Direct-Current Motors – Comprehensive Study Notes

Page 1

• Chapter reference: Theodore Wildi, Electrical Machines, Drives, and Power Systems, 6ᵗʰ Ed., Ch. 5.
• Focus: Direct-Current (DC) motors – conversion of electrical → mechanical energy.

Page 2

• Loads driven: hoists, fans, pumps, calendars, punch-presses, automobiles.
• Load types: fixed torque-speed characteristic (fan/pump) vs. highly variable (hoist/car).
• Three basic DC-motor types (classified by field connection and resulting torque-speed profile):

1. Shunt (parallel)

2. Series

3. Compound (cumulative or differential).
   • Despite AC utility supply, DC motors remain attractive where wide, efficient speed control is needed—steel mills, mines, electric traction, elevators.
   • Modern electronic drives blur AC/DC preference but millions of DC machines are still in service.

Page 3 – Counter-Electromotive Force (cemf) & Acceleration

• Any DC machine can act as generator or motor.
• Closing a DC source $Es$ across a stationary armature (resistance $R$) produces a high inrush current. • As armature conductors cut the field, an induced voltage $E</em>o = \frac{Z n \Phi}{60}$ appears (generator effect).
– $Z$ = armature conductors (lap: equal to number of conductors)
– $n$ = r/min, $\Phi$ = flux/pole (Wb).
• Polarity of $Eo$ opposes the source: hence “counter-emf”. • Net armature current $I = \frac{E</em>s - E<em>o}{R}$ (Eq 1). – At stand-still $Eo=0 \Rightarrow I\text{start}=Es/R$ (can be 20-30 × rated).
• Rising speed ↑ $E_o$ ↓ $I$ → acceleration continues until developed torque = load torque (steady-state).

Page 4 – Example 1

Given: $R = 1\,\Omega$, generator 50 V @ 500 r/min → proportional $Eo$. Connected to 150 V supply. • Start: $I = 150/1 = 150$ A. • $Eo$ values: 100 V @ 1000 r/min, 146 V @ 1460 r/min.
• $I{1000} = 50$ A; $I{1460} \approx 4$ A → torque drops sharply at light/no-load.

Page 5 – Power & Torque Equations

• Electrical input $Pa = Es I$. With $E<em>s = E</em>o + I R$ ⇒ $P<em>a = E</em>o I + I^2 R$
– $I^2 R$ = armature copper loss; $Eo I$ = converted to mechanical. • Mechanical output $P = E</em>o I$ (Eq 5).
• Torque–current–flux relation (lap-wound):
$T = \frac{Z \Phi I}{6.28}$ (Eq 6).
– Increase torque by ↑ $I$ or ↑ $\Phi$.
• Example 2: 225 kW, 250 V, 1200 r/min motor.
– Rated $I = 900$ A ($P/E$).
– Conductors/slot: 6.
– Flux/pole $\Phi = 25.7$ mWb (via Eq 6 with $T = 1791$ N·m).

Page 6 – Speed Equation

• For negligible $I R$, $Eo \approx Es$.
• Substitute into $Eo = Z n \Phi /60$ ⇒ $n = \frac{60 E</em>s}{Z \Phi}$ (Eq 7).
– Speed ∝ armature voltage; ∝ 1/flux.
• Ward-Leonard system (Fig 6): variable-voltage generator supplies motor armature; allows 0 → rated speed in both directions with regenerative capability. Generator motorized by AC supply; modern equivalent: AC→DC power electronic converters.

Page 7 – Rheostat & Armature Speed Control

• Series rheostat with armature (Fig 7) drops part of $Es$ → lower speed (below base). • Inefficient (heat loss), poor regulation; used only on small machines. • Example 3 (2000 kW, 500 V motor, $R{tot}=10$ mΩ):
a) $Es=400$ V, $Eo=380$ V ⇒ $I=2000$ A, $T=31.8$ kN·m, $n=228$ r/min.
b) $Es=350$ V (still $Eo=380$ V) ⇒ current –3000 A (reverse), braking torque 47.8 kN·m (motor acts as generator).

Page 8 – Field (Flux) Control

• Keep $Es$ constant; vary field rheostat $Rf$ to weaken $\Phi$ → speed ↑ (above base).
• Limit: ~3 : 1 speed range for shunt motors; excessive weakening risks run-away (only remanent flux left).
• Safety interlocks necessary.

Page 9 – Shunt Motor Characteristics & Example 4

• Under load: ↑ load → ↓ speed slightly (typically 10–15 % drop small motors; < 5 % large).
• Torque ∝ $I$; speed remains ~constant (Fig 8b).
• Example 4: 120 V source, 0.1 Ω armature.
– $Ix = 1$ A; $I = 50$ A. – $Eo = 115$ V.
– $P_{mech} = 5.75$ kW (≈ 7.7 hp).

Page 10 – Series Motors

• Series field in series with armature (few heavy turns).
• $\Phi \propto I$ ⇒ starting current high → very large starting torque.
• Light load → small $I$ → small $\Phi$ → speed rises dangerously (never run no-load).
• T–n & T–I curves (Fig 11): torque ~ $I^2$ at low currents (unsaturated) then ~linear; speed falls rapidly with torque.

