Here are some study notes on inductance and magnetic fields:

Inductance and Magnetic Fields

• Electromagnetism is the study of the interaction between electrically charged objects.

• A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. [1]

• Magnetic fields are generated by electric currents. [2]

• A wire carrying a current (I) generates a magnetomotive force (F), producing a magnetic field around it. [2]

• The m.m.f., measured in amperes, equals the current (I) in the wire. [2]

• Magnetic field strength (H) is a measure of the intensity of a magnetic field. It is defined as the magnetomotive force per unit length of the magnetic circuit. [2]

• The units for H are A/m. [2]

• H decreases with distance from the wire as it spreads over the increasing circumference (2πr). [3]

• For example, a straight wire carries a current of 5A. At a distance of 100mm from the wire the magnetic circuit is circular and the field is symmetrical; if r = 100mm, the path is 2πr = 0.628m. Therefore:

• H = I/l = 5/0.628 = 7.96 A/m [3]

• Magnetic flux (Φ) is a measure of the total magnetic field that passes through a given area. It is measured in webers (Wb). [3]

• Magnetic flux density (B) is a measure of the strength of the magnetic flux at a particular location. It is defined as the flux per unit area of cross-section. [4]

• It is measured in tesla (T). [4]

• One tesla is equivalent to 1 Wb/m2. [4]

• The flux density at a point is determined by the field strength and by the material present. [4]

• B = μH or B = μ0μrH where: [4]

• μ (H/m) is the permeability of the material. [4]

• μr is the relative permeability. [4]

• μ0 = 4π ⋅ 10-7 H/m is the permeability of free space. [4]

• Adding a ferromagnetic ring around a wire will increase the magnetic flux by several orders of magnitude. [5]

• This is because μr for ferromagnetic materials is 1000 or more. [5]

• Ferromagnetic material provides an easier path for magnetic field lines, concentrating and amplifying the flux compared to air or vacuum. [5]

• When a current-carrying wire is formed into a coil the magnetic field is concentrated within the coil. The magnetic field increases as more turns are added. [5]

• For a coil of N turns, the m.m.f. (F) is given by:

• F = IN [5]

• And the field strength is:

• H = IN/l [5]

• The magnetic flux produced is determined by the permeability of the material present. [6]

• A ferromagnetic material will increase the flux density. [6]

Reluctance

• Reluctance (S) is the resistance equivalent in a magnetic circuit. It is a measure of how the circuit opposes the flow of magnetic flux. [6]

• The units of reluctance are amperes per weber (A/Wb). [6]

• In a resistive circuit: R = V/I. [6]

• In a magnetic circuit: S = F/Φ, where: [6]

• F = magnetomotive force. [6]

• Φ = magnetic flux. [6]

Inductance

• Inductance is the property of a conductor by which a change in current flowing through it induces an electromotive force in both the conductor itself (self-inductance) and in any nearby conductors (mutual inductance). [7-9]

• A changing magnetic flux induces an electrical voltage (e.m.f.) in any conductor within the field. [7]

• Faraday’s law states that the magnitude of the e.m.f. induced in a circuit is proportional to the rate of change of magnetic flux linking the circuit. [7]

• Lenz’s law states that the direction of the e.m.f. is such that it tends to produce a current that opposes the change of flux responsible for inducing the e.m.f. [7]

• When a circuit forms a single loop, the e.m.f. induced is given by the rate of change of the flux. [10]

• When a circuit contains many loops the resulting e.m.f. is the sum of those produced by each loop. [10]

• Therefore, if a coil contains N loops, the induced voltage (V) is given by: [10]

• V = N dΦ/dt

• Where dΦ/dt is the rate of change of flux in Wb/s. [10]

Self-Inductance

• A changing current in a wire causes a changing magnetic field around it. [8]

• The changing magnetic field induces an e.m.f. in conductors within that field. [8]

• Therefore, when the current in a coil changes, it induces an e.m.f. in the coil which tends to oppose the change in the current. [8]

• This process is known as self-inductance, and the voltage produced across the inductor is given by: [8]

• V = L dI/dt

• Where L is the inductance of the coil (unit is the henry, H). [8]

Inductors

• An inductor is a passive electronic component designed to store energy in a magnetic field. [11]

• They are used to provide inductance. [11]

