A rectangular magnetic core shown in Fig. 8.21 has square cross section of area 16 cm2. An air gap of 2 mm is cut across one of its limbs. Find the exciting current needed in the coil having 1,000 turns wound on the core to create an air-gap flux of 4 mWb. The relative permeability of the core is 2000.
The Correct Answer and Explanation is :
Answer:
The exciting current required in the coil is 4.97 A.
Explanation:
To determine the exciting current required to establish an air-gap flux of 4 mWb in a rectangular magnetic core with a square cross-section, we follow these steps:
- Understanding the Magnetic Circuit:
The core has a square cross-section of 16 cm², and an air gap of 2 mm is introduced. The presence of the air gap significantly increases the reluctance in the magnetic circuit. The coil wrapped around the core has 1,000 turns, and the relative permeability of the core is given as 2000. - Calculating Reluctances:
Magnetic reluctance (( R )) is analogous to electrical resistance in an electrical circuit and is given by: [
R = \frac{l}{\mu A}
] where:
- ( l ) is the length of the magnetic path,
- ( \mu ) is the permeability ((\mu_0 \mu_r) for the core and (\mu_0) for the air gap),
- ( A ) is the cross-sectional area.
- Reluctance of the Air Gap: The permeability of free space is ( \mu_0 = 4\pi \times 10^{-7} ) H/m.
The reluctance of the air gap is calculated as: [
R_g = \frac{l_g}{\mu_0 A}
] - Reluctance of the Core:
Assuming a rectangular shape of the core with 20 cm × 30 cm outer dimensions, the mean magnetic path length is: [
l_c = 2 \times (0.2 + 0.3) = 1 \text{ m}
] The core permeability is: [
\mu = \mu_r \times \mu_0
] So, the reluctance of the core is: [
R_c = \frac{l_c}{\mu A}
] - Total Reluctance: [
R_{total} = R_g + R_c
]
- Finding MMF (Magnetomotive Force):
The MMF required to drive the magnetic flux is: [
\text{MMF} = \Phi \times R_{total}
] - Determining the Exciting Current:
Since MMF is given by: [
\text{MMF} = N I
] Solving for ( I ): [
I = \frac{\text{MMF}}{N}
] Substituting values, the required exciting current is 4.97 A.
This current ensures that the desired flux of 4 mWb is established across the air gap.