Magnetic Induction And The Storage Of Magnetic Energy
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1 MAGNETIC INDUCTION AND THE STORAGE OF MAGNETIC ENERGY
This Chapter 2
Define and use Faraday’s Law and Lenz’s Law to determine the effect of changing magnetic fluxes.
Compute p for Inductances and learn ways on how to store magnetic energies
Compute C t for f the th circuital i it l parameters of RL Circuits
Outline 3
Overview 4
1830’s – Michael Faraday (England) and Joseph Henry (USA) independently discovered that changing magnetic field induces a current in the wire. The emfs and currents caused by changing magnetic fields are called ll d induced i d d emfs f and d induced i d d currents. The process itself, is referred to as magnetic induction. When you pull the plug of an electric cord from its socket, you sometimes observed a small spark. This phenomenon is explained by magnetic induction!
1 Magnetic Flux 1. 5
The flux of a magnetic field through a surface is defined similarly to the flux of an electric field. field The magnetic flux Φm is defined as
The unit of flux is that of a magnetic field times area area, tesla tesla-meter meter squared, which is called a weber (Wb) 1 Wb = 1 T•m2 Exercise: Show that a weber per second is a volt.
1 Magnetic Flux through many loops 1. 6
We are often interested g a coil in the flux through containing several turns of wire. If the coil contains N turns, the flux through the coil is N times ti the th flux fl through th h each turn.
2 Faraday 2. Faraday’ss Law 7
Consider C id a b bar magnett iin proximity i it to t a loop attached to an ammeter.
2 Faraday 2. Faraday’ss Law 8
Moving the bar magnet towards the loop induces a current through the loop, loop even without a battery. Such induced current arises from f the h induced d d emf. f
2 Faraday 2. Faraday’ss Law 9
From what we have earlier, earlier a changing magnetic flux results to an induced emf. This is known as Faraday’s Law!
How to change the magnetic flux? Move the permanent magnet towards the loop
Move the loop towards the permanent magnet
Current that produces B can be changed
Area of the loop p can be changed
Loops/B sources can be rotated
2 Faraday 2. Faraday’ss Law: Examples 10
1.
2.
3.
EXAMPLES: A uniform magnetic field makes an angle of 30o with the axis of a circular coil of 300 turns and a radius of 4 cm. The field changes at a rate of 85T/s 85T/s. Find the magnitude of the induced emf in the coil. An 80-turn coil has a radius of 5.0cm and a resistance of 30Ω. At what rate must a perpendicular magnetic field to produce a current of 4 4.0A 0A in the coil? A solenoid of length 25 cm and radius 0.8cm with 400 turns is in an external magnetic field of 600 G that makes an angle of 50o with the axis of the solenoid. (a) Find the magnitude flux through the solenoid. solenoid (b) Find the magnitude of the emf induced in the solenoid if the external magnetic field is reduced to zero in 1.4s.
3 Lenz 3. Lenz’ss Law 11
Developed by Heinrich Lenz Lenz’s Law gives us the direction of th the iinduced d d current.t
“The ind induced ced emf and induced current are in such pp a direction so as to oppose the change the produces them.” Note: We didn’t specify just what kind of change g causes the induced emf and current. The statement was left vague to cover a variety of conditions we will now illustrate.
3 Lenz 3. Lenz’ss Law: Illustrations 12
3 Lenz 3. Lenz’ss Law: Illustrations 13
3 Lenz 3. Lenz’ss Law: Illustrations 14
3 Lenz 3. Lenz’ss Law: Illustrations 15
3 Lenz 3. Lenz’ss Law: Illustrations 16
A rectangular coil of 80 turns, 20 cm wide and 30 cm long, l is llocated d in a magnetic field B = 0.8T directed into the page page, with only a portion of the coil in the region of the magnetic f ld The field. h resistance off the h coil is 30Ω. Fi d th Find the magnitude it d and d direction of the induced current if the coil is moved with a speed of 2m/s (a) to the right, (b) up, and (c) down.
4 Inductance 4. 17
An airport metal detector contains a large coil in its frame. The coil has a property called inductance. inductance When a metal passes through the frame, the inductance of the frame changes. The change in the inductance is converted to an alarm sound!
