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Power, Voltage and EMF Equation of a DC Motor – Formulas

DC Motors Formulas and Back EMF Equation

EMF Equation of a DC Motor

The basic DC motor’s E.M.F equation is given below.

Eb = PΦNZ  / 60A


  • P is the number of poles
  • Ф is the Flux per pole
  • N is the Speed of motor in (RPM)
  • Z is the Number of conductors
  • A is the Number of parallel paths

In a final designed motor, the number of poles “P”, conductors “Z” and parallel paths “A” are fixed, therefore, the following quantities and parameters remains constant.

Eb ∝ ΦN

Eb = kΦN        …..        (1)

Where k is the Proportionality constant

The back EMF of DC motor equation can also be defined as 

Eb = V – IaRa        …..        (2)


Now compare both equations of (1) and (2);

kΦN = V – IaRa

k = N = V – IaRa / kΦ

The above relation shows that the speed of a DC motor can be controlled through change in voltage, flux and armature resistance.

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Power, Voltage and EMF Equation of a DC Motor -Formulas and Equations for DC Motors

Voltage Equation of a DC Motor

Input Voltage provided to the motor armature performs the following two tasks:

  • Controls the induced Back E.M.F “Eb” of the Motor.
  • Provides supply to the Ohmic IaRa drop.


V = Eb + IaRa        …..        (1)


  • E= Back E.M.F
  • IaRa   = Armature Current X Armature Resistance

The above relation is known as “Voltage Equation of the DC Motor”.

Power Equation of a DC Motor

Multiplying both sides of Voltage Equation (1) by Ia , we get the power equation of a DC motor as follow.

VIa= EbI+ Ia2 Ra        …..        (2)


  • VIInput Power supply (Armature Input)
  • EbIMechanical Power developed in Armature (Armature Output)
  • Ia2 RaPower loss in armature (Armature Copper (Cu) Loss)

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Shunt Motor:

Voltage Equation of Shunt Motor:

V = E+ Ia x Ra


  • V is the terminal voltage
  • Eis the induced back e.m.f
  • Iis the armature current
  • R­­­is the armature resistance

The Shunt Field Current:

sh ­= V / Rsh


  • Ish is the shunt field current
  • Rsh ­is the shunt field resistance

Induced Back EMF:

The armature induced voltage Eis proportional to the speed & it is given by:

E= kfΦω


  • Kis a constant based on machine construction
  • Φ is the magnetic flux
  • ω is the angular speed

Maximum Power Condition:

The output mechanical power is of shunt dc motor is maximum when the back e.m.f. produced is equal to the half of its terminal voltage i.e.

Eb = V/2

Torque & Speed:

Torque & Speed of DC shunt Motor
Machine constant

  • N = speed of the motor in RPM
  • P = No of poles
  • Z = number of armature conductors
  • A = number of armature parallel path

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Speed Regulation:

It is a term expressed in percentage that shows the change of motor speed when the load is changed.

Speed Regulation of DC Shunt Motor

  • nl = No load speed of the motor
  • Nfl = Full load speed of the motor

Input & Output Power:

Pin = VIa

Pout = T ω


  • V = terminal voltage
  • I­a = armature current
  • T = torque of the motor
  • ω = speed of the motor

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Series Motor:

Voltage Equation Of Series Motor:

V = E+ Ia Ra + IaRse

V = E+ Ia(Ra+Rse)


  • is the armature induced voltage
  • Iis the armature current
  • R­­­is the armature resistance
  • Rse is the series field resistance

Armature Induced Voltage & Torque:

The armature induced voltage Eis proportional to the speed and armature current whereas the torque Ta of series motor is directly proportional to the square of armature current & it is given by:

E= kfΦωIa

Ta = kΦ Ia2


  • Kis a constant based on machine construction
  • Φ is the magnetic flux
  • ω is the angular speed

Speed of Series Motor:

Speed of DC series motor

Input & Output Power

The input power of a series motor is given by:

Pin = VIa

The output power is given by

Pout = ωT

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Efficiency Of DC Motor:

The different motor efficiencies can be found by the following formulas and equations

Electrical Efficiency:

η=  Converted power in armature / Input electrical Power
Electrical efficiency of DC motor

Mechanical Efficiency:

η= Converted power in armature / output mechanical power
Mechanical Efficiency of DC Motor

Overall Efficiency:

η = Output mechanical Power / Input electrical Power
η = (Input Power – Total losses) / Input Power
Overall Efficiency of DC Motor

  • Pout is the useful output power
  • Pa ­­is the armature copper loss
  • Pf is the field copper loss
  • Pk is the constant losses that contains core losses & mechanical losses

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