Introduction to Electric Machines

Description: This lecture provides an overview of the course details, grading policy, and syllabus for Introduction to Electric Drives, an undergraduate-level course taught at the University of Minnesota in Spring 2025.

Description: In this introductory lecture, we will explore the earliest developments in electric motor technology, beginning with Faraday's first electromagnetic motor and Andrew Gordon's electrostatic motor, the earliest known electric motor. Along the way, we will examine key milestones that bridged these discoveries, shaping the evolution of electric motors.

Description: In this lecture, we delve deeper into the history of electric motors, examining the fundamental reasons why electrostatic motors are far less common than their electromagnetic counterparts. We also explore the mechanics of biological motors—our muscles—shedding light on their fascinating mode of operation. Through a series of live demonstrations using the Sciamble Electric Drives Lab Kit (link) and the Workbench Real-Time Control Platform (link), we analyze the key characteristics of DC, Induction, and Permanent Magnet AC (PMAC) motors.

Description: In this lecture, we systematically derive the expressions for angular speed, torque, and inertia of a rotor. We then explore the torque-speed characteristics for various loads, including flywheels, fans, and elevators. Through real-world examples, we develop an intuitive understanding of how different loads respond to torque and angular velocity variations. This session provides essential insights into load dynamics, crucial for analyzing and controlling electric drives.

Description: This lecture continues the exploration of various rotational loads through examples. Torque-speed and power vs. speed characteristics for common loads such as hoists, fans, rollers, and drills are examined. The transient speed response under a change in applied torque is analyzed using Laplace transforms. Parallels between mechanical and electrical systems are drawn, leading to the derivation of an electrical equivalent circuit for a given mechanical rotational system. A key missing element in the torque model—the torsional stiffness coefficient—is introduced, highlighting its significance in systems with large rotor lengths and high differential torques. The discussion concludes with a case study on rotor speed oscillations that lead to the 1970s Navajo coal power plant failure, illustrating the real-world impact of torsional dynamics.

Description: This lecture explores the operation of a power electronic converter used to generate variable voltage for motor control from a fixed DC voltage. The discussion begins with a simple mechanical switch circuit, systematically addressing its limitations and refining the design to arrive at the synchronous buck converter. A live demonstration with a blinking lamp provides an intuitive understanding of how the converter operates in practice. The lecture then introduces the Fourier series, the mathematical framework for decomposing repeating signals into discrete sine and cosine components, laying the foundation for further analysis of power converters.

Description: This lecture continues the exploration of power converters, beginning with a concise review of the transition from a simple mechanical switch to a synchronous buck converter topology. The Fourier coefficients of a repeating switched voltage signal are determined, illustrating how this decomposition enables the application of linear circuit analysis techniques to switched circuits. A brief historical perspective on Fourier's discovery of the Fourier series provides context for its significance in engineering analysis. The lecture then extends the synchronous buck converter, which generates only unipolar output voltage, by introducing an additional leg to form an H-bridge converter, allowing for bipolar voltage generation. The principles of H-bridge operation and the resulting output waveforms are analyzed in detail.

Description: This lecture introduces electromagnetic theory, the final fundamental component necessary for understanding motor operation. The discussion begins with Maxwell’s equations, which form the foundation of electromagnetism. Using Gauss' theorem and Stokes' theorem, along with reasonable simplifying assumptions, these equations are systematically reduced into more practical forms that will be applied throughout the course. To develop an intuitive understanding, real-world parallels are drawn, illustrating the physical significance of these equations in electrical and mechanical systems. This foundational knowledge is essential for analyzing the electromagnetic principles governing motor operation.

Description: This lecture continues the introduction to electromagnetic theory, beginning with a review of the reduced form of Maxwell's equations from the previous lecture. The discussion then progresses to the derivation of Faraday’s law, starting from the Lorentz force equation and Maxwell’s second equation. Using the simplified form of Maxwell’s equations, the lecture concludes with the derivation of the magnetic field of a solenoid.

