Unit 1
REVIEW OF PLANE TRIGONOMETRY
1. analyze a right triangle.
2. add vectors in parallelogram and polygon method. 3. use trigonometric identities.
4. analyze oblique triangles.
5. draw graphs of trigonometric functions. 6. define a complex number.
7. illustrate the graphical representation of a complex number. 8. convert complex numbers to different forms.
9. perform mathematical operations of complex numbers. 10. apply complex numbers.
11. LEARNING OUTCOMES
At the end of this unit, you are expected to
:
complex number polygon method
conjugate quadrant
cosine law rectangular form
exponential form right triangle
oblique triangle sine law
parallelogram trigonometric form
polar form vector addition
1.1 The Right Triangle
𝑠𝑖𝑛 𝐴 =𝑎𝑐 𝑐𝑜𝑠 𝐴 =𝑏𝑐 𝑡𝑎𝑛 𝐴 =𝑎𝑏 𝑐𝑠𝑐 𝐴 =𝑐 𝑎 𝑠𝑒𝑐 𝐴 = 𝑐 𝑏 𝑐𝑜𝑡 𝐴 = 𝑏 𝑎
1.1.1 Solution of a Right Triangle
Referring to Figure 1.1, the relationships of the sides a, b, and c can be expressed as 𝑐2= 𝑎2+ b2 (Pythagorean Theorem) To find c, a, and b 𝑐 = 𝑎2+ b2 a = 𝑐2− b2 b = 𝑐2− a2
𝐵
𝑐 𝑎
𝐴 𝑏 𝐶
Important Terms
Example 1.1. Solve for b and the angles A and B.
Given: 𝑎 = 12 𝑢𝑛𝑖𝑡𝑠
𝑐 = 13 𝑢𝑛𝑖𝑡𝑠
Find: side b and angle B
Known: 𝑏 = 𝑐2− a2 𝑠𝑖𝑛 𝐴 =𝑎𝑐 ; 𝐴 = sin−1 ac 𝐴 + 𝐵 = 90 ; B = 90 – A Solution: b = 132− 122 b = 5 𝑢𝑛𝑖𝑡𝑠 𝐴 = sin−1 12 13 𝐴 = 67.38 𝐵 = 90 – 67.38 𝐵 = 22.62 Answer: 𝑏 = 5 𝑢𝑛𝑖𝑡𝑠 , 𝐵 = 22.62° Example 1.2. Find the sides a and b.
𝐵
𝑐 = 13 𝑎 = 12
𝐴 𝑏 𝐶
40𝐵
𝑐 = 13 𝑎
𝐴 𝑏 𝐶
40Solution: Solving for a 𝑠𝑖𝑛 𝐴 =𝑎𝑐 𝑎 = 𝑐 sin 𝐴 = 13 sin 40 = 8.36 𝑢𝑛𝑖𝑡𝑠 To find b 𝑐𝑜𝑠 𝐴 =𝑏𝑐 𝑏 = 𝑐 cos 𝐴 = 13 cos 40 = 9.96 𝑢𝑛𝑖𝑡𝑠
Drill Problems
1. The height of the tower is unknown. There is now no way that you can climb the
tower to determine its height. The only possible way is to use a transit to find the
angle of elevation of the tower (
) and the distance (d) between the transit and the
foot of the tower. If the height of the transit is 1 meter (m), determine the height of
the tower.
Input any value of d and and click answer. Be sure you have solved the problem with the assumed values of d and before knowing the answer.
2. Find the x and y components of the force exerted by the father in pulling the cart
with his child riding.
D m ° Answer m
Input any value of F and and click answer. Be sure you have solved the problem with the assumed values of d and before knowing the answer.
1.2 Vector Addition
1.2.1 Parallelogram Method
1.2.2 Polygon (Head to Tail method) Method
F N ° Answer Fx Fy N y
F1 FT
x F2
F
Note: F1 and F2 are the given vectors, and FT is the equivalent (total or resultant) vector.
Example 1.3. Determine graphically the resultant of the following forces using: (a) parallelogram method.
(b) polygon method.
Solution:
(a) By parallelogram method, get first the resultant of F1 and F2 and call it FA. Then use again
parallelogram method to determine the resultant vector of FA and F3 to get the total or
equivalent force. y
FT
F1
F2
F1
F 2F3
(b) Using polygon method
FA
F1
FT F 2F3
F2
F1
F3
F T1.3 Formulas Involving Addition or Subtraction of Functions
sin2 + cos2 = 1 sec2 = 1 + tan2 csc2 = 1 + cot2
1.4 Sine and Cosine of the Sum or Difference of Two Angles
sin () = sin cos cos sin cos ( +) = cos cos - cos sin cos ( - ) = cos cos + cos sin sin (-A) = -sin A
cos (-A) = cos A sin (x + 90) = cos x cos (x + 90) = - sin x sin (x – 90) = - cos x sin (180 - x) = sin x cos (180 - x) = - cos x
1.5 Double Angle Formulas
sin 2 = 2 sin cos
cos 2 = cos2 – sin2 = 2cos2 – 1 = 1 – 2 sin2 tan 2 = 2 ta n 1−𝑡𝑎𝑛2 cot 2 = 𝑐𝑜𝑡2 − 1 2 cot 𝑐𝑜𝑠2 = 1 + 𝑐𝑜𝑠2 2 𝑠𝑖𝑛2 = 1 − 𝑐𝑜𝑠2 2
1.6 Oblique Triangles
b1.6.1 Sine Law 𝑎 sin 𝐴 = 𝑏 sin 𝐵 = 𝑐 sin 𝐶 1.6.2 Cosine Law 𝑎2 = 𝑏2+ � �2− 2bc cos A Example 1.4. Find the x and y components of F1.
