equivalent circuit of induction motor

• R₁ is the stator resistance.

• R₂ is the rotor resistance with respect to the stator.

• L₁ is the stator inductance.

• L₂ is the rotor inductance with respect to the stator.

• L_m is magnetizing inductance.

• s is the rotor slip.

Rotor slip s is defined in terms of the mechanical rotational speed ωm, the number of pole pairs p, and the electrical supply frequency ω by

• V_rms is the line-neutral supply voltage for a star-configuration induction motor, and the line-to-line voltage for a delta-configuration induction motor.

• n is the number of phases.

2.3 phase, 50 Hz induction motor, represented by equivalent circuit constants X1 = X2 = 0.1 ohm and R1 = R2 = 0.2 ohm is operated at half of rated voltage and frequency. Calculate the ratio of starting torque at half voltage & frequency to rated values.

Given:

f=50 hz

V=V/2

R1&R2=0.2

X1&X2=0.2

Wms=2*pi*Ns

Ns=(120*f)/P.

equattions:

1.starting torque for rated voltage and frequency.

T=(3*(V^2)*R2)/(Wms*((R1+R2)^2+(X1+X2)^2))

T=(3*(V/2)^2*0.2)/((2*3.142*120*f/P)*((0.4)^2+(0.1)^2))

Ts=(7.956e-5)*(V^2)*P .......eq1

2.starting torque for half rated voltage and frequency.

T=(3*(V/2)^2)*R2)/(Wms*((R1+R2)^2+(X1+X2)^2))

T=(3*(V/2)^2*0.2)/((2*3.142*120*f/2P)*((0.4)^2+(0.1)^2))

Ts=(3.978e-5)*(V^2)*P ........eq2

dividing eq 2 to 1 we get:

Ts=0.5.

3.

clear all 
close all
clc
n=3;
p=4;
V=230/sqrt(3);
X1=0.680;
X2=0.672;
R1=0.095;
R2=0.23;
Xm=18.7;
zeq=1i*Xm*(R1+1i*X1)/(R1+1i*(X1+Xm));
req=real(zeq);
xeq=imag(zeq);
veq=abs(V*1i*Xm/(R1+1i*(X1+Xm)));
f=linspace(10,50,4)
for b=1:4
    wms=(120*f(b))/p;
    w=(4*pi*f(b))/p;
    for h=1:200
        s(h)=h/200;
        rpm(h)=wms*(1-s(h));
        I2=abs(veq/(zeq+1i*X2+R2/s(h)));
        T(h)=n*I2^2*R2/(s(h)*w);
    end
    plot(rpm,T);
    hold on
    xlabel(\'MOTOR SPEED\')
    ylabel(\'MOTOR TORQUE\')
    legend(\'FREQUENCY-10 Hz\',\'FREQUENCY-20 Hz\',\'FREQUENCY-30 Hz\',\'FREQUENCY-40 Hz\',\'FREQUENCY-50 Hz)\')
end

4.operation of BLDC motor:

CONSTRUCTION:

BLDC motors have many similarities to AC induction motors and brushed DC motors in terms of construction and working principles respectively. Like all other motors, BLDC motors also have a rotor and a stator.

STATOR:

Similar to an Induction AC motor, the BLDC motor stator is made out of laminated steel stacked up to carry the windings. Windings in a stator can be arranged in two patterns; i.e. a star pattern (Y) or delta pattern . The major difference between the two patterns is that the Y pattern gives high torque at low RPM and the DELTA pattern gives low torque at low RPM. This is because in the DELTA configuration, half of the voltage is applied across the winding that is not driven, thus increasing losses and, in turn, efficiency and torque.

Steel laminations in the stator can be slotted or slotless as shown in Figure . A slotless core has lower inductance, thus it can run at very high speeds. Because of the absence of teeth in the lamination stack, requirements for the cogging torque also go down, thus making them an ideal fit for low speeds too (when permanent magnets on rotor and tooth on the stator align with each other then, because of the interaction between the two, an undesirable cogging torque develops and causes ripples in speed). The main disadvantage of a slotless core is higher cost because it requires more winding to compensate for the larger air gap.

Proper selection of the laminated steel and windings for the construction of stator are crucial to motor performance. An improper selection may lead to multiple problems during production, resulting in market delays and increased design costs.

ROTOR:

The rotor of a typical BLDC motor is made out of permanent magnets. Depending upon the application requirements, the number of poles in the rotor may vary. Increasing the number of poles does give better torque but at the cost of reducing the maximum possible speed.

Another rotor parameter that impacts the maximum torque is the material used for the construction of permanent magnet; the higher the flux density of the material, the higher the torque.

Working Principles and Operation

Consider the following setup of three windings in the stator designated A, B and C. replace the rotor with a single magnet.

We know that when a current is applied through a coil, a magnetic field is generated and the orientation of the field lines i.e. the poles of the generated magnet will depend on the direction of the current flowing through the coil.Using this principle, if we supply current to the coil A so that it will generate a magnetic field and attract the rotor magnet. The position of the rotor magnet will shift slightly clockwise and will align with A.If we now pass current through coils B and C one after the other , the rotor magnet will rotate in clock wise direction.

To increase efficiency, we can wind the opposite coils using a single coil so that we get double attraction. Further increasing the efficiency, we can energize two coils at the same time so that one coil will attract the magnet and the other coil will repel it. During this time, the third will be idle.For a complete 360⁰ rotation of the rotor magnet, six possible combinations of the coils A, B and C are applicable and are shown in the following timing diagram.

Base on the above diagram, we can confirm that at any time, one phase is positive, one phase is negative and the third phase is idle (or floating). So, based on the inputs from the Hall Sensors, we have two switch the phases as per the above diagram.

Driving Brushless DC Motors

If stator and rotor are essential parts of a BLDC Motor that are integral to it, then the driving electronics are equally essential. Block diagram of a typical Brushless DC Motor control or drive system is shown in the following image.

This drive circuitry is often known as Electronic Speed Controller System or simply an ESC. One common setup is called the Full Bridge Drive Circuit. It consists of an MCU with PWM outputs, six MOSFETS for the three phases of the stator windings, feedback from the Hall sensors and some power supply related components.

The MCU can be programmed to appropriately switch the MOSFETS based on the data from the Hall Sensors.