1d:I[72985,["/_next/static/chunks/3th5o5azza5j1.js","/_next/static/chunks/0953gtd9gbswy.js","/_next/static/chunks/01oly6y_17ki0.js","/_next/static/chunks/1xlwfijmtjgqj.js","/_next/static/chunks/2hzca_9a7jrui.js","/_next/static/chunks/0_v6xzhg0htsf.js","/_next/static/chunks/2x0yuymt3nfdv.js","/_next/static/chunks/1hjnhv8wc11is.js"],"default"] 1e:I[63508,["/_next/static/chunks/3th5o5azza5j1.js","/_next/static/chunks/0953gtd9gbswy.js","/_next/static/chunks/01oly6y_17ki0.js","/_next/static/chunks/1xlwfijmtjgqj.js","/_next/static/chunks/2hzca_9a7jrui.js","/_next/static/chunks/0_v6xzhg0htsf.js","/_next/static/chunks/2x0yuymt3nfdv.js","/_next/static/chunks/1hjnhv8wc11is.js"],"default"] 1f:T2b43,

A toroidal iron core used in a power inductor or transformer has a mean magnetic path length ( $l$ ) of $0.5\text{ m}$ and a cross-sectional core area ( $A$ ) of $0.002\text{ m}^2$ (or $20\text{ cm}^2$ ). The iron core material has a relative magnetic permeability ( $\mu_r$ ) of $1,500$ . A copper coil of $300$ turns ( $N$ ) is wound uniformly around the core and carries a direct current ( $I$ ) of $2\text{ A}$ . The permeability of free space is $\mu_0 = 4\pi \times 10^{-7}\text{ H/m}$ .

Calculate:

The absolute magnetic permeability of the core material ( $\mu$ )
The magnetomotive force produced by the coil ( $\mathcal{F}$ )
The magnetic reluctance of the toroidal core ( $\mathcal{R}$ )
The total magnetic flux established in the magnetic circuit ( $\Phi$ )

(Note: You can modify the variables in the input data section to explore different conditions)

20:Tf3f,

Click to view typical relative permeability values of magnetic materials (μ_r)

Relative permeability (μ_r) indicates the ease with which a magnetic material conducts magnetic flux compared to a vacuum:

Material	Approximate μ_r	Engineering Application
Air / Vacuum	1	Air-core inductors, magnetic air gaps
Ferrite Core	600 - 3,000	High-frequency switching transformers, EMI chokes
Cast Steel	1,000 - 2,000	Electrical machine frames, heavy magnetic yokes
Silicon Steel Sheet	2,000 - 8,000	Line frequency power transformers (50/60 Hz), motor stators
Mu-Metal / Permalloy	20,000 - 100,000+	Sensitive magnetic shielding against external fields

21:T4f24,

Step 1: Calculate the absolute magnetic permeability of the core ( $\mu$ ) (Evaluates the material's magnetic conductance by multiplying the permeability of free space $\mu_0$ by the relative permeability $\mu_r$ )

💡 Calculator Tip: The constant $\mu_0 = 4\pi \times 10^{-7} \approx 1.2566 \times 10^{-6}\text{ H/m}$ . Multiplying by $\mu_r = 1,500$ gives $\mu = 6\pi \times 10^{-4} \approx 1.885 \times 10^{-3}\text{ H/m}$ .

Step 2: Calculate the magnetomotive force ( $\mathcal{F}$ ) (Represents the magnetic driving potential produced by the coil turns carrying electric current)

Step 3: Calculate the magnetic reluctance of the toroidal core ( $\mathcal{R}$ ) (Calculated from the mean path length divided by the product of permeability and cross-sectional area)

💡 Calculator Tip: The denominator $\mu \cdot A = (1.885 \times 10^{-3}) \times 0.002 = 3.77 \times 10^{-6}\text{ H}\cdot\text{m}$ . Dividing $l = 0.5\text{ m}$ by this product yields $\mathcal{R} \approx 132,629\text{ A}\cdot\text{t/Wb}$ (or $1.326 \times 10^5\text{ A}\cdot\text{t/Wb}$ ).