c a
A b
F1 = 18 N FT = 30 N
F2= 20 N
B
c a
A b C
Solution:
can be found by applying cosine law:
FT2 = F12 + F22 - 2 F1 F2 cos 302 = 182 + 202 – 2(18)(20) cos = 104.15 Then = 180 - = 180 - 104.15 = 75.85 Considering F1 F1x = F1 cos = 18 cos 75.85 = 4.4 N F1y = F1 cos = 18 sin 75.85 = 17.45 N
F1 = 18 N FT = 30 N
F2= 20 N
This is also equal to F1F1 = 18 N F1y
F1x
1.7 Graphs of Trigonometric Functions
Note: It is assumed that the amplitude of these waves is 1.
Drill Problem
1. Two horses are pulling a barge along a canal as shown in the figure. The cable connected to the first horse makes an angle of 1 with respect to the direction of the canal (x-axis), while the cable attached to the second horse makes an angle of 2 . The first horse exerts a force F1
and the second horse exerts a force F2. Find the total force exerted by the two horses.
y = sin x
y = cos x
(a)
Input any value of F1 and F2 . Assume values of not less than 600 N. Also input values of 1 and 2 that are less than 90°. Solve first the problem with the assumed values before clicking the answer.
2. A school bus leaves a school at 1 PM traveling at a speed of v1 along a straight road. At 2 PM
it turns right at an angle of with respect with the first road and runs at a speed of v2. What
will be its distance from the school at 2:30 PM?
Input any value of v1 and v2 and . Solve first the problem using the assumed values before clicking
the answer.
F1 N F2 N 1 ° 1 ° Ans N
1.8 The Quadrants
Counterclockwise rotation Clockwise rotation
270 3 2 0 or 360 (0 or 2) 180 (/2) 90 (/2) II I IV III -270 −32 0 or 360 (0 or 2) -180 (-/2) -90 (-/2) II I IV III I II III IV 0 90 180 270 360 0 ð2 32 2
Problem Set No. 1
REVIEW OF PLANE TRIGONOMETRY
1. A ladder leans against a wall with its foot 10 ft from the wall. If the ladder makes an angle of 60 with the ground, how long is the ladder.
a. 20 ft b. 15 ft c. 10 ft d. 5 ft
2. In no. 1, how far above the ground is the top of the ladder. a. 12.73 ft
b. 13.72 ft c. 17.32 ft d. 19.73 ft
3. A tower 160 ft casts a shadow 92.4 ft long on the ground. Find the angle of elevation of the sun. (The angle of elevation is the angle with its rays make with level ground.)
a. 35° b. 60° c. 45° d. 75°
4. From the top of a tower at C the angle of depression of point A is 56. If the distance between point A and point B (the foot of the tower) is 180 meters, how high is the tower?
a. 168.94 m b. 178.5 m c. 230.89 m d. 266.86 m
5. A boy pushes a force of 15 lb toward north. Another boy pushes the same box toward east with a force of 20 lb. Find the resultant force.
a. 25 lb b. 30 lb c. 35 lb d. 40 lb
6. One of the diagonals of a parallelogram makes angles of 2235’ and 4848’, respectively, with two adjacent sides. If the diagonal is 18.54 ft long, find the lengths of the sides of the
parallelogram. a. 7,25 ft, 13.73 ft b. 7.51 ft, 14.72 ft c. 8.34 ft, 15.83 ft d. 9.35 ft, 16.72 ft
7. In the figure shown below, find the magnitude of F1 if F2 = 100 N and makes an angle of 30
with the respect to the x axis and the resultant of the two forces is 167 N and makes an angle of 50 with respect to the x axis.
a. 80.64 ft b. 82.64 ft c. 89.49 ft d. 90.29 ft F1 F2
1.9 Complex Numbers
y (imaginary axis) + −1 = + jA -A A x (real axis) − −1 = - j A complex plane.A complex number is one of the form x j y
that is, it is the sum of a real number and an imaginary one.
A complex number is the conjugate of another if their imaginary parts are of opposite signs. For example, -5 + j6 is the conjugate of –5 – j6.
1.9.1 Graphical Representation of Complex Numbers
Let z be the vector connecting origin and point P whose coordinates are x and y. Then, in complex form z = x + jy r y x P (x, y)
To find r and in terms of x and y:
r = 𝑥2+ 𝑦2 Note that r is the magnitude of z = tan-1𝑦𝑥
To find x and y in terms of r and : x = r cos y = r sin
1.9.2 Forms of a Complex Number
Rectangular Form: z = x + jy
Trigonometric Form: z = r cos + j r sin = r (cos + j sin )
Polar Form: z = r cjs = r
Exponential Form: z = rej , where is in radians
1.9.3 Addition and Subtraction of Complex Number
1.9.3.1 Algebraic Method
(a) Addition. To add two or more complex numbers, add the real parts and then the pure imaginary parts.
