class 34 today we will: learn about inductors and inductance learn how to add inductors in series...

Class 34Today we will:•learn about inductors and inductance•learn how to add inductors in series and parallel•learn how inductors store energy•learn how magnetic fields store energy•learn about simple LR circuits

Inductors

An inductor is a coil of wire placed in a circuit. Inductors are usually solenoidal or toroidal windings.

V

R

L

An Inductor in a DC Circuit

In a DC circuit an inductor behaves essentially like a long piece of wire – except the solenoid produces a magnetic field and the wire has a resistance.

V

R

L

An Inductor in a DC Circuit

If we bring a magnet near the inductor, an induced EMF is produced that alters the voltages and current in the circuit.

V

R

L

N

An Inductor in an AC Circuit

In an AC circuit, the current is continually changing. Hence the magnetic field in the inductor is continually changing.

R

L

An Inductor in an AC Circuit

If the magnetic field in the inductor changes, an EMF is produced. Since the inductor’s own field causes an induced EMF, this is called “self inductance.”

R

L

The Inductance of a Solenoid

We can easily calculate the EMF of a solenoidal inductor.

dt

diAnV

dt

dinAN

dt

dNV

niABA

niB

B

B

20

0

0

0

The Inductance of a Solenoid

We define inductance by the equation:

dt

diLV

dt

diAn

dt

dNV B 2

0

So, for a solenoid:

AnL 20

Units of Inductance

Inductance is in units of henrys (H).

dt

diLV

ssA

VH 1

/11

Voltages of Circuit Elements

Note that L is similar to R in these equations.

2

2

1

dt

qdL

dt

diLV

qC

V

dt

dqRRiV

Inductance in Series and Parallel

In series:

In parallel:

21

21

21

LLLdt

diL

dt

diL

dt

diL

VVV

21

21

21

21

111

LLL

L

V

L

V

L

Vdt

di

dt

di

dt

di

iii

Energy in an Inductor

Take a circuit with only an EMF and an inductor.

dt

diLVL


dt

diL

.2

1 2Lidt

ddt

diiLiP


From this we can find the energy:

dt

dULi

dt

dP 2

2

1


From this we can find the energy:

2

2

2

12

1

LiU

dt

dULi

dt

dP

Energy in a Capacitor

Recall that the energy stored in a capacitor is:

2

2

1CVU

Energy Density in a Magnetic Field

Since

2

0

20

0

2

1

2

1

Bvol

Uu

volniU

niB 0

Energy Density in an Electric Field

Recall that in an electric field:

202

1E

vol

Uu

LR Circuits

•An LR circuit consists of an inductor, a resistor, and possibly a battery. •Inductors oppose change in circuits. V

R

L

LR Circuits

•In steady state, inductors act like wires, but they significantly affect circuits when switches are opened and closed.

V

R

L

LR Circuits•When the switch is closed in this circuit, the inductor opposes current flow. Initially, it is able to keep any current from flowing.

V

R

L

LR Circuits

•Current then increases until it reaches a maximum value,

V

R

L

)(ti

t

RV /

RVi /

LR Circuits

•Current in an LR circuit is very similar to charge in an RC circuit.

V

R

L

)(ti

t

RV /

LR Circuits

eR

V 11

t

RV /)(ti

LR Circuits

•Think of the total current being a combination of the induced battery’s current and the induced current.•The current from the batter is I=V/R, and it is constant.

LR Circuits

•The induced current is initially I= – V/R.•The induced current drops to zero as time increases.

LR Circuits•If the inductor is large, it is more effective at making the induced current, so it takes a longer time for the induced current to decrease.•If the resistor is large, it is more effective at decreasing the induced current, so it takes a shorter time for the current to decrease.

