Inductors&

RL Circuits

Introduction

Inductors have a number of response characteristics similar to those of the capacitor.

The inductor exhibits its true characteristics only when a change in voltage or current is made in the network.

The Basic Inductor

• When a length of wire is formed onto a coil, it becomes a basic inductor

• Magnetic lines of force around each loop in the winding of the coil effectively add to the lines of force around the adjoining loops, forming a strong electromagnetic field within and around the coil

• The unit of inductance is the henry (H), defined as the inductance when one ampere per second through the coil, induces one volt across the coil

Symbol for the inductor.

Typical Inductors

Magnetic FieldsIn the region surrounding a permanent

magnet there exists a magnetic field, which can be represented by magnetic flux lines similar to electric flux lines.

Magnetic flux lines differ from electric flux lines in that they don’t have an origin or termination point.

Magnetic flux lines radiate from the north pole to the south pole through the magnetic bar.

Flux distribution for a permanent magnet.

Flux distribution for two adjacent, opposite poles.

Magnetic FieldsContinuous magnetic flux lines will strive to occupy as

small an area as possible.The strength of a magnetic field in a given region is directly

related to the density of flux lines in that region.If unlike poles of two permanent magnets are brought

together the magnets will attract, and the flux distribution will be as shown below.

Magnetic Fields

If like poles are brought together, the magnets will repel, and the flux distribution will be as shown.

If a nonmagnetic material, such as glass or copper, is placed in the flux paths surrounding a permanent magnet, there will be an almost unnoticeable change in the flux distribution.

Magnetic Fields

If a magnetic material, such as soft iron, is placed in the flux path, the flux lines will pass through the soft iron rather than the surrounding air because the flux lines pass with greater ease through magnetic materials than through air.

This principle is put to use in the shielding of sensitive electrical elements and instruments that can be affected by stray magnetic fields.

Magnetic Fields

Flux and Flux Density

In the SI system of units, magnetic flux is measured in webers (Wb) and is represented using the symbol .

The number of flux lines per unit area is called flux Its magnitude is determined by the following equation: density (B). Flux density is measured in teslas (T).

Flux distribution for a permanent magnet.

Digital display gaussmeter.

The Basic Inductor

• When a length of wire is formed onto a coil, it becomes a basic inductor

• Magnetic lines of force around each loop in the winding of the coil effectively add to the lines of force around the adjoining loops, forming a strong electromagnetic field within and around the coil

• The unit of inductance is the henry (H), defined as the inductance when one ampere per second through the coil, induces one volt across the coil

Magnetic flux lines around a current-carrying conductor.

Flux distribution of a single-turn coil.

Flux distribution of a current-carrying coil.

A coil of wire forms an inductor. When there is current through it, a three-dimensional electromagnetic field is created, surrounding the coil in all directions.

Electromagnet.

Inductance

Inductors are designed to set up a strong magnetic field linking the unit, whereas capacitors are designed to set up a strong electric field between the plates.

Inductance is measure in Henries (H).One henry is the inductance level that will establish a voltage of 1 volt across the coil due to a chance in current of 1 A/s through the coil.

Inductance

Inductor construction and inductance

l

ANL

2

(H) henriesin inductance

(m)length

)(m area

(t) turnsofnumber

m)(Wb/Aty permeabili

2

L

l

A

N

Factors that determine the inductance of a coil.

Thomas L. FloydElectronics Fundamentals, 6eElectric Circuit Fundamentals, 6e

Copyright ©2004 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

All rights reserved.

Defining the parameters

Physical Characteristics

• Inductance is directly proportional to the permeability of the core material

• Inductance is directly proportional to the cross-sectional area of the core

• Inductance is directly proportional to the square of the number of turns of wire

• Inductance is inversely proportional to the length of the core material

L = N2A/l

Air-core coil for Example

Winding Resistance and Capacitance

• When many turns of wire are used to construct a coil, the total resistance may be significant

• The inherent resistance is called the dc resistance or the winding resistance (RW)

• When two conductors are placed side-by-side, there is always some capacitance between them

• When many turns of wire are placed close together in a coil, there is a winding capacitance (CW)

• CW becomes significant at high frequencies

Winding resistance of a coil.

Winding capacitance of a coil.

Complete equivalent model for an inductor.