Example 1.5. Add z1 = 5 + j3 and z2 = -2 – j5 Solution: z1 + z2 = (5 + j3) + (-2 – j5)
= 3 - j2
(b) Subtraction. To subtract one complex number from another, subtract the real parts and then subtract the pure imaginary parts.
Example 1.6. Subtract z1 = 5 + j3 by z2 = -2 – j5
Solution: z1 - z2 = (5 + j3) - (-2 – j5)
= 7 + j8
(a) Addition. Use parallelogram method to find the sum of two or more complex numbers.
Example 1.7. Add z1 = 5 + j3 and z2 = -2 – j5
Solution:
First plot z1 and z2.
1. Use parallelogram method to determine the sum. 2. Count the x and y components of the sum.
As you can see the sum is 3 - j2 (b) Subtraction.
Example 1.8. Subtract z1 = 5 + j3 by z2 = -2 – j5 Solution:
1. First plot z1 and z2.
2. Take the negative of z2. This is the opposite of z2.
3. Find the difference using parallelogram method. This is similar to addition. 4. Counting the x and y components, the result is 7 + j8.
0 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7 -8 -9 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
1.9.3 Transformation of a Complex Number from One Form to Another
Example 1.9. Express each of the following in rectangular, polar and exponential forms. 0 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7 -8 -9 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
3
4
4
2
1r4
r3
r2
r1
z
2 0 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7 -8 -9 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9z
1z
3z
4Solution:
Note: Always use the smaller angle between the positive real axis (the reference) and the vector.
For z1: r1 = 42+ 52 = 6.4 For z2: r2 = 62+ 32 = 6.71 1 = tan-1 5 4 = 51.34 1 = 180 - tan -13 6 = 153.43 For z3: r3 = 42+ 52 = 6.4 For z4: r4 = 62+ 42 = 7.2 3 = 180 - tan-1 5 4 = 128.66 4 = tan -14 6 = 33.69 Answers
Rectangular Form Polar Form Exponential Form
z1 4 + j5 6.4 51.34 6.4 ej0.8961
z2 -6 + j3 6.71 153.43 6.71 ej2.6779
z3 -4 - j5 6.4 -128.66 6.4 e–j2.2455
z4 6 – j4 7.2 -33.69 7.2 e-j0.588
1.9.4 Multiplication and Division of Complex Numbers
(a) Multiplication of Complex Numbers in Rectangular Form
Example 1.10. Multiply z1 = 5 + j2 by z2 = 3 – j4 Solution: z1 z2 = (5 + j2)(3 – j4) = (5)(2) + (j2)(3) + (5)(-j4) + (j2)(-j4) = 10 + j6 – j20 – j2 8 where j2 = -1 = 10 + j6 – j20 – (-1)(8) = 18 – j14
(b) Multiplication of Complex Numbers in Polar Form Let z1 = r11 z2 = r22 z1z2 = r11 r22 = r1 r21 + 2 Example 1.11. Multiply z1 = 2045 by z2 = 2-30 Solution: z1 z2 = (2045 )( 2-30) = 20245+(-30) = 4015
(c) Division of Complex Numbers in Rectangular Form Example 1.12. Divide z1 = 5 + j2 by z2 = 3 – j4 Solution: 𝒛𝟏 𝒛𝟐 = 5 + j2 3 – j4
Rationalize, i.e., multiply the numerator and denominator by the conjugate of the denominator. The conjugate of z2 = 3 – j4 is 𝐳 = 3 + j4 𝟐
𝒛𝟏 𝒛𝟐 = 5 + j2 3 – j4 3 + j4 3 + j4 𝒛𝟏 𝒛𝟐 = 7 + 𝑗26 32+ 42 = 7 + 𝑗26 25 = 0.28 + j1.04
(d) Division of Complex Numbers in Polar Form Let z1 = r11 z2 = r22 𝒛𝟏 𝒛𝟐 = 𝑟11 𝑟22 = 𝑟1 𝑟21+ 2 Example 1.13. Divide z1 = 2045 by z2 = 2-30 Solution: 𝒛𝒛𝟏 𝟐 = 2045 2−30 = 20 2 45 − (−30) = 1075
Problem Set No. 2
COMPLEX NUMBERS 1. Add the following using algebraic and graphical methods.
(a) 8 + j4, 4 – j6 (b) -7 – j7, 8 + j3, 5 – j3 2. Perform the following using algebraic and graphical methods.
(a) (8 + j4) – (4 – j6) (b) (-7 – j7) + ( 8 + j3) – (5 – j3) 3. Convert the following to polar and exponential forms:
(a) (8 + j4) (b) (-4 + j6) (c) (-7 – j7) (d) ( 8 - j3) 4. Convert the following to rectangular and exponential forms.
