Sources and References

These notes are synthesized from standard graduate- and advanced-undergraduate references in power electronics and electric machines, and from widely used public university materials.

• Ned Mohan, Tore M. Undeland, William P. Robbins, Power Electronics: Converters, Applications, and Design, 3rd ed., Wiley. The principal text for semiconductor switches, PWM, rectifiers, and DC/DC converters.
• Robert W. Erickson and Dragan Maksimovic, Fundamentals of Power Electronics, 3rd ed., Springer. Authoritative coverage of averaged modeling, volt-second balance, and charge balance for converters.
• A. E. Fitzgerald, Charles Kingsley Jr., Stephen D. Umans, Electric Machinery, 7th ed., McGraw-Hill. Standard reference on magnetic circuits, transformers, and DC machines.
• Philip T. Krein, Elements of Power Electronics, 2nd ed., Oxford. Device-oriented introduction with strong emphasis on switching models.
• Austin Hughes and Bill Drury, Electric Motors and Drives: Fundamentals, Types and Applications, 5th ed., Newnes. Accessible coverage of DC motor drives and speed control.
• Chee-Mun Ong, Dynamic Simulation of Electric Machinery, Prentice Hall. Useful for dynamic models of transformers and DC machines.
• Ned Mohan, Electric Machines and Drives: A First Course, Wiley. A compact treatment pitched at the same level as this course.
• MIT OpenCourseWare, 6.334 Power Electronics, and 6.131 Power Electronics Laboratory, for converter analysis and laboratory practice.
• Stanford EE 292 series and Cambridge Part IIA Electrical and Information Sciences (Paper 3B2) Power Electronics and Drives notes, for transformer and drive system coverage.

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Chapter 1: Review of AC Circuit Concepts for Energy Conversion

Power electronics and electric machines operate almost exclusively with time-varying voltages and currents. Before introducing any converter or machine, we must master the tools used to describe these signals concisely: phasors, average and RMS values, complex power, and power factor correction.

Single-Phase Sinusoidal Quantities

A single-phase AC source is modeled as \( v(t) = V_m \cos(\omega t + \theta_v) \), where \( V_m \) is the peak amplitude, \( \omega = 2\pi f \) the angular frequency, and \( \theta_v \) the phase. For the North American 60 Hz grid, \( \omega \approx 377 \) rad/s. The current drawn by a linear load has the same frequency and a possibly different phase: \( i(t) = I_m \cos(\omega t + \theta_i) \).

Two scalar descriptors are fundamental. The average value of a periodic signal over one period \( T \) is

\[ V_{\text{avg}} = \frac{1}{T} \int_0^T v(t)\, dt, \]

which vanishes for a symmetric sinusoid. The root-mean-square (RMS) value is

\[ V_{\text{rms}} = \sqrt{\frac{1}{T} \int_0^T v^2(t)\, dt} = \frac{V_m}{\sqrt{2}}. \]

RMS is the voltage that, applied to a resistor, would dissipate the same average power as a DC source of that magnitude. That is why nameplate ratings on transformers, motors, and outlets are always RMS.

Phasor Notation

For a fixed frequency \( \omega \), sinusoids are in one-to-one correspondence with complex numbers through the transformation \( v(t) = \text{Re}\left[\sqrt{2}\, \tilde{V} e^{j\omega t}\right] \), where \( \tilde{V} = V_{\text{rms}} e^{j\theta_v} \) is the phasor. Derivatives become multiplications by \( j\omega \), so inductors and capacitors obey Ohm-like laws with complex impedances \( Z_L = j\omega L \) and \( Z_C = 1/(j\omega C) \). Kirchhoff’s laws apply unchanged to phasors, reducing the analysis of steady-state AC circuits to complex algebra.

Instantaneous, Active, Reactive, and Apparent Power

The instantaneous power delivered to a load is \( p(t) = v(t) i(t) \). Expanding with phasors and averaging over one period,

\[ P = V_{\text{rms}} I_{\text{rms}} \cos(\theta_v - \theta_i), \]

called the active or real power and measured in watts. The quantity \( Q = V_{\text{rms}} I_{\text{rms}} \sin(\theta_v - \theta_i) \) is the reactive power, measured in volt-amperes reactive (VAR). It represents energy that oscillates between source and load storage elements without being dissipated. Their magnitudes combine into the apparent power \( S = V_{\text{rms}} I_{\text{rms}} \) measured in volt-amperes (VA). These three obey

\[ S^2 = P^2 + Q^2, \qquad S = P + jQ = \tilde{V}\tilde{I}^*. \]

The power factor is \( \text{pf} = P/S = \cos(\theta_v - \theta_i) \), and it is leading or lagging depending on whether the current leads or lags the voltage.

