ dr r r v qs pλ qs L m pλ qr q-axis r s ω s λ qs L ls L lr ( ω s -ω r )λ qr r r v ds pλ ds L m pλ dr d-axis Fig. 3.2 Induction motor dq model in the synchronous reference frame. 3.3 The induction dq model in the synchronous frame is shown in Fig. 3.2, where ω s is the speed of the synchronously rotating frame. Assuming that the rotor winding is shorted, repeat the questions given in 3.1 and 3.2. Advanced Electromechanical Systems 24

3.4 A three phase induction motor is powered by a three phase balanced power supply. The power supply and the induction motor can be represented by masked subsystems. Assume the induction motor model is in the synchronous reference frame. The output variables of the power supply - induction motor system are stator currents, electromagnetic toque and motor speed in the stator (stationary) reference frame. Draw a system block diagram showing all the subsystem blocks including 3-phase to 2-phase transformation, stationary to rotating transformation and other transformations if required. It is not necessary to show the details of the subsystems. 3.5 Repeat Problem 3.4 except the induction motor model is in the rotor reference frame. Topic 4 Induction Motor Speed Control 4.1 A three-phase induction motor has the following nameplate data: 10hp, 60Hz, 220V and 1150rpm. The maximum torque (T max ) of the motor is 155N.m, which occurs at the motor speed of 970rpm. The starting torque of the motor is 70N.m. Assuming that the air gap flux of the motor is kept the constant, sketch to scale the toque versus speed curves at the stator frequency of 60Hz, 40Hz and 20Hz. 4.2 Assuming that the input dc voltage of a three-phase IGBT-based voltage source inverter is 200V and the duty cycle of the IGBTs is 50% (i.e., the IGBT conduction angle is 180 degrees). a) Determine the amplitude of the fundamental component, 5 th, 7 th and 11 th harmonics of the inverter output voltage (line-to-line). b) Sketch to scale the line-to-neutral voltage waveform (Phase a) and determine the amplitude of the fundamental component, 5 th, 7 th and 11 th harmonics. 4.3 Draw the inverter output voltage waveform (line-to-line) assuming that the IGBT devices in Question 4.2 have a conduction angle of 120 degrees per cycle. Assume that the inverter is loaded with a three-phase balanced resistor. Hint: the IGBT gating signals for the upper and lower IGBTs in the same inverter leg are no longer complementary. 4.4. Derive a Simulink model for a three phase voltage source inverter, assuming the conduction angle of the switching devices is 180 degrees. 4.5 Draw a block diagram (not Simulink Model) for a three-phase induction motor drive using Volts per Hertz control scheme. To ensure a constant flux operation, voltage feedback should be used. The voltage drop on the stator winding resistance at low operating frequencies should also be compensated. The drive system is implemented with a sine pulse width modulator. 4.6 Fig. 4.1 shows the block diagram of an induction motor field oriented control scheme. Derive equations and then Simulink block diagrams for the following subsystems: I s * & θ T * Resolver, Current Reference Generator and Flux & Torque Estimator. Advanced Electromechanical Systems 25

V dc ω m PI λ r T e * PI PI i T * i f * Resolver θ T i * s θ s Current Reference Generator i * as Delta Modulator G1 G3 G5 VSI θ f i as T e λ r Flux & Torque Estimator i bs i cs v as v bs Tachometer IM Fig. 4.1 Block diagram of an induction motor field oriented control. Topic 5 Theory of Synchronous Machines 5.1 Use the synchronous machine model discussed in the lecture class or given in Figure 5.5-1 (Page 202, textbook). Note: Superscripts associated with machine variables may be ignored. (a) Express flux linkages (λ qs, λ ds, λ kq1, λ kq2, λ kd, and λ fd ) in terms of machine currents (λ qs, λ ds, λ kq1, λ kq2, λ kd, and λ fd ). (b) Derive expressions for the following voltages: V qs, V ds, V kq1, V kq2, V kd, and V fd. 5.2 For small size synchronous machines, the damper windings may be omitted to reduce manufacturing cost. Assuming that for a synchronous machine, the kq2 and kd windings are not equipped, repeat questions given in 5.1. 5.3 State the reasons why the rotor reference frame is often used for synchronous machine analysis. 5.4 Briefly explain the main functions of damper windings. What are the main differences between the synchronous machine and induction machine. Advanced Electromechanical Systems 26