The Unit of Inductance 18
The Unit of Inductance is the henry(H). henr (H)
4 1 Self-Inductance 4.1 Self Inductance 19
When the switch is closed, current rises until it reaches its max value. l During the current rise, the magnetic field it produces changes thus a changing changes, magnetic flux Thus there should be an induced emf caused byy the changing magnetic flux Therefore, there is selfinduction in the circuit!
4 1 Self-Inductance 4.1 Self Inductance 20
The general relation of the self self-induced induced emf to the changing current is:
The p proportionality p y constant is the self-inductance (L) of the circuit!
By applying Faraday’s Law in reverse, we derive: Self-Inductance Self Inductance is a constant but depends on the geometry of the circuit/loop!
4 1 Self-Inductance: 4.1 Self Inductance: Example 21
1. Calculate the self-inductance of an air-core solenoid containing 300 turns t rns if the length of the solenoid is 25.0 cm and its cross-sectional area is 4.00 cm2.
2. Find the self-inductance of a solenoid of length 10 cm, area 5 cm2, and 100 turns. At what rate must g to induce an emf the current in the solenoid change of 20V?
4 2 Mutual-Inductance 4.2 Mutual Inductance 22
Figure shows two circuits. As we change the resistance in circuit 1, the current also changes Thus the magnetic field changes. it produces also changes. The changing magnetic flux induces an emf on circuit 1 and on circuit 2! Thus circuit 2 has an induced emf. This phenomenon is called, mutual induction!
Circuit 1
Circuit 2
Circuit 1
Circuit 2
4 2 Mutual-Inductance 4.2 Mutual Inductance 23
The net induced emf on circuit two is related to the changing current by:
Self-induced by 2
Mutually induced by 1 on 2
Mutual Inductance of the two circuits are equal and can be found using the transformable formula:
Mutual Inductance like Self-Inductance Self Inductance depends on the geometry of the two circuits and d the h distance between them!
4 2 Mutual Inductance: Example 4.2 24
An electric toothbrush has a base designed to hold the toothbrush handle when not in use. As shown in the Figure, the handle has a cylindrical hole that fits loosely over a matching cylinder on the base. When the handle is placed on the base, base a changing current in a solenoid inside the base cylinder induces a current in a coil inside the handle. This induced current charges the battery in the handle. We can model the base as a solenoid of length x with Nbase turns (Fig. 32.15b), carrying a current I, and having a cross sectional area A. The handle coil contains Nhandle turns and completely surrounds the base coil. Find the mutual inductance of the system.
x
5. The Storage of Magnetic Energy
25
An inductor stores magnetic energy through the current b ld up in it, just as a building capacitor stores electrical energy. energy Consider, the circuit at the right. The energy stored in an inductor carrying a current I is given by:
The magnetic energy density, uB is given by:
This is the energy that is stored in a magnetic field, regardless of the configuration!
6 RL Circuits 6. 26
RL Circuits contain a resistor and an inductor. I
flows in a single direction But changes its value, it either grows or decays
27
6. The Growth of I in RL Circuits We assume that the inductor has 0 emf initially. initially After the switch is closed, the emff off the th battery b tt equates t to the back emf of the i d inductor, and d current builds b ild according to:
As current builds up, the i d t ’ b inductor’s backk emff is i reduced to zero!
Imax is the maximum current in the circuit equivalent to ξ0/R. τ is the time constant equilvalent to L/R
28
6. The Decay of I in RL Circuits As the switches are reconfig red the Imax reconfigured, Ima current is drained by the resistor R according to: This happens because the inductor acts like a battery, with ith a llessening i currentt pump abilities! Io is the initial current in the circuit equivalent to ξ0/R. τ is the time constant equilvalent to L/R
6 RL Circuits: Examples 6. 29
1. A basic RL circuits consists of the following: a battery (ξ (ξ= 12 V), V) an inductor (L = 30 mH), mH) and a resistor (R = 6 Ω). Find the time constant, and if the switch is closed at t =0, =0 when will the current reach half its maximum value. 2. If the battery y in the example p above is carefullyy removed after the current reaches its maximum y to 10% of the value,, when will the current decay original maximum value?