Description: This lecture continues the introduction to electromagnetic theory, beginning with a review of the reduced form of Maxwell’s equations and the operation of a solenoid from the previous discussion. The analysis then extends to deriving key parameters for an air-core solenoid, such as back-EMF, inductance, energy stored, and energy density. Historical theories regarding the attraction of ferromagnetic materials to magnets are explored, followed by an examination of the behavior of an iron-core solenoid. The concept of magnetic field intensity (H) is introduced to separate material dependence from Ampère’s law. This leads to introducing the B vs. H curve, including key properties such as retentivity and coercivity. The discussion further investigates the origin of hysteresis loss, deriving its mathematical expression. Finally, the differences in the desired B-H curve characteristics for materials used in permanent magnets and those used in inductor/motor cores are examined in detail.

Description: This lecture builds on concepts from the previous discussions on electromagnetic theory to analyze magnetic circuits. The analysis begins with a C-core structure wound with a current-carrying conductor, computing the magnetic field in both the core and the airgap. Key definitions such as magnetomotive force (MMF) and reluctance are introduced, drawing parallels between electrical and magnetic circuits. The discussion is extended by replacing the wound conductor with a permanent magnet, solving for the magnetic field in the airgap. The concept of load-lines, in conjunction with the B-H curve, is introduced as a tool for determining the magnetic field of a magnet for a given reluctance. The lecture also explores the distinct characteristics of common magnet types, including Neodymium-Iron-Boron (NeFeB), Samarium-Cobalt (SmCo), and Alnico 5, highlighting differences in their behavior. Finally, strategies for minimizing magnet size—which directly impacts cost—are examined by optimizing the B-H product to achieve a desired airgap magnetic field.

Description: This lecture explores the structure and operation of a DC motor, beginning with an analysis of the torque on a current-carrying conductor loop in a magnetic field. Through this analysis, the necessity of a commutator is established, demonstrating how it enables a recurring rotational torque in the same direction when powered by DC current. The expressions for torque and back-EMF are derived based on the considered structure. The torque profile of a single loop is examined, revealing the presence of significant harmonics that contribute to vibration and noise. The improvement in torque smoothness achieved by increasing the number of coils and commutator segments is then explored. The discussion concludes with an examination of an actual DC motor, covering key components such as the stator frame, stator magnets, stacked rotor electrical steel plates, commutator, brushes, and winding distribution. The impact of using laminated plates for the rotor core is briefly discussed, highlighting its role in reducing eddy current losses.

Description: This lecture continues the discussion on DC motor construction, beginning with comparing the hypothetical air-core rotor analyzed in the previous lecture and a practical rotor wound on electrical steel (silicon steel/Si-Fe). The advantages of using a steel core are explored, including a higher magnetic field strength (due to a reduced airgap) and greater torque production (as all force acts tangentially to the rotor surface). However, this design also introduces core losses, namely, eddy current and hysteresis loss. The lecture then examines the armature winding diagram, using a simplex double-layer progressive lap winding as an example. The current paths through parallel winding paths are traced, providing insight into current distribution. The session concludes with a brief discussion on commutator action as the armature rotates, ensuring continuous torque generation in the same direction.