(a) 660 (b) 12-300 (c) 10120 (d) 9-120 5. Convert the following to rectangular and polar forms.
(a) 5ej1.45 (b) 9e-j0.9823 (c) 12ej/3 (d) 7e-j 6. Perform the following. Express answers in polar form.
(a) 3045 4-30 (b) 30ej/4 4-30 (c) 30304545 + 4 4−30−30
(d) (515 + 4e-j/3)(30 ej/4 - 490)
8. Solve this problem using complex numbers: In the figure shown below, find the magnitude of F1, its anlge with respect to the + x aixs, its x and y components , if F2 = 100 N and makes an
angle of 30 with the respect to the x axis and the resultant of the two forces is 167 N and makes an angle of 50 with respect to the x axis.
F1
Unit 2
INTRODUCTION TO ELECTRICITY
1. define electricity.
2. name some scientists who contributed to the development of electricity and electronics.
3. discuss the scientist’s contributions to electricity and electronics. 4. quote some applications of electricity and electronics.
5. identify various electrical components.
6. use metric prefixes in simplifying large and small numbers. 7. perform mathematical operations involving powers of ten and
metric prefixes.
8. discuss the difference between direct current and alternating current.
LEARNING OUTCOMES
At the end of this unit, you are expected to:
Electricity semiconductor static electricity active element dynamic electricity passive element Resistor
electrical quantities
Resistance metric prefixes
Inductor
direct current
Inductance alternating current
Transformer
2.1 Definition of Electricity
Electricity is a physical phenomenon arising from the existence and interaction of electric charge. It is a form of energy generated by friction, heat, light, magnetism, chemical reaction, etc.
2.1.1 Two Types of Electricity:
a. Static electricity – electricity at rest. It cannot flow from one place to another.
b. Dynamic electricity – also known as current electricity. Electricity in motion. It can be transmitted from one place to the other.
2.1.2 Methods of Producing Electricity
There are six methods for producing electricity: 1. Magnetism 2. Chemical action 3. Pressure 4. Heat 5. Friction 6. Light 2.1.3 Electrical Effects
With the exception of friction, electricity can be used to cause the same effects that cause it.
1. Magnetism 2. Chemical action 3. Pressure 4. Heat 5. Light
2.2 History of Electricity and Electronics
Prehistoric people experienced the properties of magnetite – permanently magnetized pieces of ore, often called lodestones. These magnetic stones were strong enough to lift pieces of iron.
The philosopher Thales of Miletus (640 – 546 B.C.) is thought to have been the first person who observed the electrical properties of amber. He noted that when amber was rubbed, it acquired the ability to pick up light objects such as straw and dry grass. He also experimented with the lodestones and knew of its power to attract iron. By the thirteenth century, floating magnets were used for compasses.
One of the first important discoveries about static electricity is attributed to William
Gilbert (1540-1603). Gilbert was an English Physician who, in a book published in 1600, described
how amber differs from magnetic loadstones in its attraction of certain materials. He found that when amber was rubbed with a cloth, it attracted only lightweight materials, whereas loadstones attracted only iron. Gilbert also discovered that other substances, such as sulfur, glass, and resin, behave as amber does. He used the Latin word electron for amber and originated the word electrical for the other substances that acted similar to amber. The word electricity was used for the first time by Sir Thomas Browne (1605-82), an English physician.
Following Gilbert’s lead, Robert Boyle published his many experimental results in 1675. Boyle was one of the early experiments with electricity in a vacuum.
Otto von Guericke (1602 – 1686) built an electrical generator and reported it in his
Experimental Nova of 1672. This device was a sulfur globe on a shaft that could be turned on its bearing . When the shaft was turned with a dry hand held on the surface, an electrical charge gathered on the globe’s surface. Guericke also noted small sparks when the globe was discharges. In his studies of attraction and gravitation, Guericke devised the first electrical generator. When a hand was held on a sulfur ball revolving in its frame, the ball attracted paper, feathers, chaff, and other light objects.
Another Englishman, Stephen Gray (1696-1736), discovered that some substances conduct electricity and some do not. Following Gray’s Lead, a Frenchman named Charles du Fay experimented with the conduction of electricity. These experiments led him, to believe that there were two kinds of electricity. He called one type vitreous electricity and the other type resinous
resinous electrify attracted each other. It is known today that two types of electrical charge exist,
positive and negative. Negative charge results from an excess of electrons in a material and positive
charge results from a deficiency of electrons.
A major advance in electrical science was made in Leyden, Holland, in 1746, when Peter van
Musschenbroek introduces a jar that served as a storage apparatus for electricity. The jar was
coated inside and out with a tinfoil, and a metallic rod was attached to the inner foil lining and passed through the lid. Leyden jar were gathered in groups (called batteries) and arranged with multiple connections, thereby further improving the discharge energy.
Benjamin Franklin (1706-90) conducted studies in electricity in the mid-1700s. He theorized
that electricity consisted of a single fluid, and he was the first to use the terms positive and negative. In his famous kite experiment, Franklin showed that lightning is electricity.
