These notes cover the full arc of ECE 260 at the University of Waterloo: from three-phase circuit fundamentals through magnetic circuit theory, transformers, DC machines, synchronous machines, and induction motors. The treatment is self-contained and derivation-focused, emphasizing the physical reasoning behind each model.

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Chapter 1: Review of AC Circuit Fundamentals

1.1 Phasor Representation and Complex Power

Alternating-current circuit analysis rests on the observation that sinusoidal steady-state responses are most economically described in the frequency domain. Given a voltage

\[ v(t) = V_m \cos(\omega t + \theta_v) \]

we associate the phasor \(\mathbf{V} = V_m \angle \theta_v\), or equivalently in polar-rectangular form,

\[ \mathbf{V} = V_m e^{j\theta_v} = V_m \cos\theta_v + j V_m \sin\theta_v. \]

The RMS (root-mean-square) phasor is more common in power engineering: \(\mathbf{V}_{\rm rms} = (V_m / \sqrt{2})\angle\theta_v\). Unless stated otherwise, phasor magnitudes in this course refer to RMS values.

For a network with voltage phasor \(\mathbf{V}\) and current phasor \(\mathbf{I}\), the complex power is

\[ S = \mathbf{V} \mathbf{I}^* = P + jQ, \]

where the asterisk denotes complex conjugation. The real part \(P = |\mathbf{V}||\mathbf{I}|\cos\phi\) is the active power (watts), and the imaginary part \(Q = |\mathbf{V}||\mathbf{I}|\sin\phi\) is the reactive power (volt-amperes reactive, VAR), where \(\phi = \theta_v - \theta_i\) is the power factor angle. The apparent power is \(|S| = |\mathbf{V}||\mathbf{I}|\) (volt-amperes, VA).

The significance of reactive power is energetic: it sloshes between the source and the reactive element each half-cycle without being consumed. However, it does burden transmission lines with current, causing resistive losses. Power factor correction — adding capacitor banks to cancel inductive reactive power — reduces these losses without changing active power delivered to the load.

The instantaneous power delivered to any one-port network is \(p(t) = v(t)i(t)\). For sinusoidal steady state with \(v(t) = V_m\cos(\omega t)\) and \(i(t) = I_m \cos(\omega t - \phi)\):

\[ p(t) = V_m I_m \cos(\omega t)\cos(\omega t - \phi) = \frac{V_m I_m}{2}\left[\cos\phi + \cos(2\omega t - \phi)\right]. \]

The time-average of the double-frequency term vanishes, leaving \(P = (V_m I_m/2)\cos\phi = V_{\rm rms} I_{\rm rms}\cos\phi\). The double-frequency oscillation has peak value \(|S|\) and represents the reactive cycling of energy. This decomposition makes precise why unity power factor is ideal from an energy perspective: all power delivered to the network is absorbed and none is returned.

1.2 Three-Phase Circuits

Most electrical power generation, transmission, and industrial motor drive is three-phase. Three-phase systems offer constant instantaneous power delivery (unlike pulsating single-phase power) and more efficient use of conductor material.

The defining property of a balanced source is that \(\mathbf{V}_a + \mathbf{V}_b + \mathbf{V}_c = 0\), which follows from the phasor identity \(1 + e^{-j2\pi/3} + e^{j2\pi/3} = 0\).

1.2.1 Wye and Delta Connections

Sources and loads are connected in either wye (Y) or delta (\(\Delta\)) configuration.

In a wye connection the three elements share a common neutral point. The line voltage (voltage between any two lines) relates to the phase voltage (voltage from line to neutral) by

\[ V_L = \sqrt{3}\, V_\phi, \qquad \angle V_L = \angle V_\phi + 30^\circ. \]

For example, \(\mathbf{V}_{ab} = \mathbf{V}_a - \mathbf{V}_b = \sqrt{3}V\angle 30^\circ\) when \(\mathbf{V}_a = V\angle 0^\circ\). The line current equals the phase current: \(I_L = I_\phi\).

In a delta connection there is no neutral. Phase voltage equals line voltage (\(V_\phi = V_L\)), but phase current relates to line current by

\[ I_L = \sqrt{3}\, I_\phi. \]

1.2.2 Balanced Circuit Analysis: Per-Phase Method

For a balanced Y-connected load with impedance \(Z_Y\) per phase, Kirchhoff’s voltage law around any loop involves only the phase voltage and one phase impedance. The neutral wire carries zero current (\(\mathbf{I}_a + \mathbf{I}_b + \mathbf{I}_c = 0\) by symmetry), so it can be removed without changing any voltage or current. This permits the per-phase analysis: solve a single-phase circuit with phase voltage \(\mathbf{V}_\phi\) and impedance \(Z_Y\), then multiply power by 3.

A \(\Delta\)-connected load with impedance \(Z_\Delta\) per phase is converted to an equivalent Y by:

\[ Z_Y = \frac{Z_\Delta}{3}. \]

This \(\Delta\)-Y (and Y-\(\Delta\)) transformation allows all analyses to be performed on an equivalent Y network without loss of generality.

1.2.3 Three-Phase Power

For a balanced three-phase system the total three-phase complex power is

\[ S_{3\phi} = 3 \mathbf{V}_\phi \mathbf{I}_\phi^* = 3 V_\phi I_\phi \angle \phi = P_{3\phi} + jQ_{3\phi}. \]

Expressing in terms of line quantities (which are directly measurable):

\[ P_{3\phi} = \sqrt{3}\, V_L I_L \cos\phi, \qquad Q_{3\phi} = \sqrt{3}\, V_L I_L \sin\phi. \]

A crucial advantage: the instantaneous three-phase power delivered to a balanced load is

\[ p_{3\phi}(t) = p_a(t) + p_b(t) + p_c(t) = 3 V_\phi I_\phi \cos\phi, \]

which is constant in time. This is why three-phase motors run smoothly without the pulsating torque characteristic of single-phase machines.

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Chapter 2: Magnetic Circuits

2.1 Fundamental Laws of Electromagnetism

The behaviour of electromagnetic devices — transformers, motors, generators — is governed by two foundational laws.

The magnetic flux density \(\mathbf{B}\) (Tesla) and field intensity \(\mathbf{H}\) are related by the constitutive relation of the medium:

\[ \mathbf{B} = \mu \mathbf{H} = \mu_r \mu_0 \mathbf{H}, \]

where \(\mu_0 = 4\pi \times 10^{-7}\) H/m is the permeability of free space and \(\mu_r\) is the relative permeability of the material. For ferromagnetic materials, \(\mu_r\) can range from several hundred to over ten thousand, though it is nonlinear and history-dependent (hysteresis).

2.2 Magnetic Circuit Model

A magnetic circuit is a structured path of ferromagnetic material (a core) in which we assume nearly all flux is confined. The analogy with electric circuits is precise:

Electric Circuit	Magnetic Circuit
EMF \(\mathcal{E}\) (V)	MMF \(\mathcal{F} = Ni\) (A-t)
Current \(I\) (A)	Flux \(\Phi\) (Wb)
Resistance \(R = \ell/(\sigma A)\) (\(\Omega\))	Reluctance \(\mathcal{R} = \ell/(\mu A)\) (A-t/Wb)
Conductance	Permeance \(\mathcal{P} = 1/\mathcal{R}\)

Applying Ampere’s law around a simple toroidal core of mean path length \(\ell_c\), cross-section \(A_c\), relative permeability \(\mu_r\), wound with \(N\) turns carrying current \(i\):

\[ H_c \ell_c = Ni \implies \Phi = B_c A_c = \mu_r \mu_0 H_c A_c = \frac{Ni}{\mathcal{R}_c}, \]

which is the magnetic circuit equivalent of Ohm’s law: \(\mathcal{F} = \Phi \mathcal{R}\).

2.2.1 Air Gaps

Many practical devices (motors, relays) have air gaps in the magnetic path. Because \(\mu_{\rm air} = \mu_0\), even a small air gap introduces a reluctance far larger than the core reluctance. For a gap of length \(\ell_g\) and area \(A_g\) (with fringing neglected):

\[ \mathcal{R}_g = \frac{\ell_g}{\mu_0 A_g}. \]

With both core and gap in series, the total MMF required is

\[ Ni = \Phi (\mathcal{R}_c + \mathcal{R}_g) = B A \left(\frac{\ell_c}{\mu_r \mu_0 A_c} + \frac{\ell_g}{\mu_0 A_g}\right). \]

For high-\(\mu_r\) materials, the air gap often dominates: \(\mathcal{R}_g \gg \mathcal{R}_c\) even when \(\ell_g \ll \ell_c\).