4. Lecture Slides Topic 1 Fig. 1-1 Waveforms of a single-phase SCR rectifier. Advanced Electromechanical Systems 27

Lecture Slides Topic 1 v d1 v a i a D1 D3 D5 v b i b v dc R L v c i c D4 D6 D2 v d 2 Fig. 1-2 Waveforms of a three-phase diode rectifier. Advanced Electromechanical Systems 28

Lecture Slides Topic 1 Fig. 1-3 Simplified Simulink model of three-phase diode rectifier. Advanced Electromechanical Systems 29

Lecture Slides Topic 1 Fig. 1-4 Typical waveforms of a three-phase SCR rectifier. Advanced Electromechanical Systems 30

Lecture Slides Topic 1 Fig. 1-5 Simulated waveforms of a three-phase SCR rectifier. Advanced Electromechanical Systems 31

Lecture Slides Topic 1 Fig. 1-6 Simulated waveforms of a DC drive with open loop control. Advanced Electromechanical Systems 32

Lecture Slides Topic 1 Fig. 1-7 Simulated waveforms of a DC drive with closed loop control. Advanced Electromechanical Systems 33

Lecture Slides Topic 3 (Free acceleration) Fig. 3-1 Simulated waveforms of a three-phase induction motor in the STATOR (STATIONARY) reference frame. Advanced Electromechanical Systems 34

Lecture Slides Topic 3 (Free acceleration) Fig. 3-2 Simulated waveforms of a three-phase induction motor in the STATOR (STATIONARY) reference frame. Advanced Electromechanical Systems 35

Lecture Slides Topic 3 (Free acceleration) Fig. 3-3 Simulated rotor speed waveform of a three-phase induction motor. Advanced Electromechanical Systems 36

Lecture Slides Topic 3 (Free acceleration) Fig. 3-4 Simulated waveforms of a three-phase induction motor in the SYNCHRONOUS reference frame. Advanced Electromechanical Systems 37

Lecture Slides Topic 3 (pu) i as (rpm) n r t (sec) (a) Simulated waveforms Stator current i as : 2.56 pu/div; Rotor speed n r : 720rpm/div Time base: 0.1sec/div (b) Measured waveforms Fig. 3-5 Simulated and measured waveforms of a three-phase induction motor during free acceleration. Advanced Electromechanical Systems 38

Lecture Slides Topic 4 T1 D T3 1 D3 G G1 3 G 5 T 5 D 5 V dc T 4 a D 4 v a T 6 b D 6 T 2 c D 2 i a i b i c n G 4 G 6 G 2 GND G 1 G 2 G 3 ωt ωt G 4 G 5 G 6 π / 3 π / 3 π 2π 3π ωt v a V dc ωt v b v c V dc ωt V dc ωt v ab V dc ωt v bc V dc ωt v ca I II III IV V VI V dc ωt Fig. 4-1 Three phase voltage source inverter with square wave operation. Advanced Electromechanical Systems 39

Lecture Slides Topic 4 T1 D T3 1 D3 G G1 3 G 5 T 5 D 5 V dc T 4 a D 4 v a T 6 b D 6 T 2 c D 2 i a i b i c n G 4 G 6 G 2 GND v v cr vma vm b vmc Vˆcr Vˆm ωt v a ( G 1 ) V dc ωt v b ( G ) 3 V dc ωt v ab v ab1 v ab = v a v b π V dc ωt 2π Fig. 4-2 Principle of sinusoidal pulse width modulation (SPWM). Advanced Electromechanical Systems 40

adlecture Slides Topic 4 (rpm) ω r * r ω r T (N m) e ( A) i as λ (Wb), r rθ )f ω(λ r θ f Fig. 4-3 Simulated waveforms of field oriented control for induction motor drives. Advanced Electromechanical Systems 41

Lecture Slides Topic 5 Fig. 5-1 Synchronous generator with a salient-pole rotor. Advanced Electromechanical Systems 42

Lecture Slides Topic 5 r kq1 i kq1 V kq1 L lkq1 i qs r s ω r λ ds L ls pλ kq1 i mq L lkq2 v qs pλ qs L mq r kq2 i kq2 q-axis pλ kq2 V kq2 r kd i kd V kd L lkd i ds r s ω r λ qs L ls pλ kd i md L lfd v ds pλ ds L md r fd i fd d-axis pλ fd V fd Fig. 5-2 dq-axis model of synchronous generator in the rotor reference frame. Advanced Electromechanical Systems 43