Chapter 28 Part I - Electric Generators, Inductance, and Transformers

Electric Generators

Principles of Electric Generators

Electric generators are devices that convert mechanical work into electrical energy, forming the backbone of global electricity production. The process typically involves rotating a coil within a magnetic field to induce an electromotive force (emf).

• Energy Sources: Common sources include natural gas, hydroelectric, nuclear, coal, solar, and wind. Most methods (except wind) use steam turbines, where water is heated to produce steam that turns a turbine.

• Angular Speed (\( \omega \)): The rotation of the generator shaft is characterized by angular speed, measured in radians per second.

• Peak emf: The maximum emf is produced when the coil is perpendicular to the magnetic field (angle = 90°).

Equations for Induced emf

• Single Loop emf: The induced emf in a single loop moving in a magnetic field is given by: $\xi = (B L v) \sin \theta$ where B is the magnetic field strength, L is the length of the conductor, v is the velocity, and \theta is the angle between the velocity and the magnetic field.

• Peak emf for One Loop: $\xi_{\text{max}} = B L v$

• Multiple Loops (Coil): For a coil with N turns and area A: $\xi = N A B \omega \sin(\omega t)$

• Peak emf for Coil: $\xi_{\text{max}} = N A B \omega$

• Angular Speed and Frequency: $\omega = 2\pi f$ where f is the frequency in Hertz (Hz).

• Frequency and Period: $f = \frac{1}{T}$ where T is the period (time for one cycle).

• Tangential Speed: $v_T = r \omega = W \pi f$ where W is the width of the coil.

• Sinusoidal emf: The induced emf varies sinusoidally with time, causing the polarity to alternate.

Back emf and Ohm's Law

• Back emf: In generators and motors, an opposing emf (back emf) is induced, following Lenz's Law. It increases with speed and reduces the net current.

• Ohm's Law with Back emf: $I = \frac{V - \xi_{\text{back}}}{R}$ where V is the applied voltage, \xi_{\text{back}} is the back emf, and R is resistance.

Mutual Inductance & Self Inductance

Mutual Inductance

Mutual inductance occurs when a changing current in one coil induces an emf in a nearby coil due to the changing magnetic flux.

• Primary and Secondary Coils: The primary coil carries current I and produces a magnetic field, which induces a changing flux through the secondary coil.

• Mutual Inductance (M): $M = \frac{N_2 \Phi_{B,2}}{I_1}$ where N_2 is the number of turns in the secondary coil, \Phi_{B,2} is the magnetic flux through the secondary, and I_1 is the current in the primary.

• Induced emf in Secondary Coil: $\xi_2 = - M \frac{dI_1}{dt}$

• Units: The unit of mutual inductance is the Henry (H), where 1 H = 1 V·s/A.

• Factors Affecting M: Geometry of coils and the properties of the ferromagnetic core.

Self Inductance

Self-inductance is the property of a coil (inductor) to induce an emf in itself due to a changing current.

• Self Inductance (L): $L = \frac{N \Phi_B}{I}$ or, for a solenoid, $L = \mu_0 n^2 A l$ where n is the number of turns per unit length, A is the cross-sectional area, and l is the length.

• Induced emf (Self-Induction): $\xi = - L \frac{dI}{dt}$

• Units: Henry (H).

• Ferromagnetic Core: Wrapping the coil around a ferromagnetic core increases the magnetic flux and thus the inductance.

Energy Stored in an Inductor

Magnetic Energy Storage

Inductors store energy in their magnetic field when current flows through them, analogous to capacitors storing energy in electric fields.

• Energy Stored: $E = \frac{1}{2} L I^2$

• Energy Density: The energy per unit volume stored in the magnetic field is: $u = \frac{1}{2} \frac{B^2}{\mu_0}$ Additional info: The formula for energy density is inferred from standard electromagnetic theory.

Transformers

Principle and Types of Transformers

A transformer is a device that changes the voltage of alternating current (AC) using the principles of mutual and self-inductance. It consists of two coils (primary and secondary) wound around a common iron core.

• Step-up Transformer: Secondary coil has more turns than the primary, increasing the output voltage.

• Step-down Transformer: Secondary coil has fewer turns than the primary, decreasing the output voltage.

Transformer Equations

• emf in Secondary Coil: $\xi_2 = - N_2 \frac{d\Phi}{dt}$

• emf in Primary Coil: $\xi_1 = - N_1 \frac{d\Phi}{dt}$

• Transformer Ratio: $\frac{\xi_2}{\xi_1} = \frac{N_2}{N_1}$ or $\frac{V_2}{V_1} = \frac{N_2}{N_1}$

• Example: If the secondary coil has 8 times more turns than the primary, the voltage increases by a factor of 8 (8:1 ratio).

Type of Transformer	Turns Ratio (N2/N1)	Voltage Output
Step-up	> 1	Increases
Step-down	< 1	Decreases

Additional info: Table inferred for clarity based on standard transformer classification.