2.2.2 Inductance

The self-inductance of a coil on a magnetic circuit is

\[ L = \frac{\lambda}{i} = \frac{N\Phi}{i} = \frac{N^2}{\mathcal{R}}. \]

This shows that inductance is inversely proportional to total reluctance: an air gap lowers the inductance of an inductor coil. The energy stored in an inductor is

\[ W = \frac{1}{2}L i^2 = \frac{1}{2} \frac{N^2}{\mathcal{R}} i^2 = \frac{1}{2} \mathcal{R} \Phi^2 = \frac{B^2 V_{\rm core}}{2\mu}, \]

which can also be written as an energy density \(w = B^2/(2\mu)\) J/m\(^3\).

2.2.3 Multiple-Winding Magnetic Circuits

When multiple coils share a single core, the superposition principle applies: the net MMF around any closed path is the algebraic sum of all MMF contributions, with sign determined by the right-hand rule (or the dot convention on the winding). For a transformer core with primary MMF \(N_1 i_1\) and secondary MMF \(N_2 i_2\) (both in the same sense for a practical transformer), the net magnetising MMF is:

\[ N_1 i_1 + N_2 i_2 = \Phi \mathcal{R}_c. \]

For an ideal transformer (\(\mathcal{R}_c \to 0\), \(\mu \to \infty\)), the left side must be zero: \(N_1 i_1 = -N_2 i_2\), which is the current transformation law (the negative sign reflecting that one winding’s MMF opposes the other’s to maintain zero net MMF in the ideal core).

2.3 Hysteresis and Core Losses

Real ferromagnetic materials are not linear: the \(B\)-\(H\) relationship exhibits hysteresis. When an AC excitation drives the core around a hysteresis loop, energy is dissipated each cycle equal to the area enclosed by the \(B\)-\(H\) loop:

\[ W_{\rm hys} = \oint H\, dB \cdot V_{\rm core} \quad \text{(J per cycle)}. \]

Hysteresis power loss scales as

\[ P_{\rm hys} = k_h f B_m^n, \]

where \(k_h\) is a material constant, \(f\) is frequency, \(B_m\) is peak flux density, and the Steinmetz exponent \(n \approx 1.6\text{–}2.0\) for most silicon steels.

Eddy currents arise because Faraday’s law applies to any closed conducting path in the core itself: a time-varying flux induces circulating currents in the conducting material, dissipating power as \(I^2 R\). Eddy current losses scale as

\[ P_{\rm eddy} = k_e f^2 B_m^2, \]

where \(k_e\) depends on material resistivity and lamination thickness. To reduce eddy currents, transformer and machine cores are built from thin laminations of silicon steel (0.3–0.6 mm thick), separated by insulating varnish. The square dependence on \(f\) makes eddy losses more severe in high-frequency applications.

Total core loss (also called iron loss) is

\[ P_{\rm core} = P_{\rm hys} + P_{\rm eddy} = k_h f B_m^n + k_e f^2 B_m^2. \]

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Chapter 3: Transformers

3.1 The Ideal Transformer

A transformer is two (or more) magnetically coupled coils wound on a common core. In the ideal model, we assume: (1) infinite core permeability so the magnetizing current is zero, (2) zero winding resistance, (3) no leakage flux — all flux produced by one winding links the other.

Let the primary have \(N_1\) turns and the secondary \(N_2\) turns. With all flux \(\Phi\) common to both windings:

\[ e_1 = N_1 \frac{d\Phi}{dt}, \qquad e_2 = N_2 \frac{d\Phi}{dt}. \]

Dividing:

\[ \frac{V_1}{V_2} = \frac{N_1}{N_2} = a, \]

where \(a\) is the turns ratio. By conservation of power (\(v_1 i_1 = v_2 i_2\) for an ideal lossless transformer):

\[ \frac{I_1}{I_2} = \frac{N_2}{N_1} = \frac{1}{a}. \]

An impedance \(Z_L\) connected across the secondary appears as \(Z_{\rm ref} = a^2 Z_L\) when referred to the primary. This impedance transformation is the transformer’s key utility in power transmission and electronics matching.

3.2 Practical Transformer: Equivalent Circuit

A real transformer differs from the ideal in four respects:

1. Primary resistance \(R_1\): resistive voltage drop in the primary winding.
2. Primary leakage reactance \(X_1 = \omega L_{l1}\): due to flux that links only the primary.
3. Core loss resistance \(R_c\): accounts for hysteresis and eddy losses.
4. Magnetizing reactance \(X_m = \omega L_m\): the inductive component of the exciting current needed to establish the core flux.
5. Secondary leakage reactance \(X_2\) and secondary resistance \(R_2\): analogous quantities on the secondary side.

The equivalent circuit referred to the primary has the following series impedance for the primary: \(R_1 + jX_1\), followed by the shunt branch \(R_c \| jX_m\) connected to the primary-voltage node, and then the secondary impedances referred to the primary: \(R_2' = a^2 R_2\), \(X_2' = a^2 X_2\), driving the referred load \(Z_L' = a^2 Z_L\).

The approximate equivalent circuit (for most power-transformer calculations) moves the shunt branch entirely to the primary terminal, giving a simple series circuit:

\[ \mathbf{V}_1 \approx \mathbf{E}_1, \quad \mathbf{V}_2' = \mathbf{V}_1 - \mathbf{I}_2'(R_{eq} + jX_{eq}), \]

where \(R_{eq} = R_1 + a^2 R_2\) and \(X_{eq} = X_1 + a^2 X_2\) are the total equivalent series resistance and reactance referred to the primary.

3.2.1 Derivation of the Equivalent Circuit

The derivation of the transformer equivalent circuit follows directly from Faraday’s law and the KVL applied to each winding. For the primary winding:

\[ v_1 = R_1 i_1 + \frac{d\lambda_1}{dt} = R_1 i_1 + L_{11}\frac{di_1}{dt} + M\frac{di_2}{dt}, \]

where \(L_{11} = L_{l1} + L_m\) is the total primary self-inductance (\(L_{l1}\) = leakage, \(L_m\) = mutual) and \(M\) is the mutual inductance. In the phasor domain at frequency \(\omega\):

\[ \mathbf{V}_1 = R_1 \mathbf{I}_1 + j\omega L_{l1} \mathbf{I}_1 + j\omega L_m (\mathbf{I}_1 + \mathbf{I}_2/a) \cdot a, \]

where the magnetising branch current is \(\mathbf{I}_m = \mathbf{I}_1 - (-\mathbf{I}_2/a)\). Identifying \(jX_1 = j\omega L_{l1}\) and \(jX_m = j\omega L_m\), and separating core loss, gives the familiar T-circuit equivalent with the shunt branch. This rigorous derivation confirms why the shunt branch appears between the two series impedance halves (the exact circuit) rather than at the primary terminal (the approximate circuit).

3.3 Open-Circuit and Short-Circuit Tests

The four parameters \(R_c\), \(X_m\), \(R_{eq}\), \(X_{eq}\) are determined experimentally.

3.4 Voltage Regulation

Voltage regulation (VR) measures how much the secondary terminal voltage changes between no-load and full-load:

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

Using the approximate equivalent circuit, the no-load secondary voltage equals the primary voltage referred: \(V_{2,\rm NL} = V_1 / a\). The full-load secondary voltage drops due to impedance:

\[ \frac{V_1}{a} = \frac{V_2}{a} + I_2(R_{eq,s} + jX_{eq,s}), \]

where \(R_{eq,s} = R_{eq}/a^2\) and \(X_{eq,s} = X_{eq}/a^2\) are secondary-referred equivalent impedances. An approximate formula for VR at power factor angle \(\phi\) is:

\[ \text{VR} \approx \epsilon_R \cos\phi + \epsilon_X \sin\phi, \]

where \(\epsilon_R = R_{eq} I_{\rm rated} / V_{\rm rated}\) (per-unit resistance drop) and \(\epsilon_X = X_{eq} I_{\rm rated} / V_{\rm rated}\) (per-unit reactance drop). Lagging power factor increases VR (secondary voltage droops); leading power factor can yield negative VR (voltage rises under load).

3.5 Transformer Efficiency

Transformer efficiency at output power \(P_{\rm out} = V_2 I_2 \cos\phi_2\) is:

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

where \(P_{\rm core} = V_1^2 / R_c\) (essentially constant if the primary voltage is held constant) and \(P_{\rm cu} = I_1^2 R_{eq}\) (varies as the square of load current). Maximum efficiency occurs when copper loss equals core loss: \(P_{\rm cu} = P_{\rm core}\), which determines the optimal load fraction.

