FUNDAMENTALS OF MAGNETIC FIELDS AND INDUCTANCE

Q2. What is Resistance of a Conductor?

A2. The effective resistance of a conductor can be expressed in terms of its power loss and current.

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R = Ploss / I²

Where:

• R = effective resistance in ohms (Ω)

• Ploss = power loss in the conductor in watts (W)

• I = current flowing through the conductor in amperes (A)

 

The effective resistance depends on how current is distributed throughout the conductor. When current is uniformly distributed, the effective resistance corresponds to the DC resistance. Under AC conditions, current distribution can change, affecting the effective resistance.

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For overhead lines, the main losses include conductor I² losses, corona losses and leakage losses through insulators. Under normal operating conditions, leakage through insulators is generally small.​​

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Q3. What is Magnetic Flux and Magnetic Flux Density?

A3. When current flows through a conductor, it produces a magnetic field in the form of closed loops around the conductor. The direction of the magnetic field can be determined using the right-hand rule.

Point your right thumb in the direction of conventional current. Your curled fingers indicate the direction of the magnetic field surrounding the conductor.

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Magnetic Flux Density

Magnetic flux density, represented by , describes the magnetic flux passing through a unit area perpendicular to the field.

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B = Φ/A

Where:

• B= magnetic flux density in tesla (T)

• Φ= magnetic flux in weber (Wb)

• A= area perpendicular to the magnetic field in square metres (m²)

For a uniform magnetic field perpendicular to the surface, this equation gives the flux density directly. More generally, magnetic flux is calculated from the surface integral of the normal component of B.

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The relationship between magnetic flux density B and magnetic field intensity in a linear medium is:

​B = μ*H

 

Here, μ is the permeability of the medium, measured in henries per metre (H/m). For a non-magnetic medium such as air, permeability is approximately equal to the permeability of free space, μ₀.

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Q4. What is Biot-Savart Law?

A4. The Biot–Savart law determines the magnetic flux density produced at a point by a small current-carrying element.

 

dB = (μ/4*μ) * (I*dl*sin θ)/r²

Where:

• dB= small contribution to magnetic flux density (T)

• μ= permeability of the medium (H/m)

• I= current in the conductor (A)

• dl= infinitesimal length of the current element (m)

• r= distance from the current element to the observation point (m)

• θ= angle between the current element and the direction toward the observation point

The total magnetic field is obtained by integrating the contributions from all current elements, taking their directions into account.

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Magnetic Flux Density Due to a Long Straight Conductor

For a very long, straight conductor carrying current I, integration of the Biot–Savart law gives:

B = μ*I/(2π*r)

 

The corresponding magnetic field intensity is: H = I/2π*r

 

These equations show that the magnetic field around a long straight conductor:

• Increases in proportion to the current I.

• Decreases as the distance r from the conductor increases.

• Depends on the permeability of the surrounding medium when expressed as flux density B.

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​Q5. What is Ampere's Circuital Law? 

A5. ​Ampere’s circuital law relates the magnetic field around a closed path to the total current enclosed by that path. ∮ H.dl = Ienclosed

 

If the closed path encloses turns, each carrying current in the same direction, then: ∮ H.dl = N.I

 

For a long straight conductor, choose a circular path of radius . Since has the same magnitude everywhere along the circle, H*(2π*r) = I

 

Therefore, H = I / (2π*r) and, for a linear medium, B = μ.I / (2π*r)

 

Thus, Ampere’s circuital law provides a convenient way to calculate the magnetic field when the current distribution and geometry have sufficient symmetry.

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Q6. What is Inductors and Inductance?

A6. An inductor is a component that stores energy in a magnetic field. When current flows through a coil or conductor, magnetic flux is produced. If the flux links the circuit, a change in current changes the flux linkages and induces an electromotive force (EMF).

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Flux Linkages

If a coil has turns and each turn is linked by the same magnetic flux , the total flux linkages are:

 

λ = N.Φ

Where:

• λ= flux linkages in weber-turns (Wb-turn)

• N= number of turns

• Φ= magnetic flux linking each turn (Wb)​

 

Self-Inductance

Self-inductance is the ratio of the flux linkages produced by a circuit to the current flowing through that circuit, for a linear magnetic system.

 

L = λ / I = N.Φ / I

Where:

• λ = self-inductance in henries (H)

• N = flux linkages (Wb-turn)

• I = current (A)

 

For a magnetic system whose permeability is not constant, the incremental inductance is expressed as:

L = dλ/dI

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The unit of inductance is the henry (H). A circuit has an inductance of one henry(H) when a current change of one ampere per second produces an induced voltage of one volt, under the passive sign convention.

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Induced Voltage Due to Changing Flux Linkages

Faraday’s law states that a changing magnetic flux linkage induces a voltage.

For a single circuit, e = - dλ/dt

 

For a linear inductor with constant , this becomes:e = - L.dI/dt

 

The minus sign represents Lenz’s law: the induced voltage acts in a direction that opposes the change producing it,

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Q7. What is Mutual Inductance?

A7. Mutual inductance occurs when a change in current in one circuit produces a change in magnetic flux linking another circuit.

If current I₂ in circuit 2 produces flux linkages λ₁₂ in circuit 1, the mutual inductance is: M₁₂ = λ₁₂ / I₂

 

For a linear magnetic system, mutual inductance is measured in henries (H). The induced voltage in circuit 1 due to changing current in circuit 2 is: e₁ = - M₁₂ dI₂ / dt

 

For sinusoidal steady-state analysis, the mutual-inductance voltage is represented by the phasor relationship: v₁ = j.ω.M₁₂.I₂

 

The sign or phase relationship depends on the chosen reference directions and dot convention.

 

Difference between self-inductance and mutual inductance: self-inductance relates a circuit’s own current to its own flux linkages, whereas mutual inductance relates the current in one circuit to the flux linkages it produces in another circuit.

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Note: μ₀ = 4π × 10⁻⁷ H/m