Energy storage and conversion to heat in an inductor.

Symbols for fixed and variable inductors.

Thomas L. FloydElectronics Fundamentals, 6eElectric Circuit Fundamentals, 6e

Copyright ©2004 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

All rights reserved.

Inductor symbols.

Thomas L. FloydElectronics Fundamentals, 6eElectric Circuit Fundamentals, 6e

Copyright ©2004 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

All rights reserved.

Relative sizes of different types of inductors: (a) toroid, high-current; (b) phenolic (resin or plastic core); (c) ferrite core.

Variable inductors with a typical range of values from 1 H to 100 H; commonly used in oscillators and various RF circuits such as CB transceivers, televisions, and radios.

Molded inductor color coding.

Digital reading inductance meter.

Self-Inductance

• Inductance is a measure of a coil’s ability to establish an induced voltage as a result of a change in its current, and that induced voltage is in a direction to oppose that change in current

• An inductor stores energy in the magnetic field created by the current:

W = 1/2 LI2

Faraday’s and Lenz’s Laws

• Faraday’s law:– The amount of voltage induced in a coil is directly

proportional to the rate of change of the magnetic field with respect to the coil

• Lenz’s law:– When the current through a coil changes and an

induced voltage is created as a result of the changing electromagnetic field, the direction of the induced voltage is such that it always opposes the change in current

Induced voltage is created by a changing magnetic field.

Induced Voltage

If a conductor is moved through a magnetic field so that it cuts magnetic lines of flux, a voltage will be induced across the conductor.

Induced Voltage

Faraday’s law of electromagnetic inductionThe greater the number of flux lines cut per unit time (by

increasing the speed with which the conductor passes through the field), or the stronger the magnetic field strength (for the same traversing speed), the greater will be the induced voltage across the conductor.

If the conductor is held fixed and the magnetic field is moved so that its flux lines cut the conductor, the same effect will be produced.

Induced Voltage

Faraday’s law of electromagnetic induction

If a coil of N turns is placed in the region of the changing flux, as in the figure below, a voltage will be induced across the coil as determined by Faraday’s Law.

Induced Voltage

Lenz’s law

An induced effect is always such as to oppose the cause that produced it.

Induced Voltage

• The inductance of a coil is also a measure of the change in flux linking a coil due to a change in current through the coil

–N is the number of turns, is the flux in webers, and i is the current through the coil

Induced Voltage

The larger the inductance of a coil (with N fixed), the larger will be the instantaneous change in flux linking the coil due to the instantaneous change in the current through the coil.

The voltage across an inductor is directly related to the inductance L and the instantaneous rate of change through the coil. The greater the rate of change of current through the coil, the greater the induced voltage.

V) (volts, dt

diLv L

L

R-L Transients: The Storage Phase

The changing voltage and current that result during the storing of energy in the form of a magnetic field by an inductor in a dc circuit.

The instant the switch is closed, inductance in the coil will prevent an instantaneous change in the current through the coil.

The potential drop across the coil VL, will equal the impressed voltage E as determined by Kirchhoff’s voltage law.

R-L Transients: The Storage Phase

An ideal inductor (Rl = 0 ) assumes a short-circuit equivalent in a dc network once steady-state conditions have been established.

The storage phase has passed and steady-state conditions have been established once a period of time equal to five time constants has occurred.

The current cannot change instantaneously in an inductive network.

The inductor takes on the characteristics of an open circuit at the instant the switch is closed.

The inductor takes on the characteristics of a short circuit when steady-state conditions have been established.

Initial Values

The drawing of the waveform for the current iL from the initial value to a final value.

Initial Values• Since the current through a coil cannot change

instantaneously, the current through a coil will begin the transient phase at the initial value established by the network before the switch was closed

• The current will then pass through the transient phase until it reaches the steady-state (or final) level after about 5 time constants

• The steady-state level of the inductor current can be found by substituting its short-circuit equivalent (or Rl for the practical equivalent)

R-L Transients: The Release Phase

In R-L circuits, the energy is stored in the form of a magnetic field established by the current through the coil.

An isolated inductor cannot continue to store energy since the absence of a closed path would cause the current to drop to zero, releasing the energy stored in the form of a magnetic field.

Energy storage and conversion to heat in an inductor.