3.5.1 Condition for Maximum Efficiency

Since \(P_{\rm core}\) is constant and \(P_{\rm cu} = (S_{\rm out}/S_{\rm rated})^2 P_{\rm cu,FL}\) varies quadratically with load fraction \(x = S_{\rm out}/S_{\rm rated}\), we maximise efficiency by minimising total loss for a given output power:

\[ \frac{d}{dx}\left(\frac{P_{\rm core} + x^2 P_{\rm cu,FL}}{x}\right) = 0 \implies -\frac{P_{\rm core}}{x^2} + P_{\rm cu,FL} = 0 \implies x^* = \sqrt{\frac{P_{\rm core}}{P_{\rm cu,FL}}}. \]

At the optimal load fraction \(x^*\), the copper loss equals the core loss. For the transformer in the previous example: \(x^* = \sqrt{1.1/1.6} = 0.829\), so maximum efficiency occurs at 82.9% of full load (unity pf): \(\eta_{\rm max} = 0.829\times 100/(0.829\times 100 + 1.1 + 1.1) = 82.9/85.1 = 97.4\%\) — the same as full-load (by coincidence of numbers in that example, since \(x^*\) happened to be close to 1 there).

Practical transformers in distribution systems are typically designed for maximum efficiency at 50–70% of rated load, matching the typical daily average loading profile.

3.6 The Per-Unit System

Large power systems contain transformers at many voltage levels. Repeated referral of impedances becomes tedious and error-prone. The per-unit (pu) system resolves this by normalising all quantities to a chosen base.

For a single-phase system, choose base apparent power \(S_{\rm base}\) (VA) and base voltage \(V_{\rm base}\) (V). The remaining base quantities follow:

\[ I_{\rm base} = \frac{S_{\rm base}}{V_{\rm base}}, \qquad Z_{\rm base} = \frac{V_{\rm base}^2}{S_{\rm base}}. \]

The per-unit value of any quantity is its actual value divided by the appropriate base:

\[ V_{\rm pu} = \frac{V}{V_{\rm base}}, \quad I_{\rm pu} = \frac{I}{I_{\rm base}}, \quad Z_{\rm pu} = \frac{Z}{Z_{\rm base}}. \]

The elegant result for transformers is that the per-unit turns ratio is unity — the ideal transformer disappears from the equivalent circuit. The per-unit equivalent impedance of a transformer is the same whether referred to primary or secondary (provided consistent base voltage is used on each side). Transformer nameplate data expressed in per-unit is independent of the actual voltage level, making system-level calculations straightforward.

For three-phase systems, choose \(S_{\rm base,3\phi}\) and \(V_{\rm base,LL}\) (line-to-line). Then:

\[ I_{\rm base} = \frac{S_{\rm base,3\phi}}{\sqrt{3}\, V_{\rm base,LL}}, \qquad Z_{\rm base} = \frac{V_{\rm base,LL}^2}{S_{\rm base,3\phi}}. \]

3.7 Three-Phase Transformer Connections

Three-phase transformers are formed either from three single-phase units or from a single three-limb/five-limb core. The primary and secondary can each be connected in wye (Y) or delta (\(\Delta\)), yielding four common configurations: Y-Y, Y-\(\Delta\), \(\Delta\)-Y, \(\Delta\)-\(\Delta\).

The Y-\(\Delta\) and \(\Delta\)-Y connections introduce a 30\(^\circ\) phase shift between primary and secondary line voltages. This is why transformers used in power grids are assigned a clock-hour designation (e.g., Dy1 means delta primary, wye secondary, 30\(^\circ\) lag). The \(\Delta\) winding allows circulating third-harmonic currents within the delta loop, preventing third-harmonic voltages from appearing on the line — a practical advantage for reducing distortion.

The Y-Y connection is simpler but requires that the neutral be properly connected and grounded to suppress triplen harmonics and to maintain voltage symmetry under unbalanced loads.

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Chapter 4: Energy and Co-Energy in Magnetic Systems

The per-unit system is especially powerful when comparing machine parameters across different ratings. For example, a synchronous machine’s per-unit synchronous reactance \(X_S^{\rm pu}\) is directly comparable to that of any other synchronous machine regardless of MW rating or voltage level, because both are normalised to their own machine base. This normalisation reveals the intrinsic electromagnetic design, independent of physical scale.

Changing base: if a component is specified in per-unit on base \(S_{\rm base,old}\), \(V_{\rm base,old}\), and we wish to convert to base \(S_{\rm base,new}\), \(V_{\rm base,new}\):

\[ Z_{\rm pu,new} = Z_{\rm pu,old} \cdot \frac{S_{\rm base,new}}{S_{\rm base,old}} \cdot \left(\frac{V_{\rm base,old}}{V_{\rm base,new}}\right)^2. \]

This change-of-base formula is indispensable in system-level analysis where a common system base (e.g., 100 MVA, 230 kV) is chosen and all component data must be converted accordingly.

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Chapter 4: Energy and Co-Energy in Magnetic Systems

4.1 Energy Stored in the Magnetic Field

For a singly-excited magnetic system (one coil, one mechanical degree of freedom \(x\)), the energy input from the electrical source in time \(dt\) is

\[ dW_e = e\, i\, dt = i\, d\lambda. \]

This energy is partitioned between the magnetic field energy and the mechanical work:

\[ dW_e = dW_f + dW_m, \]

where \(W_f\) is the field energy and \(dW_m = F\, dx\) is mechanical work. The field energy stored in terms of flux linkage and current is:

\[ W_f(\lambda, x) = \int_0^\lambda i(\lambda', x)\, d\lambda'. \]

For a linear system (constant permeability), \(\lambda = L(x) i\), so \(i = \lambda / L(x)\) and

\[ W_f = \frac{\lambda^2}{2L(x)} = \frac{1}{2}L(x)i^2. \]

The energy balance equation \(dW_e = dW_f + dW_m\) is the magnetic analogue of the first law of thermodynamics. In a lossless conservative system, \(W_f\) is a state function of \((\lambda, x)\) — it depends only on the current state, not the path taken. This means the mechanical work done moving from position \(x_1\) to \(x_2\) at constant flux linkage \(\lambda\) equals the decrease in field energy:

\[ W_m = W_f(\lambda, x_1) - W_f(\lambda, x_2) \quad \text{(constant } \lambda \text{)}. \]

The mechanical force at constant flux linkage is therefore:

\[ F\big|_{\lambda} = -\frac{\partial W_f(\lambda, x)}{\partial x}\bigg|_{\lambda}. \]

The negative sign: the system does mechanical work by decreasing field energy (at constant flux). For constant current operation, a different calculation applies — hence the need for co-energy.

4.2 Co-Energy and Force Calculation

The co-energy is the Legendre dual of the field energy:

\[ W_f'(i, x) = i\lambda - W_f = \int_0^i \lambda(i', x)\, di'. \]

For a linear system, \(W_f = W_f' = L(x)i^2/2\).

For a multiply-excited system (such as a machine with stator and rotor windings), the torque is:

\[ T = \left.\frac{\partial W_f'(i_1, i_2, \theta)}{\partial \theta}\right|_{i_1, i_2 = \rm const}, \]

where \(\theta\) is the angular position of the rotor. This formulation is the theoretical foundation for all rotating machine torque production.

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Chapter 5: DC Machines

5.1 Construction and Operating Principle

A DC machine consists of two principal parts: the stator (stationary part, also called the field structure) and the rotor (rotating part, also called the armature). The stator carries the field winding, which creates a stationary magnetic flux in the machine. The rotor carries the armature winding and is connected to an external circuit via a commutator — a mechanical rectifier formed by segmented copper bars and carbon brushes.

The commutator is the defining feature of DC machines. It ensures that, regardless of rotor position, the armature current in each conductor under a north pole always flows in the same direction, and likewise for conductors under south poles. This produces a unidirectional torque.

The generated (back) EMF of a DC machine is:

\[ E_A = K\Phi\omega_m = K\Phi n, \]

where \(\Phi\) is the flux per pole, \(\omega_m\) is the rotor angular velocity (rad/s), \(n\) is speed in appropriate units, and \(K\) is a machine constant depending on the number of poles, armature conductors, and parallel paths. More explicitly,

\[ K = \frac{N_c P}{2\pi a}, \]

where \(N_c\) is the total number of armature conductors, \(P\) is the number of poles, and \(a\) is the number of parallel paths.

The electromagnetic torque developed by the armature is:

\[ T_{\rm em} = K\Phi I_A, \]

where \(I_A\) is the armature current. The direction of \(T_{\rm em}\) relative to rotation determines motor versus generator action.

