TEL305 — Electrical Machines: Lecture Note

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Course code: TEL 305 Course title: Electrical Machines Course credit load: 3 Units Series 1: Introduction to Machinery Principles and DC Machinery Fundamentals • Part 1.1: Foundations of Machinery Principles o Sub Part 1.1.1: Rotational Motion o Sub Part 1.1.2: Newton's Law and Power Relationships o Sub Part 1.1.3: The Magnetic Field • Part 1.2: Magnetic Circuits and Electromagnetic Induction o Sub Part 1.2.1: Magnetic Circuits with an Air Gap o Sub Part 1.2.2: Faraday's Law o Sub Part 1.2.3: Production of Induced Force on a Current-Carrying Wire • Part 1.3: Induced Voltage and the Linear DC Machine o Sub Part 1.3.1: Induced Voltage on a Conductor Moving in a Magnetic Field o Sub Part 1.3.2: The Linear DC Machine o Sub Part 1.3.3: The Simple Rotating Loop Between Curved Pole Faces • Part 1.4: Commutation and DC Machine Construction o Sub Part 1.4.1: Commutation o Sub Part 1.4.2: Construction of a DC Machine o Sub Part 1.4.3: Simple Armature Winding • Part 1.5: Armature Reaction and Real Machine Equations o Sub Part 1.5.1: Armature Reaction o Sub Part 1.5.2: Interpoles, Compensating Winding, and Brush Shifting o Sub Part 1.5.3: Internal Generated Voltage and Induced Torque Equations

Introduction This is the opening Series of TEL 305: Electrical Machines. This Series lays the foundation for the entire course, introducing the mechanical and magnetic principles common to every electrical machine, before turning to the fundamentals of the DC machine, the simplest and most instructive machine with which to build your understanding of electromechanical energy conversion. The aim of this Series is to equip you with a sound understanding of rotational motion, Newton's law, and the magnetic field; magnetic circuits with an air gap and Faraday's law; induced voltage, induced force, and the linear DC machine; commutation and the construction of a DC machine; and armature reaction and the equations governing a real DC machine's internal generated voltage and induced torque. We shall commence this Series by examining the foundations of machinery principles: rotational motion, Newton's law, and the magnetic field. Next, we will consider magnetic circuits with an air gap, Faraday's law, and the production of induced force. Following that, our attention turns to induced voltage, the linear DC machine, and the simple rotating loop. We will then examine commutation and the construction of a DC machine. Finally, we will conclude by examining armature reaction and the equations of a real DC machine. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1: explain the fundamental concepts of rotational motion, Newton's law, and the magnetic field underlying machinery principles, • SLO 1.2: analyse a magnetic circuit with an air gap and apply Faraday's law to determine the induced force on a current-carrying wire, • SLO 1.3: derive the induced voltage on a moving conductor and describe the linear DC machine and the simple rotating loop, • SLO 1.4: describe commutation and the construction of a DC machine, including simple armature winding, • SLO 1.5: explain armature reaction and interpoles, and derive the internal generated voltage and induced torque equations of a real DC machine.

1.1 Foundations of Machinery Principles 1.1.1 Rotational Motion Rotational motion describes the motion of every rotating electrical machine, and is characterised by the angular position, θ, measured in radians, the angular velocity, ω = dθ/dt, measured in radians per second, and the angular acceleration, α = dω/dt, the rotational analogues of linear position, velocity, and acceleration. Angular velocity is related to the more familiar rotational speed, n, measured in revolutions per minute, by ω = 2πn/60, a conversion used constantly throughout this course, since machine nameplates and datasheets are conventionally rated in revolutions per minute, while the torque and power equations developed in this Part require angular velocity in radians per second. Worked example: A motor's shaft rotates at n = 1500 rpm. Its angular velocity is ω = 2π(1500)/60 = 157.1 rad/s. Application: Correctly converting between revolutions per minute and radians per second is an essential, recurring calculation throughout this course, since a small error in this conversion propagates directly into every subsequent torque and power calculation for a rotating machine. Fill in the blank: Angular velocity is defined as ω = d________/dt. The conversion from rpm to rad/s is ω = 2πn/________. 1.1.2 Newton's Law and Power Relationships Newton's law for rotational motion states that the net torque applied to a rotating body equals the product of its moment of inertia, J, and its resulting angular acceleration, τnet = Jα, directly analogous to the familiar linear form, F = ma, with torque replacing force, moment of inertia replacing mass, and angular acceleration replacing linear acceleration. The mechanical power associated with a rotating shaft is the product of the applied torque and the resulting angular velocity, P = τω, a relationship that directly parallels the electrical power relationship P = vi already familiar from circuit theory, and that is used throughout this course to relate a machine's mechanical output to its electrical input, or vice versa. Worked example: A motor develops a torque of 10 N·m at a shaft speed of ω = 150 rad/s. The mechanical power delivered is P = τω = 10 × 150 = 1500 W. Application: The power relationship P = τω is the essential bridge between a machine's mechanical behaviour, torque and speed, and its electrical behaviour, voltage and current, and underlies every efficiency and rating calculation performed throughout the remainder of this course. Fill in the blank: Newton's law for rotational motion is τnet = Jα, where J is the ________ of inertia. Mechanical power is given by P = τ × ________.