• The inductance of a coil depends on its dimensions and the materials around which it is formed. [11]

• L = μ0AN2/l [11]

• The inductance is greatly increased through the use of a ferromagnetic core: [11]

• L = μ0μrAN2/l [11]

Inductors in Series and Parallel

• When several inductors are connected together, their effective inductance can be calculated in the same way as for resistors – provided that they are not linked magnetically. [11]

Relationship Between Voltage and Current in an Inductor

• Consider a circuit with an inductor that is initially un-energised. [12]

• The current through it will be zero. [12]

• When the switch is closed at t = 0: [12]

• I is initially zero. [12]

• Therefore VR is initially 0. [12]

• Therefore VL is initially V. [12]

• As the inductor is energised: [12]

• I increases. [12]

• VR increases. [12]

• Therefore VL decreases. [12]

• This is exponential behaviour. [12]

• The time taken for the current to rise to a certain value (i.e. to approach its steady-state condition) is determined by L/R. [13]

• This value is the time constant, T (Greek tau). [13]

• τ = L/R [13]

Sinusoidal Voltages and Currents

• Consider the application of a sinusoidal current to an inductor: [14]

• V = L dI/dt [14]

• Voltage is directly proportional to the time differential of the current. [14]

• Since the differential of a sine wave is a cosine wave, the voltage is phase-shifted by 90° with respect to the current. [14]

• The phase-shift is in the opposite direction to that in a capacitor: in an inductor the current lags the voltage. [14]

Energy Storage in an Inductor

• Inductors store energy within a magnetic field. [14]

• It can be calculated in a similar manner to the energy stored in a capacitor. [14]

• In a small amount of time, dt, the energy added to the magnetic field is the product of the instantaneous voltage, the instantaneous current and the time. [14]

• Energy added = vi dt = L di/dt i dt = Li di [14]

• Thus, when the current is increased from zero to I: [14]

• E = L∫0Ii di = 1/2 LI2 [14]

• For example, an inductor of 10 mH with 5 A of current passing through it: [15]

• E = 1/2 LI2 = 1/2 ⋅ 10-2 ⋅ 52 = 125 mJ [15]

Mutual Inductance

• If two conductors are linked magnetically, a changing current in one conductor creates a changing magnetic flux, inducing a voltage in the second conductor. [15]

• Mutual inductance is quantified in a similar way to self-inductance. If a current (I1) flows in one circuit, the voltage induced in a second circuit (V2) is given by: [9]

• V2 = M dI1/dt

• Where M is the mutual inductance between the two circuits. [9]

Mutual Inductance and Transformers

• Mutual inductance is often applied to the interaction of coils in a transformer. [9]

• In a transformer, a changing current in one coil (the primary) is used to induce a changing current in a second coil (the secondary). [9]

• The coupling coefficient measures the fraction of flux from one coil that links with another. [9]

• A coupling coefficient value of 1 means total flux linkage, while 0 means no linkage. [16]

• To increase coupling between coils, you can: [16]

• Move the coils closer together. [16]

• Wrap one coil around the other. [16]

• Add a ferromagnetic core. [16]

• The best coupling is achieved by using a continuous ferromagnetic loop, which increases both the inductance and flux linkage between the coils. [16]

Transformer – Basic Operation

• A transformer has a primary coil (input) and a secondary coil (output). [17]

• These coils are wound on a ferromagnetic core to achieve maximum magnetic coupling (ideal coupling coefficient = 1). [17]

• An alternating voltage (V₁) applied to the primary creates a magnetic field, inducing a corresponding alternating voltage in the secondary. [17]

Transformer

• The voltage ratio between primary and secondary is determined by the turns ratio (N₂/N₁): [17]

• V₂/V₁ = N₂/N₁ [17]

Transformer - Key Points

• AC Only: Transformers only work with alternating currents; DC will not induce voltage. [18]

• Power Conservation: No energy source other than the input signal. The output power is almost equal to the input power: [18]

• V₁I₁ = V₂I₂ [18]

• Load Effect: Adding a load to the secondary circuit decreases output voltage as current increases due to opposing magnetic flux. [18]

Transformer - Applications

• Step-down transformers are widely used in power supplies for low-voltage electronics. [18]

• Transformers provide electrical isolation, preventing direct electrical connection between circuits. [18]