Self-Inductance and Mutual Inductance

Electromagnetic induction is a fundamental principle in physics and electrical engineering. While Faraday’s Law explains how changing magnetic fields induce EMF, self-inductance and mutual inductance describe how circuits and coils interact with their own or neighboring magnetic fields. These concepts are critical for understanding transformers, inductors, electrical circuits, and energy storage in magnetic fields.

This post explores self-inductance and mutual inductance in detail, including their definitions, mathematical formulations, applications, and examples.

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1. Introduction

When a current flows through a conductor, it produces a magnetic field. If this current changes with time, the magnetic field also changes, which can induce an electromotive force (EMF).

There are two possibilities:

1. Self-Induction: The coil induces an EMF in itself due to a changing current.
2. Mutual Induction: A changing current in one coil induces an EMF in a neighboring coil.

Understanding these phenomena is essential in circuit design, power systems, and modern electronics.

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2. Self-Inductance

2.1 Definition

Self-inductance is the property of a coil (or circuit) by which a changing current induces an EMF in itself, opposing the change in current. This is a direct consequence of Faraday’s Law and Lenz’s Law.

• Symbol: LLL
• Unit: Henry (H)
• Formula for induced EMF:

E=−LdIdt\mathcal{E} = – L \frac{dI}{dt}E=−LdtdI​

Where:

• E\mathcal{E}E = induced EMF
• LLL = self-inductance of the coil
• III = current through the coil

Key Point: The negative sign indicates that the induced EMF opposes the change in current (Lenz’s Law).

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2.2 Factors Affecting Self-Inductance

1. Number of Turns (N): More turns increase magnetic flux linkage.
2. Core Material: Ferromagnetic cores (iron) increase inductance due to higher permeability.
3. Coil Geometry: Longer coils with larger cross-sectional areas have higher inductance.
4. Length of the Coil: Inductance decreases with increasing length if other factors are constant.

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2.3 Magnetic Flux Linkage

The flux linkage (λ\lambdaλ) is the total magnetic flux linked with all turns of a coil: λ=NΦB\lambda = N \Phi_Bλ=NΦB​

Where NNN = number of turns, ΦB\Phi_BΦB​ = magnetic flux through one turn.

Self-inductance is defined as: L=λI=NΦBIL = \frac{\lambda}{I} = \frac{N \Phi_B}{I}L=Iλ​=INΦB​​

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2.4 Energy Stored in an Inductor

A coil with self-inductance LLL stores energy in its magnetic field when current flows: U=12LI2U = \frac{1}{2} L I^2U=21​LI2

• Energy is proportional to the square of current.
• Crucial in energy storage applications and transient analysis.

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2.5 Practical Examples of Self-Inductance

1. Inductors in Circuits: Smooth current variations in DC circuits.
2. Transformers: Each coil exhibits self-inductance.
3. Energy Storage: Magnetic energy storage in inductors.
4. Switching Circuits: Protect components from sudden current changes using self-inductance.

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3. Mutual Inductance

3.1 Definition

Mutual inductance is the property by which a changing current in one coil induces EMF in a neighboring coil due to the changing magnetic flux linking both coils.

• Symbol: MMM
• Unit: Henry (H)
• Induced EMF in secondary coil:

E2=−MdI1dt\mathcal{E}_2 = – M \frac{dI_1}{dt}E2​=−MdtdI1​​

Where:

• E2\mathcal{E}_2E2​ = induced EMF in the second coil
• I1I_1I1​ = current in the first coil

Key Concept: Mutual inductance depends on coil geometry, number of turns, core material, and relative position of the coils.

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3.2 Mutual Inductance Formula

If coil 1 has N1N_1N1​ turns and coil 2 has N2N_2N2​ turns, the mutual inductance is: M=N2Φ21I1M = \frac{N_2 \Phi_{21}}{I_1}M=I1​N2​Φ21​​

Where Φ21\Phi_{21}Φ21​ is the flux through coil 2 due to current I1I_1I1​ in coil 1.

Reciprocity Principle: M12=M21=MM_{12} = M_{21} = MM12​=M21​=M

• Mutual inductance is symmetric.
• The induced EMF depends on the rate of change of current in the primary coil.

