Synchronous, Stepper and AC Motors Formulas and Equations

AC Machine Formulas & Equations for Stepper and Synchronous Motors

Following are the useful equations and AC motors formulas while designing and analyzing synchronous motors, stepper motors and other related AC machines.

Synchronous Machine:

Speed of Synchronous Machine:

Synchronous machine are designed to be operated at synchronous speed, which is given by:

Where

• N­_s is the synchronous speed
• f is the line voltage frequency
• P is the number of poles in a machine

Synchronous Motor:

Voltage Equation of Synchronous Motor:

V = E_b + I_a(R_a + jX_s)

Where

• V = voltage applied
• E_b = Back emf
• I_a = Armature current
• R_a = Armature resistance
• X_s = synchronous reactance

Resultant Voltage:

The difference between the voltage applied V and back EMF is known as resultant voltage E_R

E_R = V – E_b

E_R = I_a(R_a + jX_s)

Internal Angle:

It is the angle by which the armature current I_a lags behind the resultant voltage in armature E_R, and it is given by;

Back EMF Generated:

E_b = K_aφ_aN_s

Where

• K_a = constant of the armature winding
• φ_a = magnetic Flux per pole of the rotor
• N_s = synchronous speed of the rotor

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Different Excitations:

• E_b = V             Normal Excitation                            Lagging Power Factor
• E_b < V             Under-Excitation                              Lagging Power Factor
• E_b > V             Over- Excitation                                Leading Power Factor

Input Power:

The input power of synchronous motor is given by:

Where

Φ is the angle between V and I_a

Mechanical Power In Rotor:

Where

• α is the load angle between E­_b & V
• Φ is the angle between V & I_a
• T_g is gross torque produced
• N_s is the synchronous speed

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Stepper Motors Formulas

Step Angle:

Where

• β = step angle, the angle of rotation of the shaft with each pulse.
• N_s = number of stator poles or teeth
• N_r = number of rotor poles or teeth

Resolution Of Stepper Motor:

The number of steps required to complete one revolution, its given by;

The higher the resolution, the higher the accuracy of the stepper motor.

Motor Speed:

Where

• n = motor speed in revolution per second
• f = stepping pulse frequency

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