5.2 Equivalent Circuit and Voltage Equation

The armature circuit of a DC machine is simply the back-EMF source in series with the armature resistance \(R_A\):

\[ V_T = E_A + I_A R_A \quad \text{(motor, current flows into armature)}, \]\[ V_T = E_A - I_A R_A \quad \text{(generator, current flows out of armature)}. \]

Here \(V_T\) is the terminal voltage. For a motor, current is positive into the positive terminal, so \(I_A = (V_T - E_A)/R_A\). Since \(E_A = K\Phi\omega_m\), for motor operation \(E_A < V_T\) (back-EMF is less than supply), while for generator operation \(E_A > V_T\).

Power balance for a motor:

\[ P_{\rm in} = V_T I_A = E_A I_A + I_A^2 R_A = P_{\rm em} + P_{\rm cu,A}, \]

where \(P_{\rm em} = E_A I_A = T_{\rm em} \omega_m\) is the power converted from electrical to mechanical, and \(P_{\rm cu,A} = I_A^2 R_A\) is armature copper loss. Additional losses (friction, windage, stray load losses, field copper loss) are subtracted to obtain shaft output power.

5.3 Types of DC Machines

5.3.1 Separately Excited DC Motor

The field winding is supplied from an independent DC source. Field current \(I_F\) (and hence \(\Phi\)) is controlled independently of the armature. This gives maximum flexibility:

\[ \omega_m = \frac{V_T - I_A R_A}{K\Phi} = \frac{V_T}{K\Phi} - \frac{R_A}{(K\Phi)^2} T_{\rm em}. \]

Speed control methods:

• Armature voltage control (below base speed): varies \(V_T\) while holding \(\Phi\) at rated value. Provides constant-torque characteristic (armature current and hence torque capacity unchanged). Speed is proportional to \(V_T\).
• Field weakening (above base speed): reduces \(\Phi\) while \(V_T\) is at rated. Since \(E_A = K\Phi\omega_m\) must remain approximately \(\leq V_T\), reducing \(\Phi\) allows \(\omega_m\) to increase. However, torque capacity decreases since \(T_{\rm em} = K\Phi I_A\) and \(I_A\) is limited. This gives constant-power operation above base speed.

5.3.2 Shunt DC Motor

The field winding is connected in parallel (shunt) with the armature across the same supply \(V_T\). Field current \(I_F = V_T / R_F\) is essentially constant (assuming \(V_T\) is constant), so flux is nearly constant. The motor behaves similarly to a separately excited motor:

\[ \omega_m \approx \frac{V_T}{K\Phi} - \frac{R_A}{(K\Phi)^2} T_{\rm em}. \]

The speed-torque curve is nearly flat (slightly drooping due to the \(I_A R_A\) drop and the slight effect of armature reaction on flux). Shunt motors are used where nearly constant speed under varying load is desired: fans, pumps, machine tools.

5.3.3 Series DC Motor

The field winding carries the full armature current: \(I_F = I_A\). If the magnetic circuit is unsaturated, \(\Phi \propto I_A\), so

\[ T_{\rm em} = K\Phi I_A \propto I_A^2. \]

The speed-torque relation becomes:

\[ \omega_m = \frac{V_T}{\sqrt{K_{\rm series} T_{\rm em}}} - \frac{(R_A + R_S)}{K_{\rm series}}, \]

which is a hyperbolic curve: very high torque at low speed (since high \(I_A\) gives both high \(\Phi\) and high current), very high speed at light load. Warning: a series motor must never be operated unloaded, as speed can rise dangerously high (theoretically infinite for an ideal series motor). Series motors are used in traction (electric trains, trams), cranes, and starter motors — applications requiring very high starting torque.

5.3.4 DC Generator Operation

When the machine’s rotor is driven by a prime mover (turbine, engine) at speed \(\omega_m > E_A/K\Phi\) relative to the terminal voltage, armature current reverses direction and the machine delivers electrical power to the load. The terminal voltage of a separately excited generator is:

\[ V_T = E_A - I_A R_A - V_{\rm brush}, \]

where \(V_{\rm brush} \approx 2\) V accounts for the brush contact voltage drop. For a shunt generator, the field is self-excited from the output: the machine builds up voltage through a bootstrap process relying on residual magnetism in the core.

5.3.5 Speed-Torque Characteristic Summary

The natural (full-voltage, no external resistance) speed-torque curves of the main DC motor types differ markedly:

• Separately excited / shunt: Nearly flat. Speed drops only slightly from no-load to full-load (typically 5–10% regulation). Suitable for constant-speed drives.
• Series: Steep hyperbolic curve. Enormous starting torque, but speed rises dangerously at light load. Suitable for traction (speed automatically limited by mechanical load).
• Compound (both shunt and series field windings): Intermediate behaviour, combining moderate speed regulation with improved starting torque compared to pure shunt. Cumulative compound (fields aiding) is common in applications like conveyors.

The speed regulation of a DC motor is defined analogously to a transformer:

\[ \text{SR} = \frac{n_{\rm NL} - n_{\rm FL}}{n_{\rm FL}} \times 100\%. \]

A shunt motor might have SR of 5–10%; a series motor SR is effectively infinite (speed goes to infinity at zero load — hence the warning never to run one unloaded).

5.4 DC Motor Starting

At standstill (\(\omega_m = 0\)), there is no back-EMF, so the starting current would be \(I_{A,\rm start} = V_T / R_A\), which can be 10–25 times rated current. To limit starting current, a resistance \(R_{\rm ext}\) is inserted in series with the armature. As speed builds up and \(E_A\) increases, sections of the starting resistance are switched out (traditionally via a drum controller, now by power electronics). Full-voltage starting of small motors is permissible when \(I_{\rm start} < 6 I_{\rm rated}\); larger machines require reduced-voltage or current-limited starting.

5.4.1 DC Generator Characteristics

A separately excited DC generator has its terminal voltage determined by:

\[ V_T = E_A - I_A R_A = K\Phi\omega_m - I_A R_A. \]

As load (armature current) increases, the terminal voltage droops due to the armature resistance drop and (for non-compensated machines) due to armature reaction weakening the flux. The external characteristic (plot of \(V_T\) vs. \(I_A\)) is nearly linear with a small negative slope for a separately excited generator.

A self-excited shunt generator must first build up its terminal voltage from residual magnetism. At startup, the small residual flux induces a small EMF, which drives a small field current, which strengthens the flux, which increases the EMF — a positive-feedback bootstrap process. Build-up succeeds only if: (1) residual magnetism exists, (2) field circuit polarity is correct (field current must aid residual flux), and (3) field resistance is below the critical resistance (the slope of the magnetisation curve at the operating point).

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Chapter 6: Rotating Magnetic Field and AC Machine Fundamentals

6.1 Rotating Magnetic Field

The key physical concept underlying all AC rotating machines is the production of a spatially rotating magnetic field from stationary, sinusoidally distributed windings excited by polyphase currents.

Consider three identical sinusoidally distributed windings placed 120\(^\circ\) apart in space around a circular stator, excited by balanced three-phase currents:

\[ i_a(t) = I_m \cos(\omega t), \quad i_b(t) = I_m \cos\!\left(\omega t - \frac{2\pi}{3}\right), \quad i_c(t) = I_m \cos\!\left(\omega t + \frac{2\pi}{3}\right). \]

Each winding produces a sinusoidally distributed MMF in the air gap. The MMF of winding \(a\) is

\[ \mathcal{F}_a(\theta_s, t) = \frac{N_1}{2} I_m \cos(\omega t) \cos\theta_s, \]

where \(\theta_s\) is the spatial angle. Adding the contributions of all three phases (using the identity for the sum of three cosine products):

\[ \mathcal{F}_{\rm total}(\theta_s, t) = \frac{3}{2}\frac{N_1}{2}I_m \cos(\omega t - \theta_s). \]

This is a traveling wave in space: a sinusoidal MMF distribution that rotates at angular velocity \(\omega\) rad/s (electrical). The peak of the MMF wave sweeps around the air gap at this angular velocity, creating a uniformly rotating magnetic field. This is the AC machine’s analogue of the DC machine’s stationary main-field flux.

The fact that \(\mathcal{F}_{\rm total} \propto \cos(\omega t - \theta_s)\) — a wave moving in the \(+\theta_s\) direction with angular velocity \(\omega\) — is a deep result that can also be understood by analogy: three phasors of equal magnitude separated by \(120^\circ\) in both time phase and space phase always produce a resultant of constant magnitude and uniform angular velocity. This is the AC machine counterpart of the DC machine’s commutated armature MMF, which is held stationary by the commutator. In the AC machine, the stationarity in the rotor reference frame is achieved by electromagnetic induction rather than mechanical switching.

