Calculation of Power Factor

To calculate the power factor, we need to find the total impedance of the motor. The synchronous impedance per phase is given as 0.5+j4 ohm. Since it is a 3-phase motor, the total impedance will be three times the synchronous impedance.

Z = 3 * (0.5+j4) = 1.5+j12 ohm

The power factor can be calculated using the formula:

power factor = cos(θ) = Re(Z) / |Z|

where Re(Z) is the real part of the impedance and |Z| is the magnitude of the impedance.

Re(Z) = 1.5 ohm
|Z| = sqrt((1.5)^2 + (12)^2) = sqrt(226.5) ohm

power factor = 1.5 / sqrt(226.5) = 0.099

Calculation of Current

The current flowing through the motor is given by the formula:

I = V / Z

where V is the rated voltage and Z is the total impedance.

I = 400 V / (1.5+j12) ohm

To simplify the calculation, we can convert the impedance to polar form:

Z = sqrt((1.5)^2 + (12)^2) * e^(arctan(12/1.5))

I = 400 V / (sqrt(226.5) * e^(arctan(12/1.5)))

I = 400 / (sqrt(226.5) * e^(arctan(8)))

Now, we can calculate the magnitude and angle of the current:

|I| = 400 / (sqrt(226.5)) = 26.25 A
θ = arctan(8) = 80.54 degrees

Calculation of Active Power

The active power can be calculated using the formula:

P = 3 * |I|^2 * Re(Z) * power factor

P = 3 * (26.25)^2 * 1.5 * 0.099 = 4357.22 W

Calculation of Reactive Power

The reactive power can be calculated using the formula:

Q = 3 * |I|^2 * Re(Z) * sqrt(1 - power factor^2)

Q = 3 * (26.25)^2 * 1.5 * sqrt(1 - 0.099^2) = 12425.62 VAR

Calculation of Apparent Power

The apparent power can be calculated using the formula:

S = sqrt(P^2 + Q^2)

S = sqrt((4357.22)^2 + (12425.62)^2) = 13100.49 VA

Calculation of Power Factor Angle

The power factor angle can be calculated using the formula:

θ = arccos(power factor)

θ = arccos(0.099) = 84.12 degrees

Conclusion

- The power factor of the motor is 0.099, which indicates a lagging power factor.
- The current flowing through the motor is 26.25 A with an angle of 80.54 degrees.
- The active power consumed by the motor is 4357.22