Description: This lecture consists of two parts. The first part provides a quick overview of the winding diagram for a DC motor armature, followed by an exploration of how current commutes as the rotor rotates. The discussion examines the effects of proper vs. improper commutation. This sets the stage for the following lecture, where methods to limit sparking due to improper commutation will be introduced. The second part serves as a midterm review, covering key topics discussed so far. The first topic is mechanical systems, where the torque vs. speed relationship is analyzed for different types of loads. The speed response is solved for various torque profiles, drawing parallels between mechanical torque equations and electrical network equations. A brief discussion on electromotive force follows, clarifying how emf differs from the terminal voltage of a motor. The process of determining EMF polarity as the armature coil rotates is explained, along with why an increase in terminal voltage leads to higher rotor speed, reinforcing the connection between electrical and mechanical systems. The review then moves on to the synchronous buck converter, beginning with an overview of how the converter operates and the reasons for using semiconductor switches instead of mechanical switches. The justification for replacing the bottom diode with a switch is discussed, followed by an examination of gate signal patterns for the two switches. The previous analysis of switched voltage across an RL load is now extended to include RC and RLC loads, broadening the understanding of converter behavior in different circuit configurations. The final section of the review covers electromagnetic theory, starting with the four fundamental equations necessary to describe motor operation. A step-by-step approach is taken to solving simple magnetic circuits, drawing direct parallels with electrical circuit analysis. The discussion then extends to more complex magnetic circuits, exploring methods for solving the B-field in circuits with both permanent magnets and electromagnets.

Description: This lecture examines armature reaction in DC motors and its effects on performance. The discussion begins with the delayed commutation caused by the shifting of the magnetic neutral axis, followed by an analysis of the demagnetizing effect of the cross-magnetizing armature reaction. The destabilization due to the demagnetizing effect of the direct-magnetizing armature reaction is also explored, particularly in cases where the brushes are intentionally or unintentionally shifted from the geometric neutral axis. To address these issues, several mitigation techniques are introduced, including shifting the brushes, the use of interpoles, and compensation windings. As a transition to the next lecture, the fundamental relationships between electromagnetic torque and current, as well as back-EMF and speed, are derived using concepts from electromagnetic theory. These equations provide the foundation for integrating electrical and mechanical system dynamics, paving the way for a deeper analysis of DC motor operation and control in the upcoming discussion.

Description: This lecture begins with a brief review of armature reaction in DC motors and the various compensation methods used to mitigate its adverse effects, including brush shifting, interpoles, and compensation windings. The focus then shifts to modeling the DC motor using numerical simulation. A complete electromechanical model is developed by combining the previously derived relationships for electromagnetic torque, current, back-EMF, and speed with the electrical dynamics of the armature circuit and the mechanical dynamics of the rotor-load system. This model forms the basis for analyzing and controlling motor behavior under various operating conditions.

Description: This lecture provides an overview of techniques for estimating DC motor parameters. The discussion then turns to the design of a closed-loop controller for maintaining constant motor speed under varying load conditions and changing motor parameters. Real-world control strategies are examined to build intuition, which is then supported by a mathematical analysis of system stability and performance. These concepts form the foundation for implementing robust speed control in practical motor drive applications.

Description: This lecture continues with examples to develop an understanding of the Nyquist stability criterion. Various transfer functions are analyzed to determine stability based on the encirclement of the -1 + j0 point in the F(s)-plane. For systems where the numerator order is less than the denominator, a connection is drawn between the Nyquist plot and the Bode plot, providing insight into frequency-domain interpretation of stability. For open-loop stable systems, the phase margin is introduced as a quantitative measure of system stability derived from the Bode plot. In the next lecture, these foundational concepts will be used to design a PI controller for the DC motor.

Description: This lecture begins with a review of the Nyquist stability criterion and the use of phase margin as a quantitative measure of system stability. These concepts provide the foundation for controller design in dynamic systems. The focus then shifts to the control of DC motor torque (current), speed, and position using a PI controller. The process of selecting PI controller gains is explored in detail, with an emphasis on achieving a desired phase margin and bandwidth to ensure both stability and responsive performance across different control objectives.

Description: In the first section, the DC motor controller design analysis is completed by examining the computation of steady-state error in the absence of an integrator in the PI controller. The effects of saturation due to real-world limits are discussed, showing how such constraints can lead to limit cycles in motor current and speed. The concept of anti-windup is introduced as a method to mitigate these effects. The second section provides a concise review of AC circuit analysis, serving as essential background for upcoming lectures on AC machines. The use of phasors for steady-state analysis is revisited, along with a discussion of real and reactive power. The lecture concludes with an explanation of why the three-phase system is the most widely adopted configuration in power generation, transmission, and motor applications.

