16.3 Three-Phase & Single-Phase Induction Motors

16.3 Three-Phase & Single-Phase Induction Motors

Operating Principles & Slip Mechanics

The three-phase induction motor is the most widely utilized electric motor in modern industrial plants due to its rugged self-starting construction, high reliability, and low cost. It operates on the principle of electromagnetic induction without requiring electrical connections to the rotor.

Creation of Stator Rotating Magnetic Field (RMF)

When balanced 3-phase AC currents flow through stator windings physically displaced by $120^\circ$ electrical around the stator frame, they establish a rotating magnetic field of constant magnitude $B_{\text{net}} = 1.5 B_m$ rotating at synchronous speed:

$$n_s = \frac{120 f}{P} \quad [\text{rpm}]$$

Induction of Rotor Current & Slip Definition

The relative motion between the stator RMF ($n_s$) and the stationary rotor conductors ($n_r = 0$) cuts rotor bars, inducing an electromotive force ($e_2$) by Faraday's law. Because rotor bars are short-circuited by end-rings, heavy rotor currents ($i_2$) flow. Interaction between $i_2$ and the RMF produces electromagnetic torque ($T_e$) accelerating the rotor in the direction of the RMF (Lenz's law).

The fractional difference between synchronous speed $n_s$ and actual rotor speed $n_r$ is defined as slip ($s$):

$$s = \frac{n_s - n_r}{n_s} \quad \implies \quad n_r = (1 - s) n_s$$

• At standstill ($n_r = 0$): Slip $s = 1.0$ ($100\%$ slip).
• At synchronous speed ($n_r = n_s$): Slip $s = 0$. (Flux cutting ceases, $e_2 = 0, i_2 = 0, T_e = 0$; hence an induction motor can never reach $n_s$).
• Rotor frequency: $f_r = s f$.
• Running induced rotor EMF: $E_{2r} = s E_2$.
• Running rotor reactance: $X_{2r} = s X_2$.

---

Per-Phase Equivalent Circuit & Power Flow Balance

To analyze performance, the rotor circuit operating at frequency $f_r$ is transformed to the primary stator frequency $f$ by dividing the rotor parameters by $s$:

$$\mathbf{I}_2' = \frac{E_2'}{\sqrt{\left(\frac{R_2'}{s}\right)^2 + (X_2')^2}}$$

The total equivalent rotor resistance is split into two components:

$$\frac{R_2'}{s} = R_2' + \left( \frac{1 - s}{s} \right) R_2'$$

• $R_2'$ represents actual physical rotor winding $I^2 R$ copper losses.
• $\left( \frac{1 - s}{s} \right) R_2'$ represents the electrical equivalent of the mechanical shaft load.

                  INDUCTION MOTOR PER-PHASE EQUIVALENT CIRCUIT

        I1 ───►   R1         X1                  R'2        X'2       ───► I'2
     ───────────█████──────UUUUUU───────┬───────█████──────UUUUUU──────────────┐
                                        │                                      │
                                   I0   │                                     ███  [(1-s)/s] R'2
                                    ┌───┴───┐                                 ███  (Mechanical
                                 Ic │       │ Im                               │    Load)
                                   ███     UUU                                 │
                                Rc ███  Xm UUU                                 │
                                    │       │                                  │
     ───────────────────────────────┴───────┴──────────────────────────────────┴┘