6.2 Synchronous Speed

The rotating field completes one full revolution per electrical cycle. For a \(P\)-pole machine, one electrical cycle corresponds to \(1/(P/2)\) mechanical revolutions. Thus the synchronous speed (in rev/min) is:

\[ n_s = \frac{120 f}{P}, \]

where \(f\) is the supply frequency (Hz) and \(P\) is the number of poles. At 60 Hz (North American standard), a 2-pole machine runs at 3600 rpm, a 4-pole at 1800 rpm, a 6-pole at 1200 rpm.

In radians per second (mechanical):

\[ \omega_s = \frac{4\pi f}{P} = \frac{2\pi n_s}{60}. \]

6.3 Slip

For an induction motor, the rotor runs at a mechanical speed \(\omega_m\) slightly less than synchronous speed \(\omega_s\) (the rotor cannot catch up with the rotating field — it must lag to sustain induced currents and hence torque). The slip is defined as:

\[ s = \frac{\omega_s - \omega_m}{\omega_s} = \frac{n_s - n_r}{n_s}. \]

At standstill (start): \(s = 1\). At synchronous speed: \(s = 0\) (no torque). Rated operation typically: \(s = 0.01\text{–}0.05\) (1–5%). For generator operation of an induction machine, \(s < 0\) (machine runs above synchronous speed). For plugging (electromagnetic braking), \(s > 1\) (current in rotor reversal brakes the motor).

The frequency of rotor currents and voltages is:

\[ f_r = s f. \]

At rated slip of 3%, rotor currents are at \(0.03 \times 60 = 1.8\) Hz — nearly DC. At standstill they are at line frequency (60 Hz).

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Chapter 7: Induction Motors

7.1 Construction

The three-phase induction motor consists of:

• Stator: laminated iron core with three-phase windings embedded in slots, connected to the supply.
• Rotor: either a squirrel-cage rotor (aluminium or copper bars shorted by end rings, no external connection) or a wound rotor (three-phase winding brought out to slip rings for external resistance insertion).

The squirrel-cage motor is the workhorse of industry: robust, cheap, essentially maintenance-free. The wound-rotor motor allows speed and torque control by inserting external resistance in the rotor circuit — important historically, now largely supplanted by variable-frequency drives.

7.2 Equivalent Circuit

The induction motor is a transformer with a short-circuited secondary (the rotor) that can move. The equivalent circuit per phase, referred to the stator, has:

• Stator resistance \(R_1\) and leakage reactance \(X_1\) in series.
• Shunt branch: core-loss resistance \(R_c\) and magnetising reactance \(X_m\).
• Rotor branch: resistance \(R_2/s\) and leakage reactance \(X_2\) in series.

The rotor branch impedance \(R_2/s\) deserves careful interpretation. The actual rotor resistance is \(R_2\); the remaining \(R_2(1-s)/s\) represents the mechanical power converted per phase divided by rotor current squared. Splitting:

\[ \frac{R_2}{s} = R_2 + R_2\frac{1-s}{s}. \]

The term \(R_2\) represents rotor copper loss; \(R_2(1-s)/s\) represents the air-gap power converted to mechanical work.

7.3 Power Flow and Efficiency

Define the air-gap power \(P_{\rm AG}\) as the power crossing the air gap (from stator to rotor). From the equivalent circuit:

\[ P_{\rm AG} = 3 I_2^2 \frac{R_2}{s}. \]

The rotor copper loss is:

\[ P_{\rm cu,r} = 3 I_2^2 R_2 = s P_{\rm AG}. \]

The power converted to mechanical form (electromagnetic power):

\[ P_{\rm mech} = P_{\rm AG} - P_{\rm cu,r} = (1-s) P_{\rm AG} = 3 I_2^2 R_2 \frac{1-s}{s}. \]

The shaft output power is:

\[ P_{\rm out} = P_{\rm mech} - P_{\rm F\&W} - P_{\rm stray}, \]

where \(P_{\rm F\&W}\) is friction and windage loss and \(P_{\rm stray}\) is stray load loss. The developed electromagnetic torque is:

\[ T_{\rm em} = \frac{P_{\rm mech}}{\omega_m} = \frac{P_{\rm AG}}{\omega_s} = \frac{3}{\omega_s} \cdot \frac{I_2^2 R_2}{s}. \]

This last form is useful: torque is \(P_{\rm AG}/\omega_s\), irrespective of slip.

7.4 Torque-Speed Characteristic

From the equivalent circuit, the rotor current (neglecting stator impedance and shunt branch for clarity) is:

\[ I_2 \approx \frac{V_\phi}{\sqrt{(R_2/s)^2 + X_2^2}}. \]

Substituting into the torque formula:

\[ T_{\rm em} = \frac{3}{\omega_s} \cdot \frac{V_\phi^2 R_2/s}{(R_2/s)^2 + X_2^2}. \]

This function of slip \(s\) exhibits:

• At small slip (near synchronous speed): \(T \approx (3V_\phi^2 / \omega_s)(s/R_2)\) — torque is approximately proportional to slip (linear region).
• At large slip (near standstill): \(T \approx (3V_\phi^2 / \omega_s)(R_2 / s X_2^2)\) — torque decreases as slip increases.
• Maximum (pullout) torque at \(s_{\rm max} = R_2 / X_2\):

\[ T_{\rm max} = \frac{3}{2\omega_s} \cdot \frac{V_\phi^2}{X_2}. \]

Note that \(T_{\rm max}\) is independent of rotor resistance \(R_2\). Increasing \(R_2\) (external resistance in a wound-rotor motor) shifts the peak torque to higher slip, allowing the motor to develop maximum torque at starting — a valuable technique.

The typical induction motor torque-speed curve passes through: starting torque (\(s=1\), moderate), an unstable region of increasing torque, a maximum (pullout) torque, and then stable decreasing torque back to zero at synchronous speed. Normal operating region is to the right of the pullout torque (high speed, low slip), where the slope \(dT/d\omega_m\) is negative (stable operation).

7.4.1 Effect of Rotor Resistance on Torque-Speed Curve

One of the most instructive observations in induction motor theory is that changing the rotor resistance \(R_2\) shifts the torque-speed curve horizontally without changing the maximum torque value. More precisely, the slip at maximum torque is \(s_{\rm max} = R_2/X_2\), so doubling \(R_2\) doubles the slip at maximum torque but leaves \(T_{\rm max}\) unchanged. This is because:

\[ T_{\rm max} = \frac{3}{2\omega_s} \cdot \frac{V_\phi^2}{X_2}, \]

which depends only on stator voltage and rotor leakage reactance, not \(R_2\).

For a wound-rotor motor, inserting external resistance \(R_{\rm ext}\) in the rotor circuit replaces \(R_2\) by \(R_2 + R_{\rm ext}\). By choosing \(R_{\rm ext}\) such that \(s_{\rm max} = (R_2 + R_{\rm ext})/X_2 = 1\), the maximum torque occurs at standstill — the motor starts with its maximum possible torque. This is enormously beneficial for loads requiring high starting torque (crushers, large compressors). As the motor accelerates, the external resistance is progressively removed (in steps or continuously via a liquid rheostat) so that the operating point tracks near maximum torque throughout the acceleration.

7.5 Starting Methods

At starting (\(s=1\)), the starting current of an induction motor is typically 5–7 times full-load current, because the rotor impedance \(R_2 + jX_2\) is small compared to the voltage. This large inrush can disturb the supply voltage. Starting methods include:

1. Direct-on-line (DOL): Full voltage applied. Simple, gives maximum starting torque. Acceptable only for small motors or stiff supplies.
2. Star-delta starting: Motor starts in Y connection (phase voltage = \(V_L/\sqrt{3}\)), reducing starting current and torque by factor 3. Switched to delta at running speed. Cheap but causes a current transient on switching.
3. Autotransformer starting: Voltage reduced by an autotransformer tap (typically 65–80% of line voltage). Current and torque both reduced by the square of the voltage ratio. Smoother than star-delta.
4. Variable-frequency drive (VFD): Ramps up frequency and voltage together, maintaining constant V/f ratio (hence constant flux). Provides smooth starting with very low inrush current. Now the dominant method for large motors.
5. Wound-rotor with external resistance: Inserts resistance in rotor circuit to increase \(s_{\rm max}\) to 1, giving maximum torque at starting. Resistance gradually cut out as motor accelerates.

7.6 Speed Control

For an induction motor the relationship \(n_r = n_s(1-s) = (120f/P)(1-s)\) shows three knobs: frequency \(f\), number of poles \(P\), and slip \(s\).