Inductors in DC Circuits

• When there is constant current in an inductor, there is no induced voltage

• There is a voltage drop in the circuit due to the winding resistance of the coil

• Inductance itself appears as a short to dc

R-L Transients: The Release Phase

Analyzing the R-L circuit in the same manner as the R-C circuit.When a switch is closed, the voltage across the resistor

R2 is E volts, and the R-L branch will respond in the change in the current di/dt of the equation vL = L(di/dt) would establish a high voltage vL across the coil.

Average Induced Voltage

For inductors, the average induced voltage is defined by

V) (volts, av t

iLv L

L

Steady State Conditions

An inductor can be replaced by a short circuit in a dc circuit after a period of time greater than five time constants have passed.

Assuming that all of the currents and voltages have reached their final values, the current through each inductor can be found by replacing each inductor with a short circuit.

Energy Stored by an Inductor

The ideal inductor, like the ideal capacitor, does not dissipate the electrical energy supplied to it. It stores the energy in the form of a magnetic field.

Self-Inductance

• Inductance is a measure of a coil’s ability to establish an induced voltage as a result of a change in its current, and that induced voltage is in a direction to oppose that change in current

• An inductor stores energy in the magnetic field created by the current:

W = 1/2 LI2

RL Time Constant

• Because the inductor’s basic action opposes a change in its current, it follows that current cannot change instantaneously in an inductor

= L/Rwhere: is in seconds (s)

L is in henries (H)

R is in ohms ()

Energizing Current in an Inductor• In a series RL circuit, the current will increase to

approximately 63% of its full value in one time-constant () interval after the switch is closed

• The current reaches its final value in approximately 5

Energizing current in an inductor.

De-energizing Current in an Inductor

• In a series RL circuit, the current will decrease to approximately 63% of its fully charged value one time-constant () interval after the switch is closed

• The current reaches 1% of its initial value in approximately 5; considered to be equal to 0

De-energizing current in an inductor.

Induced Voltage in the Series RL Circuit

• At the instant of switch closure, the inductor effectively acts as an open with all the source voltage across it

• During the first 5 time constants, the current is building up exponentially, and the induced coil voltage is decreasing

• The resistor voltage increases with current• After 5 time constants, all of the source voltage is

dropped across the resistor and none across the coil

Demonstrating the effect of opening a switch in series with an inductor with a steady-state current.

Inductors in DC Circuits

• When there is constant current in an inductor, there is no induced voltage

• There is a voltage drop in the circuit due to the winding resistance of the coil

• Inductance itself appears as a short to dc

Exponential Formulas

• The general formulas for RL circuits are:

v =VF+ (Vi - VF)e-Rt/L

i =IF+ (Ii - IF)e-Rt/L

Where VF and IF are final values of voltage and current, Vi and Ii are initial values of voltage and current, v and i are instantaneous values of induced voltage or current at time t

The power curve for an inductive element under transient conditions.

Inductors in Series and in Parallel

Inductors, like resistors and capacitors, can be placed in series

Increasing levels of inductance can be obtained by placing inductors in series

Series Inductors

• When inductors are connected in series, the total inductance increases

LT = L1 + L2 + L3 + … + Ln

Inductors in series.

Inductors in Series and in Parallel

Inductors, like resistors and capacitors, can be placed in parallel. Decreasing levels of inductance can be obtained by

placing inductors in parallel.

Parallel Inductors

• When inductors are connected in parallel, the total inductance is less than the smallest inductance

1/LT = 1/L1 + 1/L2 + 1/L3 + … + 1/Ln

Inductors in parallel.

Inductive Reactance

• Inductive reactance is the opposition to sinusoidal current, expressed in ohms

• The inductor offers opposition to current, and that opposition varies directly with frequency

• The formula for inductive reactance, XL, is:

XL = 2f L

The current in an inductive circuit varies inversely with the frequency of the source voltage

Phase Relationship of Current and Voltage in an Inductor

• The current lags inductor voltage by 90• The curves below are for a purely inductive circuit

Quality Factor (Q) of a Coil

• The quality factor (Q) is the ratio of the reactive power in the inductor to the true power in the winding resistance of the coil or the resistance in series with the coil

Q = (reactive power) / (true power)

Q = XL/RW

An inductor used as an RF choke to minimize interfering signals on the power supply line

Checking a coil by measuring the resistance