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3.3 Factors Affecting Mutual Inductance

1. Number of Turns: More turns increase flux linkage.
2. Core Material: Ferromagnetic cores enhance magnetic coupling.
3. Distance Between Coils: Closer coils have stronger mutual coupling.
4. Orientation: Coaxial and parallel coils maximize mutual flux linkage.

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3.4 Energy Consideration

The energy stored in two magnetically coupled coils is: U=12L1I12+12L2I22+MI1I2U = \frac{1}{2} L_1 I_1^2 + \frac{1}{2} L_2 I_2^2 + M I_1 I_2U=21​L1​I12​+21​L2​I22​+MI1​I2​

• L1,L2L_1, L_2L1​,L2​ = self-inductances of the coils
• MMM = mutual inductance
• I1,I2I_1, I_2I1​,I2​ = currents in the coils

Significance: Total energy includes contributions from self and mutual inductances.

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3.5 Practical Examples of Mutual Inductance

1. Transformers: Step-up or step-down voltages using mutual induction.
2. Coupled Circuits: Signal transfer in electronics.
3. Induction Motors: Rotor currents induced by stator fields.
4. Wireless Power Transfer: Mutual induction between coils in charging pads.

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4. Comparison Between Self and Mutual Inductance

Feature	Self-Inductance	Mutual Inductance
Definition	EMF induced in a coil by its own changing current	EMF induced in one coil by changing current in another coil
Symbol	L	M
Depends on	Coil turns, geometry, core material	Number of turns in both coils, distance, orientation, core
Energy	12LI2\frac{1}{2} L I^221​LI2	MI1I2M I_1 I_2MI1​I2​ (part of total energy)
Application	Inductors, energy storage, DC smoothing	Transformers, coupled circuits, induction motors

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5. Inductance in Solenoids and Toroids

5.1 Solenoid Self-Inductance

For a long solenoid: L=μ0N2AlL = \mu_0 \frac{N^2 A}{l}L=μ0​lN2A​

Where:

• NNN = number of turns
• AAA = cross-sectional area
• lll = length of solenoid
• μ0\mu_0μ0​ = permeability of free space

With a ferromagnetic core: L=μN2AlL = \mu \frac{N^2 A}{l}L=μlN2A​

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5.2 Toroid Self-Inductance

For a toroidal coil: L=μN2A2πrL = \frac{\mu N^2 A}{2 \pi r}L=2πrμN2A​

• rrr = mean radius of toroid
• Toroids confine magnetic flux, reducing energy losses.

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5.3 Mutual Inductance Between Coaxial Coils

For two coaxial solenoids with length lll, radii r1,r2r_1, r_2r1​,r2​, and turns N1,N2N_1, N_2N1​,N2​: M=μ0N1N2AlM = \mu_0 \frac{N_1 N_2 A}{l}M=μ0​lN1​N2​A​

Where AAA is the overlapping cross-sectional area.

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6. Applications in Electrical Engineering

6.1 Transformers

• Step-up and step-down AC voltages using mutual inductance.
• Turns ratio determines voltage transformation:

V2V1=N2N1\frac{V_2}{V_1} = \frac{N_2}{N_1}V1​V2​​=N1​N2​​

• Energy conservation: V1I1=V2I2V_1 I_1 = V_2 I_2V1​I1​=V2​I2​ (ideal transformer)

6.2 Inductors

• Self-inductance stores energy in magnetic field.
• Used in filters, chokes, and smoothing circuits.

6.3 Coupled Circuits

• Mutual inductance enables signal transfer between circuits without direct connection.
• Basis for RF circuits, wireless communication.

6.4 Energy Storage and Magnetic Braking

• Inductors absorb energy from changing currents.
• Mutual induction can create eddy currents for braking mechanisms.

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7. Time-Dependent Current Effects

• In AC circuits, inductance opposes current changes.
• Inductive reactance:

XL=2πfLX_L = 2 \pi f LXL​=2πfL

• In coupled circuits, mutual inductance introduces phase shifts and affects resonance.

