Given: A magnetic core with two parallel outer legs and a center leg. Center leg has an air gap of length ( l_g = 1 ) mm. Neglect fringing. Mean path lengths: center ( l_c = 0.2 ) m, outer legs ( l_o = 0.4 ) m each. Cross-section ( A = 4 ) cm² all legs. ( \mu_r = 2000 ) for iron. Coil on center leg: ( N=1000, I=1 ) A. Find flux in center leg.
Solution (abbreviated):
Answer: Typical result — center leg flux ≈ 0.85 mWb (depends on exact dimensions).
A concise guide to create a PDF titled "Magnetic Circuits — Problems and Solutions" that students or instructors can use. Includes suggested structure, sample problems with worked solutions, notation, and formatting tips.
To solve magnetic circuits, it is helpful to compare them to electric circuits:
| Electric Circuit | Magnetic Circuit | | :--- | :--- | | Electromotive Force (EMF), $V$ (Volts) | Magnetomotive Force (MMF), $F$ (Ampere-turns) | | Current, $I$ (Amperes) | Magnetic Flux, $\phi$ (Webers) | | Resistance, $R$ ($\Omega$) | Reluctance, $\mathcalR$ (Ampere-turns/Weber) | | Conductivity, $\sigma$ | Permeability, $\mu$ |
Problem Statement: A magnetic core is made of an iron alloy with a constant relative permeability ($\mu_r$) of 1000. The core has a mean length of $50 , \textcm$ and a cross-sectional area of $10 , \textcm^2$. A coil with $500$ turns is wound around the core.
Solution:
Step 1: Calculate the Reluctance ($\mathcalR$) of the core. First, determine the absolute permeability $\mu$: $$ \mu = \mu_0 \mu_r = (4\pi \times 10^-7) \times 1000 = 4\pi \times 10^-4 , \textH/m $$
Convert dimensions to meters: $$ l = 50 , \textcm = 0.5 , \textm $$ $$ A = 10 , \textcm^2 = 10 \times 10^-4 , \textm^2 = 0.001 , \textm^2 $$
Calculate Reluctance: $$ \mathcalR = \fracl\mu A = \frac0.5(4\pi \times 10^-4)(0.001) $$ $$ \mathcalR = \frac0.51.256 \times 10^-6 \approx 398,100 , \textAt/Wb $$
Step 2: Apply Hopkinson’s Law to find MMF ($NI$). $$ NI = \phi \mathcalR $$ $$ NI = (0.005) \times (398,100) $$ $$ NI \approx 1990.5 , \textAmpere-turns $$
Step 3: Calculate Current ($I$). $$ I = \fracNIN = \frac1990.5500 $$ $$ \boxedI \approx 3.98 , \textA $$
Given: Symmetrical three-limb core (like transformer). Center limb has coil N=300 turns, length of outer limbs = 0.6 m each, center limb length = 0.2 m, all limbs A=0.001 m². μ_r = 2000 constant. Current I=3 A. Find flux in each outer limb. Neglect leakage.
Solution:
Answer: Flux in each outer limb = 2.26 mWb.
A single closed path for flux, often with different materials (e.g., air gap + iron core). Given: Dimensions, number of turns, current, and B-H curve. Find: Flux or current.