Chapter Six is pretty much a straight forward chapter. 1. 2. 3. 4. 5 5. 6.
7.
8.
Sources of AC R in AC L in i AC C in AC LC in AC The Series RLC in AC Resonance in AC T Transformers f
1
IN THIS FINAL CHAPTER, YOU SHOULD BE ABLE TO… Understand the sources of alternating currents
Understand the operating principles of transformers
Analyze the behavior of specific f R, L, C combinations
Analyze A l the th behaviors b h i of R, L, and C if alternating currents flow through them
Define phasors and discover its importance when analyzing for the behaviors of R, L, and C
2
TIME-VARYING VALUES |
|
To identify time varying values, we use lower case letters!!
To identify fixed values we shall use values, upper case letters
3
WHY STUDY ALTERNATING CURRENTS? |
|
|
More than 99% of the electrical energy used today is produced by electrical generators in the form of alternating current (ac). AC’s advantage over DC because electrical energy can be transported over long distances at very high voltage and low currents to reduce energy losses due to Joule heat! AC can then be transformed, with almost no energy loss loss, to lower and safer voltages and correspondingly higher currents for everyday use!
4
ALTERNATING CURRENTS |
|
|
Alternating Currents – are currents whose value vary periodically over time! Alternating Currents, in general are sinusoidal in nature and generally supply alternating voltages of the form: Because voltage changes, it is positive ½ the period, and negative ½ the period! i d!
In our country most AC’s have frequencies of 60 Hz or angular q y of 377 rad/s frequency
5
1. AC SOURCES |
|
|
There are many kinds off AC sources!! The most common probably are the AC outlets in our homes! But how do we actually actua yp produce oduce alternating currents?
6
2. RESISTORS IN AC Consider the circuit to the right. i ht | The instantaneous voltage and current through the resistor are given by: |
|
Notice that vR and iR are in phase phase” with one “in another!
7
2. RESISTORS IN AC |
|
There is still power loss i resistors in i t when h current passes through th them because b off the th voltage drop! This power has 3 forms: y
Instantaneous
y
Average
y
Maximum
8
ROOT MEAN SQ QUARED ((RMS)) VALUES |
|
|
|
Most AC ammeters and voltmeters are designed to measure rms values of currents and voltages, instead off the h maximum i values. l So there S h is i a necessity i to interconvert between rms and maximum values! Simple Rule: the rms value is the maximum value over the square root of 2! Example:
9
EXERCISES |
1. Find Pav in terms of Irms and R
|
2. Find Pav in terms of ξmax and Imax
|
3 Find Pav in terms of ξrms and Irms 3.
|
4 Find 4. Fi d Irms in i terms t off ξrms and dR
|
5. A 12-Ω resistor is connected across a sinusoidal emf that has a peak value of 48V. Find (a) the rms current, (b) the average power, (c) the maximum power.
10
3. INDUCTORS IN AC Consider the circuit to the right. i ht | The instantaneous voltage and current are given by: |
|
Notice that vL and iL are out of phase, phase iL lagging vL by π/2 rads χL is i called ll d inductive i d i reactance, it i has the unit of ohms! This means that Thi th t inductors i d t reactt differently to current by offering resistance!
11
4. CAPACITORS IN AC Consider the figure to the right. i ht | The instantaneous voltage and current are given by: |
|
Notice that vC and iC are out of phase phase”, iC leading “out vC with π/2 radians. χC is i called ll d capacitive i i reactance, it i has the unit of ohms! This means that Thi th t capacitors it reactt differently to current by offering resistance!
12
L,, C IN AC: EXAMPLES 1. A 40mH inductor is placed across an ac generator that has a maximum emf of 120V. Find the inductive reactance and the maximum current when the frequency is (a) 60 Hz (b) 2000 Hz H
|
What can you conclude about the relation of inductive reactance and current?
2. A 20-μF capacitor is placed across a generator that has a maximum emf of 100V. Find the capacitive reactance and the maximum current when the frequency is (a) 60 Hz (b) 5000 Hz
|
What can you conclude about the relation of capacitive reactance and 13 current?
