To calculate the approximate voltage induced and its frequency in the given scenarios, we need to consider the slip of the induction motor. The slip (s) is defined as the difference between synchronous speed (Ns) and the actual rotor speed (Nr), divided by the synchronous speed:
s = (Ns - Nr) / Ns
The synchronous speed (Ns) of an induction motor can be calculated using the formula:
Ns = (120 * f) / P
Where:
f = supply frequency (Hz)
P = number of poles
In this case, the motor has 6 poles and is connected to a 60 Hz supply, so the synchronous speed is:
Ns = (120 * 60) / 6
Ns = 1200 rpm
Now, we can calculate the approximate rotor speed (Nr) in rpm for each scenario.
(a) At 300 rpm:
Nr = 300 rpm
(b) At 1000 rpm:
Nr = 1000 rpm
(c) At 1500 rpm:
Nr = 1500 rpm
Next, we can calculate the slip (s) for each scenario using the formula mentioned earlier.
(a) At 300 rpm:
s = (1200 - 300) / 1200
s = 0.75
(b) At 1000 rpm:
s = (1200 - 1000) / 1200
s = 0.1667
(c) At 1500 rpm:
s = (1200 - 1500) / 1200
s = -0.25
The slip values help us determine the approximate voltage induced in the rotor (Vr) and its frequency (fr).
Vr = s * Vs
Where:
Vs = supply voltage
fr = s * f
Where:
f = supply frequency
Given that the voltage induced in the rotor bars is 4 V when the rotor is locked, we can use this value to find the approximate voltage induced and its frequency for each scenario.
(a) At 300 rpm:
Vr = 0.75 * 4 V
Vr = 3 V
fr = 0.75 * 60 Hz
fr = 45 Hz
(b) At 1000 rpm:
Vr = 0.1667 * 4 V
Vr ≈ 0.667 V
fr = 0.1667 * 60 Hz
fr ≈ 10 Hz
(c) At 1500 rpm:
Vr = -0.25 * 4 V
Vr = -1 V
fr = -0.25 * 60 Hz
fr = -15 Hz
Therefore, the approximate voltage induced and its frequency for each scenario are as follows:
(a) At 300 rpm: Vr ≈ 3 V, fr = 45 Hz
(b) At 1000 rpm: Vr ≈ 0.667 V, fr ≈ 10 Hz
(c) At 1500 rpm: Vr = -1 V, fr = -15 Hz (negative sign indicates a reversal of direction)