LR Circuits

•We’ll show next time that the inductive time constant is:

R

L

Class 35Today we will:•learn more about LR circuits•learn the LR time constant•learn about LC circuits and oscillation•learn about phase angles

LR Circuits

•An LR circuit consists of an inductor, a resistor, and possibly a battery. •Inductors oppose change in circuits. V

R

L

LR Circuits

eR

V 11

t

RV /)(ti

The Definition of Inductance

We define inductance by the equation:

dt

diLV

LR Circuits

•We’ll show that the inductive time constant is:

R

L

Kirchoff’s Loop Law

•The voltage around the loop at any given time must be zero.•It’s important to be careful of signs.

V

R

L


•As the current increases, the inductor opposes the increase. It produces an EMF in opposition to the battery.

V

R

L


V

R

L

0 iRdt

diLV


V

R

L

0 iRdt

diLV

0

0

iRdt

diLV

dt

di


0

0

iRdt

diLV

dt

di

R

L

VeeR

VL

VeVeR

VLV

eVeR

VLV

eR

Vti

tt

tt

tt

t

0

0

01

1)(

//

//

//

/

Another LR Circuit

•We can also make an LR circuit in which the current decreases.•At t=0, the switch is moved from 1 to 2.

V

R

L

2

1

Another LR Circuit

•The current is initially RVi /

/)( teR

Vti

V

R

L

2

1


•We apply Kirchoff’s Laws again.•This time the induced EMF pushes charge forward.

V

R

L

2

1


0 iRdt

diL

V

R

L

2

1


0 iRdt

diL

0

0

iRdt

diL

dt

di

V

R

L

2

1


0 iRdt

diL

R

L

eR

VRe

R

VL

eR

Vti

tt

t

0

)(

//

/

V

R

L

2

1


0 iRdt

diL

R

L

eR

VRe

R

VL

eR

Vti

tt

t

0

)(

//

/

)(ti

t

V

R

L

2

1


)(ti

t

eR

V 1

RV /

LC Circuits

•An LC circuit consists of an inductor and a capacitor.

L

C

LC Circuits•Initially the capacitor is charged.•No current flows as the inductor initially prevents it.•Energy is in the electric field of the capacitor.

L

C

LC Circuits•The charge reduces.•Current increases.•Energy is shared by the capacitor and the inductor.

L

C

i

LC Circuits•The charge goes to zero.•Current increases to a maximum value.•Energy is in the magnetic field of the inductor.

LC

i

LC Circuits•The inductor keeps current flowing.•The capacitor begins charging the opposite way.•The capacitor and inductor share the energy.

LC

i

LC Circuits

•The capacitor becomes fully charged.•Current stops.•All the energy is in the capacitor.

LC

LC Circuits•The capacitor begins to discharge.•Current flows counterclockwise.•The capacitor and inductor share the energy.

LC

i

LC Circuits

•Without energy loss, current would continue to oscillate forever.

LC

i


•First, we’ll be really sloppy about signs.

L

C

i


•First, we’ll be really sloppy about signs.•Know this!

L

C

i


L

C

iktkt

Lc

BeAetq

tqLCdt

qd

dt

qdL

dt

diL

C

q

VV

)(

)(1

2

2

2

2


L

C

iktkt

Lc

BeAetq

tqLCdt

qd

dt

qdL

dt

diL

C

q

VV

)(

)(1

2

2

2

2 But that’s not oscillatory!


L

C

i

tBtAtq

tqLCdt

qd

cossin)(

)(1

2

2


tCVtq cos)( 0

If the charge is maximum Q=CV0 at t=0, then we have:

tBtAtq cossin)(

Oscillating Frequency

LC

tCVLC

tCV

tCVtq

tqLCdt

qd

1

cos1

cos

cos)(

)(1

002

0

2

2

Being Careful about Signs

•Whenever we have AC circuits, we have to be really careful about signs. Signs tell us directions, and directions can be confusing.

