16.2 Synchronous Motors & Special AC Machines

16.2 Synchronous Motors & Special AC Machines

Fundamental Operation & Phasor Dynamics

A synchronous motor is a doubly-excited AC machine that converts three-phase electrical power into mechanical shaft power. Its stator is connected to a 3-phase AC supply creating a rotating magnetic field (RMF) at synchronous speed $n_s = \frac{120 f}{P}$, while its rotor is energized with direct current (DC) to create fixed magnetic poles.

Magnetic Locking & Zero Speed Regulation

Once pulled into synchronism, the salient or non-salient rotor poles lock magnetically with the opposite-polarity rotating poles of the stator RMF. Consequently, the motor rotates strictly at synchronous speed under all steady-state load conditions from no-load up to breakdown pull-out torque:

$n_r = n_s = \frac{120 f}{P} \quad \implies \quad \text{Speed Regulation } = \frac{n_{\text{nl}} - n_{\text{fl}}}{n_{\text{fl}}} \times 100\% = 0\%$

Per-Phase Motor Voltage Equation

In motor operation, active electrical power enters the stator terminals against the internal counter-EMF (back-EMF) $\mathbf{E}_p$ induced by the rotating DC rotor flux. The per-phase phasor relationship is:

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

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

where $\mathbf{V}_p$ is the applied phase voltage and $\mathbf{E}_p$ is the excitation back-EMF (whose magnitude is directly proportional to DC field current $I_f$).

                      SYNCHRONOUS MOTOR PHASOR DIAGRAMS (PER-PHASE)

  (A) Under-Excited (Lagging PF)     (B) Normal Excitation (Unity PF)    (C) Over-Excited (Leading PF)
           Vp                                 Vp                                 Vp
           ▲                                  ▲                                  ▲
           │  / Ef                            │  / Ef                            │  / Ef
           │ /                                │ /                                │ /
           │/ δ                               │/ δ                               │/ δ
   ────────┼─────────►                 ───────┼─────────►                 ───────┼─────────►
          /│                                  │                                 /│
         / │                                  │                                / │
        /  │ Ia (lags)                        │ Ia (in phase)                 /  │ Ia (leads)
       ▼   ▼                                  ▼                              ▼   

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Excitation Control: V-Curves & Synchronous Condensers

A unique operational feature of synchronous motors is that varying the DC field current $I_f$ alters the operating power factor without changing mechanical shaft speed or real power output.

Three Excitation Regimes at Constant Mechanical Load

1. Under-Excitation ($E_p \cos \delta < V_p$):
   • The rotor field flux is insufficient to maintain stator voltage balance, forcing the stator to draw lagging reactive magnetizing current ($+Q$) from the AC line.
   • Armature current $\mathbf{I}_a$ lags terminal voltage $\mathbf{V}_p$ (motor behaves like an inductive load).
2. Normal Excitation ($E_p \cos \delta = V_p$):
   • Field current $I_f$ is adjusted such that stator current $\mathbf{I}_a$ is in phase with $\mathbf{V}_p$ (unity power factor, $\cos \theta = 1.0$).
   • Armature current magnitude $I_a$ reaches its absolute minimum for the given active power load.
3. Over-Excitation ($E_p \cos \delta > V_p$):
   • Excessive rotor field flux produces a back-EMF exceeding terminal voltage requirements. To counter this, the stator draws leading current (supplying $-Q$ into the line).
   • Armature current $\mathbf{I}_a$ leads terminal voltage $\mathbf{V}_p$ (motor behaves like a capacitive bank).

V-Curves & Inverted V-Curves

• V-Curves: Plots of armature current $I_a$ vs field current $I_f$ for various constant load power levels ($0%, 50%, 100%$ full load). The curves form characteristic "V" shapes, with the minimum point representing unity power factor.
• Inverted V-Curves: Plots of power factor $\cos \theta$ vs field current $I_f$, forming inverted "$\Lambda$" peak curves reaching $1.0$ at normal excitation.

Industrial Application: Synchronous Condensers

An over-excited synchronous motor operating completely un-loaded (zero shaft output power, $P \approx 0$) acts purely as a adjustable 3-phase capacitive bank supplying leading reactive kVAR to the power system. In industrial plants dominated by lagging induction motors, synchronous condensers are installed to improve overall plant power factor to $0.95\text{ lagging}$ or unity, avoiding utility low-power-factor penalties and reducing transmission line losses.