Power Factor Correction

Utilities penalize loads with a poor power factor because the current required to deliver a given active power is inflated by \( 1/\text{pf} \), wasting capacity in the generation and distribution system. Industrial loads dominated by induction motors are typically inductive, with \( \text{pf} \approx 0.7\text{-}0.85 \) lagging. Correction is achieved by placing a capacitor bank in parallel with the load. To raise the power factor from \( \cos\theta_1 \) to \( \cos\theta_2 \), the capacitor must supply the reactive power

\[ Q_C = P\left(\tan\theta_1 - \tan\theta_2\right), \]

which determines the capacitance \( C = Q_C/(\omega V_{\text{rms}}^2) \).

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Chapter 2: Magnetic Circuits and Electromechanical Energy Conversion

Electric machines and transformers route magnetic flux through structured iron cores. Analyzing these structures with a magnetic circuit analogy, loosely parallel to electric circuits, is the quickest path to quantitative design.

Ampere’s Law and Magnetomotive Force

For a ferromagnetic core of mean length \( \ell \) and cross-sectional area \( A \), carrying an enclosed current \( Ni \) (a coil of \( N \) turns and current \( i \)), Ampere’s law gives the field intensity \( H\ell = Ni \). The product \( \mathcal{F} = Ni \) is the magnetomotive force (MMF). The flux density is \( B = \mu H \), where \( \mu = \mu_r \mu_0 \) with \( \mu_0 = 4\pi \times 10^{-7} \) H/m. The flux linking the core is \( \Phi = BA \).

Assuming flux is confined to the core, we obtain a scalar circuit relation

\[ \mathcal{F} = \Phi \mathcal{R}, \qquad \mathcal{R} = \frac{\ell}{\mu A}, \]

where \( \mathcal{R} \) is the reluctance of a section, analogous to electrical resistance. Reluctances in series add; in parallel, inverse reluctances (permeances) add. An air gap, with \( \mu = \mu_0 \), typically dominates total reluctance because \( \mu_r \) in silicon steel is 2000-6000 under normal operating flux densities.

Saturation, Hysteresis, and Core Losses

Real ferromagnetic materials are nonlinear: beyond a knee near 1.5-1.8 T, increasing \( H \) produces little additional \( B \) as magnetic domains become fully aligned. The \( B\text{-}H \) curve also traces a hysteresis loop under AC excitation; its enclosed area equals the energy dissipated per cycle per unit volume. Summed across the core, hysteresis loss scales as \( P_h = k_h f B_m^n \) with \( n \) roughly 1.6-2.2 (Steinmetz).

A second core loss mechanism, eddy currents, arises because the alternating flux induces voltages around closed paths in the conducting core. The resulting circulation produces Joule heating proportional to \( f^2 B_m^2 \) per unit volume, controlled by laminating the core perpendicular to the flux path. The sum of hysteresis and eddy losses is measured on a per-kilogram basis and listed on material datasheets as the core loss at a given frequency and flux density.

Faraday’s Law and Induced Voltages

A coil embracing a time-varying flux experiences an induced EMF

\[ e(t) = N \frac{d\Phi}{dt} = \frac{d\lambda}{dt}, \]

where \( \lambda = N\Phi \) is the flux linkage. This relation is the engine of every transformer, generator, and motor. For a sinusoidal flux \( \Phi(t) = \Phi_m \sin(\omega t) \) linking \( N \) turns, the RMS induced voltage is \( E = 4.44 f N \Phi_m \), a formula frequently used to size iron cores.

Coenergy, Force, and Torque

For a linear magnetic system with inductance \( L(x) \) that depends on a mechanical coordinate \( x \), the stored energy at current \( i \) is \( W_f = \tfrac{1}{2} L(x) i^2 \). The mechanical force produced on the movable member is

\[ F = \frac{\partial W_f'}{\partial x}\bigg|_{i}, \]

where \( W_f' \) is the magnetic coenergy, equal to \( W_f \) for linear systems. For rotational systems the same formula yields electromagnetic torque as \( T = \partial W_f'/\partial \theta \) at fixed current. This single expression unifies relays, solenoids, stepper motors, and rotating machines.

