chapter 2p3

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Overview

We continue our study of energy storage elements with a discussion of inductors. Inductors, likeresistors and capacitors, are passive two-terminal circuit elements. That is, no external power supplyis necessary to make them function. Inductors commonly consist of a conductive wire wrappedaround a core material; inductors store energy in the form of a magnetic field set up around thecurrent-carrying wire.

In this chapter, we describe physical properties of inductors and provide a mathematical model for anideal inductor. Using this ideal inductor model, we will develop mathematical relationships for theenergy stored in an inductor and governing relations for series and parallel connections of inductors.The chapter will conclude with a brief discussion of practical (non-ideal) inductors.

Before beginning this chapter, you shouldbe able to:

After completing this chapter, you should beable to:

Evaluate integral and differentialrelations

Sketch both the integral and derivativeof a given time function

Define voltage and current in terms ofelectrical charge (Chapter 1.1)

Determine equivalent resistance ofseries and parallel combinations ofresistors (Chapter 1.5)

Write the circuit symbol for an inductor

State the mechanism by which an inductorstores energy

State from memory the voltage-currentrelationship for an inductor in bothdifferential and integral form

State from memory the response of aninductor to constant voltages andinstantaneous current changes

Write the mathematical expressiondescribing energy storage in an inductor

Determine the equivalent inductance ofseries and parallel combinations ofinductors

Sketch a circuit describing a non-idealinductor

This chapter requires:

N/A


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2.3: Inductors

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Inductors:

Passing a current through a conductive wire will create a magnetic fieldaround the wire. Thismagnetic field is generally thought of in terms of as forming closed loops of magnetic fluxaround thecurrent-carrying element. This physical process is used to create inductors. Figure 1 illustrates a

common type of inductor, consisting of a coiled wire wrapped around a core material. Passing acurrent through the conducting wire sets up lines of magnetic flux, as shown in Figure 1; the inductorstores energy in this magnetic field. The inductanceof the inductor is a quantity which tells us howmuch energy can be stored by the inductor. Higher inductance means that more energy can bestored by the inductor. Inductance has units of Henrys, abbreviated H.

The amount of inductance an inductor has is governed by the geometry of the inductor and theproperties of the core material. These effects can be complex; rather than attempt a comprehensivediscussion of these effects, we will simply claim that, in general, inductance is dependent upon thefollowing parameters:

The number of times the wire is wrapped around the core. More coils of wire results in ahigher inductance.

The core materials type and shape. Core materials are commonly ferromagnetic materials,since they result in higher magnetic flux and correspondingly higher energy storage. Air,however, is a fairly commonly used core material presumably because of its readyavailability.

The spacing between turns of the wire around the core.

Figure 1. Wire-wrapped inductor with applied current through conductive wire.

We will denote the total magnetic flux created by the inductor by , as shown in Figure 1. For a linearinductor, the flux is proportional to the current passing through the wound wires. The constant ofproportionality is the inductance, L:

)()( tLit = (1)


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2.3: Inductors



Voltage is the time rate of change of magnetic flux, so

dt

tdtv

)()(

= (2)

Combining equations (1) and (2) results in the voltage-current relationship for an ideal inductor:

dt

tdiLtv

)()( = (3)

The circuit symbol for an inductor is shown in Figure 2, along with the sign conventions for thevoltage-current relationship of equation (3). The passive sign convention is used in the voltage-current relationship, so positive current is assumed to enter the terminal with positive voltage polarity.

Figure 2. Inductor circuit symbol and voltage-current sign convention.

Integrating both sides of equation (3) results in the following form for the inductors voltage-currentrelationship:

)()(1

)(0

0

tidvL

ti

t

t

+= (4)

In equation (4), i(t0) is a known current at some initial time t0and is used as a dummy variable ofintegration to emphasize that the only t which survives the integration process is the upper limit ofthe integral.

Important result:

The voltage-current relationship for an ideal inductor can be stated in either differential or

integral form, as follows:

dt

tdiLtv

)()( =

)()(1

)(0

0

tidvL

ti

t

t

+=


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2.3: Inductors



Example 1: A circuit contains a 100mH inductor. The current as a function of time through theinductor is measured and shown below. Plot the voltage across the inductor as a function of time.

In the time range 0


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2.3: Inductors



Example 2: If the voltage as a function of time across an inductor with inductance L =10 mH is asshown below, determine the current as a function of time through the capacitor. Assume that thecurrent through the capacitor is 0A at time t = 0.

Voltage,

V

At time t = 0, the current is given to be 0A.

In the time period 0


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2.3: Inductors



It is often useful, when analyzing circuits containing inductors, to examine the circuits response toconstant operating conditions and to instantaneous changes in operating condition. We examine theinductors response to each of these operating conditions below:

Inductor response to constant voltage:

If the current through the inductor is constant, equation (3) indicates that the voltage acrossthe inductor is zero. Thus, if the current through the inductor is constant, the inductor isequivalent to a short circuit.

Inductor response to instantaneous current changes:

If the current through the inductor changes instantaneously, the rate of change of current isinfinite. Thus, by equation (3), if we wish to change the current through an inductorinstantaneously, we must supply infinite voltage to the inductor. This implies that infinitepower is available, which is not physically possible. Thus, in any practical circuit, the currentthrough an inductor cannot change instantaneously.

