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InductorsA coil of wire can create a magnetic field if a

current is run through it. If that current

changes (as in the AC case), the magnetic field

created by the coil will change. Will this

changing magnetic field through the coil causea voltage to be created across the coil? YES!

This is called self-inductance and is the basis

behind the circuit element called the inductor.

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Inductors

Vinductor = -L dI /dt

&ere the minus sign means that when the

current is increasing, the voltage across the

inductor will tend to o!!ose the increase,

and it also means when the current is

decreasing, the voltage across the inductorwill tend to o!!ose the decrease.

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Units: Henry

'rom Vinductor = -L dI /dt

# has units of olt *Am!sec+ which is

called a &enry"

1 Henry = Volt-sec / Am .

This is the same unit we had for mutual

inductance before (recall the transformer).

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Solenoid tye inductor

'or a ca!acitor, we started with a !arallel

!late configuration since the !arallel !lates

!rovided a uniform electric field betweenthe !lates.

In the same way, we will start with a device

that !rovides a uniform magnetic fieldinside the device" a solenoid.

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olenoid inductor

solenoid = µnI where n -#ength .

ecall 'araday/s #aw" V = d/dt "∫∫ •dA # .

ince the I in the 0 is a constant with res!ect to the variable of

integration (dA), we have"

V = L dI/dt where # ∫∫ (µn) dA , and since 0 is uniformover the area for the solenoid (and in the same direction asarea)"

# (µn)A. 0ut the area is for each of - loo!s, so" # µ -1A#ength where Aeach loo! π 1.

L =µπ

$%& % / Len't( )

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i2e of a &enry

'rom the inductance of a solenoid"L =

µπ

$%& % / Len't(

we see that with vacuum inside the solenoid, µ becomes µo which has a value of 3π 4 5678 T7mA, a rather smallnumber. will normally be less than a meter, and so 1

will also ma%e # small.

&owever, - can be large, and µ can be a lot larger than µo if we use a magnetic material.

Altogether, a (enry is a rat(er lar'e inductance.

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Inductors

Lsolenoid = µπ$%& % / Len't( )

As we indicated before, the value of the

inductance deends only on the sha!e (,

#ength, -) and materials (µ).The inductance relates the voltage across it to

the changing current through it"

V = - L dI/dt . We again use #en2/s #aw to

give us the direction (sign) of the voltage.

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Ener'y Stored in an Inductor

We start from the definition of voltage" V =

*E/+ (or 9: ;). 0ut since the voltage

across an inductor is related to the currentchange, we might e4!ress ; in terms of I"

I = d+/dt, or d; I dt. Therefore, we have"

:stored Σ ;i i ∫ d; ∫ I dt and now weuse VL = L dI/dt to get"Estored ∫ (# dIdt) I dt ∫ # I dI ,1/%LI%.

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&e.ie of Ener'y in 0ircuits

There is ener'y stored in a ca!acitor (that has

:lectric 'ield)" Estored = ,1/%0V% .

ecall the is related to :.

There is ener'y stored in an inductor (that has

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eview of Circuit :lements

esistor" V& = & I where I = ∆+/∆t

Ca!acitor" V0 = ,1/0+ (from 0 = +/V)

Inductor" VL = -L ∆I/∆t

We can ma%e an analogy with mechanics"

+ is li%e = t is li%e t,

I =∆

+/∆

t is li%e . =∆

/∆

t=∆

I/∆

t is li%e a =∆

./∆

t

V is li%e 2= 0 is li%e 1/3 (s!ring)=

& is li%e air resistance, L is li%e m.

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&L 0ircuit

What ha!!ens when we have a resistor in series with aninductor in a circuit?

2rom t(e mec(anical analo'y, this should be li%e having

a mass with air resistance. If we have a constant force

(li%e gravity), the ob$ect will accelerate u! to a terminals!eed (due to force of air resistance increasing u! to the

!oint where it balances the gravity).

Σ2=ma → − bv − mg ma, or

m d./dt 4 5. = -m'

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# Circuit (cont.)

If we connect the resistor and the inductor to a

battery and then turn the switch on, from

the mechanical analogy we would e4!ectthe current (which is li%e velocity) to begin

to increase until it reaches a constant

amount.

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# Circuit 7 ;ualitative loo%

'rom the circuit !oint of view, initially we have 2erocurrent so there is no (voltage dro! across the

resistor). Thus the full voltage of the battery is trying

to change the current, hence # battery, and so

dIdt battery #.

&owever, as the current increases, there is more

voltage dro! across the resistor, , which reduces

the voltage across the inductor (# battery 7 ),and hence reduces the rate of change of the current>

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# Circuit"

6ifferential E+uationTo see this behavior ;uantitatively, we need to

get an e;uation.

