Electromagnetic Induction and Alternating Currents

18. Electromagnetic Induction

Electromagnetic Induction is a phenomenon where a changing magnetic field through a coil of wire induces an electromotive force (EMF) across the coil. This concept is fundamental to the operation of many electrical devices, including generators and transformers.

18.1 Faraday's Laws; Induced Electric Fields

Faraday's First Law: Whenever a conductor is placed in a varying magnetic field, an electromotive force (EMF) is induced. If the conductor circuit is closed, a current is induced.

Faraday's Second Law: The magnitude of the induced EMF is equal to the rate of change of magnetic flux through the circuit.

Mathematically, this is expressed as:

EMF = -N (d /dt)

Where:

• EMF is the induced electromotive force.
• N is the number of turns in the coil.
• d /dt is the rate of change of magnetic flux.
• The negative sign indicates the direction of the induced EMF (Lenz's Law).

Induced Electric Fields: A changing magnetic field also produces a non-conservative electric field. This induced electric field is responsible for driving the induced current in a conductor. Unlike electrostatic fields, induced electric fields are not derivable from a scalar potential and form closed loops.

18.2 Lenz's Law, Motional Electromotive Force

Lenz's Law: This law states that the direction of the induced current or EMF is always such that it opposes the change in magnetic flux that produced it. This is a direct consequence of the conservation of energy.

Example: If a north pole of a magnet is moved towards a coil, the induced current in the coil will create a north pole to oppose the motion. If the north pole is moved away, the induced current will create a south pole to attract it back.

Motional Electromotive Force: When a conductor moves through a magnetic field, a magnetic force acts on the free charges within the conductor, causing them to accumulate at the ends of the conductor. This charge separation creates an electric field, leading to a potential difference across the conductor, known as motional EMF.

For a conductor of length L moving with velocity v perpendicular to a magnetic field B, the motional EMF is:

EMF = B L v

18.3 A.C. Generators; Eddy Currents

A.C. Generators (Alternators): An AC generator converts mechanical energy into electrical energy. It works on the principle of electromagnetic induction. A coil is rotated in a uniform magnetic field, causing the magnetic flux through the coil to change continuously. This changing flux induces an alternating EMF and hence an alternating current.

The instantaneous EMF generated is given by:

EMF = N B A sin( t)

Where N is the number of turns, B is the magnetic field strength, A is the area of the coil, and is the angular velocity.

Eddy Currents: These are circulating currents induced in bulk conductors when they are subjected to changing magnetic fields. While useful in applications like magnetic braking and induction furnaces, eddy currents can also cause energy loss in transformers and other AC devices due to heating. They are typically minimized by using laminated cores.

18.4 Self-inductance and Mutual Inductance

Self-inductance: When the current in a coil changes, the magnetic flux associated with the coil itself also changes, inducing an EMF in the same coil. This phenomenon is called self-induction, and the property of the coil that opposes this change is called self-inductance (L).

The self-induced EMF is given by:

EMF = -L (dI/dt)

The unit of inductance is Henry (H).

Mutual Inductance: When the current in one coil changes, it can induce an EMF in a nearby, coupled coil. This phenomenon is called mutual induction, and the property that quantifies this coupling is mutual inductance (M).

The mutually induced EMF in coil 2 due to a changing current in coil 1 is:

EMF2 = -M (dI1/dt)

Mutual inductance is also measured in Henrys.

18.5 Energy Stored in an Inductor

An inductor stores energy in its magnetic field when current flows through it. The energy (U) stored in an inductor is proportional to the square of the current (I) and its self-inductance (L).

U = (1/2) L I^2

This energy is released when the current through the inductor decreases, which is why inductors can resist sudden changes in current.

18.6 Transformer

A transformer is a device that changes (transforms) AC voltages from one level to another using the principle of mutual induction. It consists of two coils, a primary coil and a secondary coil, wound around a common soft iron core.

For an ideal transformer, the ratio of voltages is proportional to the ratio of the number of turns:

Vp/Vs = Np/Ns

Where Vp and Vs are the primary and secondary voltages, and Np and Ns are the number of turns in the primary and secondary coils, respectively.

If Ns > Np, it's a step-up transformer; if Ns < Np, it's a step-down transformer.

19. Alternating Currents

Alternating Current (AC) is an electric current that periodically reverses direction, in contrast to direct current (DC), which flows only in one direction. AC is the primary form in which electric power is delivered to businesses and residences.

19.1 Peak and RMS Value of AC Current and Voltage

AC voltage and current vary sinusoidally with time:

V = V0 sin( t)

I = I0 sin( t + )

• V0 and I0 are the peak (or maximum) values of voltage and current.
• is the angular frequency.
• is the phase difference between voltage and current.

The Root Mean Square (RMS) value is a measure of the effective value of an AC voltage or current, equivalent to the DC value that would produce the same average power dissipation in a resistive load.

VRMS = V0 / 2

IRMS = I0 / 2

19.2 AC through a Resistor, a Capacitor and an Inductor

The behavior of AC in different circuit components:

• Resistor (R): In a purely resistive circuit, voltage and current are in phase. The opposition to current flow is resistance (R).
• Capacitor (C): In a purely capacitive circuit, the current leads the voltage by 90 degrees ( /2 radians). The opposition to current flow is capacitive reactance (Xc = 1/( C)).
• Inductor (L): In a purely inductive circuit, the voltage leads the current by 90 degrees ( /2 radians). The opposition to current flow is inductive reactance (XL = L).

19.3 Phasor Diagram

Phasor diagrams are graphical representations used to analyze AC circuits. Voltages and currents are represented as rotating vectors (phasors) whose lengths correspond to their peak or RMS values and whose angles represent their phase relative to a reference. This simplifies the analysis of phase relationships in complex AC circuits.

19.4 Series Circuits Containing Combination of Resistance, Capacitance and Inductance

In RLC series circuits, the total opposition to current flow is called impedance (Z), which is the vector sum of resistance and reactances.

Z = (R^2 + (XL - Xc)^2)

The phase angle ( ) between the total voltage and current is given by:

tan( ) = (XL - Xc) / R

19.5 Series Resonance, Quality Factor

Series Resonance: Occurs in an RLC series circuit when the inductive reactance equals the capacitive reactance (XL = Xc). At resonance, the impedance is at its minimum (equal to R), leading to maximum current for a given voltage. The resonant frequency (f0) is:

f0 = 1 / (2 (LC))

Quality Factor (Q-factor): A dimensionless parameter that describes how underdamped an oscillator or resonator is. In an RLC series circuit, it indicates the sharpness of the resonance peak. A higher Q-factor means a narrower bandwidth and more selective circuit.

Q = (1/R) (L/C) or Q = XL / R at resonance.

19.6 Power in AC Circuits: Power Factor

In AC circuits, power can be categorized into three types:

• Apparent Power (S): The product of RMS voltage and RMS current (measured in VA). S = VRMS IRMS
• Reactive Power (Q): Power exchanged between the source and reactive components (inductors and capacitors), not contributing to useful work (measured in VAR). Q = VRMS IRMS sin( )
• True (or Average) Power (P): The actual power consumed by the circuit and converted into heat or work (measured in Watts). P = VRMS IRMS cos( )

Power Factor (PF): The ratio of true power to apparent power. It is equal to the cosine of the phase angle ( ) between voltage and current.

PF = cos( ) = P / S

A power factor close to 1 indicates efficient use of power, while a low power factor indicates a large reactive power component, leading to inefficiencies.