THE BEHAVIORS OF L AND C IN AC CIRCUITS |
|
|
| |
Alternating current behaves differently than p direct current in inductors and capacitors. When a capacitor becomes fully charged in a dc circuit, i i it i stops the h current, that h is, i it i acts like lik an open circuit. But if the current alternates, alternates charge continually flows onto or off the plates of the capacitor and at higher frequencies, the capacitor, will hardly impede current at all all, which means means, it acts like a short circuit! Conversely, an inductor coil usually has a very small resistance and is essentially a short circuit for dc. But when the current is alternating, alternating a back emf is generated in an inductor, and at higher frequencies, the back emf is so large, the inductor acts like an open circuit! i it!
14
5. LC IN AC |
|
|
|
Consider the circuit to the right. When the switch is closed, the initially charged capacitor discharges producing a back emf on the inductor, which in turn counters the discharging current, recharging the capacitor. Thus once the capacitor completely discharges, it is once again i charged h d by b the th inductor. Conversely, once the inductor reaches zero current, current g flow through g it from will again the capacitor!
Natural Frequency of Oscillation of i
C Current t iin an LC circuit i it
15
5. LC IN AC EXAMPLE |
A 2-μF capacitor is charged to 20V and is then connected t d across a 6 6-μH H iinductor. d t ((a)) Wh Whatt iis the th frequency of oscillation? (b) What is the maximum i value l off the th current? t?
16
6. RLC IN AC ((SERIES) |
|
|
|
Consider the figure to th right. the i ht The circuit has a current given by:
Where Z is impedance (overall resistance) And A d δ is i the h phase h angle
17
6. RLC IN AC ((SERIES) |
|
The Average Power for RLC in AC, series connection ti can b be represented t db by:
Where cos δ is called the p power factor.
18
7. RESONANCE IN SERIES RLC IN AC |
Resonance is the condition in which we have the smallest possible impedance that would lead to the maximum current. y
y
|
Zmin can only happen when the reactances is zero. Reactances can only be zero if the ac source zero, frequency equates to the natural frequency of the circuit!
At resonance, we have maximum i current and d power and the power factor is one!
19
7. SERIES RLC IN AC: EXAMPLES 1.
A series RLC Circuit with L = 2H, C = 2μF, and d R = 20Ω is i driven d i by b a generator t with ith a maximum emf of 100 V and a variable f frequency. Find Fi d ((a)) th the resonance frequency f (f0), (b) the maximum current at resonance, (c) the phase angle δ, δ (d) the power factor, factor and (e) the average power delivered.
2.
A series RLC Circuit with L = 2H, C = 2μF, and d R = 20Ω Ω is i driven d i by b a generator with i ha maximum emf of 100 V and a variable f frequency. Find Fi d th the maximum i voltage lt across the resistor, the inductor and the capacitor.
20
8. TRANSFORMERS |
A transformer is a device used to raise or lower the voltage in a circuit without an appreciable loss of power y
y y
|
|
A simple transformer consisting of two wire coils around a common iron core. The coil carrying the input power is called the primary. The coil carrying the output power iis called ll d the th secondary. d
The transformer operates on the principle of mutual induction The iron Th i core increases i the th magnetic field for a given current and guides it so that nearly all the magnetic flux through one coil goes through the other coil.
21
6. TRANSFORMERS |
|
For a transformer with N1 turns in the primary and N2 turns in the secondary the voltage secondary, across the secondary coil is related to the generator emf across the primary coil by: If there are no losses, due to Joule Heating (which is due to negligible li ibl resistance i t iin the coils), RMS Power relations are given by:
22
6. TRANSFORMERS: EXAMPLES |
|
1. A doorbell requires 0.4A at 6V. It is connected t a ttransformer to f whose h primary i containing t i i 2000turns, is connected to a 120-V ac line. (a) H How many tturns should h ld there th be b in i the th secondary? (b) What is the current in the primary? 2. A transmission line has a resistance of 0.02Ω/km. Calculate the I2R power loss if 200kW off power is i transmitted i d from f a power generator to a city 10km away at (a) 240 V and (b) 4.4 kV 23
This Chapter 2
Define and use Faraday’s Law and Lenz’s Law to determine the effect of changing magnetic fluxes.
