# 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

Where;

• 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)

Where;

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.

Related Posts: ### 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.

i.e.

V = Eb + IaRa        …..        (1)

Where

• 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)

Where,

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

Related Posts:

### Shunt Motor:

#### Voltage Equation of Shunt Motor:

V = E+ Ia x Ra

Where

• 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

Where

• 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

Where

• 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:

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

Related Posts:

#### Speed Regulation:

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

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

#### Input & Output Power:

Pin = VIa

Pout = T ω

Where

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

Related Posts:

### Series Motor:

#### Voltage Equation Of Series Motor:

V = E+ Ia Ra + IaRse

V = E+ Ia(Ra+Rse)

Where

• 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

Where

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

#### Speed of 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

Related Posts:

### 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 #### Mechanical Efficiency:

η= Converted power in armature / output mechanical power #### Overall Efficiency:

η = Output mechanical Power / Input electrical Power
η = (Input Power – Total losses) / Input Power Where

• 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

Related Formulas and Equations Posts:

### Electrical Technology

1. R. Prabhakar Rao says:

I need power knowledge for the selection of motors and VFD.

2. Ganesh says:

Hello sir I want the full equations for DC Generator, Motors and transformers as well…..

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