Resistors

•Choose a direction for positive current.

i

Resistors

•Choose a direction for positive current. •We use V=iR to give the voltage across a resistor. •We call V positive when the voltage pushes current in the negative direction.

i 0iRVR

i

Resistors

•Choose a direction for positive current. •We use V=iR to give the voltage across a resistor. •We call V positive when the voltage pushes current in the negative direction.

i 0iRVR

i

positive voltage

Resistors

•Choose a direction for positive current. •We use V=iR to give the voltage across a resistor. •We call V positive when the voltage pushes current in the negative direction.•We call V negative when the voltage pushes current in the positive direction.

i 0iRVR

i

Resistors

•Choose a direction for positive current. •We use V=iR to give the voltage across a resistor. •We call V positive when the voltage pushes current in the negative direction.•We call V negative when the voltage pushes current in the positive direction.

i 0iRVR

i

negative voltage

Resistors

t

)(ti

)(tVR

•The voltage and current are in phase.•The phase angle is 0°.

Capacitors


i

Capacitors

•Choose a direction for positive current. •If the left plate is positive, q>0 and V>0.•If the left plate is negative, q<0 and V<0.

i

i voltage positive

voltage negative

Capacitors

•We plot charge and voltage as a function of time.

t

)(tVC

)(tq

Capacitors

i

i q becoming more positive

•If current is flowing right, dq/dt>0.•If current is flowing left, dq/dt<0.

i

i

q becoming more positive

Capacitors

i

i q becoming more positive

•If current is flowing right, dq/dt>0.•If current is flowing left, dq/dt<0.

i

i

q becoming more positive

dt

dqi

Capacitors

•When q (or V) increases, the current is positive.•When q (or V) decreases, the current is negative.

t

)(tq

)(tVC

)(ti

Capacitors

t

)(ti

•The voltage lags the current.•The phase angle is −90°.

−90°

)(tVC

Inductors


i

Inductors

•Choose a direction for positive current.•We need to consider current in both directions.

•In each direction, we need to consider current

increasing and decreasing.

i

Inductors

•In each direction, we need to consider current increasing and decreasing.

t)(tiincreasingi

dt

di

i

0

0

0iinduced

Inductors

• The induced EMF opposes the current.• The voltage positive because it is pushing current in the negative direction.

i 0

dt

diLVL

i 0dt

di

positive voltage

Inductors


t)(ti

decreasingidt

di

i

0

0

0iinduced

Inductors

•The induced EMF aids the current.•The voltage is positive because it is pushing current in the negative direction.

i 0

dt

diLVL

i 0dt

di

positive voltage

Inductors


t)(tidecreasingi

dt

di

i

0

0

0iinduced

Inductors

•The induced EMF aids the current.•The voltage is negative because it is pushing current in the positive direction.

i 0

dt

diLVL

i 0dt

di

negative voltage

Inductors


t)(ti

increasingidt

di

i

0

0

0iinduced

Inductors

•The induced EMF opposes the current.•The voltage is negative because it is pushing current in the positive direction.

i 0

dt

diLVL

i 0dt

di

negative voltage

Inductors

•The voltage is negative when di/dt is negative.•The voltage is positive when di/dt is positive.

t

)(ti

)(tVL

Inductors

t

)(ti

•The voltage leads the current.•The phase angle is +90°.

)(tVL

+90°

Voltage and Current in an LC Circuit

t

)(ti

+90°

)(tVC)(tVL

−90°

Class 36Today we will:•learn about phasors•define capacitive and inductive reactance•learn about impedance•apply Kirchoff’s laws to AC circuits

Voltage and Current in AC Circuits

ELI the ICE MAN

t

)(ti

+90°

)(tVC)(tVL

−90°


ELI the ICE MAN

t

)(ti

+90°

)(tVC)(tVL

−90°

In an inductor (L) EMF leads I ELI


ELI the ICE MAN

t

)(ti

+90°

)(tVC)(tVL

−90°

In a capacitor (C) I leads EMF ICE

Representing an AC Current

•If a current varies sinusoidally, we usually represent it graphically.

t

)(ti0i

Representing an AC Current

•But we can also represent it with an arrow that goes up and down in time.

Representing an AC Current•An arrow that goes up and down sinusoidally is just the y component of a vector that rotates at a constant angular frequency.•The length of the arrow is and the angle of the vector is .

0it

0i

Phasors•A phasor is a vector that represents a sinusoidal function. Its magnitude is the amplitude of the sine wave (it’s maximum value) and its angle is the phase angle (the argument of the sine function).

