10 Inductor Principles (66-67)(1)(1)

Inductors

Definition of Inductor

An inductor is a circuit element that stores energy in a magnetic field due to the flow of current through a multi-turn coil of wire. Inductors are commonly used in various electrical applications including filters, transformers, and energy storage devices. They impede changes in current, making them essential in managing electrical circuits' stability.

Principles Governing Inductors

The behavior of inductors can be explained by:

• Faraday’s Law of Induction: Indicates that a changing magnetic field can induce an electromotive force (emf). It describes how the induced emf in a closed loop is proportional to the rate of change of the magnetic flux.

• Lenz’s Law: States that the direction of the induced emf will oppose the change in current that created it, which means that if the current increases, the induced emf will act to decrease it and vice versa. This property is critical for energy conservation in circuits.

Key Characteristics

• An electric current flowing through an inductor generates a magnetic field, which expands and collapses as the current increases and decreases.

• Changes in this current will induce a back emf, opposing the change in current. This inherent characteristic of inductors is used in applications like audio equipment where signal integrity is essential.

• Magnitude of Induced emf: Relates directly to the rate of change of current; faster changes yield greater induced emf. The relationship is described mathematically as:

  [ \text{emf} = -L \frac{di}{dt} ]

• The direction of induced emf always RESISTS the change in current, which can be demonstrated by considering the circuit during transient conditions, such as the switching on and off of a current source.

Unit of Inductance

The unit of inductance is the Henry (H), named after Joseph Henry, an American scientist. A 1 Henry inductor will produce an induced emf of 1 volt when the current changes at a rate of 1 Ampere per second. Inductors can be found in various sizes and types, including air-core, iron-core, and ferrite-core, which all affect their inductance value and application.

Practical Considerations

In practice, the behavior of real inductors also includes resistance due to the wire used in construction, which means that they are not perfect energy storage devices.

• Therefore, a practical inductor requires an equivalent circuit that includes both a resistor (representing winding resistance) and an inductor. This models how real-world devices behave and is essential for accurate circuit simulation and design.

Energy Stored by an Inductor

Energy Storage Principle

An inductor can store energy due to the induced emf created by the magnetic field. The energy stored in an inductor is represented by the equation:

[ W = \frac{1}{2} LI^2 ]This formula highlights how the energy stored is proportional to the inductance and the square of the current, indicating that even small increases in current can lead to significantly higher energy storage.

Inductors in Series

For inductors connected in series, the total voltage is the sum of individual voltages:

[ V_s = V_1 + V_2 + V_3 ]

The combined inductance can be expressed mathematically:

[ L_s = L_1 + L_2 + L_3 ]This arrangement magnifies the overall inductance, and thus the overall effects of back emf are also compounded.

The current through an inductor in series is given by:

[ L_s \left( \frac{di}{dt} \right) = L_1 \left( \frac{di}{dt} \right) + L_2 \left( \frac{di}{dt} \right) + L_3 \left( \frac{di}{dt} \right) ]

Inductors in Parallel

For inductors connected in parallel, the total current is the sum of the individual currents:

[ I_P = I_1 + I_2 + I_3 ]

The effective inductance for two inductors in parallel can be derived from:

[ \frac{1}{L_P} = \frac{1}{L_1} + \frac{1}{L_2} + ... ]

This formula shows how combining inductors in parallel decreases the overall inductance since each inductor allows the flow of current, reducing the overall back emf produced.

The energy stored in the inductor is also represented as:

[ W = \frac{1}{2} LI^2 ](Joules)This consistent equation demonstrates that regardless of connection type, the fundamental properties governing inductor behavior remain the same, yet their application can vary significantly based on configuration.