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

Electrical Technology

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_aR_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_aR_a

k = N = V – I_aR_a / kΦ

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

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_aR_a drop.

i.e.

V = E_b + I_aR_a ….. (1)

Where

E_b= Back E.M.F
I_aR_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_bI_a+ I_a² R_a ….. (2)

Where,

VI_a= Input Power supply (Armature Input)
E_bI_a= Mechanical Power developed in Armature (Armature Output)
I_a² R_a = Power loss in armature (Armature Copper (Cu) Loss)

Voltage Equation of Shunt Motor:

V = E_b+ I_a x R_a

Where

V is the terminal voltage
E_bis the induced back e.m.f
I_ais the armature current
R_ais the armature resistance

The Shunt Field Current:

I_sh= V / R_sh

Where

I_shis the shunt field current
R_shis the shunt field resistance

Induced Back EMF:

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

E_b= k_fΦω

Where

K_fis 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:

And

Where

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

Speed Regulation:

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

Where

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

Voltage Equation Of Series Motor:

V = E_a+ I_a R_a + I_aR_se

V = E_a+ I_a(R_a+R_se)

Where

E_ais the armature induced voltage
I_ais the armature current
R_ais the armature resistance
R_se is the series field resistance

Armature Induced Voltage & Torque:

The armature induced voltage E_ais 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_fΦωI_a

T_a = k_fΦ I_a²

Where

K_fis 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

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_out is the useful output power
P_ais the armature copper loss
P_f is the field copper loss
P_k is the constant losses that contains core losses & mechanical losses