Any circuit which allows an instantaneous change in the current through an ideal inductor isnot physically realizable. We may sometimes assume, for mathematical convenience, that anideal inductors current changes suddenly; however, it must be emphasized that thisassumption requires an underlying assumption that infinite power is available and is thus notan allowable operating condition in any physical circuit.

Energy Storage:

The instantaneous power dissipated by an electrical circuit element is the product of the voltage andcurrent:

)()()( titvtp = (5)

Using equation (3) to write the voltage in equation (5) in terms of the inductors current:

dt

tditiLtp

)()()( = (6)

As was previously done for capacitors, we integrate the power with respect to time to get the energystored in the inductor:

Important Inductor Properties:

Inductors can be replaced by short-circuits, under circumstances when all operatingconditions are constant.

Currents through inductors cannot change instantaneously. No such requirement is

placed on voltages.


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2.3: Inductors



=

t

L dtdt

diLitW

)()()(

Which, after some manipulation (comparable to the approach taken when we calculated energy

storage in capacitors), results in the following expression for the energy stored in an inductor:

)(2

1)(

2 tLitWL = (7)

Inductors in Series:

Consider the series connection of Ninductors shown in Figure 3.

Figure 3. Series connection of N inductors.

Applying Kirchoffs voltage law around the loop results in:

)()()()(21

tvtvtvtvNL++= (8)

Using equation (3) to write the inductor voltage drops in terms of the current through the loop gives:

( )dt

tdiLLL

dt

tdiL

dt

tdiL

dt

tdiLtv

N

N

)(

)()()()(

21

21

L

L

++=

+++=

Using summation notation results in

dt

tdiLtv

N

k

k

)()(

1

=

=

(9)

This is the same equation that governs the circuit of Figure 4, if


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2.3: Inductors



=

=

N

k

keq LL1

(10)

Thus, the equivalent inductance of a series combination of inductors is simply the sum of theindividual inductances. This result is analogous to the equations which provide the equivalentresistance of a series combination of resistors.

Figure 4. Equivalent circuit to Figure 3.

Inductors in Parallel:

Consider the parallel combination of Ninductors, as shown in Figure 5.

LNi(t) L1 L2

i1(t) i 2(t) i N(t)+

-

v(t)

Figure 5. Parallel combination of I inductors.

Applying Kirchoffs current law at the upper node results in:

)()()()(21

titititi NL++= (11)

Using equation (4) to write the inductor currents in terms of their voltage drops gives:

[ ]

)()(111

)()()()(1

)(1

)(1

)()(1

)()(1

)()(1

)(

0

21

00201

21

002

2

01

1

0

000

000

tidvLLL

tititidvL

dvL

dvL

tidv

L

tidv

L

tidv

L

ti

t

tN

N

t

tN

t

t

t

t

N

t

tN

t

t

t

t

+

+++=

++++

++=

+++

++

+=

L

LL

L


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2.3: Inductors



This can be re-written using summation notation as

)()(1

)(0

10

tidvL

ti

t

t

N

k k

+

=

=

(12)

This is the same equation that governs the circuit of Figure 4, if

=

=

N

k keq LL 1

11(13)

For the special case of two inductors L1 and L2 in series, equation (13) simplifies to

21

21

LL

LLLeq

+= (14)

Equations (13) and (14) are analogous to the equations which provide the equivalent resistance ofseries combinations of resistors.

Practical Inductors:

Most commercially available inductors are manufactured by winding wire in various coil configurationsaround a core. Cores can be a variety of shapes; Figure 1 in this chapter shows a core which isbasically a cylindrical bar. Toroidal cores are also fairly common a closely wound toroidal core hasthe advantage that the magnetic field is confined nearly entirely to the space inside the winding.

Inductors are available with values from less than 1 micro-Henry (1H = 10-6 Henries) up to tens ofHenries. A 1H inductor is very large; inductances of most commercially available inductors aremeasured in millihenries (1mH = 10-3 Henries) or microhenries. Larger inductors are generally usedfor low-frequency applications (in which the signals vary slowly with time).

Summary: Series and Parallel Inductors

The equivalent inductance of a series combination of inductors L1, L2, , LN is governed by arelation which is analogous to that providing the equivalent resistance of a seriescombination of resistors:

=

=

N

k

keq LL1

The equivalent inductance of a parallel combination of inductors L1, L2, , LN is governed bya relation which is analogous to that providing the equivalent resistance of a parallelcombination of resistors:

=

=

N

k keq LL 1

11


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2.3: Inductors



Attempts at creating inductors in integrated-circuit form have been largely unsuccessful; thereforemany circuits which are to implemented as integrated circuits do not include inductors. Inclusion ofinductance in the analysis stage of these circuits may however, be important. Since any current-carrying conductor will create a magnetic field, the stray inductanceof supposedly non-inductive

circuit elements can become an important consideration in the analysis and design of a circuit.

Practical inductors, unlike the ideal inductors discussed in this chapter, dissipate power. Anequivalent circuit model for a practical inductor is generally created by placing a resistance in serieswith an ideal inductor, as shown in Figure 6.

Figure 6. Equivalent circuit model for a practical inductor.

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