0i

sin)( 0iti

sin0i

Phasors

•Phasors are useful because we can add two sine waves with the same frequency by adding their phasors.

ytiti

xtitiiiadding

ytixtii

ytixtiiphasors

tiitiifunctions

ˆsinsin

ˆcoscos:

ˆsinˆcos

ˆsinˆcos:

sin,sin:

2010

201021

20202

10101

202101

•The y component of the sum of the phasors equals the sum of the functions.

Phasors

•Doesn’t this just make things harder???

Phasors

•Doesn’t this just make things harder???

•No – because phasor diagrams are a lot easier than algebra and trig identities.

•Let’s look at some examples…

Resistors and Phasors•First, we wish to find the maximum voltage across a resistor in terms of the maximum current.•Note these correspond to the length of the phasors.•All we need for this is Ohm’s Law: RiVR 00

Resistors and Phasors

•Now we can make a voltage phasor and a current phasor for the resistor.•The voltage is in phase with the current, so the phasors point in the same way.

iRV2Rif

Resistor Phasors in Time

Inductors and Phasors

•As before, we wish to find the maximum voltage across an inductor in terms of the maximum current.

00

0

0

)cos()(

)sin()(

LiVSo

tLidt

diLtVThen

titiLet

L

L

Inductive Reactance

•Comparing this result to Ohm’s Law, we see that has a role that is similar to resistance – it relates voltage to current.•We define this to be the “inductive reactance” and give it a symbol .

LX

XiV

L

LL

00

L

LX

Inductive Reactance

•This makes sense, because an inductor offers more opposition to the current if the frequencey is large and if the inductance is large.

LX

XiV

L

LL

00


•We again want a voltage phasor and a current phasor.•The voltage leads the current by 90°, so the phasors are:

i

2LXif

LV

Capacitors and Phasors

•We find the maximum voltage across a capacitor in terms of the maximum current.

00

0

0

1

)cos(11)(

)(

)sin()(

iC

VSo

dt

dqias

tiC

idtCC

tqtVThen

titiLet

C

C

Capacitive Reactance

•Comparing this result to Ohm’s Law, we see that is similar to resistance.•We define this to be the “capacitive reactance” and give it a symbol .

CX

XiV

C

CC

1

00

C/1

CX

Capacitive Reactance

•This makes sense because a capacitor offers more resistance to current flow if it builds up a big charge. If the frequency is large or the capacitance is large, it is hard to build up much charge.

CX

XiV

C

CC

1

00


•We again want a voltage phasor and a current phasor.•The voltage lags the current by 90°, so the phasors are:

i

2CXif

CV

Circuit Rules for AC Circuits

If two circuit elements are in parallel:


If two circuit elements are in parallel:1) their voltage phasors are the same.

CL VV

Li

Ci


If two circuit elements are in parallel:1) their voltage phasors are the same.2) their current phasors add to give the

total current.

CL VV

Li

Ci

toti


If two circuit elements are in series:


If two circuit elements are in series:1) their current phasors are the same.

LV

CL ii

CV


If two circuit elements are in series:1) their current phasors are the same.2) their voltage phasors add to give the

total voltage.

LV

CL ii

totV

CV

An Example

We know ε0, ω, the resistances, the capacitance, and the inductance.

1R

2R

L

C

An Example

Label the currents.

1R

2R

L

C

2i

3i

1i

An Example

We don’t know the currents, but we know the relationships between current and voltage.

1R

2R

L

C

2i

3i

1i

An Example

1R

2R

L

C

11010 RiVR

2i

3i

1i

An Example

1R

2R

L

C

22020 RiVR

2i

3i

1i

An Example

1R

2R

L

C

LL XiV 200

2i

3i

1i

An Example

1R

2R

L

C

CC XiV 300

2i

3i

1i

An Example

LRLR ZiV 200

We can combine circuit elements and calculatethe total impedance of these elements.

1R

LRZ

C

2i

3i

1i

An Example

1i

We can combine all the circuit elements into asingle impedance.