Compute p for Inductances and learn ways on how to store magnetic energies
Compute C t for f the th circuital i it l parameters of RL Circuits
Outline 3
Overview 4
1830’s – Michael Faraday (England) and Joseph Henry (USA) independently discovered that changing magnetic field induces a current in the wire. The emfs and currents caused by changing magnetic fields are called ll d induced i d d emfs f and d induced i d d currents. The process itself, is referred to as magnetic induction. When you pull the plug of an electric cord from its socket, you sometimes observed a small spark. This phenomenon is explained by magnetic induction!
1 Magnetic Flux 1. 5
The flux of a magnetic field through a surface is defined similarly to the flux of an electric field. field The magnetic flux Φm is defined as
The unit of flux is that of a magnetic field times area area, tesla tesla-meter meter squared, which is called a weber (Wb) 1 Wb = 1 T•m2 Exercise: Show that a weber per second is a volt.
1 Magnetic Flux through many loops 1. 6
We are often interested g a coil in the flux through containing several turns of wire. If the coil contains N turns, the flux through the coil is N times ti the th flux fl through th h each turn.
2 Faraday 2. Faraday’ss Law 7
Consider C id a b bar magnett iin proximity i it to t a loop attached to an ammeter.
2 Faraday 2. Faraday’ss Law 8
Moving the bar magnet towards the loop induces a current through the loop, loop even without a battery. Such induced current arises from f the h induced d d emf. f
2 Faraday 2. Faraday’ss Law 9
From what we have earlier, earlier a changing magnetic flux results to an induced emf. This is known as Faraday’s Law!
How to change the magnetic flux? Move the permanent magnet towards the loop
Move the loop towards the permanent magnet
Current that produces B can be changed
Area of the loop p can be changed
Loops/B sources can be rotated
2 Faraday 2. Faraday’ss Law: Examples 10
1.
2.
3.
EXAMPLES: A uniform magnetic field makes an angle of 30o with the axis of a circular coil of 300 turns and a radius of 4 cm. The field changes at a rate of 85T/s 85T/s. Find the magnitude of the induced emf in the coil. An 80-turn coil has a radius of 5.0cm and a resistance of 30Ω. At what rate must a perpendicular magnetic field to produce a current of 4 4.0A 0A in the coil? A solenoid of length 25 cm and radius 0.8cm with 400 turns is in an external magnetic field of 600 G that makes an angle of 50o with the axis of the solenoid. (a) Find the magnitude flux through the solenoid. solenoid (b) Find the magnitude of the emf induced in the solenoid if the external magnetic field is reduced to zero in 1.4s.
3 Lenz 3. Lenz’ss Law 11
Developed by Heinrich Lenz Lenz’s Law gives us the direction of th the iinduced d d current.t
“The ind induced ced emf and induced current are in such pp a direction so as to oppose the change the produces them.” Note: We didn’t specify just what kind of change g causes the induced emf and current. The statement was left vague to cover a variety of conditions we will now illustrate.
3 Lenz 3. Lenz’ss Law: Illustrations 12
3 Lenz 3. Lenz’ss Law: Illustrations 13
3 Lenz 3. Lenz’ss Law: Illustrations 14
3 Lenz 3. Lenz’ss Law: Illustrations 15
3 Lenz 3. Lenz’ss Law: Illustrations 16
A rectangular coil of 80 turns, 20 cm wide and 30 cm long, l is llocated d in a magnetic field B = 0.8T directed into the page page, with only a portion of the coil in the region of the magnetic f ld The field. h resistance off the h coil is 30Ω. Fi d th Find the magnitude it d and d direction of the induced current if the coil is moved with a speed of 2m/s (a) to the right, (b) up, and (c) down.
4 Inductance 4. 17
An airport metal detector contains a large coil in its frame. The coil has a property called inductance. inductance When a metal passes through the frame, the inductance of the frame changes. The change in the inductance is converted to an alarm sound!