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Chapter 3: Single-Phase Transformers

A transformer is two or more coils magnetically coupled through a shared iron core. Its role is to trade voltage against current at nearly constant power, connect subsystems at different voltage levels, and provide galvanic isolation. It is the cleanest setting in which to exercise the magnetic circuit ideas of the previous chapter.

Ideal Transformer

Consider primary and secondary windings of \( N_1 \) and \( N_2 \) turns on a common core with negligible reluctance and zero losses. Applying Faraday’s law to each winding and Ampere’s law to the core yields

\[ \frac{v_1}{v_2} = \frac{N_1}{N_2} = a, \qquad \frac{i_1}{i_2} = \frac{N_2}{N_1} = \frac{1}{a}. \]

The ratio \( a \) is the turns ratio. Impedances on the secondary side reflect to the primary as \( Z_1' = a^2 Z_2 \). An ideal transformer thus serves as a lossless impedance scaler, preserving instantaneous power.

Equivalent Circuit of a Real Transformer

Actual transformers depart from the ideal through finite winding resistances, leakage flux that does not fully link the other winding, finite core permeability, and core losses. The standard approximate equivalent circuit, referred to the primary side, is a series impedance \( R_{eq1} + jX_{eq1} \) representing the sum of primary and reflected secondary resistances and leakage reactances, plus a shunt branch across the source containing a core-loss resistance \( R_c \) in parallel with a magnetizing reactance \( X_m \). Because the shunt branch draws only a few percent of rated current, it is often moved to the input terminals for hand calculations with little error.

Open-Circuit and Short-Circuit Tests

Two laboratory tests separate the parameters. In the open-circuit test, the secondary is left open and rated voltage is applied to the low-voltage side. The measured current is the magnetizing current, and the measured power equals the core loss at rated flux. Knowing \( V \), \( I_{oc} \), and \( P_{oc} \) directly gives \( R_c \) and \( X_m \). In the short-circuit test, the secondary is shorted and a small voltage is applied until rated current flows; the shunt branch is negligible at the reduced flux, so the measurement yields \( R_{eq} \) and \( X_{eq} \).

Voltage Regulation and Efficiency

Voltage regulation measures the change in secondary terminal voltage between no-load and full-load operation, expressed as

\[ \text{VR} = \frac{V_{2,\,\text{NL}} - V_{2,\,\text{FL}}}{V_{2,\,\text{FL}}} \times 100\%. \]

Good distribution transformers maintain regulation below roughly 2-3% at unity power factor. Efficiency is

\[ \eta = \frac{P_{\text{out}}}{P_{\text{out}} + P_{\text{cu}} + P_{\text{core}}}, \]

where copper losses are \( P_{\text{cu}} = I^2 R_{eq} \) and scale with the square of loading, while core losses are essentially constant. Maximum efficiency occurs when copper and core losses are equal, and transformers are deliberately designed so that this condition falls in the middle of their typical load range.

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Chapter 4: DC Machines

DC machines were the first practical electric motors and remain ubiquitous in servos, traction, and low-cost actuators because their torque and speed are directly proportional to easily controlled terminal quantities.

Construction and Back-EMF

A DC machine consists of a stator producing a stationary field flux \( \Phi_f \) (from either permanent magnets or a separately excited field winding) and a rotor carrying armature conductors connected to a segmented commutator. Brushes against the commutator inject current into whichever coils currently span the magnetic neutral axis, so the torque on the rotor remains unidirectional despite AC currents in individual coils.

The induced armature voltage, or back-EMF, is

\[ E_a = K \Phi_f \omega_m, \]

and the electromagnetic torque is

\[ T_e = K \Phi_f I_a, \]

where \( K \) is a construction constant and \( \omega_m \) the rotor mechanical speed. With a constant field, these become \( E_a = K_e \omega_m \) and \( T_e = K_t I_a \), with \( K_e = K_t \) in SI units — the same constant appears in both equations, a direct consequence of energy conservation.

Steady-State Circuit Model

Treating the armature as a resistance \( R_a \) in series with the back-EMF source, the terminal equation is

\[ V_t = E_a + I_a R_a = K_e \omega_m + I_a R_a. \]

Solving for speed at a given load torque \( T_L = K_t I_a \) gives the familiar torque-speed characteristic

\[ \omega_m = \frac{V_t}{K_e} - \frac{R_a}{K_e K_t} T_L. \]

The line has a no-load speed \( V_t/K_e \) and a stall torque \( K_t V_t/R_a \). Its slope — the speed droop per newton-metre — is set entirely by \( R_a \), so low-loss armatures give stiff speed regulation.