Z

An Example

1i

We can combine all the circuit elements into asingle impedance.

Z

Zi100

An Example

We’ll start with the inductor.

1R

2R

L

C

2i

3i

1i

LiXiV LL 20200

An Example

Draw the current in an arbitrary direction – along the x-axis will do – with an arbitrary length.

1R

2R

L

C

2i

3i

1i

LiXiV LL 20200

2i

An Example

We know the direction of the voltage phasor from “ELI.”

1R

2R

L

C

2i

3i

1i

LiXiV LL 20200

2i

LV

An Example

We know the magnitude of the voltage phasor is ωL times i20.

1R

2R

L

C

2i

3i

1i

LiXiV LL 20200

2i

LV

An Example

Now we add the resistor.

1R

2R

L

C

2i

3i

1i

2200 RiVR

2i

LV

An Example

The voltage phasor is in the same direction as the current. It’s magnitude comes from:

1R

2R

L

C

2i

3i

1i

2200 RiVR

2i

LV

RV

An Example

We add the voltages of the two series elements.

1R

2R

L

C

2i

3i

1i

2200 RiVR

2i

LV

RV

LRV

An Example

Let’s look at the math.

2i

LV

RV

LRV

An Example

The LR voltage phasor has two components:

LLR

LRLR

XiVRiV

yVxVV

2002200

00

,

ˆˆ

2i

LV

RV

LRV

An Example

The length of the LR voltage phasor is:

22220

20

200

2002200

00

,

ˆˆ

LLRLR

LLR

LRLR

XRiVVV

XiVRiV

yVxVV

2i

LV

RV

LRV

Impedance

Impedance is the combined “effective resistance” of circuit elements. It is given the symbol Z.

LRLR ZiV 200

2i

LV

RV

LRV

An Example

We can define a total impedance for L and R:

LR

LLRLR

LLR

LRLR

Zi

XRiVVV

XiVRiV

yVxVV

20

22220

20

200

2002200

00

,

ˆˆ

2i

LV

RV

LRV

An Example

We can define a total impedance for L and R:

222 LLR XRZ

2i

LV

RV

LRV

An ExampleWe can calculate the phase angle of the LR phasor:

2

1

0

01 tantanR

X

V

V L

R

LLR

2i

LV

RV

LRV

LR

Impedance

Whenever we know a voltage and a current we can find an impedance. Because of differences in phases, impedances are not added as easily as in DC circuits.

An Example

Now let’s continue the problem, without filling in all the mathematical details.

An Example

We replace the inductor and resistor with a single impedance:

1R

LRZ

C

2i

3i

1i

2i

CLR VV

An Example

The LR voltage phasor is the same as the C voltage phasor, since LR and C are in parallel.

2i

CLR VV

1R

LRZ

C

2i

3i

1i

An Example

We can find the direction and magnitude of .

2i

CLR VV

3i

1R

LRZ

C

2i

3i

1i

An Example

The direction of i3 is given by “ICE.”

2i

CLR VV

3i

1R

LRZ

C

2i

3i

1i

An Example

The magnitude of i3 is given by:

2i

C

LR

LRCLRC

X

Zii

ZiXiVV

2030

203000

CLR VV

3i

1R

LRZ

C

2i

3i

1i

An Example

The total current through LRC is .

2i

3i

32 iiiLRC

LRCi

CLR VV

1R

LRZ

C

2i

3i

1i

An Example

By Kirchoff’s junction rule, we have .

2i

3i

LRCii

1

1i

CLR VV

1R

LRZ

C

2i

3i

1i

An Example

We can replace LRC with a single impedance.

2i

3i

1i

CLR VV

1R

LRCZ

1i

An Example

Now we can find the voltage across the resistor

with Ohm’s Law. 11010 RiVR

1i

CLRC VV

RV

1R

LRCZ

1i

An Example

Finally, we can add the two voltages to get the

EMF. 1R

2R

L

C

2i

3i

1i

LRCV

1RV

11010 RiVR

Class 37Today we will: • study the series LRC circuit in detail.• learn the resonance condition• learn what happens at resonance• calculate power in AC circuits

•Assume we know the maximum voltage of the AC power source, .