We can get a differential e;uation by using the0onser.ation of Ener'y (Σi 6)"

V5attery - Vresistor - Vinductor = 7.

*This loo%s li%e an ordinary algebraic e;uation.0ut all the /s are not constants" we have

relations for resistor and inductor .+

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# Circuit"

ifferential :;uation

V5attery - Vresistor - Vinductor = 7)

With V5attery = constant, Vresistor = I& , and

Vinductor = L dI/dt, we have the differentiale;uation (for I(t))"

V5attery - I& - L dI/dt = 7. This can be rewritten as"I& 4 L dI/dt = V5attery

which is an inhomogeneous first order differentiale;uation, $ust li%e we had for the mechanical analogy"

m d./dt 4 5. = -m' .

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# Circuit (cont.)

I& 4 L dI/dt = V5attery

The (omo'eneous e;uation is" LdI/dt 4 &I = 7.

This has a dying e4!onential solution"

IH,t = Io e-,&/L t .

The in(omo'eneous e;uation is"L dI/dt 4 &I = V5attery .

This has the sim!le solution" II,t = V5attery / & .

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# Circuit (cont.)

I,t I&(t) @ II(t) Io e-,&/L t @ V5attery / & .

To find the com!lete solution (that is, find Io ,

we a!!ly the initial conditions"

I(t6) 6 Io @ battery so Io 7 battery .

Therefore, we have"

I,t = "V5attery/"1 - e-,&/L t # .

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# Circuit

I,t = "V5attery/"1 - e-,&/L t # .

As time goes on, the current does increase andfinally reaches the value battery which is

what it would be without the inductor !resent.The gra!h on the ne4t slide shows this function

when V5attery = %9 volts= & = %9 Ω, and L = 1

H. -ote that the ma4 current is 5 Am!, andthe time scale is in milliseconds.

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# Circuit" I(t) versus t

I ( t ) v s t f o r R L C i r c u i t

0

0 . 2

0 . 4

0 . 6

0 . 8

1

1 . 2

1 6

1

1

1

6

2

1

2

6

3

1

3

6

4

1

4

6

t in m i l l i s e c o n d s

I i n

a m

p s

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& L and 0 in a circuit

We have already considered an C circuit (in

9art 1) and an # circuit ($ust now). We

loo%ed at these cases for a circuit in which

we had a battery and then threw the switch.

&owever, the main reason these circuit

elements are im!ortant is in AC circuits. We

loo% ne4t at the case of the three elements in

a series circuit with an AC voltage a!!lied.

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#C Circuit" scillations

$eton;s Second La" Σ ' ma can be writtenas" Σ ' 7 ma 6 . With a s!ring and resistance,this becomes" -3 -5. -ma = 7

This is li%e the e;uation we get from0onser.ation of Ener'y when we have aca!acitor, resistor and inductor" Σ 6 , or,1/0< 4 &I 4 L dI/dt = 7 ,

where V0 = ,1/0

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#C Circ it"

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#C Circuit"

6ifferential E+uation

tarting with Conservation of :nergy

(ΣVi = 7) and using I , V0 =

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#C Circuit"

ifferential :;uation,1/0< 4 &d

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#C Circuit

When we substitute this e4!ression

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#C Circuit

0y !lacing a resistor in the circuit, the

differential e;uation becomes a little harder.

We need to consider either both sines andcosines, or we need to consider e4!onentials

with imaginary numbers in the e4!onent.

This can be done, and reasonable solutionscan be found, but we will not !ursue that

here. We will !ursue an alternative way.

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#C Circuit"

ImedanceAn alternative way of considering the #C

Circuit is to use the conce!t of imedance.

The idea of im!edance is that all three of thema$or circuit elements im!ede the flow of

current.

A resistor obviously limits the current in acircuit. 0ut a ca!acitor and an inductor also

limit the current in an AC circuit.

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#C Circuit

The basic idea we will !ursue is that in a

series circuit,

a) the same current flows through all of theelements" I = Io sin,ωt = and

b) the voltages at any instant add u! to 2ero

around the circuit"

V& ,t 4 V0,t 4 VL,t = VA0,t .

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0aaciti.e &eactance

'or a ca!acitor" V0 = ,1/0

-ote that the constant (5ωC) acts $ust li%e . We call

this the ca!acitive reactance,>0 = ,1/ω0 .