Step 4: Calculate the total magnetic flux ( $\Phi$ ) using Hopkinson's Law (The magnetic equivalent of Ohm's Law: Magnetic Flux = Magnetomotive Force / Magnetic Reluctance)

22:T5cce,

From the calculated parameters of the toroidal magnetic circuit, the following core engineering insights emerge:

Electrical Circuit Analogy (Hopkinson's Law vs. Ohm's Law):
- Magnetomotive Force ( $\mathcal{F}$ ): Evaluates to . It acts analogously to voltage ( $V$ ) in an electric circuit, serving as the primary source of effort driving magnetic lines of flux. When engineers need to increase $\mathcal{F}$ without increasing excitation current (which would increase thermal dissipation and copper loss), they increase the coil turn count ( $N$ ).
- Magnetic Reluctance ( $\mathcal{R}$ ): Evaluates to . It acts analogously to electrical resistance ( $R$ ). Selecting a ferromagnetic material with $\mu_r = 1,500$ reduces reluctance dramatically compared to air, confining virtually all magnetic lines of force inside the toroidal core and minimizing stray leakage flux into surrounding equipment.
- Magnetic Flux ( $\Phi$ ): Evaluates to (or ). It acts analogously to electric current ( $I$ ), representing the total closed-loop magnetic flux circulating through the core.
Magnetic Flux Density and Saturation Check: Dividing the magnetic flux by the core cross-sectional area yields the flux density: $B = \frac{\Phi}{A} = \frac{4.524 \times 10^{-3}\text{ Wb}}{0.002\text{ m}^2} \approx 2.26\text{ T (Tesla)}$ In practical electrical machine design, standard electrical steels saturate around $B_{sat} \approx 1.5 - 1.8\text{ T}$ . A theoretical operating point of $2.26\text{ T}$ indicates that the core is driven well into magnetic saturation. In saturation, differential permeability drops sharply toward that of air ( $\mu_0$ ), causing excitation current to surge and generating severe harmonic distortion. To restore linear operation below $1.5\text{ T}$ , the design engineer must enlarge the core area ( $A$ ) or decrease the excitation current ( $I$ ).
Air Gap Sensitivity Analysis: If a microscopic air gap of just $1\text{ mm}$ ( $l_g = 0.001\text{ m}$ ) is cut into this toroid, despite being only $0.2\%$ of the physical path length, the air gap reluctance is: $\mathcal{R}_{gap} = \frac{0.001}{(4\pi \times 10^{-7}) \times 0.002} \approx 397,887\text{ A}\cdot\text{t/Wb}$ This single gap adds three times more reluctance than the entire iron core combined! As a result, total flux would plummet to roughly one quarter of its original value. This illustrates why transformer cores require tightly clamped, precision-machined joints without unintended air gaps.

For preliminary engineering use only. Results must be independently verified by a licensed professional engineer before use in design, construction, or fabrication. This tool does not constitute engineering services and provides no certification of code compliance.

12:["$","main",null,{"className":"max-w-6xl mx-auto py-12 px-6","children":[["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"Article\",\"headline\":\"Calculation Example: Magnetic Circuit Basics\",\"description\":\"Worked calculation example for magnetomotive force, reluctance, and magnetic flux in a toroidal core\",\"keywords\":\"\",\"url\":\"https://wisawakorn.com/en/knowledge/calculations/magnetic-circuits-basics-example\",\"mainEntityOfPage\":{\"@type\":\"WebPage\",\"@id\":\"https://wisawakorn.com/en/knowledge/calculations/magnetic-circuits-basics-example\"},\"inLanguage\":\"en-US\"}"}}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https://wisawakorn.com/en\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Knowledge\",\"item\":\"https://wisawakorn.com/en/knowledge\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Calculations\",\"item\":\"https://wisawakorn.com/en/knowledge/calculations\"},{\"@type\":\"ListItem\",\"position\":4,\"name\":\"Calculation Example: Magnetic Circuit Basics\",\"item\":\"https://wisawakorn.com/en/knowledge/calculations/magnetic-circuits-basics-example\"}]}"}}],["$","div",null,{"className":"grid grid-cols-1 lg:grid-cols-12 gap-12","children":[["$","div",null,{"className":"lg:col-span-8 xl:col-span-9","children":[["$","$L1d",null,{"text":"Back"}],["$","header",null,{"className":"mb-10","children":["$","h1",null,{"className":"text-4xl sm:text-5xl font-bold text-gray-900 dark:text-gray-100 tracking-tight mb-4","children":"Calculation Example: Magnetic Circuit Basics"}]}],["$","div",null,{"className":"prose prose-lg dark:prose-invert max-w-none","children":["$","$L1e",null,{"htmlContent":"

This example demonstrates the step-by-step procedure for analyzing a basic magnetic circuit using a toroidal iron core. It applies Hopkinson's Law to calculate the magnetomotive force, magnetic reluctance, and total circulating magnetic flux.