1.1.3 The Magnetic Field The magnetic field is the fundamental medium through which every electrical machine converts energy between its electrical and mechanical forms, and is produced whenever electric current flows through a conductor, most usefully concentrated and shaped, for machine purposes, by winding that conductor into a coil around a core of ferromagnetic material. Ampere's law relates the magnetic field intensity, H, circulating around a closed path to the current enclosed by that path, ∑Hl = Inet, forming the basis for analysing the magnetic circuits examined in the next Part of this Series, exactly as Kirchhoff's voltage law relates voltage around a closed electrical loop to the sources within it. Worked example: A coil of 100 turns carries a current of 2 A around a magnetic core of mean path length 0.5 m. Applying Ampere's law, the magnetic field intensity is H = NI/l = (100 × 2)/0.5 = 400 A·t/m. Application: Understanding the magnetic field as the shared medium linking a machine's electrical windings to its mechanical rotor is the single most important conceptual foundation for the entire study of electrical machines, since every machine examined throughout this course, however different its construction, converts energy by this same fundamental mechanism. Fill in the blank: A magnetic field is produced whenever electric ________ flows through a conductor. Ampere's law states that ∑Hl = ________. Figure 1.1: Rotational motion of a shaft, and the magnetic field around a current-carrying conductor.

Pilot questions 1.1 1. State the relationship between angular velocity and rotational speed in rpm. (Linked to SLO 1.1) 2. State Newton's law for rotational motion. (Linked to SLO 1.1) 3. State the formula for mechanical power in terms of torque and angular velocity. (Linked to SLO 1.1) 4. State Ampere's law. (Linked to SLO 1.1) 5. Explain how a magnetic field is produced by a current-carrying conductor. (Linked to SLO 1.1) 1.2 Magnetic Circuits and Electromagnetic Induction 1.2.1 Magnetic Circuits with an Air Gap A magnetic circuit is a closed path, usually formed from a high-permeability ferromagnetic core, along which magnetic flux, Φ, is guided, exactly as a conducting wire guides electric current around an electrical circuit, and every practical electrical machine's magnetic circuit necessarily includes one or more small air gaps, the narrow physical clearance separating a machine's stationary and rotating parts. The reluctance, R, of a section of a magnetic circuit, R = l/(μA), is the magnetic analogue of electrical resistance, and, because air's permeability μ is far lower than that of a ferromagnetic core, even a very short air gap typically contributes the great majority of a machine's total magnetic circuit reluctance. Worked example: A magnetic circuit has a core reluctance of Rcore = 10,000 A·t/Wb and an air gap reluctance of Rgap = 200,000 A·t/Wb. The air gap, though physically very short, accounts for the vast majority of the total series reluctance, Rtotal = 10,000 + 200,000 = 210,000 A·t/Wb. Application: Because the air gap dominates a machine's total magnetic circuit reluctance, machine designers work hard to keep the air gap as short as mechanically practical, since even a small increase in air gap length significantly increases the magnetomotive force, and hence the exciting current, required to establish a given amount of useful magnetic flux. Fill in the blank: Reluctance is the magnetic analogue of electrical ________. An air gap typically contributes the majority of a machine's

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