Types and Characteristics

DC machines are classified by how the field is connected. In separately excited and permanent-magnet machines the field is independent of armature current, producing the straight torque-speed line above. In shunt machines the field winding sits across the armature terminals, giving similar behavior at rated voltage. In series machines the field winding carries the armature current, so \( \Phi_f \propto I_a \) and \( T \propto I_a^2 \); speed rises sharply at light load but enormous starting torque is available, making them the historical choice for streetcar and locomotive traction. Compound machines combine shunt and series fields to tailor the curve.

Starting, Losses, and Speed Control

Directly connecting a stationary DC machine to rated voltage draws \( V_t/R_a \), often 10-20 times rated current, risking commutator damage. Classical starters insert external resistance that is progressively shorted out as the motor accelerates and back-EMF grows. Losses include armature copper loss \( I_a^2 R_a \), field copper loss, core loss in the rotating armature iron, brush-contact loss, and mechanical friction and windage. Efficiency at full load is typically 80-92%.

Three levers control speed. Armature voltage control varies \( V_t \), sweeping the torque-speed line vertically without changing its slope — the preferred method for speeds below base, because flux and hence torque capacity stay at rated values. Field weakening reduces \( \Phi_f \) above base speed, trading torque for speed at constant power. Armature resistance control inserts external \( R \), wasteful but simple. Modern DC drives implement armature voltage control electronically through a power converter, which is the subject of the next two chapters.

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Chapter 5: Power Electronic Converters

Power electronics replaces the rheostats, contactors, and rotating converters of an earlier era with solid-state switches that process energy at efficiencies routinely exceeding 95%. The discipline rests on a single observation: an ideal switch is either open (no current) or closed (no voltage), and so dissipates no power. Real switches approach this ideal closely enough that the analysis is dominated by topology and timing rather than device dissipation.

Power Semiconductor Devices

The devices used as switches in this course include:

• The power diode, a two-terminal uncontrolled switch that conducts in the forward direction and blocks in reverse.
• The thyristor (SCR), a four-layer device that latches on when a gate pulse is applied with forward voltage and turns off only when its current falls to zero; used in phase-controlled rectifiers and high-power DC transmission.
• The power BJT and MOSFET, three-terminal fully controllable switches driven by base current or gate voltage respectively. MOSFETs dominate below roughly 200 V because their on-resistance is low and they switch in tens of nanoseconds.
• The IGBT, which marries a MOSFET gate with a BJT-like output stage, owning the 600-1700 V range in motor drives and traction.

A switch has four figures of merit: voltage rating, current rating, switching speed, and on-state voltage drop. No single device wins on all four, so converter design is in part the art of choosing the right device for a given frequency and power.

PWM and the Duty Cycle

Pulse-width modulation (PWM) drives a switch with a rectangular waveform of fixed frequency \( f_s = 1/T_s \) and variable duty cycle \( D \), defined as the fraction of the period during which the switch is closed. The moving average of the switched signal is \( D \) times its peak, so PWM synthesizes an arbitrary low-frequency voltage from a fixed DC source simply by varying the duty cycle and letting an \( LC \) filter remove the switching harmonics.

The Buck Converter

A buck (step-down) converter places a controlled switch between a DC source \( V_{\text{in}} \) and an inductor-capacitor low-pass filter, with a freewheeling diode from ground to the inductor input. When the switch is on, the inductor voltage is \( V_{\text{in}} - V_o \) and its current ramps up. When the switch is off, the diode conducts, the inductor voltage is \( -V_o \), and its current ramps down. In steady state, the volt-seconds applied to the inductor must sum to zero over one period (inductor volt-second balance):

\[ (V_{\text{in}} - V_o) D T_s + (-V_o)(1-D) T_s = 0 \quad \Longrightarrow \quad V_o = D V_{\text{in}}. \]

The output is a DC voltage smaller than the input, continuously adjustable by the duty cycle. Neglecting losses, energy conservation forces \( I_o = I_{\text{in}}/D \), just as an ideal transformer would.