$\text{Plant } Q_{\text{new}} = Q_{\text{load}} - Q_{\text{sync}} = P_{\text{plant}} \tan \theta_{\text{target}}$

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Developed Mechanical Power, Torque & Pull-Out Limits

The internal gross mechanical power developed per phase ($P_{\text{mech,p}}$) by the rotor converted from electrical power is:

$P_{\text{mech,3}\phi} = 3 E_p I_a \cos(\theta + \delta) = \frac{3 V_p E_p}{Z_s} \cos(\theta_s - \delta) - \frac{3 E_p^2 R_a}{Z_s^2}$

where $\theta_s = \arctan\left(\frac{X_s}{R_a}\right)$ is the synchronous impedance angle and $\delta$ is the power/torque angle.

When armature resistance is negligible ($R_a \approx 0, Z_s \approx X_s, \theta_s \approx 90^\circ$):

$P_{\text{mech,3}\phi} = \frac{3 V_p E_p}{X_s} \sin \delta$

$T_{\text{dev}} = \frac{P_{\text{mech,3}\phi}}{\omega_s} = \frac{3 V_p E_p}{\omega_s X_s} \sin \delta$

Breakdown (Pull-Out) Torque Limit

As mechanical load on the motor shaft increases, the torque angle $\delta$ retards (increases in magnitude). Maximum developed mechanical power occurs at:

$\text{For } R_a = 0: \quad \delta_{\text{max}} = 90^\circ$

$\text{For non-zero } R_a: \quad \delta_{\text{max}} = 180^\circ - \theta_s = 90^\circ + \alpha \quad \left( \text{where } \tan \alpha = \frac{R_a}{X_s} \right)$

$P_{\text{mech,max}} = 3 \left[ \frac{V_p E_p}{Z_s} - \frac{E_p^2 R_a}{Z_s^2} \right]$

If shaft load torque exceeds the maximum pull-out torque, magnetic locking snaps, the rotor loses synchronism, and heavy pulsing currents flow until protection relays trip the machine.

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Starting Mechanics & Hunting Suppression

The Starting Problem

At standstill ($n_r = 0$), the stator RMF sweeps past the stationary rotor poles at full synchronous speed ($3600\text{ rpm}$ or $1800\text{ rpm}$). Within one half-cycle ($1/120\text{ s}$ at $60\text{ Hz}$), the torque direction reverses from forward to backward. Owing to high rotor mass inertia, the average starting torque is strictly zero ($T_{\text{start}} = 0$).

Practical Starting Methods

1. Amortisseur (Damper) Windings: Heavy copper bars embedded in rotor pole faces and short-circuited by end-rings (identical to a squirrel-cage rotor). The motor is started un-excited as an induction motor. When speed accelerates to $95% - 98%$ of synchronous speed, DC field excitation is energized, creating magnetic pull that locks the rotor into synchronism.
2. Auxiliary Pony Motor: A small induction motor coupled to the main shaft accelerates the rotor to synchronous speed before applying DC excitation and closing the main stator breaker.
3. Variable Frequency Drive (VFD): The stator frequency $f$ is started at $0\text{ Hz}$ and smoothly ramped up, keeping stator RMF locked to the rotor from standstill.

The Hunting Phenomenon & Damper Winding Action

When mechanical load suddenly changes or grid frequency fluctuates, the rotor torque angle $\delta$ overshoots its new steady-state equilibrium value, causing the rotor to oscillate back and forth around $\omega_s$. This dynamic instability is called hunting.

Damper Winding Hunting Suppression: During steady-state synchronous rotation ($n_r = n_s$), damper bars rotate at the exact speed of the RMF, so no EMF or current is induced in them. When hunting occurs ($n_r \neq n_s$), relative motion between the damper bars and the RMF induces heavy eddy currents in the damper winding. By Lenz's law, these currents create damping torque that rapidly dissipates oscillation energy as $I^2 R$ heat, suppressing hunting within a few cycles.

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Special AC Machines & Drive Systems