EMF and Torque Equation of DC Machine — Derivation & Significance
The EMF equation and torque equation are the two fundamental expressions that govern the operation of every DC machine — whether it's a generator producing voltage or a motor developing mechanical force. Understanding these equations helps you predict machine performance, design armature windings, and select the right DC machine for any application.
EMF Equation of DC Machine
When the armature of a DC machine rotates in a magnetic field, an EMF is induced in the armature winding according to Faraday's law of electromagnetic induction. In a DC generator, this is called the generated EMF. In a DC motor, it is called the back EMF. The expression is identical in both cases.
Where:
- E = Induced EMF (Volts)
- Φ = Useful flux per pole (Webers)
- N = Speed of armature rotation (RPM)
- P = Total number of poles
- Z = Total number of armature conductors
- A = Number of parallel paths (A = P for lap winding, A = 2 for wave winding)
Derivation of EMF Equation
Let the armature rotate at n = N/60 revolutions per second. Since flux per pole is Φ, each conductor cuts a total flux of P × Φ in one complete revolution.
EMF per conductor = P × Φ / (1/n) = n × P × Φ
The number of armature conductors in each parallel path = Z/A. The total EMF is determined by conductors connected in series in any one path:
E = n × P × Φ × (Z/A)
E = (N/60) × P × Φ × Z / A
E = (Φ × N × P × Z) / (60 × A)
Significance of EMF Equation
- For a given machine, P, Z, and A are constants. Therefore E ∝ Φ × N — EMF is directly proportional to flux and speed.
- In a generator, increasing field current (Φ) or prime mover speed (N) increases the generated voltage.
- In a motor, back EMF opposes the supply voltage and regulates armature current automatically.
- The equation helps determine the number of conductors needed during armature design.
Torque Equation of DC Machine
When a current-carrying armature conductor is placed in a magnetic field, it experiences a force (Lorentz force) that produces torque on the armature shaft. The torque equation of a DC machine is:
T = 0.159 × Z × Φ × P × Ia / A N·m
Where:
- T = Torque developed by armature (N·m)
- Ia = Armature current (Amperes)
- Φ = Flux per pole (Wb)
- P, Z, A = Same as in EMF equation
Derivation of Torque Equation
Let B = Φ/a be the flux density, where a = cross-sectional area of flux path. Current in each conductor i = Ia/A. The force on one conductor:
Torque due to one conductor = F × r = B × i × l × r. For Z total conductors:
The cross-sectional area of the flux path at radius r is: a = (2πrl) / P
Substituting:
T = (Z × Φ × P × Ia) / (2π × A)
T = 0.159 × Z × Φ × P × Ia / A N·m [since 1/2π = 0.159]
Torque in Series vs Shunt Machines
Since P, Z, and A are machine constants:
- DC Series Motor: Flux Φ is proportional to armature current (before saturation). Therefore T ∝ Ia² — gives high starting torque, ideal for traction and cranes.
- DC Shunt Motor: Flux Φ is practically constant (separate field winding). Therefore T ∝ Ia — gives linear torque-current relationship, ideal for constant-speed applications.
EMF vs Torque — Key Differences
Numerical Example
Problem: A 4-pole DC motor has 500 armature conductors, wave winding (A=2), flux per pole = 0.03 Wb, and runs at 1200 RPM with armature current of 20 A. Find the back EMF and torque developed.
E = (0.03 × 1200 × 4 × 500) / (60 × 2)
E = 72000 / 120 = 600 V
T = 0.159 × Z × Φ × P × Ia / A
T = 0.159 × 500 × 0.03 × 4 × 20 / 2
T = 95.4 N·m
Verification using power method: Power = E × Ia = 600 × 20 = 12000 W. Torque = P / ω = 12000 / (2π × 1200/60) = 12000 / 125.66 = 95.5 N·m ✓
Frequently Asked Questions
1. What is the difference between generated EMF and back EMF?
Generated EMF is the voltage produced in a DC generator's armature. Back EMF is the voltage induced in a DC motor's armature that opposes the supply voltage. Both use the same formula E = ΦNPZ/60A, but their role differs — generated EMF drives current to the load, while back EMF limits armature current in a motor.
2. Why is torque proportional to Ia² in a series motor?
In a DC series motor, the field winding carries the full armature current, so flux Φ is directly proportional to Ia (before magnetic saturation). Since T ∝ Φ × Ia, substituting Φ ∝ Ia gives T ∝ Ia². This quadratic relationship provides very high starting torque.
3. What happens to EMF if the number of parallel paths increases?
EMF is inversely proportional to A (number of parallel paths). A lap winding (A = P) produces lower EMF but higher current capacity compared to a wave winding (A = 2) which produces higher EMF but lower current per path.
4. Can torque be developed without armature current?
No. The torque equation T = 0.159 × ZΦPIa/A clearly shows that torque is zero when armature current Ia = 0. A current-carrying conductor in a magnetic field is essential for force production (Lorentz force principle).
5. How are EMF and torque equations related to each other?
They are connected through power balance. In a motor: Mechanical power = E × Ia = T × ω. Therefore T = E × Ia / ω. Both equations share the same machine constants (P, Z, A, Φ), confirming that a machine capable of generating high EMF will also develop proportionally high torque.