","components":[{"id":"__component_2__","type":"ExampleSection","props":{"title":"1. Problem Statement"},"children":"$1f"},{"id":"__component_3__","type":"IVPRenderer","props":{"id":"electrical-magnetic-circuits-engine"},"children":""},{"id":"__component_6__","type":"ExampleValue","props":{"type":"USER_INPUT","label":"Mean magnetic path length (l)","value":"0.5","unit":"m","variableId":"l"},"children":""},{"id":"__component_7__","type":"ExampleValue","props":{"type":"USER_INPUT","label":"Core cross-sectional area (A)","value":"0.002","unit":"m2","variableId":"a"},"children":""},{"id":"__component_8__","type":"ExampleValue","props":{"type":"USER_INPUT","label":"Relative magnetic permeability (μ_r)","value":"1500","unit":"unitless","variableId":"mur"},"children":""},{"id":"__component_9__","type":"ExampleValue","props":{"type":"USER_INPUT","label":"Number of coil turns (N)","value":"300","unit":"turns","variableId":"n"},"children":""},{"id":"__component_10__","type":"ExampleValue","props":{"type":"USER_INPUT","label":"Electric current (I)","value":"2","unit":"A","variableId":"i"},"children":""},{"id":"__component_5__","type":"ExampleDataGroup","props":{},"children":"

"},{"id":"__component_4__","type":"ExampleSection","props":{"title":"2. Input Data"},"children":"$20"},{"id":"__component_12__","type":"ResultExplanation","props":{"formulaId":"magnetic-permeability-formula","lang":"en","inputVariables":"{\"mu0\":{\"value\":\"1.257e-6\",\"unit\":\"H/m\",\"format\":\"scientific\"},\"mur\":{\"value\":1500,\"unit\":\"unitless\"}}","result":"{\"value\":\"1.885e-3\",\"unit\":\"H/m\",\"format\":\"scientific\"}","interactiveResult":"mu","interactiveMap":"{\"mu0\":\"mu0\",\"mur\":\"mur\",\"mu\":\"mu\"}","substitutionTemplate":"\\mu = [mu0] \\times [mur]"},"children":""},{"id":"__component_13__","type":"ResultExplanation","props":{"formulaId":"magnetomotive-force-formula","lang":"en","inputVariables":"{\"n\":{\"value\":300,\"unit\":\"turns\"},\"i\":{\"value\":2,\"unit\":\"A\"}}","result":"{\"value\":600,\"unit\":\"A·t\"}","interactiveResult":"f","interactiveMap":"{\"n\":\"n\",\"i\":\"i\",\"f\":\"mmf\"}","substitutionTemplate":"\\mathcal{F} = [n] \\times [i]"},"children":""},{"id":"__component_14__","type":"ResultExplanation","props":{"formulaId":"magnetic-reluctance-formula","lang":"en","inputVariables":"{\"l\":{\"value\":0.5,\"unit\":\"m\"},\"mu\":{\"value\":\"1.885e-3\",\"unit\":\"H/m\",\"format\":\"scientific\"},\"a\":{\"value\":0.002,\"unit\":\"m2\"}}","result":"{\"value\":132629.12,\"unit\":\"A·t/Wb\"}","interactiveResult":"r","interactiveMap":"{\"l\":\"l\",\"mu\":\"mu\",\"a\":\"a\",\"r\":\"reluctance\"}","substitutionTemplate":"\\mathcal{R} = \\frac{[l]}{[mu] \\times [a]}"},"children":""},{"id":"__component_15__","type":"ResultExplanation","props":{"formulaId":"magnetic-flux-hopkinson-formula","lang":"en","inputVariables":"{\"f\":{\"value\":600,\"unit\":\"A·t\"},\"r\":{\"value\":132629.12,\"unit\":\"A·t/Wb\"}}","result":"{\"value\":\"4.524e-3\",\"unit\":\"Wb\",\"format\":\"scientific\"}","interactiveResult":"phi","interactiveMap":"{\"f\":\"mmf\",\"r\":\"reluctance\",\"phi\":\"flux\"}","substitutionTemplate":"\\Phi = \\frac{[f]}{[r]}"},"children":""},{"id":"__component_11__","type":"ExampleSection","props":{"title":"3. Result Calculation"},"children":"$21"},{"id":"__component_17__","type":"DynamicText","props":{"variableId":"mmf","isResult":"true","fallback":"600","unit":"A·t"},"children":""},{"id":"__component_18__","type":"DynamicText","props":{"variableId":"reluctance","isResult":"true","fallback":"132629","unit":"A·t/Wb"},"children":""},{"id":"__component_19__","type":"DynamicText","props":{"variableId":"flux","isResult":"true","fallback":"0.004524","conversionFactor":"1000","unit":"mWb"},"children":""},{"id":"__component_20__","type":"DynamicText","props":{"variableId":"flux","isResult":"true","fallback":"0.004524","unit":"Wb"},"children":""},{"id":"__component_16__","type":"ExampleSection","props":{"title":"4. Engineering Significance"},"children":"$22"},{"id":"__component_1__","type":"WorkedExample","props":{"title":"Toroidal Magnetic Circuit Calculation Under Coil Excitation","id":"example-1"},"children":"