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8. Practical Examples and Problem Solving

Example 1: Self-Inductance of Solenoid

• Solenoid: 1000 turns, length 0.5 m, area 0.01 m²
• Find LLL (air core).

L=μ0N2Al=4π×10−710002⋅0.010.5≈0.025 HL = \mu_0 \frac{N^2 A}{l} = 4\pi \times 10^{-7} \frac{1000^2 \cdot 0.01}{0.5} \approx 0.025 \text{ H}L=μ0​lN2A​=4π×10−70.510002⋅0.01​≈0.025 H

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Example 2: Mutual Inductance Between Coils

• Coil 1: 200 turns, Coil 2: 100 turns, area 0.02 m², length 0.1 m, permeability μ0\mu_0μ0​
• Find MMM:

M=μ0N1N2Al=4π×10−7200⋅100⋅0.020.1≈5×10−3 HM = \mu_0 \frac{N_1 N_2 A}{l} = 4\pi \times 10^{-7} \frac{200 \cdot 100 \cdot 0.02}{0.1} \approx 5 \times 10^{-3} \text{ H}M=μ0​lN1​N2​A​=4π×10−70.1200⋅100⋅0.02​≈5×10−3 H

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Example 3: Energy Stored in Two Coupled Coils

• L1=2 H,L2=1 H,M=0.5 H,I1=3 A,I2=2 AL_1 = 2 \text{ H}, L_2 = 1 \text{ H}, M = 0.5 \text{ H}, I_1 = 3 \text{ A}, I_2 = 2 \text{ A}L1​=2 H,L2​=1 H,M=0.5 H,I1​=3 A,I2​=2 A

U=12L1I12+12L2I22+MI1I2U = \frac{1}{2} L_1 I_1^2 + \frac{1}{2} L_2 I_2^2 + M I_1 I_2U=21​L1​I12​+21​L2​I22​+MI1​I2​ U=0.5(2)(9)+0.5(1)(4)+0.5(3)(2)(0.5?)?U = 0.5(2)(9) + 0.5(1)(4) + 0.5(3)(2)(0.5?)?U=0.5(2)(9)+0.5(1)(4)+0.5(3)(2)(0.5?)?

Let’s calculate carefully:

• First term: 0.5⋅2⋅9=90.5 \cdot 2 \cdot 9 = 90.5⋅2⋅9=9 J
• Second term: 0.5⋅1⋅4=20.5 \cdot 1 \cdot 4 = 20.5⋅1⋅4=2 J
• Third term: MI1I2=0.5⋅3⋅2=3M I_1 I_2 = 0.5 \cdot 3 \cdot 2 = 3MI1​I2​=0.5⋅3⋅2=3 J

Total Energy: U=9+2+3=14 JU = 9 + 2 + 3 = 14 \text{ J}U=9+2+3=14 J

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9. Advantages and Limitations

Advantages:

• Fundamental in AC and DC circuit design.
• Enables efficient energy transfer in transformers.
• Essential for electromagnetic devices and sensors.

Limitations:

• Energy losses occur due to resistance and eddy currents.
• Mutual inductance depends on precise alignment of coils.
• High inductance can limit rapid current changes.

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10. Advanced Concepts

10.1 Coupling Coefficient (k)

• Represents the fraction of magnetic flux of one coil linking the other:

k=ML1L2k = \frac{M}{\sqrt{L_1 L_2}}k=L1​L2​​M​

• 0≤k≤10 \le k \le 10≤k≤1
• k = 1 for perfect coupling, k < 1 in practical circuits.

10.2 Transformers and Resonance

• Mutual inductance plays a key role in tuned transformers and resonant circuits, allowing selective frequency transmission.

10.3 Superconducting Inductors

• High inductance with negligible resistance.
• Stores large energy efficiently for magnetic energy storage systems.

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11. Applications in Modern Technology

1. Electrical Power Systems: Transformers, inductors, and chokes.
2. Electronic Devices: Wireless charging, RF coils, coupling circuits.
3. Industrial Systems: Induction motors, magnetic brakes.
4. Medical Equipment: MRI coils use mutual and self-inductance principles.
5. Energy Storage: SMES (Superconducting Magnetic Energy Storage).