Stages of Power Flow in 3-Phase Induction Motors

1. Stator Electrical Input Power: $P_{\text{in}} = \sqrt{3} V_L I_L \cos \theta$
2. Stator Losses: Stator Copper Loss $P_{\text{cu1}} = 3 I_1^2 R_1$, Stator Core Loss $P_i$
3. Air-Gap Power ($P_{\text{ag}}$): Power transferred electromagnetically from stator to rotor across the air gap: $$P_{\text{ag}} = P_{\text{in}} - P_{\text{cu1}} - P_i = 3 (I_2')^2 \frac{R_2'}{s}$$
4. Rotor Copper Loss ($P_{\text{cu2}}$): Internal ohmic heating in rotor conductors: $$P_{\text{cu2}} = 3 (I_2')^2 R_2' = s P_{\text{ag}}$$
5. Gross Mechanical Power Developed ($P_{\text{mech}}$): Power converted into rotational mechanical form: $$P_{\text{mech}} = P_{\text{ag}} - P_{\text{cu2}} = (1 - s) P_{\text{ag}} = 3 (I_2')^2 R_2' \left( \frac{1 - s}{s} \right)$$
6. Net Shaft Power ($P_{\text{shaft}}$): Output power available at the mechanical shaft coupling: $$P_{\text{shaft}} = P_{\text{mech}} - P_{\text{rotational}} \quad (P_{\text{rotational}} = P_{\text{friction}} + P_{\text{windage}} + P_{\text{stray}})$$

Fundamental Power Ratio Mandate: $P_{\text{ag}} : P_{\text{cu2}} : P_{\text{mech}} = 1 : s : (1 - s)$ Rotor Conversion Efficiency: $\eta_{\text{rotor}} = \frac{P_{\text{mech}}}{P_{\text{ag}}} = 1 - s$

---

Torque-Slip Characteristics & NEMA Motor Design Classes

The developed electromagnetic torque as a function of slip $s$ is:

$$T_e = \frac{P_{\text{ag}}}{\omega_s} = \frac{3}{\omega_s} \left[ \frac{V_1^2 \left(\frac{R_2'}{s}\right)}{\left(R_1 + \frac{R_2'}{s}\right)^2 + (X_1 + X_2')^2} \right]$$

                    3-PHASE INDUCTION MOTOR TORQUE-SLIP CURVE

    Torque T ▲
             │                   Breakdown / Peak Torque Tmax
             │                        ┌───────┐
             │                       ╱         ╲
             │                      ╱           ╲
             │                     ╱             ╲
   Tstart ───┼─► █                ╱               ╲
             │   █               ╱                 ╲   Low-Slip Normal
             │   █              ╱                   ╲  Operating Region (Linear)
             │   █             ╱                     ───────────
             └───┴────────────┴──────────────────────────┴──────────────► Slip s
             s=1.0          s = s_max                  s=0
          (Standstill)    (Breakdown Slip)         (Synchronous Speed)

Peak Breakdown (Pull-Out) Torque ($T_{\text{max}}$)

By differentiation $\frac{dT_e}{ds} = 0$, maximum torque and its corresponding breakdown slip $s_{\text{max}}$ are:

$$s_{\text{max}} = \frac{R_2'}{\sqrt{R_1^2 + (X_1 + X_2')^2}} \approx \frac{R_2'}{X_1 + X_2'}$$ $$T_{\text{max}} = \frac{3}{2 \omega_s} \left[ \frac{V_1^2}{R_1 + \sqrt{R_1^2 + (X_1 + X_2')^2}} \right] \approx \frac{3}{2 \omega_s} \left[ \frac{V_1^2}{2(X_1 + X_2')} \right]$$

Critical Engineering Principle: Peak breakdown torque magnitude $T_{\text{max}}$ is completely independent of rotor resistance $R_2'$. However, the slip $s_{\text{max}}$ at which peak torque occurs is directly proportional to $R_2'$. Increasing rotor resistance shifts the peak torque toward standstill ($s = 1.0$), augmenting starting torque without altering maximum torque capacity.