Charles Augustin de Coulomb (1736-1806), a French physicist, in 1785 proposed the laws that govern the attraction and repulsion between electrically charged bodies. Today, the unit of electrical charge is called the coulomb.
Luigi Galvani (1737-98) experimented with current electricity in 1786. Galvani was a
professor of anatomy at the University of Bologna in Italy. Electrical current was once known as
galvanism in his honor.
In 1800, Alessandro Volta (1745-1827), an Italian professor of physics, discovered that the chemical action between moisture and two different metals produced electricity. Volta constructed the first battery, using copper and zinc plates separated by paper that had been moistened with a salt solution. This battery, called the voltaic pile, was the first source of steady electric current. Today, the unit of electrical potential energy is called the volt in honor of Professor Volta.
A Danish scientist, Hans Christian Oersted (1777-1851), is credited with the discovery of electromagnetism, in 1820. He found that electrical current flowing through a wire caused the needle of a compass to move. This finding showed that a ,magnetic field exists around a current-carrying conductors and that the field is produced by the current.
The modern unit of electrical current is the ampere (also called amp) in honor of the French physicist André Ampère (1775-1836). In 1820, Ampère measured the magnetic effect of an electrical current. He found that two wires carrying current can attract and repel each other, just as magnets can. By 1822, Ampère had developed the fundamental laws that are basic to the study of electricity.
One of the most well known and widely used laws in electrical circuits today is Ohm’s law. It was formulated by Georg Simon Ohm (1787-1854), a German teacher, in 1826. Ohm’s law gives us the relationship among the three important electrical quantities of resistance, voltage, and current.
Although it was Oersted who discovered electromagnetism, it was Michael Faraday (1791-1867) who carried the study further. Faraday was an English physicist who believed that electricity could produce magnetic effects, then magnetism could produce electricity. In 1831 he found that a moving magnet cause an electric current in a coil of wire placed within the field of the magnet. This effect, known today as electromagnetic induction, is the basic principle of electric generators ands transformers.
Joseph Henry (1797-1878), an American physicist, independently discovered the same
principle in 1831, and it is in his honor that the unit of inductance is called the henry. The unit of capacitance, the farad, is named in honor of Michael Faraday.
A paper published by James Prescott Joule in 1841 claimed the discovery of the relationship between a current and the heat or energy produced, which today we call Joule’s law. The unit of energy is called the joule in his honor.
In the 1860s, James Clerk Maxwell (1831-79), a Scottish Physicist, produced a set of mathematical equations that expressed the laws governing electricity and magnetism. These formulas are known as Maxwell’s equations. Maxwell also predicted that electromagnetic waves (radio waves) that travel at the speed of light in space could be produced.
It was left to Heinrich Rudolph Hertz (1857-94), a German physicist, to actually produce these waves that Maxwell predicted. Hertz performed this work in the late 1880s. Today, the unit of frequency is called the hertz.
The Beginning of Electronics
The early experiments in electronics involved electric currents in glass tubes. One of the first to conduct such experiments was a German named Heinrich Geissler (1814-79). Geissler found that when he removed most of the air from a glass tube, the tube glowed when an electrical potential was placed across it.
Around 1878, Sir William Crookes (1832-1919), a British scientist, experimented with tubes similar to those of Geissler. In his experiments, Crookes found that the current in the tubes seemed to consist of particles.
Thomas Edison (1847 – 1931), experimenting with the carbon-filament light bulb that he had
invented, made another important finding. He inserted a small metal plate in the bulb. When the plate was positively charged, a current flowed from the filament to the plate. This device was the first thermionic diode. Edison patented it but never used it.
The electron was discovered in the 1890s. The French physicist Jean Baptiste Perrin (1870 – 1942) demonstrated that the current in a vacuum tube consists of negatively charged particles. Some of the properties of these particles were measured by Sir Joseph Thomson (1856 – 1940), a British physicist, in experiments he performed between 1895 and 1897. These negatively charged particles later became known as electrons. The charge on the electron was accurately measured by an American physicist, Robert A. Millikan (1868 – 1953), in 1909. As a result of these discoveries, electrons could be controlled, and the electronic age was ushered in.
Putting the Electron to Work
A vacuum tube that allowed electrical current in only one direction was constructed in 1904 by John
A. Fleming, a British scientist. The tube was used to detect electromagnetic waves. Called the
Major progress in electronics, however, awaited the development of a device that could boost, or amplify, a weak electromagnetic wave or radio signal. This device was the audion, patented in 1907 by Lee de Forest, an American. It was a triode vacuum tube capable of amplifying small electrical signals.
Two other Americans, Harold Arnold and Irving Langmuir, made great improvements in the triode tube between 1912 and 1914. About the same time, de Forest and Edwin Armstrong, an electrical engineer, used the triode tube in an oscillator circuit. In 1914, the triode was incorporated in the telephone system and made the transcontinental telephone network possible.
The tetrode tube was invented in 1916 by Walter Schottky, a German. The tetrode, along with the pentode (invented in 1926 by Tellegen, a Dutch engineer), provided great improvements over the triode. The first television picture tube, called the kinescope, was developed in the 1920s by
Vladimir Zworykin, an American researcher.