The Unit of Inductance 18
The Unit of Inductance is the henry(H). henr (H)
4 1 Self-Inductance 4.1 Self Inductance 19
When the switch is closed, current rises until it reaches its max value. l During the current rise, the magnetic field it produces changes thus a changing changes, magnetic flux Thus there should be an induced emf caused byy the changing magnetic flux Therefore, there is selfinduction in the circuit!
4 1 Self-Inductance 4.1 Self Inductance 20
The general relation of the self self-induced induced emf to the changing current is:
The p proportionality p y constant is the self-inductance (L) of the circuit!
By applying Faraday’s Law in reverse, we derive: Self-Inductance Self Inductance is a constant but depends on the geometry of the circuit/loop!
4 1 Self-Inductance: 4.1 Self Inductance: Example 21
1. Calculate the self-inductance of an air-core solenoid containing 300 turns t rns if the length of the solenoid is 25.0 cm and its cross-sectional area is 4.00 cm2.
2. Find the self-inductance of a solenoid of length 10 cm, area 5 cm2, and 100 turns. At what rate must g to induce an emf the current in the solenoid change of 20V?
4 2 Mutual-Inductance 4.2 Mutual Inductance 22
Figure shows two circuits. As we change the resistance in circuit 1, the current also changes Thus the magnetic field changes. it produces also changes. The changing magnetic flux induces an emf on circuit 1 and on circuit 2! Thus circuit 2 has an induced emf. This phenomenon is called, mutual induction!
Circuit 1
Circuit 2
Circuit 1
Circuit 2
4 2 Mutual-Inductance 4.2 Mutual Inductance 23
The net induced emf on circuit two is related to the changing current by:
Self-induced by 2
Mutually induced by 1 on 2
Mutual Inductance of the two circuits are equal and can be found using the transformable formula:
Mutual Inductance like Self-Inductance Self Inductance depends on the geometry of the two circuits and d the h distance between them!
4 2 Mutual Inductance: Example 4.2 24
An electric toothbrush has a base designed to hold the toothbrush handle when not in use. As shown in the Figure, the handle has a cylindrical hole that fits loosely over a matching cylinder on the base. When the handle is placed on the base, base a changing current in a solenoid inside the base cylinder induces a current in a coil inside the handle. This induced current charges the battery in the handle. We can model the base as a solenoid of length x with Nbase turns (Fig. 32.15b), carrying a current I, and having a cross sectional area A. The handle coil contains Nhandle turns and completely surrounds the base coil. Find the mutual inductance of the system.
x
5. The Storage of Magnetic Energy
25
An inductor stores magnetic energy through the current b ld up in it, just as a building capacitor stores electrical energy. energy Consider, the circuit at the right. The energy stored in an inductor carrying a current I is given by:
The magnetic energy density, uB is given by:
This is the energy that is stored in a magnetic field, regardless of the configuration!
6 RL Circuits 6. 26
RL Circuits contain a resistor and an inductor. I
flows in a single direction But changes its value, it either grows or decays
27
6. The Growth of I in RL Circuits We assume that the inductor has 0 emf initially. initially After the switch is closed, the emff off the th battery b tt equates t to the back emf of the i d inductor, and d current builds b ild according to:
As current builds up, the i d t ’ b inductor’s backk emff is i reduced to zero!
Imax is the maximum current in the circuit equivalent to ξ0/R. τ is the time constant equilvalent to L/R
28
6. The Decay of I in RL Circuits As the switches are reconfig red the Imax reconfigured, Ima current is drained by the resistor R according to: This happens because the inductor acts like a battery, with ith a llessening i currentt pump abilities! Io is the initial current in the circuit equivalent to ξ0/R. τ is the time constant equilvalent to L/R
6 RL Circuits: Examples 6. 29
1. A basic RL circuits consists of the following: a battery (ξ (ξ= 12 V), V) an inductor (L = 30 mH), mH) and a resistor (R = 6 Ω). Find the time constant, and if the switch is closed at t =0, =0 when will the current reach half its maximum value. 2. If the battery y in the example p above is carefullyy removed after the current reaches its maximum y to 10% of the value,, when will the current decay original maximum value?