• Variable frequency (VFD): Most flexible and efficient. Varying \(f\) varies \(n_s\); maintaining V/f constant keeps flux (and hence torque capability) constant. Now essentially standard for variable-speed drive applications.
• Pole-changing: A motor with two independent winding sets can switch between, e.g., 4-pole and 8-pole operation, giving two discrete speeds. Simple and cheap; limited flexibility.
• Slip control via rotor resistance (wound rotor): External resistance increases slip at a given torque, lowering speed. Wastes energy; useful primarily for cranes and hoists needing smooth control at reduced speed for short durations.

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Chapter 8: Synchronous Machines

8.1 Construction

A synchronous machine has its rotor driven (or driving) at exactly synchronous speed \(n_s = 120f/P\). The rotor carries a DC field winding that creates poles of alternating polarity. As the rotor turns, a rotating magnetic flux is produced — not by three-phase currents in the stator, but mechanically. The three-phase stator winding cuts this rotating flux and has sinusoidal EMFs induced in it.

Two rotor types are common:

• Round rotor (cylindrical): Slots machined uniformly around the rotor periphery; field winding distributed in slots. Used in high-speed turbogenerators (2-pole, 3600 rpm at 60 Hz). The air gap is essentially uniform, so the machine is called non-salient.
• Salient pole rotor: Discrete projecting poles. Used in slow-speed, large-diameter machines (hydro generators, synchronous motors with many poles). The air gap is non-uniform — wider between poles — giving position-dependent reluctance.

8.2 Equivalent Circuit (Round-Rotor Machine)

Under balanced three-phase operation, the round-rotor synchronous machine is analysed on a per-phase basis. The field winding on the rotor carries DC current \(I_F\) and produces the field flux \(\Phi_F\). As the rotor turns at synchronous speed, the flux linkage with each stator phase varies sinusoidally, inducing the internal generated voltage (back-EMF):

\[ E_A = K \Phi_F \omega_s, \]

in phasor form \(\mathbf{E}_A = E_A \angle\delta\), where \(\delta\) is the power angle (also called the torque angle or load angle) — the angle by which \(\mathbf{E}_A\) leads the terminal voltage \(\mathbf{V}_\phi\).

The per-phase equivalent circuit is simply the internal EMF \(\mathbf{E}_A\) in series with the synchronous impedance \(Z_S = R_A + jX_S\):

\[ \mathbf{V}_\phi = \mathbf{E}_A - \mathbf{I}_A Z_S, \]

where the sign convention is for generator operation (current \(\mathbf{I}_A\) flows out of the machine to the load). For motor operation, the convention reverses: \(\mathbf{E}_A = \mathbf{V}_\phi + \mathbf{I}_A Z_S\) with current flowing into the machine.

The synchronous reactance \(X_S = X_{ar} + X_l\) includes the armature reaction reactance \(X_{ar}\) (which models the effect of stator MMF on rotor flux) and the stator leakage reactance \(X_l\). In per-unit, \(X_S\) typically ranges from 0.8 to 1.5 for large machines.

8.3 Power and Torque in Synchronous Machines

Neglecting stator resistance (\(R_A \approx 0\)), which is a good approximation for large machines where \(X_S \gg R_A\), the three-phase power delivered by a generator is:

\[ P = \frac{3 V_\phi E_A}{X_S} \sin\delta. \]

This is the fundamental power-angle relation for a synchronous machine. Key observations:

• \(P\) is maximum at \(\delta = 90^\circ\): this is the static stability limit. Beyond \(90^\circ\), increasing \(\delta\) decreases \(P\) — the machine loses synchronism.
• For a given terminal voltage \(V_\phi\) and synchronous reactance \(X_S\), the maximum deliverable power is \(P_{\rm max} = 3V_\phi E_A / X_S\), controlled by the field excitation \(E_A\).
• The electromagnetic torque is \(T_{\rm em} = P / \omega_s\).

The three-phase reactive power is:

\[ Q = \frac{3 V_\phi}{X_S}(E_A \cos\delta - V_\phi). \]

8.4 Voltage Regulation and the Phasor Diagram

For a synchronous generator supplying a load at power factor \(\cos\phi\), the phasor diagram gives:

\[ \mathbf{E}_A = \mathbf{V}_\phi + j X_S \mathbf{I}_A \quad (R_A \text{ neglected}). \]

Voltage regulation is defined as in transformers: \(\text{VR} = (|E_A| - V_\phi)/V_\phi \times 100\%\). At lagging pf, \(|E_A| > V_\phi\) and VR is positive (overexcited condition). At leading pf (capacitive load), the armature reaction aids the field flux: \(|E_A| < V_\phi\) and VR is negative (underexcited condition, the terminal voltage is actually higher than the open-circuit voltage — an unusual situation).

8.4.1 Parallel Operation of Synchronous Generators

In a power system, many synchronous generators operate in parallel on a common bus. Connecting a new generator to an infinite bus (a bus whose voltage and frequency are fixed by the large system) requires synchronising — the incoming generator’s terminal voltage must match the bus in magnitude, frequency, phase, and phase sequence. This is verified with a synchroscope or sync-check relay.

Once synchronised and carrying no load, the generator floats at synchronous speed with \(\delta = 0\) (no torque angle, no active power). To load the generator, the prime mover input (steam valve opening, water gate position) is increased. This advances the rotor, increasing \(\delta\) and hence active power. To adjust reactive power (and hence power factor), the field excitation \(I_F\) is changed. These two controls — prime mover throttle for \(P\), field rheostat for \(Q\) — are decoupled in a well-designed governor/AVR (automatic voltage regulator) system.

Droop control: in a real system (not an infinite bus), increasing generator output raises the shared bus frequency slightly. This is intentional — generators use a droop characteristic so that increased load is shared automatically among all online generators in proportion to their droop settings, without any communication.

8.5 V-Curves

For a synchronous motor operating at fixed shaft load (fixed \(P\)), varying the field current \(I_F\) (and hence \(E_A\)) changes the reactive power exchange with the network. The plot of armature current \(|I_A|\) versus field current \(I_F\) at constant output power forms a family of V-shaped curves — the V-curves of the synchronous motor.

• At the bottom of each V (minimum \(|I_A|\)): unity power factor operation. The machine exchanges no reactive power.
• Left of minimum (underexcited, low \(I_F\)): the machine absorbs reactive power, behaving as an inductor from the grid’s perspective. \(I_A\) leads \(E_A\) and lags \(V_\phi\) — lagging pf.
• Right of minimum (overexcited, high \(I_F\)): the machine supplies reactive power, behaving as a capacitor. Leading pf.

This property — controllable reactive power output by adjusting field excitation — makes the synchronous motor (or synchronous condenser, a synchronous motor running at no load) a valuable reactive power compensator in power systems.

8.6 Salient Pole Machines and Reluctance Torque

A salient pole machine has a non-uniform air gap: the reluctance varies with the angular position of the rotor. The two-reaction theory (Blondel, 1923) resolves the armature MMF into a component along the direct axis (d-axis, aligned with rotor poles) and a component along the quadrature axis (q-axis, aligned between poles). Different synchronous reactances apply to each axis: \(X_d > X_q\) because the d-axis path passes through iron (low reluctance), while the q-axis path passes through the wider inter-pole gap.

The power-angle relation for a salient-pole machine (generator convention, neglecting \(R_A\)) is:

\[ P = \frac{3 V_\phi E_A}{X_d}\sin\delta + \frac{3 V_\phi^2}{2}\left(\frac{1}{X_q} - \frac{1}{X_d}\right)\sin 2\delta. \]

The second term is the reluctance power (or reluctance torque), which arises purely from the saliency (\(X_d \neq X_q\)) and is independent of field excitation \(E_A\). A salient-pole machine develops reluctance torque even with zero field current. The peak reluctance torque occurs at \(\delta = 45^\circ\) and is proportional to \(V_\phi^2(1/X_q - 1/X_d)X_dX_q/(2\cdot)...\) — the saliency ratio \((X_d - X_q)/X_q\) governs its relative importance.

8.6.1 Salient Pole Phasor Diagram

For a salient-pole synchronous generator supplying load at power factor \(\cos\phi\) lagging, the phasor diagram must account for the different d- and q-axis reactances. The procedure is:

1. Define the q-axis direction: it leads \(\mathbf{E}_A\) by 90\(^\circ\). Its direction is found by: \(\mathbf{E}_A = \mathbf{V}_\phi + jX_q \mathbf{I}_A + R_A \mathbf{I}_A\) (using \(X_q\) for the whole current initially).
2. Resolve \(\mathbf{I}_A\) into d-axis component \(I_d\) (along the negative d-axis) and q-axis component \(I_q\) (along the q-axis).
3. Compute \(|E_A| = V_\phi\cos\delta + X_d I_d - R_A I_q\) (or the full phasor expression).