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oltage across the ca!acitor

V0 = -V0o cos,ωt

I = Io sin,ωt

V0o = Io>0 where >0 = 1/ω0

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Inducti.e &eactance

'or an inductor, VL = L dI/dt, if I = Iosin,ω

tthen dI/dt = Ioω cos,ωt) -ote that cosine isE6o different than sine 7 we say it is E6o out of !hase. This means that VL is 7o out of (ase

it( t(e current)

ince VL = L dI/dt, VL = ωLIo cos,ωt , and wesee that the constant ω# acts $ust li%e . We

call this the inductive reactance,>L = ωL .

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oltage across the inductor

VL = VLo cos,ωt

I = Io sin,ωt

VLo = Io>L where >L = ωL

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L&0 in series

-ote that in a series combination, the current

must be the same in all the elements, while

the voltage adds. &owever, the voltagemust add to 2ero across a com!lete circuit at

every instant of time. 0ut since the

voltages are out of !hase with each other,the am!litudes of the voltages (and hence

the rms voltages) will not add u! to 2ero>

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#C in series

'or" I = Io sin,ωt

V& = I& = &Io sin,ωt

V0 = ,1/08< = -,1/ω0 Io cos,ωt

VL = L8dI/dt = ,ωL Io cos,ωt

V& 4 V0 4 VL = VA0 =

Io "& sin,ωt ? ,1/ω0 cos,ωt 4 ωL cos,ωt#

= Io @ sin,ωt4 = Vo sin,ωt4

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Imedance

In 17 s!ace, 4 and y are E6o a!art. We combine thetotal s!ace se!aration by the 9ythagorean

Theorem" r *41@y1+51 .

If we add u! the /s, this is e;uivalent to adding u!

the reactances. 0ut we must ta%e the !hases into

account. 'or the total im!edance, F, we get " Vrms

= Irms @ where @ = "& % 4 ,ωL - 1/ω0%#1/% .

(-ote that we had to subtract G# from GC because of thesigns involved.)

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&esonance

Vrms = Irms@ where @ = "& % 4 ,ω

L-1/ω

0%#1/%

-ote that when ,ωL - 1/ω0 = 7, F is smallest

and so I is biggest> This is the condition for

resonance. Thus whenω = [1

L0#1/%, we haveresonance. This is the same result we would

get using the differential e;uation route.

-ote that this is e;uivalent to the resonance of as!ring when ω = [3 m#1/%.

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V,t .ersus Vrms

2rom 0onser.ation of Ener'yΣ

Vi = 7) B(is (oldstrue at e.ery time. Thus"

V0,t 4 V& ,t 4 VL,t = Vosin,ωt4θο . 0ut due to the

!hase differences, C will be ma4imum at a different

time than , etc.

&owever, when we deal with rms voltages, we ta%e

into account the !hase differences by using the

9ythagorean Theorem. Thus,V0-rms 4 V&-rms 4 VL-rms C VA0-rms .

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$et *oer 6eli.ered

What about energy delivered?'or a resistor, electrical energy is changed into heat, so the

resistor will remove !ower from the circuit.

2or an inductor and a caacitor t(e ener'y is

merely stored for use later) Hence t(e a.era'e

oer used is Dero for both of these. -ote that

9ower IH, but I and are out of !hase by E6o

for both the inductor and ca!acitor, so on average

there is 2ero !ower delivered.

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-et 9ower elivered

Thus, even though we have V=I@ as a

generali2ed hm/s #aw, !ower is still"

*a.' = I%& (not 9I1F).

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Fa'netism in Fatter

ust as materials affect the electric fields in s!ace,

so do materials affect the magnetic fields in

s!ace.ecall that we described the effect of materials on

the electric fields with the dielectric constant, J.

This measured the stretchabilityD of the electric

charges in the materials. This stretching due toa!!lied electric fields caused electric fields itself.

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Two main effects

Atoms have electrons that orbitD the !ositive

nuclei. These orbitingD electrons act li%e

little current loo!s and can create smallmagnets.

:ffect 5" A diamagnetic effect similar to the

dielectric effect.:ffect 1" An aligning effect.

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iamagnetic :ffect

:ffect 5" 0y #en2/s law, when the a!!lied

magnetic field changes, there is a tendency in

the circuit to resist the change. This effecttends to create a magnetic field o!!osing the

change. In this effect, the material acts to

reduce any a!!lied e4ternal magnetic field.

This is similar to the dielectric effect that leads

to the dielectric constant for electric fields.

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Aligning :ffect

:ffect 1" ince the normal currentsD due to

the orbitingD electrons act li%e tiny

magnets, these magnets will tend to alignwith an e4ternal magnetic field. This

tendency to align will tend to add to any

e4ternal a!!lied magnetic field.&eat tends to destroy this ordering tendency.

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-et esult

When both of these effects (#en2/s law and aligning)are combined, we find three different ty!es of

results"

5. iamagnetic"

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'urther elations

0 µ& µo(& @

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