Chapter Six is pretty much a straight forward chapter. 1. 2. 3. 4. 5 5. 6.
7.
8.
Sources of AC R in AC L in i AC C in AC LC in AC The Series RLC in AC Resonance in AC T Transformers f
1
IN THIS FINAL CHAPTER, YOU SHOULD BE ABLE TO… Understand the sources of alternating currents
Understand the operating principles of transformers
Analyze the behavior of specific f R, L, C combinations
Analyze A l the th behaviors b h i of R, L, and C if alternating currents flow through them
Define phasors and discover its importance when analyzing for the behaviors of R, L, and C
2
TIME-VARYING VALUES |
|
To identify time varying values, we use lower case letters!!
To identify fixed values we shall use values, upper case letters
3
WHY STUDY ALTERNATING CURRENTS? |
|
|
More than 99% of the electrical energy used today is produced by electrical generators in the form of alternating current (ac). AC’s advantage over DC because electrical energy can be transported over long distances at very high voltage and low currents to reduce energy losses due to Joule heat! AC can then be transformed, with almost no energy loss loss, to lower and safer voltages and correspondingly higher currents for everyday use!
4
ALTERNATING CURRENTS |
|
|
Alternating Currents – are currents whose value vary periodically over time! Alternating Currents, in general are sinusoidal in nature and generally supply alternating voltages of the form: Because voltage changes, it is positive ½ the period, and negative ½ the period! i d!
In our country most AC’s have frequencies of 60 Hz or angular q y of 377 rad/s frequency
5
1. AC SOURCES |
|
|
There are many kinds off AC sources!! The most common probably are the AC outlets in our homes! But how do we actually actua yp produce oduce alternating currents?
6
2. RESISTORS IN AC Consider the circuit to the right. i ht | The instantaneous voltage and current through the resistor are given by: |
|
Notice that vR and iR are in phase phase” with one “in another!
7
2. RESISTORS IN AC |
|
There is still power loss i resistors in i t when h current passes through th them because b off the th voltage drop! This power has 3 forms: y
Instantaneous
y
Average
y
Maximum
8
ROOT MEAN SQ QUARED ((RMS)) VALUES |
|
|
|
Most AC ammeters and voltmeters are designed to measure rms values of currents and voltages, instead off the h maximum i values. l So there S h is i a necessity i to interconvert between rms and maximum values! Simple Rule: the rms value is the maximum value over the square root of 2! Example:
9
EXERCISES |
1. Find Pav in terms of Irms and R
|
2. Find Pav in terms of ξmax and Imax
|
3 Find Pav in terms of ξrms and Irms 3.
|
4 Find 4. Fi d Irms in i terms t off ξrms and dR
|
5. A 12-Ω resistor is connected across a sinusoidal emf that has a peak value of 48V. Find (a) the rms current, (b) the average power, (c) the maximum power.
10
3. INDUCTORS IN AC Consider the circuit to the right. i ht | The instantaneous voltage and current are given by: |
|
Notice that vL and iL are out of phase, phase iL lagging vL by π/2 rads χL is i called ll d inductive i d i reactance, it i has the unit of ohms! This means that Thi th t inductors i d t reactt differently to current by offering resistance!
11
4. CAPACITORS IN AC Consider the figure to the right. i ht | The instantaneous voltage and current are given by: |
|
Notice that vC and iC are out of phase phase”, iC leading “out vC with π/2 radians. χC is i called ll d capacitive i i reactance, it i has the unit of ohms! This means that Thi th t capacitors it reactt differently to current by offering resistance!
12
L,, C IN AC: EXAMPLES 1. A 40mH inductor is placed across an ac generator that has a maximum emf of 120V. Find the inductive reactance and the maximum current when the frequency is (a) 60 Hz (b) 2000 Hz H
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What can you conclude about the relation of inductive reactance and current?
2. A 20-μF capacitor is placed across a generator that has a maximum emf of 100V. Find the capacitive reactance and the maximum current when the frequency is (a) 60 Hz (b) 5000 Hz
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What can you conclude about the relation of capacitive reactance and 13 current?