The internal voltage is:

\[ E_A = (V_\phi \cos\delta + R_A I_d - X_q I_q) + j(V_\phi \sin\delta + X_d I_d + R_A I_q)... \]

More compactly, with \(R_A = 0\):

\[ |E_A| = V_\phi\cos\delta + X_d I_d, \]\[ 0 = V_\phi\sin\delta - X_q I_q, \]

from which \(I_q = V_\phi\sin\delta / X_q\) and \(I_d = (|E_A| - V_\phi\cos\delta)/X_d\). Given measured \(V_\phi\), \(I_A\), \(\phi\), the power angle \(\delta\) is determined by:

\[ \tan\delta = \frac{X_q I_A \cos\phi - R_A I_A \sin\phi}{V_\phi + X_q I_A \sin\phi + R_A I_A \cos\phi}. \]

(For generator at lagging pf; signs adjust for leading or motor convention.)

8.7 Starting Synchronous Motors

A synchronous motor is not inherently self-starting when connected directly to the supply, because a single-frequency rotating field interacting with a stationary rotor produces a torque that averages to zero over a full cycle (the field rotates at 3600 rpm at 60 Hz, far too fast for the rotor inertia to follow). Starting methods:

1. Damper winding (amortisseur): Short-circuited conductors embedded in the pole faces. These act as a squirrel cage during starting, allowing induction motor action to accelerate the rotor to near synchronous speed. The field winding is shorted (through a resistor) during starting to prevent damaging voltages induced by the rotating stator field. At near-synchronous speed, DC field excitation is applied and the rotor is pulled into synchronism.
2. Variable-frequency starting (VFD): The supply frequency is ramped up slowly from near-zero, allowing the rotor to follow. Preferred for large synchronous motors (compressors, pumps) where DOL starting is impractical.
3. Pony motor: An auxiliary motor brings the synchronous machine to near-synchronous speed before field excitation is applied.

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Chapter 9: Single-Phase Induction Motors

9.1 Double Revolving Field Theory

A single-phase stator winding excited by alternating current produces a pulsating standing-wave MMF, not a rotating field. The standing wave can be decomposed into two rotating waves of half amplitude travelling in opposite directions:

\[ \mathcal{F}(\theta_s, t) = F_m \cos\theta_s \cos\omega t = \frac{F_m}{2}\cos(\theta_s - \omega t) + \frac{F_m}{2}\cos(\theta_s + \omega t). \]

The forward-rotating component has amplitude \(F_m/2\) and rotates at \(+\omega_s\); the backward-rotating component has amplitude \(F_m/2\) and rotates at \(-\omega_s\). Each component alone would produce an induction motor torque. The net torque at any slip \(s\) (defined relative to the forward field) is:

\[ T_{\rm net}(s) = T_f(s) - T_b(s), \]

where \(T_f(s)\) is the torque due to the forward field at slip \(s\), and \(T_b(s)\) is the torque due to the backward field, which operates at slip \((2-s)\) relative to the rotor. At standstill (\(s=1\)): \(T_f(1) = T_b(1)\) (forward and backward torques are equal and opposite — net torque is zero). This is why a single-phase induction motor has no starting torque.

Once the motor is turning in either direction (say, forward), \(s < 1\) and \((2-s) > 1\): the forward component produces more torque than the backward component. The net forward torque accelerates the rotor, and the motor runs. The backward component causes a performance penalty (increased losses, reduced efficiency) compared to a three-phase motor.

9.2 Starting Methods

Since the single-phase induction motor cannot self-start, an auxiliary means must create a rotating field at starting:

1. Split-phase motor: A high-resistance auxiliary winding displaced 90° in space is connected in parallel with the main winding. The different L/R ratios cause the auxiliary current to lead the main current in time, creating an elliptical (approximately) rotating field sufficient to start the motor. The auxiliary winding is disconnected by a centrifugal switch at ~75% of synchronous speed.

2. Capacitor-start motor: A capacitor in series with the auxiliary winding shifts the auxiliary current nearly 90° ahead of the main current. This produces nearly circular rotating field at starting, giving higher starting torque than the split-phase design. Widely used for loads requiring high starting torque (compressors, pumps).

3. Capacitor-start, capacitor-run motor: Two capacitors — a large electrolytic for starting (switched out) and a smaller oil-filled capacitor that remains in circuit permanently. The run capacitor improves running efficiency and power factor.

4. Permanent-split-capacitor (PSC) motor: A single capacitor remains in circuit at all times. No centrifugal switch; simple and reliable. Used in fans, small pumps where starting torque requirement is modest.

5. Shaded-pole motor: A copper band (shading coil) short-circuited around part of each stator pole. The shading coil delays the flux in the shaded portion, creating a weak rotating field. Very low efficiency and starting torque; used only in very small ratings (fans, timers).

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Chapter 10: Variable-Frequency Drives and Power Electronics Basics

10.1 Motivation

Fixed-frequency AC supply systems impose a fixed synchronous speed on induction and synchronous motors. Historically, speed control required lossy methods (rotor resistance, gear boxes, throttling). Variable-frequency drives (VFDs) — also called adjustable-speed drives (ASDs) or inverter drives — allow continuous, efficient speed control by supplying the motor with variable-frequency, variable-voltage AC.

10.2 Basic VFD Topology

A standard VFD has three stages:

1. Rectifier: Converts the fixed-frequency AC supply (e.g., 60 Hz three-phase) to DC. Typically a diode bridge or active front-end (AFE) rectifier. The DC bus voltage is approximately \(\sqrt{2}\) times the line-to-line RMS voltage for a diode rectifier.

2. DC link (bus): A large capacitor bank filters the rectified DC and stores energy, providing a stiff voltage source for the inverter. Some drives use a DC link inductor (current-source inverter topology).

3. Inverter: Converts DC back to AC at the desired frequency and voltage. Uses IGBTs (insulated-gate bipolar transistors) switched at high frequency (typically 2–20 kHz) in a three-phase bridge configuration. The switching pattern is controlled by pulse-width modulation (PWM).

10.2.1 IGBT Switching and PWM

The heart of a modern inverter is the IGBT (Insulated Gate Bipolar Transistor) — a voltage-controlled device combining the ease of gate drive of a MOSFET with the low on-state voltage drop of a bipolar transistor. In a three-phase voltage-source inverter (VSI), six IGBTs are arranged in three half-bridge legs, each controlled by complementary signals. The output AC voltage waveform is synthesised using pulse-width modulation (PWM).

In sinusoidal PWM, a sinusoidal reference waveform at the desired output frequency \(f_{\rm out}\) is compared with a triangular carrier wave at a much higher frequency \(f_c\) (the switching frequency, typically 2–20 kHz). Whenever the reference exceeds the carrier, the upper IGBT in the leg is turned on (positive output); otherwise the lower IGBT is on. The resulting output is a high-frequency train of narrow pulses whose average over each carrier cycle is proportional to the desired sinusoid.

The fundamental output voltage is:

\[ V_{1,\rm out} = m_a \cdot \frac{V_{\rm dc}}{2}, \]

where \(m_a = \hat{v}_{\rm ref}/\hat{v}_{\rm carrier}\) is the modulation index (\(0 \leq m_a \leq 1\) for linear modulation). The output spectrum contains the fundamental plus harmonics at multiples of \(f_c\) and their sidebands (\(f_c \pm 2f_{\rm out}\), \(2f_c \pm f_{\rm out}\), etc.). Because \(f_c \gg f_{\rm out}\), these harmonics are at high frequency and are attenuated by the motor’s own inductance. The motor sees essentially a sinusoidal fundamental with small high-frequency ripple.

10.3 Volts-per-Hertz Control

The most basic control strategy is constant V/f (volts-per-hertz) control. The motor’s air-gap flux is proportional to \(V_\phi / f\) (from Faraday’s law, \(E_A = 4.44 N f \Phi_{\rm max}\) for the fundamental). To keep flux (and hence torque capability) constant as speed varies, the inverter output voltage is proportional to frequency:

\[ \frac{V_\phi}{f} = \text{const} = \frac{V_{\rm rated}}{f_{\rm rated}}. \]

At very low frequencies, the stator resistance drop becomes significant relative to the applied voltage; a voltage boost is applied to compensate. Above base frequency, the voltage is clamped at rated value (the inverter cannot exceed supply voltage), and the machine operates in field-weakening mode with reduced flux and torque.