During World War II, several types of microwave tubes were developed that made possible modern microwave radar and other communications systems. In 1939, the magnetron was invented in Britain by Henry Boot and John Randall. In the same year, the klystron microwave tube was developed by two Americans, Russell Varian and his brother Sigurd Varian. The traveling-wave tube was invented in 1943 by Rudolf Komphner, an Austrian-American.
The Computer
The computer probably has had more impact on modern technology than any other single type of electronic system. The first electronic digital computer was completed in 1946 at the University of Pennsylvania. It was called the Electronic Numerical Integrator and Computer (ENIAC). One of the most significant developments in computers was the stored program concept, developed in the 1940s by John von Neumann, an American mathematician.
Solid State Electronics
The crystal detectors used in the early radios were the forerunners of modern solid state devices. However, the era of solid state electronics began with the invention of the transistor in 1947 at Bell Labs. The inventors were Walter Brattain, John Bardeen, and William Shockley.
In the early 1960s, the integrated circuit was developed. It incorporated many transistors and other components on a single small chip of semiconductor material. Integrated circuit technology continues to be developed and improved, allowing more complex circuits to be built on smaller chips.
The introduction of the microprocessor in the early 1970s created another electronics revolution: the entire processing portion of a computer placed on a single, small, silicon chip. Continued development brought about complete computers on a single chip by the late 1970s.
Major Events in Electrical Sciences and Engineering
1675 Robert Boyle was published Production of Electricity. 1746 The Leyden jar was demonstrated in Holland.
1750 Benjamin Franklin invented the lightning conductor. 1767 Joseph Priestley published the Present State of Electricity.
1786 Luigi Galvani observed electrical convulsion in the legs of dead frogs. 1800 Alessandro Volta announced the voltaic pile.
1801 Henry Moyes was the first to observe an electric arc between carbon rods. 1820 Hans Oersted discovered the deflection of a magnetic needle by current on a
wire.
1821 Michael Faraday produced magnetic rotation of a conductor and magnet- the first electric motor.
1825 Andre-Marie Ampere defined electrodynamics.
1828 Joseph Henry produced silk-covered wire and more powerful electromagnets. 1831 Michael Faraday discovered electromagnetic induction and carried out
experiments with an iron ring and core. He also experimented with a magnet and rotating disk.
1836 Samuel Morse devised a simple relay.
1836 Electric light from batteries was shown at Paris Opera.
1841 James Joule stated the relation between current and energy produced. 1843 Morse transmitted telegraph signals from England to France.
1850 First channel telegraph signals from Baltimore to Washington, D.C. 1858 Atlantic telegraph cable was completed and the first message sent.
1861 Western Union established telegraph service from New York to San Francisco. 1862 James Clerk Maxwell determined the ohm.
1873 Maxwell published Treatise on Electricity and Magnetism. 1874 Alexander Graham Bell invented the telephone.
1877 Thomas Edison invented the telephone. 1877 Edison Electric Light Company was formed.
1881 First hydropower station was brought into use at Niagara, New York.
1881 Edison constructed the first electric power station at Pearl Street, New York. 1883 Overhead trolley electric railways were started at Portrush and Richmond,
Virginia.
1884 Philadelphia electrical exhibition was held.
1885 The American Telephone and Telegraph Company was organized. 1886 H. Hallerith introduced his tabulating machine.
1897 J.J. Thomson discovered the electron.
1898 Guglielmo Marconi transmitted radio signals from South Foreland to Wimereux, England.
1904 John Ambrose Fleming invented the thermionic diode.
Computers Communications
Medicine Automation
Consumer Products
2.4 Circuit Components and Measuring Instruments
These can be the carbon-composition type or wound with special resistance wire. Their function is to limit the amount of current or divide the voltage in a circuit.
Capacitors
A capacitor is constructed of two conductor plates separated by an insulator (called a dielectric). Its basic function is to concentrate the electric field of voltage across the dielectric. As a result, the capacitor can accumulate and store electric charge from the voltage source. Furthermore, the dielectric can discharge the stored energy when the charging source is replaced by a conducting path.
When ac voltage is applied, the capacitor charges and discharges as the voltage varies. The practical application of this effect is the use of capacitors to pass an ac signal but to block a steady dc voltage.
Capacitors Inductors
Inductors
An inductor is just a coil of wire. Its basic function is to concentrate the magnetic field of electric current in the coil. Most important, an induced voltage is generated when the current with its associated magnetic field changes in value or direction. Inductors are often called chokes.
In the practical application of a choke, the inductor can pass a steady current better than alternating current. The reason is that a steady current cannot produce induced voltage. Note that the effect of a choke, passing a steady current, is the opposite of that of a coupling capacitor, which blocks dc voltage.
A transformer consists of two or more coil windings in the same magnetic field. Induced voltage is produced when the current changing in any winding. The purpose of a transformer is to increase or decrease the amount of ac voltage coupled between the windings. However, the transformer operates only with alternating current.
Transformers Semiconductor Devices
Semiconductor Devices
These include diode rectifiers and transistor amplifiers, either as separate, discrete components or as part of an IC chip. A diode has two electrodes; the transistor has three. In addition, the silicon controlled rectifier (SCR) and triac are used for power-control circuits.