"},{"id":"__component_0__","type":"InteractiveVisualization","props":{"id":"electrical-magnetic-circuits-engine","version":"1.0.0","initialValues":"{\"l\":0.5,\"a\":0.002,\"mur\":1500,\"n\":300,\"i\":2}","hideControls":"true","hideRenderer":"true"},"children":"

"}]}]}],"$L23"]}],"$L24"]}]]}] 25:I[22016,["/_next/static/chunks/3th5o5azza5j1.js","/_next/static/chunks/0953gtd9gbswy.js","/_next/static/chunks/01oly6y_17ki0.js","/_next/static/chunks/1xlwfijmtjgqj.js","/_next/static/chunks/2hzca_9a7jrui.js","/_next/static/chunks/0_v6xzhg0htsf.js","/_next/static/chunks/2x0yuymt3nfdv.js","/_next/static/chunks/1hjnhv8wc11is.js"],""] 26:I[58681,["/_next/static/chunks/3th5o5azza5j1.js","/_next/static/chunks/0953gtd9gbswy.js","/_next/static/chunks/01oly6y_17ki0.js","/_next/static/chunks/1xlwfijmtjgqj.js","/_next/static/chunks/2hzca_9a7jrui.js","/_next/static/chunks/0_v6xzhg0htsf.js","/_next/static/chunks/2x0yuymt3nfdv.js","/_next/static/chunks/1hjnhv8wc11is.js"],"default"] 23:["$","section",null,{"aria-label":"Related Content","className":"mt-16 pt-8 border-t border-gray-200 dark:border-gray-800","children":[["$","h2",null,{"className":"text-2xl font-bold mb-6 text-gray-900 dark:text-gray-100","children":"Related Content"}],["$","div",null,{"className":"mb-8","children":[["$","h3",null,{"className":"text-lg font-semibold mb-4 text-gray-700 dark:text-gray-300","children":"Articles"}],["$","div",null,{"className":"grid grid-cols-1 md:grid-cols-2 gap-4","children":[["$","$L25","magnetic-circuits-basics",{"href":"/en/knowledge/calculations/magnetic-circuits-basics","className":"block p-4 bg-gray-50 dark:bg-[#111] rounded-xl border border-gray-200 dark:border-gray-800 hover:border-red-500 dark:hover:border-red-500 transition-colors focus:outline-none focus:ring-2 focus:ring-red-500","children":[["$","h4",null,{"className":"font-bold text-gray-900 dark:text-gray-100 mb-1","children":"Magnetic Circuits Basics"}],["$","p",null,{"className":"text-sm text-gray-600 dark:text-gray-400 line-clamp-2","children":"Foundational calculation methods for magnetic flux, magnetomotive force, and reluctance in simple magnetic circuits."}]]}]]}]]}],false]}] 24:["$","aside",null,{"className":"lg:col-span-4 xl:col-span-3 space-y-8","children":["$","div",null,{"className":"sticky top-8","children":["$","$L26",null,{}]}]}]