10.4 Energy Savings with VFDs

For centrifugal loads (fans, pumps, compressors) the mechanical power required scales as the cube of speed:

\[ P_{\rm mech} \propto n^3. \]

A throttling valve wastes energy at partial flow; reducing motor speed with a VFD instead reduces input power dramatically. Operating at 80% of full speed requires only \(0.8^3 = 51.2\%\) of full-power — nearly half the energy. This is the primary economic driver for VFD adoption in HVAC and pumping systems.

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Chapter 11: Measurement and Performance Assessment

The energy savings from VFD use on centrifugal loads is the largest single energy efficiency opportunity in industry. In pumping systems, a two-speed motor switch (which can only operate at 100% or 50% of synchronous speed) gives \(P \propto (50\%)^3 = 12.5\%\) power at low speed — already 7.5\(\times\) saving. A VFD gives continuously variable savings proportional to the cube of the speed ratio. In large HVAC systems (chiller plants, cooling towers), VFD retrofits routinely achieve 30–50% reduction in annual electrical energy consumption.

Regenerative braking is another VFD advantage: when the motor decelerates (or when a machine operates as a generator, such as an elevator descending), the VFD can return energy to the DC bus. If an active front-end rectifier (AFE, or regenerative front end) is used, this energy is returned to the AC supply rather than dissipated in a braking resistor. This is critical in applications with frequent acceleration-deceleration cycles (lifts, cranes, rail transit).

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Chapter 11: Measurement and Performance Assessment

11.1 No-Load and Blocked-Rotor Tests (Induction Motor)

The induction motor equivalent circuit parameters are determined by tests analogous to the transformer open-circuit and short-circuit tests.

11.2 Efficiency and Performance Standards

Motor efficiency is regulated internationally. The IEC 60034-30-1 standard defines efficiency classes IE1 (standard), IE2 (high), IE3 (premium), and IE4 (super-premium). In Canada and the US, NEMA Premium (roughly IEC IE3) has been mandatory for most industrial motors since 2010–2016.

For a three-phase induction motor at rated conditions, typical efficiency ranges from ~90% for a 1 hp motor to ~96% for a 500 hp motor. The major losses are:

• Stator copper loss: \(3 I_1^2 R_1\)
• Core loss: \(P_{\rm core}\)
• Air-gap (rotor copper) loss: \(s P_{\rm AG}\)
• Friction and windage: \(P_{\rm F\&W}\)
• Stray load losses: \(\sim 0.5\%\) of output (assigned by standard in the absence of direct measurement)

The overall efficiency:

\[ \eta = \frac{P_{\rm out}}{P_{\rm in}} = \frac{P_{\rm out}}{P_{\rm out} + P_{\rm losses}}. \]

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Chapter 12: Integrated System Perspective

11.3 Synchronous Generator Tests

For a synchronous generator, the key tests are:

• Open-circuit test (OCC — open-circuit characteristic): Drive the machine at synchronous speed, vary \(I_F\) from zero to above rated, measure the terminal voltage \(V_{\rm oc}\). The resulting \(V_{\rm oc}\) vs. \(I_F\) curve is the OCC. At low \(I_F\) it is linear (the air-gap line, slope = \(E_A/I_F = K\omega_s\)); at higher \(I_F\) it saturates as the core approaches saturation.

• Short-circuit test (SCC — short-circuit characteristic): With terminals short-circuited, vary \(I_F\) and measure \(I_A\). The SCC is linear (no saturation, since \(V=0\) and the machine flux is depressed by armature reaction). The unsaturated synchronous reactance is:

\[ X_S = \frac{V_{\rm oc}({\rm air\text{-}gap\, line})}{I_{\rm sc}} \bigg|_{I_F = \rm same}, \]

evaluated at the same field current on the linear (air-gap line) portion of the OCC.

The saturated synchronous reactance (used at rated excitation) uses the rated voltage on the actual OCC (not the air-gap line) divided by the same short-circuit current. Because the OCC is nonlinear, \(X_S\) is smaller in the saturated case — typically 15–20% smaller.

12.1 Power System Structure and Machine Roles

Electric power systems consist of generation, transmission, and distribution. Synchronous generators convert mechanical shaft power (steam turbine, gas turbine, hydro turbine) to three-phase AC power at the generation voltage level (typically 11–25 kV). Step-up transformers (Y-\(\Delta\) or Y-Y) raise the voltage to transmission levels (115–765 kV) to minimise \(I^2R\) losses over long distances. Step-down transformers reduce voltage in stages for sub-transmission (26–69 kV), primary distribution (4–35 kV), and secondary distribution (120/240 V or 208/120 V in North America).

Induction motors constitute by far the largest fraction of electrical load: approximately 70% of industrial electricity consumption. Transformers, motors, and generators are all described by equivalent circuits derived from the same underlying physics — Ampere’s and Faraday’s laws applied to magnetically coupled, mechanically coupled electromagnetic systems.

The transformation from generation to load involves a cascade of energy conversion steps, each governed by equivalent circuit models derived in this course:

\[ \text{Shaft power} \xrightarrow{\text{Synchronous generator}} \text{AC power at } V_{\rm gen} \xrightarrow{\text{Step-up transformer}} \text{AC power at } V_{\rm tx} \xrightarrow{\text{Transmission line}} \cdots \xrightarrow{\text{Step-down transformer}} \text{AC at } V_{\rm dist} \xrightarrow{\text{Motor}} \text{Shaft power.} \]

The same fundamental law — Faraday’s induction — governs the transformer, the generator, and the motor. The same phasor analysis handles each stage. The per-unit system unifies the numerical analysis across all voltage levels. This integrated picture is the core of power engineering education.

12.2 Connecting the Threads: Unified Machine Theory

All rotating machines, at their core, produce torque by the interaction of two magnetic fields. The condition for average (non-zero) torque production is that the stator and rotor field must rotate at the same speed relative to each other — they must be synchronised in the average. This is achieved by:

• DC machines: The commutator mechanically rectifies the armature current to keep the armature MMF stationary in space, always at 90° to the main field MMF, regardless of rotor speed.
• Induction machines: The rotor currents are induced at slip frequency \(sf\), which rotates relative to the rotor at speed \(sn_s\). Since the rotor itself turns at \((1-s)n_s\), the rotor MMF rotates at \(sn_s + (1-s)n_s = n_s\) relative to the stator — always synchronous with the stator field.
• Synchronous machines: The rotor field is created by DC excitation on a rotor spinning at synchronous speed. The rotor field is inherently synchronous with the stator field; the angle \(\delta\) between them determines the torque.

12.3 Choosing an Electric Machine

The selection of a machine for a given application involves tradeoffs:

Machine Type	Speed Control	Starting Torque	Maintenance	Efficiency	Typical Application
Squirrel-cage induction	VFD or pole-switching	Moderate	Very low	High	Most industrial drives
Wound-rotor induction	Rotor resistance, VFD	High	Moderate	Lower (with resistance)	Cranes, hoists
DC shunt/separately excited	Excellent (armature V or field)	Moderate	High (brushes)	Good	Rolling mills, older traction
DC series	Limited (resistance)	Very high	High (brushes)	Good	Traction, starters
Synchronous	VFD	Poor (no self-start)	Low (brushless)	Very high	Large compressors, power factor correction
Synchronous reluctance	VFD	Moderate	Very low	High	Pumps, fans (competing with IE4)

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Sources and Further Reading

Chapman, S. J. Electric Machinery Fundamentals, 5th ed. McGraw-Hill, 2011. — The primary textbook for this course; comprehensive treatment of DC machines, transformers, induction motors, and synchronous machines at the undergraduate level.

Fitzgerald, A. E., Kingsley, C., and Umans, S. D. Electric Machinery, 7th ed. McGraw-Hill, 2014. — The classic graduate-level reference; more rigorous treatment of two-reaction theory, energy methods, and generalised machine theory.

MIT OpenCourseWare 6.685 Electric Machines, Prof. James Kirtley. Open lecture notes and problem sets. Freely available at ocw.mit.edu. Excellent on energy and co-energy methods, doubly-excited systems, and machine design.

NPTEL Electric Machines lecture series. National Programme on Technology Enhanced Learning, Government of India. Covers equivalent circuits, testing, and performance of all major machine types; freely available online.

Boldea, I. and Nasar, S. A. The Induction Machine Handbook. CRC Press, 2002. — Comprehensive reference on induction machine theory, design, and drives; useful for deeper study of torque-speed characteristics and vector control.

Kundur, P. Power System Stability and Control. McGraw-Hill, 1994. — Chapter on synchronous machine modelling in power system context; standard reference for salient-pole two-axis models.