Page 11 – Series Motor Speed Control & Example 5

• Increase speed: shunt a resistor across series field (diverts current → weaker $\Phi$).
• Decrease speed: insert resistor in series with entire circuit.
• Example 5 (15 hp, 240 V, 1780 r/min, full-load 54 A):
– For 24 N·m torque → per-unit torque 0.4 ⇒ speed 1.4 pu (2492 r/min), current 0.6 pu (32.4 A).
– Efficiency 80.5 %.

Page 12 – Applications

• Series: traction (trains, trams), cranes, hoists—need high start torque & variable speed.
• Compound Motors: contain both shunt & series fields; cumulative compound gives better start torque than shunt, limited run-away risk unlike pure series.
– Speed drop 10–30 % from no-load→full-load.
– Differential compounding (series opposes shunt) rarely used (instability).

Page 13 – Reversal & Starting

• Reverse rotation by:
– Reversing armature leads or
– Reversing both fields.
• Starting: high inrush must be limited (1.5–2 × rated).
– Starting rheostat or modern solid-state starters accomplish this.

Page 14 – Face-Plate Starter (Manual)

• Resistive segments $R1$–$R4$ switched out step-by-step as motor accelerates.
• Holding coil (interlock) drops out on supply loss or field loss → returns handle to “off”.

Page 15 – Stopping / Braking

• Problems with large inertia loads; purely coasting may take > 1 h.
• Two electrodynamic methods:

1. Dynamic braking – disconnect armature from supply, connect to resistor: generates reverse torque proportional to $I_2$.

2. Plugging – reverse supply polarity with series resistor: $Eo+Es$ produces large reverse current/torque; must open circuit at zero speed to avoid reverse run-up.
   • Speed–time comparison (Fig 18): plugging stops in ≈ 2 $T_o$; dynamic braking asymptotic.

Page 16 – Dynamic Braking Calculations

• Half-time (mechanical) constant $T<em>o = \frac{J n</em>1^2}{131.5 P<em>1}$ – $J$ kg·m², $n1$ initial r/min, $P1$ initial braking W. – Conventional time constant $T = To/0.693$.
• Example 6: 225 kW, $J=177$ kg·m², $n1=1280$ r/min, $R=0.2$ Ω → $P1=220.5$ kW ⇒ $To=10$ s. – Speed halves every 10 s; down to 20 r/min in 60 s. – Windage-only braking ⇒ $To≈276$ s.
• Plugging (Example 7): with $R=0.4$ Ω, same initial $To$, stopping time $t</em>s = 2 T_o = 20\text{ s}$.

Page 17 – Armature Reaction

• Armature mmf distorts/shifts main flux: neutral zone shifts opposite rotation, causes sparking; flux under one pole tip saturates → net field weakening (~10 % in large machines).
• Mitigation: stabilized shunt winding (few series turns).

Page 18 – Commutating Poles (Interpoles)

• Narrow poles between main poles, series-connected with armature, mmf ≈ armature mmf (slightly higher).
• Provide localized reversing flux that cancels $L\,di/dt$ in commutating coil → spark-free commutation independent of load.

Page 19 – Compensating Windings

• Heavy bars inset in main pole faces and series with armature.
• Produce mmf equal/opposite to armature mmf across entire pole face. Advantages:

1. Shorter air gap → smaller field copper.

2. Armature inductance ↓ 4–5× → rapid current response.

3. Peak torque capability 3–4× rated without flux-weakening.
   • Essential for large (>100 kW) rapidly reversing steel-mill & rolling-mill motors.

Page 20 – Variable-Speed Theory (Per-Unit)

Circuit (Fig 25): rated quantities = 1 pu.
• Constant-torque region (0 ≤ n ≤ 1):
– Keep $\Phi=1$, $Ia=1$, vary $Ea=n$.
– Output torque $T=\Phi I<em>a = 1$. • Constant-horsepower region (n > 1): – Hold $Ea=1$, $I_a=1$, weaken field $\Phi=1/n$.
– Torque $T=1/n$; mechanical power $P = nT =1$ (rated).
• Practical limitations (Fig 29):
– High-speed: commutation & centrifugal stresses.
– Low-speed: cooling losses; if self-ventilated, permissible torque tapers to ~0.25 pu at stall. External blowers restore ideal curve.

Page 21 – Permanent-Magnet (PM) DC Motors

• Replace shunt field with ceramic or rare-earth magnets:
– Higher efficiency (no field copper loss).
– Compact (no field coils/iron).
– Large effective air-gap → minimal armature-reaction distortion, low inductance, excellent dynamic response.
• Ideal < ≈ 5 hp but sizes up to 30 hp exist with NdFeB or SmCo magnets.
• Drawback: cost of magnets; inability to speed-up by field weakening.
• Example unit (Fig 30): 1.5 hp, 90 V, 2900 r/min; armature 73 mm × 115 mm, 20 slots, 40 bars, $R_a(20°C)=0.34\,\Omega$.

Page 22 – Review & Problem Themes

• Topics for practice: motor types, cemf origin, speed control methods (armature vs. field), starting resistors, reversing, braking (dynamic vs plugging), armature reaction, compensating means, variable-speed operation, PM motor temperature effects.