A 3-phase 50Hz, 8 pole induction motor has an effective stator to rotor turns ratio of 4:2, rotor resistance 0.2ohms and rotor standstill reactance 0.3ohms. Calculate the starting torque and maximum torque.
At full load the slip is 0.08. Calculate the torque, mechanical output power and actual rotor speed.
To calculate the starting torque of the induction motor, we need to find the equivalent rotor impedance. The equivalent rotor impedance is the sum of the rotor resistance and reactance.
Rotor impedance (Zr) = Rotor resistance (Rr) + Rotor reactance (Xr)
Zr = 0.2ohms + 0.3ohms
Zr = 0.5ohms
To calculate the starting torque (Ts), we can use the formula:
Ts = (1 / s) * ( (s * Es^2) / (Zr + [(s^2) * Zr]))
Where:
s = slip
Es = stator voltage per phase
Zr = rotor impedance
In this case, we have a stator to rotor turns ratio of 4:2, so the stator voltage per phase (Es) is equivalent to the effective rotor voltage per phase multiplied by the turns ratio.
Es = 2 * (Effective rotor voltage per phase)
As we are not given the effective rotor voltage, we cannot calculate the starting torque without additional information.
Moving onto the maximum torque, let's calculate that.
The maximum torque (Tmax) occurs at the slip (s) where the rotor resistance is equal to the rotor reactance. In this case, the rotor resistance is 0.2ohms and the rotor reactance is 0.3ohms.
Therefore, to find the maximum torque, the slip (s) can be calculated using the formula:
s = (Zr / Xr)^(1/2)
s = (0.2ohms / 0.3ohms)^(1/2)
s = (2/3)^(1/2)
s = 0.816
To calculate the maximum torque (Tmax):
Tmax = (3 * Es^2) / (2 * (s * (Zr + Xr)))
Where:
s = slip
Es = stator voltage per phase
Zr = rotor impedance
Xr = rotor reactance
Again, we require the effective rotor voltage per phase to calculate the maximum torque.
Moving on to full load conditions, where the slip is given as 0.08, we can calculate the torque, mechanical output power, and actual rotor speed.
To calculate the torque at full load, we can use the formula:
Torque = (3 * Es^2 * Rr) / (s * (s * Xr + Rr)^2)
To calculate the mechanical output power, we can use the formula:
Output Power = (3 * Es^2 * Rr) / (s * (s * Xr + Rr))
To calculate the actual rotor speed at full load, we can use the formula:
Rotor Speed = (1 - s) * Synchronous Speed
Where:
s = slip at full load
Es = stator voltage per phase
Rr = rotor resistance
Xr = rotor reactance
Synchronous Speed = (120 * Frequency) / Number of poles
As we are not provided with the stator voltage per phase, we cannot calculate the torque, mechanical output power, and actual rotor speed at full load.
To calculate the starting torque and maximum torque of the induction motor, we can use the following formulas:
Starting Torque (Ts) = (3 * V^2 * R2) / (s * (R2^2 + s^2 * X2^2))
Maximum Torque (Tmax) = (3 * V^2) / (2 * ω * ((R2 / ω)^2 + (X2)^2))
Where:
V = supply voltage
R2 = rotor resistance
X2 = rotor standstill reactance
s = slip
ω = angular speed = 2π * frequency
Given information:
Frequency (f) = 50 Hz
Pole pairs (P) = 8
Stator to rotor turns ratio (N1:N2) = 4:2
R2 = 0.2 Ω
X2 = 0.3 Ω
1. Calculate the angular speed (ω):
ω = 2π * f / P
= 2π * 50 / 8
≈ 39.27 rad/s
2. Calculate the supply voltage (V):
The effective stator to rotor turns ratio is given as 4:2, which means that N1/N2 = 4/2.
V1/V2 = N1/N2
V2 = V1 * (N2/N1)
V2 = V1 * (2/4) = V1/2
From this, we can assume V1 = V and V2 = V/2.
3. Calculate the starting torque (Ts):
s = 1 (at start)
Ts = (3 * V^2 * R2) / (s * (R2^2 + s^2 * X2^2))
= (3 * V^2 * 0.2) / (1 * (0.2^2 + 1^2 * 0.3^2))
= (0.6 * V^2) / (0.04 + 0.09)
= (0.6 * V^2) / 0.13
4. Calculate the maximum torque (Tmax):
s = 0 (at maximum torque)
Tmax = (3 * V^2) / (2 * ω * ((R2 / ω)^2 + (X2)^2))
= (3 * V^2) / (2 * 39.27 * ((0.2 / 39.27)^2 + (0.3)^2))
= (3 * V^2) / (2 * 39.27 * (0.0001 + 0.09))
= (3 * V^2) / (2 * 39.27 * 0.0901)
= (3 * V^2) / (2 * 35.93)
5. At full load with slip (s = 0.08):
For the torque at full load, we can substitute the value of s = 0.08 in the formula for starting torque (Ts). Using the same formula:
T = (3 * V^2 * R2) / (s * (R2^2 + s^2 * X2^2))
= (3 * V^2 * 0.2) / (0.08 * (0.2^2 + 0.08^2 * 0.3^2))
= (0.6 * V^2) / (0.064 + 0.08^2 * 0.09)
= (0.6 * V^2) / (0.064 + 0.064 * 0.09)
= (0.6 * V^2) / (0.064 + 0.00576)
= (0.6 * V^2) / 0.06976
6. Calculate the mechanical output power (Pout) at full load:
Pout = Tmax * ω
= ((3 * V^2) / (2 * 35.93)) * 39.27
7. Calculate the actual rotor speed (Nactual) at full load:
Nactual = (1 - s) * Ns
= (1 - 0.08) * 60 * f / P
Note: Ns = synchronous speed = 120 * f / P
Let's calculate the values step by step.