Active and Passive Elements
Active elements - are capable of delivering power to some external device.
Examples: dependent and independent voltage and current sources
Passive elements – are capable of receiving power. They are able to store to store finite amounts of energy and then return that energy later to various external devices. Examples are resistors, inductors, and capacitors.
Electronic Instruments
Typical instruments include the power supply, for providing voltage and current; the voltmeter, for measuring voltage; the ammeter, for measuring current; the ohmmeter, for measuring resistance; the wattmeter, for measuring power; and the oscilloscope for observing and measuring AC voltages.
2.5 Electrical Quantities and Units with SI symbols.
Quantity Symbol Unit Symbol
Capacitance C farad F Charge Q coulomb C Conductance G Siemen S Current
I
ampere A Energy W Joule J Frequency F hertz Hz Impedance Z ohm Inductance L henry H Power P Watt W Reactance X ohm Resistance R ohm Voltage V Volt V2.6 Metric Prefixes
Power of Ten Value Metric Prefix Metric Symbol
109 one billion giga G
106 one million mega M
103 one thousand kilo k
10-3 one-thousandth milli m
10-6 one-millionth micro
10-9 one-billionth nano n
Problem Set No. 3
METRIC PREFIXES
I. Express each of the following as quantity having a metric prefix:
1) 31 𝑥 10−3𝐴 a. 0.31 mA b. 3.1 mA c. 31 mA d. 310 mA 2) 5.5 𝑥 103 𝑉 a. 5.5 kV b. 55 kV c. 550 kV d. 5.5 MV 3) 200 𝑥 10−12 𝐹 a. 200 pF b. 200 nF c. 200 µF d. 2000 pF 4) 0.000003 𝐹 a. 3 µF b. 30 µF c. 3 nF d. 30 nF 5) 3,300,000 a. 3.3 kΩ b. 33 kΩ c. 3.3 MΩ d. 33 MΩ 6) 350 𝑥 10−9𝐴 a. 35 nF b. 350 nF c. 3.5 pF d. 35 pF
II. Express each quantity as a power of ten: 7) 5 𝐴 a. 5 x 10-3 A b. 5 x 10-6 A c. 5 x 10-9 A d. 50 x 10-3 A 8) 43 𝑚𝑉 a. 43 x 10-3 V b. 43 x 10-6 V c. 43 x 10-9 V d. 43 x 10-12 V 9) 275 𝑘 a. 275 x 106 V b. 275 x 103 V c. 275 x 10-3 V d. 275 x 10-6 V 10) 10 𝑀𝑊 a. 10 x 1012 W b. 10 x 109 W c. 10 x 106 W d. 10 x 103 W
III. Add the following quantities:
11) 6 𝑚𝐴 + 3 A = _________ mA a. 6.03
b. 60.03 c. 6.003 d. 6.3
12) 550 𝑚𝑉 + 3.2 V = _________ V a. 375 b. 37.5 c. 3.75 d. 0.375 13) 12 𝑘 + 6800 = ________ k 14) 1880 15) 188 16) 18.8 17) 1.88 18) 15 𝑀𝑊 + 7500 𝑘𝑊 = __________ MW a. 0.0225 b. 0.225 c. 2.25 d. 22.5
2.7 Comparison of AC and DC
Direct Current. The DC electricity, flows in one direction. The flow is said to be from negative to positive. The normal source of a DC electricity, is the dry cell or storage battery.
Alternating Current. The AC electricity constantly reverses its direction of flow. It is generated by machine called generator. This type of current is universally accepted because of its limited number of applications with the following advantages.
1. It is easily produced. 2. It is cheaper to maintain.
3. It could be transformed into higher voltage.
4. It could be distributed to far distance with low voltage drop. 5. It is more efficient compared with the direct current. Comparison of DC Voltage and AC Voltage
DC Voltage AC Voltage
Fixed Polarity Reverses polarity
Can be steady or vary in magnitude Varies between reversals in polarity Steady value cannot be stepped up or down by
a transformer
Can be stepped up or down for electric power distribution
Easier to measure Easier to amplify
Heating effect is the same for direct or alternating current
AC
Objective Test No. 1
INTRODUCTION TO ELECTRICITY 1. Which of the following is not an electrical quantity?
time power current voltage
2. The unit of current is volt
watt joule ampere
3. The unit of voltage is ohm
volt watt farad
4. The unit of resistance is ohm hertz henry ampere 5. 15,000 W is the same as 15 W 15 mW 15 kW 15 MW
6. The quantity 4.7 x 103 is the same as 0.0047
470 4700 47,000
7. The quantity 56 x 10-3 is the same as 0.056
0.560 560 56,000
8. The number 3,300,000 can be expressed as 3.3 x 10-6
3.3 x 106 3.3 x 109 3.3 x 1012
9. Ten milliamperes can be expressed as 10 A
10 mA 10 kA 10 MA
10. Five thousand volts can be expressed as 5 kV
50 MV 500 kV 5000 kV
11. Twenty million ohms can be expressed as 20
20 m 20 M 20 MW
12. Hertz is the unit of time
power frequency inductance
13. An oscilloscope is usually used to measure rms voltage
average voltage maximum voltage effective voltage 14. A step-down transformer,
lowers both the voltage and current. increases both the voltage and current. lowers the voltage and increases the current. lowers the current and increases the voltage. 15. The prefix pico means
10-15 of a unit 10-12 of a unit 10-9 of a unit 10-6 of a unit
Objective Test No. 2
INTRODUCTION TO ELECTRICITY
Note: Answers of some items may not be found in this text. Look for answers in other references.
