A simple motor has N=250 turns of wire on a coil measuring 12 cm \times \, 4 cm. It's resistance is R=10 ohms. The magnetic field is B=1 T.
(a) How much current (in Ampere) in the coil is needed to produce a maximum torque of 40 Nm?
b) What maximum EMF (in Volts) is needed to drive the motor at 50 Hz if the current is constant?
(a)
Torque is
M=p⒨•B•sinα
p⒨=NIA
If M(max) =sin α=1
M(max)=NIAB,
I= M(max)/NAB =
=40/250•0.12•0.04•1=33.3 A
(b)
ℰ=-dΦ=-d(NBAsinωt)/dt=
=-NBAωcosωt
ℰ(max)=NBAω=2πNBAf=
=2π•250•1•0.12•0.04•50=377 V
Thanks
To answer these questions, we need to use the formulas related to the torque and electromagnetic force (EMF) in a motor. Here's how we can approach each question:
(a) How much current (in Ampere) in the coil is needed to produce a maximum torque of 40 Nm?
The torque produced by a motor is given by the equation:
Torque (τ) = N * B * A * I
Where:
- N is the number of turns of wire in the coil
- B is the magnetic field strength
- A is the area of the coil
- I is the current flowing through the coil
In this case, we are given:
- N = 250 turns
- B = 1 T (Tesla)
- A = 12 cm * 4 cm = 48 cm² = 0.0048 m²
- Torque (τ) = 40 Nm
To find the current (I), we rearrange the equation as follows:
I = τ / (N * B * A)
Plugging in the given values:
I = 40 Nm / (250 * 1 T * 0.0048 m²)
Simplifying the expression:
I ≈ 33.33 A
Therefore, approximately 33.33 Amperes of current is needed to produce a maximum torque of 40 Nm.
(b) What maximum EMF (in Volts) is needed to drive the motor at 50 Hz if the current is constant?
The formula for calculating the EMF (E) in a motor is:
E = 2πfNBA
Where:
- f is the frequency of the coil rotation (in Hz)
- N, B, and A have the same meaning as before
In this case, we are given:
- f = 50 Hz
- N = 250 turns
- B = 1 T (Tesla)
- A = 12 cm * 4 cm = 48 cm² = 0.0048 m²
Plugging in the given values:
E = 2π * 50 Hz * 250 * 1 T * 0.0048 m²
Simplifying the expression:
E ≈ 75.4 V
Therefore, the maximum EMF needed to drive the motor at 50 Hz, with a constant current, is approximately 75.4 Volts.