# 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

**E _{b} = V – I_{a}R_{a} ….. (2)**

Where;

- V is the supply voltage
- I
_{a}is the Armature current - R
_{a}is the Armature resistance

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

**kΦN = V – I_{a}R_{a}**

**k = N = V – I _{a}R_{a} / 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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**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 “E
_{b}” of the Motor. - Provides supply to the Ohmic I
_{a}R_{a}drop.

i.e.

**V = E _{b} + I_{a}R_{a} ….. (1)**

Where

- E
_{b }= Back E.M.F - I
_{a}R_{a }= 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 I_{a} , we get the power equation of a DC motor as follow.

**VI _{a}= E_{b}I_{a }+ I_{a}^{2} R_{a} ….. (2)**

Where,

**VI**Input Power supply (Armature Input)_{a }=**E**Mechanical Power developed in Armature (Armature Output)_{b}I_{a }=**I**Power loss in armature (Armature Copper (Cu) Loss)_{a}^{2}R_{a}=

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**Shunt Motor:**

**Voltage Equation of Shunt Motor:**

**V = E _{b }+ I_{a} x R_{a}**

Where

- V is the terminal voltage
- E
_{b }is the induced back e.m.f - I
_{a }is the armature current - R
_{a }is the armature resistance

**The Shunt Field Current:**

**I _{sh }= V / R_{sh}**

Where

- I
_{sh }is the shunt field current - R
_{sh }is the shunt field resistance

**Induced Back EMF:**

The armature induced voltage E_{b }is proportional to the speed & it is given by:

**E _{b }= k_{f}Φω**

Where

- K
_{f }is 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.

**E _{b} = 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

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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.

- N
_{nl }= No load speed of the motor - N
_{fl }= Full load speed of the motor

**Input & Output Power:**

**P _{in }= VI_{a}**

**P _{out }= T**

**ω**

Where

- 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 _{a }+ I_{a} R_{a} + I_{a}R_{se}**

**V = E _{a }+ I_{a}(R_{a}+R_{se})**

Where

- E
_{a }is the armature induced voltage - I
_{a }is the armature current - R
_{a }is the armature resistance - R
_{se}is the series field resistance

**Armature Induced Voltage & Torque:**

The armature induced voltage E_{a }is proportional to the speed and armature current whereas the torque T_{a} of series motor is directly proportional to the square of armature current & it is given by:

**E _{a }= k_{f}ΦωI_{a}**

**T _{a} = k_{f }Φ I_{a}^{2}**

Where

- K
_{f }is 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:

**P _{in }= VI_{a}**

The output power is given by

**P _{out }= 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_{e }= Converted power in armature / Input electrical Power

**Mechanical Efficiency:**

η_{m }= Converted power in armature / output mechanical power

**Overall Efficiency:**

η = Output mechanical Power / Input electrical Power

η = (Input Power – Total losses) / Input Power

Where

**P**is the useful output power_{out}**P**_{a}_{ }is the armature copper loss**P**is the field copper loss_{f}**P**is the constant losses that contains_{k}**core losses**&**mechanical losses**

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

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