NEMA Motor Design Classifications

---

3-Phase Motor Starting Methods

Across starting, locked-rotor currents reach $500\% - 800\%$ of rated full-load current. Standard starting schemes reduce line voltage or adjust impedance:

1. Direct-On-Line (DOL) Starting: $100\%$ full voltage applied. Max starting torque ($100\% T_{\text{start,DOL}}$), but severe inrush current.
2. Star-Delta ($\text{Y-}\Delta$) Starter: Motor windings are connected in Star during starting, then switched to Delta during running. $$V_{\text{phase,Y}} = \frac{V_{\text{line}}}{\sqrt{3}} \quad \implies \quad I_{\text{line,start,Y}} = \frac{1}{3} I_{\text{line,start,DOL}}, \quad T_{\text{start,Y}} = \frac{1}{3} T_{\text{start,DOL}}$$
3. Autotransformer Starter: Taps provide reduced fraction $k$ (e.g. $k = 0.80$ or $80\%$) of line voltage. $$I_{\text{motor,start}} = k I_{\text{DOL}}, \quad I_{\text{line,start}} = k^2 I_{\text{DOL}}, \quad T_{\text{start}} = k^2 T_{\text{start,DOL}}$$
4. Rotor Resistance Starting (Wound-Rotor Motors): External resistors connected to rotor slip rings add $R_{\text{ext}}'$ to rotor circuit, setting $s_{\text{max}} = 1.0$ to achieve $T_{\text{start}} = T_{\text{max}}$ while maintaining low starting current.

---

Single-Phase Induction Motors & Starting Topologies

Single-phase induction motors are widely used in home appliances, HVAC, and commercial equipment where 3-phase power is unavailable.

Double-Revolving Field Theory

A single-phase pulsating magnetic flux $B(t) = B_m \cos(\omega t)$ does not rotate naturally. According to Ferrari's double-revolving field theory, a pulsating field resolves into two counter-rotating fields of equal magnitude $B_f = B_b = \frac{B_m}{2}$ rotating at $+n_s$ and $-n_s$.

$$s_f = s, \quad s_b = 2 - s$$

At standstill ($n_r = 0, s = 1.0$), forward slip $s_f = 1.0$ and backward slip $s_b = 1.0$. The forward torque and backward torque are exactly equal and opposite ($T_f = T_b$), resulting in zero net starting torque ($T_{\text{start}} = 0$).

                  SINGLE-PHASE INDUCTION MOTOR STARTING TOPOLOGIES

   (A) Split-Phase               (B) Capacitor-Start            (C) Permanent-Split Capacitor
     Main Winding                  Main Winding                    Main Winding
   ┌───███████───┐               ┌───███████───┐                 ┌───███████───┐
   │             │               │             │                 │             │
 L ├─████─┬─  ──┤ N            L ├─████─┬─  ──┤ N              L ├─████─┬─  ──┤ N
   │ Aux  │ Centrifugal          │ Aux  │ Centrifugal            │ Aux  │
   │ Wind └─o/ o─┘               │ Wind ├──┤├───o/ o─┘           │ Wind ├──┤├──┐
   │                             │      │ C_start                │      │ C │
   └─────────────────────────────┴──────┴────────────────────────┴──────┴───┴──┘

Single-Phase Starting Methods

1. Split-Phase Motor: Aux winding (high resistance, low reactance) in parallel with main winding (low resistance, high reactance) creates a $\sim 30^\circ$ phase shift. Centrifugal switch opens aux winding at $75\%$ speed. Moderate starting torque.
2. Capacitor-Start Motor: Heavy-duty electrolytic capacitor in series with aux winding creates a near $90^\circ$ phase shift. Centrifugal switch disconnects aux circuit at $75\%$ speed. High starting torque ($300\% - 450\%$).
3. Permanent-Split Capacitor (PSC) Motor: Run capacitor remains permanently in series with aux winding. No centrifugal switch. Quiet operation, low starting torque, high running power factor.
4. Two-Value (Capacitor-Start, Capacitor-Run) Motor: Employs large start capacitor for initial high torque and small run capacitor for high running efficiency.
5. Shaded-Pole Motor: Salient poles fitted with a copper shorting ring (shading coil) across part of each pole face. Delays flux in shaded segment, creating a weak rotating field. Lowest starting torque and efficiency ($20\% - 40\%$); used in small desk fans and blowers.