16.1 Synchronous Generators (Alternators)

16.1 Synchronous Generators (Alternators)

Synchronous Speed & Stator Winding Architecture

A synchronous generator (or alternator) converts mechanical energy supplied by a prime mover (such as a steam turbine, hydro turbine, or diesel engine) into three-phase electrical power at a precise constant frequency. Unlike induction machines, the electrical frequency generated by a synchronous generator is rigidly locked to its mechanical speed of rotation.

Synchronous Speed Equation

The fundamental relationship connecting rotor mechanical speed $n_s$ (in revolutions per minute, rpm), system electrical frequency $f$ (in Hertz, Hz), and total field pole count $P$ is:

$$n_s = \frac{120 f}{P} \quad \implies \quad \omega_s = \frac{4 \pi f}{P} \text{ rad/s}$$

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Armature Winding Factors & Generated RMS EMF

Stator armature windings of modern alternators are distributed double-layer 3-phase windings placed in slots around the stator periphery. To improve wave shape purity and economize end-turn copper, windings utilize fractional-pitch chording and distributed coils.

1. Pitch Factor (Chording Factor, $k_p$)

When the coil span (distance between two coil sides) equals full pole pitch (180° electrical), induced voltages in both coil sides are exactly in phase and add directly ($k_p = 1.0$). If a coil is short-chorded by an angle $\alpha$ (electrical degrees), the induced EMFs in the two sides have a phase displacement of $\alpha$.

$$k_p = \cos\left(\frac{\alpha}{2}\right)$$

where $\alpha = (1 - \text{coil pitch fraction}) \times 180^\circ$.

Key Benefit of Fractional Pitching: Short-chording by $\alpha = 180^\circ / n$ completely eliminates the $n$-th harmonic voltage component. For example, a $\frac{5}{6}$ fractional pitch has $\alpha = 30^\circ$, giving $k_{p1} = \cos(15^\circ) = 0.9659$ for the fundamental, while for the 3rd harmonic ($n=3$), $k_{p3} = \cos(3 \times 15^\circ) = \cos(45^\circ) = 0.7071$. This significantly suppresses triplen harmonic distortions.

2. Distribution Factor (Breadth Factor, $k_d$)

Because armature coils of each phase are distributed across $m$ slots per pole per phase rather than concentrated in a single slot, the induced EMFs in adjacent coils are phase-displaced by slot electrical angle $\gamma$.

$$m = \frac{S}{P \times m_{\text{phase}}}, \quad \gamma = \frac{P \times 180^\circ}{S} = \frac{180^\circ}{m \times m_{\text{phase}}}$$ $$k_d = \frac{\sin\left(\frac{m \gamma}{2}\right)}{m \sin\left(\frac{\gamma}{2}\right)}$$

where $S$ is total stator slots, $P$ is total poles, and $m_{\text{phase}} = 3$ for three-phase alternators.

Combined Winding Factor ($k_w$) & RMS EMF Equation

The total winding factor is the product of pitch factor and distribution factor:

$$k_w = k_p \times k_d$$

Combining Faraday's law with stator winding distribution, the RMS induced electromotive force per phase ($E_p$) is:

$$E_p = 4.44 k_p k_d f N_p \Phi_m = 4.44 k_w f N_p \Phi_m$$

where:

• $f$ = system frequency in Hz
• $N_p$ = series armature turns per phase ($N_p = Z_p / 2$)
• $Z_p$ = total armature conductors per phase
• $\Phi_m$ = maximum magnetic flux per pole in Webers (Wb)
• Line-to-line terminal voltage for wye connection: $V_L = \sqrt{3} E_p$

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Armature Reaction & Synchronous Impedance Equivalent Circuit

When an alternator supplies load current $\mathbf{I}_a$, the armature current creates its own magnetic field (armature MMF). The interaction between the armature MMF and the main rotor field MMF is called armature reaction.

                    SYNCHRONOUS GENERATOR PER-PHASE EQUIVALENT CIRCUIT

            + ┌──────────┐    Ra        jXl       jXar      + ───► Ia
              │          │  █████     UUUUUU     UUUUUU       │
              │  Ef / Ep ├────┴─────────┴──────────┴──────────┤
           Ef │ (Internal│            └─────┬──────┘          │ Vt / Vp (Terminal Phase Voltage)
              │   EMF)   │                  │                 │
            - └──────────┘               jXs (Synchronous     │
                                           Reactance)         - 
     ─────────────────────────────────────────────────────────┴────

Nature of Armature Reaction across Load Power Factors

Synchronous Reactance ($X_s$) and Internal Voltage Equation

To account for armature leakage flux ($X_l$) and armature reaction flux ($X_{ar}$), both reactances are combined into a single parameter called synchronous reactance:

$$X_s = X_l + X_{ar}$$ $$Z_s = R_a + j X_s = \sqrt{R_a^2 + X_s^2} \angle \theta_s$$

The fundamental phasor equation for a synchronous generator per phase is:

$$\mathbf{E}_p = \mathbf{V}_p + \mathbf{I}_a \mathbf{Z}_s = \mathbf{V}_p + \mathbf{I}_a (R_a + j X_s)$$

where $\mathbf{E}_p$ is the internal excitation voltage, $\mathbf{V}_p$ is terminal phase voltage, and $\mathbf{I}_a$ is phase armature current.