Description: This lecture examines the structure of a sinusoidally distributed winding in the stator of an AC motor. The analysis begins with the magnetic field produced by a single-phase coil, demonstrating that it does not generate a net rotating field in the airgap. To address this limitation, the concept of using multiple coils shifted in space and supplied with currents shifted in time is introduced. This arrangement enables the creation of a rotating magnetic field, which is fundamental to the operation of AC machines. The lecture provides both physical insight and mathematical reasoning behind this critical phenomenon in AC motor operation.

Description: This lecture continues the analysis of the magnetic field distribution in the air gap of an AC motor caused by sinusoidally distributed stator windings. The focus shifts to polyphase machines, examining how the combined effect of multiple phases produces a rotating magnetic field. The concept of space vectors is introduced as a tool to track the instantaneous peak of the airgap field over time. This formulation is then extended to provide an intuitive graphical representation of three-phase quantities, such as currents and voltages, laying the groundwork for further analysis of AC motors.

Description: This lecture begins the derivation of key electromagnetic relationships for the Permanent Magnet AC (PMAC) motor, including expressions for inductance, electromagnetic torque, and back-EMF.  The next lecture will build on this foundation by analyzing the steady-state operation of the PMAC motor, using the derived expressions to examine performance under various operating conditions.

Description: This lecture is a continuation of the three-part series on deriving key parameters for Permanent Magnet AC (PMAC) motors. Building on the previous session, the analysis further develops the mathematical expressions for self- and mutual inductances, as well as the electromagnetic torque.

Description: This lecture begins with a review of the derivations for inductance, back-EMF, and electromagnetic torque in Permanent Magnet AC (PMAC) motors. To build a more intuitive understanding of motor operation, the analysis is then reframed using space vector representation, offering a graphical and conceptual view of the rotating magnetic field and its interaction with the rotor. The lecture concludes with a hardware demonstration, reinforcing the key concepts covered over the past three sessions through practical observation.

Description: This lecture covers two topics outside the core syllabus related to Permanent Magnet AC (PMAC) motors. The first part focuses on the relationship between power, load, and torque angle, and the second part introduces the modeling of the PMAC motor in the abc-domain.

Description: This lecture begins with a review of PMAC motor operation using space vector representation. The discussion then transitions to an introduction to induction motor operation, starting with an analysis of the motor's structure and the derivation of how sinusoidally distributed stator currents induce currents in the rotor bars. The lecture concludes with a review of transformer operation, laying the foundation for the development of the equivalent circuit of the induction motor in the following lecture.

Description: This lecture continues the exploration of induction motor operation by further developing the equivalent circuit model, drawing direct parallels with the behavior of a transformer. The model is used to represent the electrical interactions between the stator and rotor under steady-state conditions. The session concludes with a space vector analysis for the special case in which the rotor is stationary. In the following lecture, the analysis will be extended to derive the torque expression for a rotating rotor and to modify the equivalent circuit accordingly to reflect rotor motion.

Description: This lecture generalizes the transformer-based equivalent circuit model of the induction motor from the stationary to the rotating rotor case. Using fundamental circuit analysis techniques, the model is then systematically reduced to a simpler form. In this process, the behavior of rotor currents and voltages is examined in greater detail, along with the interactions among stator and rotor space vectors. Using the finalized equivalent circuit, the torque-speed characteristics of the induction motor are derived, providing key insight into motor performance under varying load conditions. The lecture concludes with a hardware demonstration that highlights the robustness and reliability of induction motors, underscoring their reputation as the workhorses of industry.

Description: The first half of this lecture focuses on the motor's behavior in generation and regenerative braking modes, analyzing the regions of stable and unstable operation based on the value of slip. The second half of the lecture examines the starting mechanism of a single-phase induction motor and how, after startup, it generates a continuous, non-zero torque during normal operation.