The Boost Converter

A boost converter swaps the positions of inductor and switch, placing the inductor directly across the source with the switch to ground. Closing the switch charges the inductor from the source; opening it forces the inductor current to continue through a diode into the output, adding the source voltage to the inductor’s EMF. Volt-second balance on the inductor gives

\[ V_o = \frac{V_{\text{in}}}{1 - D}, \]

so the output always exceeds the input and diverges as \( D \to 1 \) (limited in practice by losses).

The Buck-Boost and Beyond

A buck-boost converter reverses the diode and swaps the inductor so that the output polarity is inverted and its magnitude is \( D/(1-D) \) times the input. It can step up or step down. These three — buck, boost, buck-boost — are the canonical second-order DC/DC topologies and collectively illustrate the volt-second balance technique, which also governs Ćuk, SEPIC, and Zeta converters covered in textbook treatments.

Continuous vs Discontinuous Conduction

The analyses above assume the inductor current never reaches zero within a cycle, i.e. continuous conduction mode (CCM). At light load the current can ramp down to zero before the period ends, entering discontinuous conduction mode (DCM); the diode becomes reverse-biased, the inductor floats, and the conversion ratio starts depending on load. Converter designers size the inductor so that CCM is maintained at the minimum expected load, or deliberately exploit DCM for specific benefits.

Losses and Efficiency

Non-ideal switches dissipate power through on-state conduction loss and switching-transition loss. Conduction loss is \( I^2 R_{\text{on}} \) for a MOSFET or \( V_{\text{ce,sat}} I \) for an IGBT. Switching loss equals the energy dissipated during one on-off cycle times the switching frequency. Raising \( f_s \) shrinks the passive components but increases switching loss, which sets the upper limit on how fast a given device can be operated.

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Chapter 6: DC Motor Control Using Power Electronics

The final module of the course joins the previous two: it uses a power electronic converter to drive a DC machine over a wide range of speeds and torques.

Single-Quadrant Chopper Drive

The simplest motor drive is a buck converter feeding the armature of a permanent-magnet DC motor. By adjusting the duty cycle, the drive sets the armature voltage to

\[ V_a = D V_{\text{in}}, \]

and the machine settles to a speed \( \omega_m = (V_a - I_a R_a)/K_e \) determined by the load torque. Because both \( V_a \) and \( I_a \) are positive, the converter operates in the first quadrant of the torque-speed plane only: it drives the motor forward but cannot brake it.

Four-Quadrant H-Bridge Drive

A full H-bridge consists of four controlled switches arranged so that either polarity of armature voltage and either direction of armature current can be produced. Combining the four sign possibilities yields the four quadrants: forward motoring, forward braking (regeneration), reverse motoring, and reverse braking. Two PWM strategies are common. In bipolar PWM, diagonal pairs of switches are driven by complementary pulses, producing an armature voltage that toggles between \( +V_{\text{in}} \) and \( -V_{\text{in}} \) with average \( (2D-1)V_{\text{in}} \). In unipolar PWM, the two legs are switched at different duty cycles so that the armature sees \( 0, +V_{\text{in}}, \) or \( -V_{\text{in}} \); the switching ripple doubles in effective frequency, halving the required inductance for a given current ripple.

Current and Speed Control Loops

Industrial DC drives use a cascaded controller: an inner loop regulates armature current (and hence torque) with a bandwidth of roughly a kilohertz, while an outer loop regulates speed with a bandwidth one decade lower. The inner loop limits current during startup and disturbance recovery, preventing both commutator damage and excessive torque on the mechanical load. Anti-windup must be included in the outer loop because the inner current reference saturates during large speed changes.

A simple small-signal model of the machine and load is a first-order system with time constant \( \tau_m = J R_a/(K_e K_t + B R_a) \), where \( J \) is rotor inertia and \( B \) friction. The electrical time constant \( \tau_e = L_a/R_a \) sets the achievable current-loop bandwidth.

Regenerative Braking

When the motor decelerates or is driven by its load, the machine operates as a generator: the back-EMF exceeds the applied \( V_a \), armature current reverses, and mechanical energy is converted back into electrical energy. With an H-bridge and a bus capable of absorbing energy — for example a battery in an electric vehicle — this power is regenerated and used to recharge the source. When the source cannot absorb power, a brake chopper dissipates the energy in a resistor. Regenerative braking is one of the defining economic advantages of electronic drives over purely mechanical braking, and it ties the whole sequence of topics together: magnetic energy conversion in the machine, power electronic switching in the converter, and closed-loop control in the drive.