•We wish to find the current and the voltages across each circuit element.

The Series LRC Circuit

R

L

i

C

0

•The current phasor is the same everywhere in the series LRC circuit.

•The voltage phasors add:


0 CLR VVV

R

L

i

C

•The current phasor is the same everywhere in the series LRC circuit.

•The voltage phasors add:


R

L

i

C Zi

C

iXiV

LiXiV

RiV

CC

LL

R

00

000

000

00

0 CLR VVV


R

L

i

C

RiVR 00 CC XiV 00

i

LL XiV 00

•We can then draw phasors:


R

L

i

C

RiVR 00

CL VV

CC XiV 00

Zi00

LL XiV 00

•The total EMF of the resistor, capacitor, and inductor is the vector sum of the three voltages.

i


R

L

i

C

RCX

Z

LX

•Since is a common factor in all the voltages, we can divide it out and simplify the math a little.

CL XX

0i

i


R

L

i

C

•Let’s redo this diagram step by step. Be sure you know this well!


R

L

i

C

i

•Start with the current direction. It’s easiest just to put this on the +x axis.


R

L

i

C

LX

•Put along the +y axis.LX

i


R

L

i

CCX

LX

•Put along the --y axis.CX

i


R

L

i

C

RCX

Z

LX

•Put along the +x axis.R

i


R

L

i

C

RCX

LX

•Now we add all the impedances as vectors.

•First, add and .

CL XX

LX CX

i


R

L

i

C

RCX

LX

•Then add R to get Z.

CL XX Z

i


R

L

i

C

RCX

LX

CL XX Z

i

LRCV

•We know . and,0 00 ZiVLRC


R

L

i

C

RCX

LX

CL XX

Z

i

LRCV

•We know . and,0 00 ZiVLRC


R

L

i

C

RCX

LX

•We can find and :

CL XX

Z 0i

Zi

RXXZ CL

00

22

Z

i

Varying the Frequency

•As we vary the frequency, the phasor diagram changes.

•At low frequency, the capacitive reactance is large.

•At high frequency, the inductive reactance is large.

Varying the Frequency

•In this animation, we keep the maximum EMF constant and increase the frequency of the AC power supply.

LX

i

CX

Resonance

•Resonance is where the inductive and capacitive reactances are equal.

Resonance


•The resonant frequency is:

LC

CLXX CL

1

1

Resonance


•The resonant frequency is:

•The impedance is minimum and the current is maximum.

RRXXZ CL 22

LC

CLXX CL

1

1

Power in AC Circuits

•At any given instant, the power dissipated by a resistor is

•Over a full cycle of period T, the average power dissipated in the resistor is:

tRiRi

tVtiP

220

2 sin

)()(

00020

202

20

2

1

2

1

2sin

)(

ViViP

T

T

Ridtt

T

Ri

T

dttP

PTt

t

Tt

t

Power in AC Circuits

•The power provided by a voltage source is:

cos2

1

sincoscossinsin

)sin(sin

)()()(

00

00

00

iP

ttti

tti

ttitP

rms Currents and Voltages

•We want to define an “effective AC current.”



•The effective current is less than the maximum current.




•The average current is zero.




•The average current is zero.

•Absolute values are awkward mathematically.


•So, we take the square root of the average value of the current squared.

•This is called the rms or “root mean square” current.

•Similarly:

0

0

220

2

1sin

1iti

Ti

T

rms

02

1VVrms


•Normally, when we specify AC voltages or currents, we mean the rms values.

•Standard US household voltage is 110-120V rms.

Power in Terms of rms Quantities

•When we rewrite our power equations in terms of rms currents and voltages, they become similar to DC formulas.

Resistor

Power supply:

R

VRiViViP rms

rmsrmsrms

22

002

1

coscos2

100 rmsrmsiiP

class 34 today we will: learn about inductors and inductance learn how to add inductors in series...

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