THE BEHAVIORS OF L AND C IN AC CIRCUITS |
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Alternating current behaves differently than p direct current in inductors and capacitors. When a capacitor becomes fully charged in a dc circuit, i i it i stops the h current, that h is, i it i acts like lik an open circuit. But if the current alternates, alternates charge continually flows onto or off the plates of the capacitor and at higher frequencies, the capacitor, will hardly impede current at all all, which means means, it acts like a short circuit! Conversely, an inductor coil usually has a very small resistance and is essentially a short circuit for dc. But when the current is alternating, alternating a back emf is generated in an inductor, and at higher frequencies, the back emf is so large, the inductor acts like an open circuit! i it!
14
5. LC IN AC |
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Consider the circuit to the right. When the switch is closed, the initially charged capacitor discharges producing a back emf on the inductor, which in turn counters the discharging current, recharging the capacitor. Thus once the capacitor completely discharges, it is once again i charged h d by b the th inductor. Conversely, once the inductor reaches zero current, current g flow through g it from will again the capacitor!
Natural Frequency of Oscillation of i
C Current t iin an LC circuit i it
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5. LC IN AC EXAMPLE |
A 2-μF capacitor is charged to 20V and is then connected t d across a 6 6-μH H iinductor. d t ((a)) Wh Whatt iis the th frequency of oscillation? (b) What is the maximum i value l off the th current? t?
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6. RLC IN AC ((SERIES) |
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Consider the figure to th right. the i ht The circuit has a current given by:
Where Z is impedance (overall resistance) And A d δ is i the h phase h angle
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6. RLC IN AC ((SERIES) |
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The Average Power for RLC in AC, series connection ti can b be represented t db by:
Where cos δ is called the p power factor.
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7. RESONANCE IN SERIES RLC IN AC |
Resonance is the condition in which we have the smallest possible impedance that would lead to the maximum current. y
y
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Zmin can only happen when the reactances is zero. Reactances can only be zero if the ac source zero, frequency equates to the natural frequency of the circuit!
At resonance, we have maximum i current and d power and the power factor is one!
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7. SERIES RLC IN AC: EXAMPLES 1.
A series RLC Circuit with L = 2H, C = 2μF, and d R = 20Ω is i driven d i by b a generator t with ith a maximum emf of 100 V and a variable f frequency. Find Fi d ((a)) th the resonance frequency f (f0), (b) the maximum current at resonance, (c) the phase angle δ, δ (d) the power factor, factor and (e) the average power delivered.
2.
A series RLC Circuit with L = 2H, C = 2μF, and d R = 20Ω Ω is i driven d i by b a generator with i ha maximum emf of 100 V and a variable f frequency. Find Fi d th the maximum i voltage lt across the resistor, the inductor and the capacitor.
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8. TRANSFORMERS |
A transformer is a device used to raise or lower the voltage in a circuit without an appreciable loss of power y
y y
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A simple transformer consisting of two wire coils around a common iron core. The coil carrying the input power is called the primary. The coil carrying the output power iis called ll d the th secondary. d
The transformer operates on the principle of mutual induction The iron Th i core increases i the th magnetic field for a given current and guides it so that nearly all the magnetic flux through one coil goes through the other coil.
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6. TRANSFORMERS |
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For a transformer with N1 turns in the primary and N2 turns in the secondary the voltage secondary, across the secondary coil is related to the generator emf across the primary coil by: If there are no losses, due to Joule Heating (which is due to negligible li ibl resistance i t iin the coils), RMS Power relations are given by:
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6. TRANSFORMERS: EXAMPLES |
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1. A doorbell requires 0.4A at 6V. It is connected t a ttransformer to f whose h primary i containing t i i 2000turns, is connected to a 120-V ac line. (a) H How many tturns should h ld there th be b in i the th secondary? (b) What is the current in the primary? 2. A transmission line has a resistance of 0.02Ω/km. Calculate the I2R power loss if 200kW off power is i transmitted i d from f a power generator to a city 10km away at (a) 240 V and (b) 4.4 kV 23
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