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Synchronous Impedance Testing & Short-Circuit Ratio (SCR)

Equivalent circuit parameters ($R_a, X_s, Z_s$) are determined via two standard laboratory tests conducted at synchronous speed:

1. Open-Circuit Test (OCC)

• Stator terminals are left open ($I_a = 0$).
• Field current $I_f$ is gradually varied from 0 up to saturation while recording line-to-line open-circuit voltage $V_{\text{oc}}$.
• At low $I_f$, the curve is linear (Air-Gap Line); at high $I_f$, magnetic saturation bends the curve.

2. Short-Circuit Test (SCC)

• Stator terminals are short-circuited through ammeters ($V_p = 0$).
• Field current $I_f$ is gradually increased while recording short-circuit armature line current $I_{\text{sc}}$.
• The SCC is a straight line because heavy demagnetizing armature reaction keeps the net core flux low, preventing magnetic saturation.

Unsaturated Synchronous Impedance Calculation

For a specific field current $I_f$ within the linear unsaturated region:

$$Z_{s(\text{unsat})} = \frac{V_{\text{oc,line}} / \sqrt{3}}{I_{\text{sc}}} \quad [\Omega/\text{phase}]$$ $$X_s = \sqrt{Z_s^2 - R_a^2}$$

where $R_a$ is the effective AC armature resistance per phase (measured using DC resistance $R_{\text{dc}}$ and corrected for skin effect: $R_{a,\text{ac}} \approx 1.2 - 1.5 R_{\text{dc}}$).

Short-Circuit Ratio (SCR)

Short-Circuit Ratio (SCR) is defined as the ratio of field current required to produce rated voltage on open circuit to the field current required to produce rated current on short circuit:

$$\text{SCR} = \frac{I_{f(\text{for rated } V_{\text{oc}})}}{I_{f(\text{for rated } I_{\text{sc}})}}$$ $$\text{SCR} \approx \frac{1}{X_{s(\text{pu})}(\text{unsaturated})}$$

• High SCR (e.g. $1.0 - 1.5$ for salient-pole hydro generators) indicates a larger physical air gap, lower synchronous reactance $X_s$, superior voltage stability, and greater steady-state power capacity, but higher physical machine size and cost.
• Low SCR (e.g. $0.5 - 0.7$ for steam turbo-generators) indicates higher $X_s$, smaller air gap, and reduced physical size.

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Two-Reaction Theory for Salient-Pole Alternators

Cylindrical-rotor generators have uniform air gaps, so reluctance is independent of rotor position ($X_d = X_q = X_s$). In contrast, salient-pole generators feature non-uniform air gaps (small air gap along the pole axis, large air gap between poles).

According to Blondel's Two-Reaction Theory, armature current $\mathbf{I}_a$ is resolved into two orthogonal components:

1. Direct-axis current ($I_d$): Acts along the magnetic pole axis (path of minimum reluctance, high reactance $X_d$).
2. Quadrature-axis current ($I_q$): Acts perpendicular to the pole axis (path of maximum reluctance, low reactance $X_q$).

$$\text{Since air gap is minimum along d-axis: } X_d > X_q$$

Total Active Power Output (Salient Pole)

$$P_{3\phi} = \frac{3 V_p E_p}{X_d} \sin \delta + \frac{3 V_p^2}{2} \left( \frac{1}{X_q} - \frac{1}{X_d} \right) \sin 2\delta$$

• The first term $\frac{3 V_p E_p}{X_d} \sin \delta$ is the excitation power produced by field flux.
• The second term $\frac{3 V_p^2}{2} \left( \frac{1}{X_q} - \frac{1}{X_d} \right) \sin 2\delta$ is the reluctance power, which exists due to rotor saliency even if field excitation fails ($E_p = 0$).
• Maximum power transfer occurs at power angle $\delta < 90^\circ$ (typically around $60^\circ - 75^\circ$).

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Parallel Operation & Synchronization Protocols

To parallel an oncoming generator with an active power grid (infinite bus), four rigid synchronization conditions must be fulfilled:

1. Equal Terminal Voltage Magnitudes: $V_{\text{gen}} = V_{\text{bus}}$ (verified via voltmeter or AVR setpoint).
2. Equal Frequencies: $f_{\text{gen}} = f_{\text{bus}}$ (verified via frequency meter or prime mover governor speed).
3. Identical Phase Sequence: A-B-C matching (verified during installation using phase-rotation meter).
4. Identical Phase Angle Alignment: $\delta = 0^\circ$ at breaker closing (verified using synchroscope or 3-lamp method).

Grid Control Dynamics

• Adjusting prime mover governor mechanical power alters active power output ($P$) and system frequency ($f$).
• Adjusting generator field excitation ($I_f$) alters reactive power output ($Q$) and bus terminal voltage ($V$).