1. Discovered the difference between conductors and insulators in 1729. Luigi Galvani
Stephen Gray Alessandro Volta
Gottfried Wilhelm Leibniz
2. Discovered Galvanic action in 1780. Luigi Galvani
Stephen Gray Alessandro Volta
Gottfried Wilhelm Leibniz
3. Invented the electric dry cell in 1800. Luigi Galvani
Stephen Gray Alessandro Volta
Gottfried Wilhelm Leibniz
4. Discovered electromagnetism and invented in 1820. J W Ritter
Luigi Galvani William Herschel Hans Christian Oersted
5. Discovered thermoelectricity in 1821. J W Ritter
T J Seebeck Stephen Gray
Hans Christian Oersted
6. Developed Fourier analysis in 1828. T J Seebeck
Stephen Gray
Hans Christian Oersted Jean-Baptiste-Joseph Fourier
7. Formulated Ohm’s Law in 1826. T J Seebeck
George S. Ohm
Jean-Baptiste-Joseph Fourier
8. Discovered electromagnetic induction in 1831. T J Seebeck
George S. Ohm Michael Faraday
Jean-Baptiste-Joseph Fourier 9. Invented the transformer in 1831.
T J Seebeck George S. Ohm Michael Faraday
Jean-Baptiste-Joseph Fourier
10. Formulated the law of electrolysis in 1834. George S. Ohm
Charles Babbage Michael Faraday
Jean-Baptiste-Joseph Fourier 11. Invented the electric motor in 1837.
Samuel Morse Charles Babbage Michael Faraday Thomas Davenport
12. Invented the magnetohydrodynamic battery in 1839. Samuel Morse
Charles Babbage Michael Faraday Thomas Davenport
13. Discovered the photovoltaic effect in 1839. Samuel Morse
Michael Faraday Edmond Becquerel Thomas Davenport 14. Invented the fuel cell in 1839.
William Grove Michael Faraday Edmond Becquerel Thomas Davenport
15. Invented the differential resistance measurer in 1843. John Herschel
Charles Wheatstone Edmond Becquerel 16. Formulated KCL and KVL in 1845. John Herschel George S. Ohm Gustav Kirchhoff Charles Wheatstone
17. Invented submarine cable insulation in 1847. Charles Wheatstone
Gustav Kirchhoff George S. Ohm Werner Siemens
18. Discovered magnetostriction in 1847. James Prescott Joule
Gustav Kirchhoff George S. Ohm Werner Siemens
19. Invented the lead acid cell in 1860. Plante
Michael Faraday George Boole J P Reis
20. Invented the Leclanche in 1868. Plante
Georges Leclanche James Clerk Maxwell J P Reis
21. Invented the induction motor in 1888. Henry Hunnings
Nikola Tesla
Augustus Desire Waller Thomas Alva Edison 22. Discovered electron in 1897.
Joseph John Thomson Guglielmo Marconi Wilhelm Rontgen Almon Brown Strowger 23. Invented Nickel-iron cell in 1900.
Guglielmo Marconi Thomas Alva Edison
Almon Brown Strowger Joseph John Thomson
24. Invented synchronous induction motor in 1902. Ernst Danielson
Heaviside and Kennely Thomas Alva Edison Max Planck 25. Discovered superconductivity in 1911. Lee De Forest Thadius Cahill Alessandro Artom Kamerlingh Onnes 26. Discovered neutron in 1932. James Chadwick Tellegen and Hoist H S Black Julius Lilienfield 27. Conceptualized FET in 1935. James Chadwick Westinghouse Co. E H Armstrong Oskar Heil
28. Invented Digital Voltmeter in 1952. M V Wilkes J I Nishizawa Yoshire Nakamats Andy Kay 29. Invented LED in 1960. Leo Esaki J I Nishizawa
Allen and Gibbons Andy Kay
30. Invented IC in 1958. G T Wright
Johnson and deLoach Jack Kilby
Direction: Search for names of scientist who contributed to the development of electricity, electrical
components and instruments found in this puzzle. Encircle the name or word vertically, horizontally, backward, upward or downward.
C A B A D E T S R E O N A I T S I R H C S N A H O Q W E H R T T U C Y R Y Q O O N P H A S D F G H J U K H W L Z A L E S S A N D R O V O L T A Z X C V B N M Q V M W E R R L L Y V P M U I P A S D T L U I O P T P A Q S M D F I L E U C G A I J O S E P H H E N R Y H J K Y K L E Z T N E W O D S C C B D F G H J K S L R E S I S T O R T A R A S X J Z R I H E Z X C V B N O I M Q C E D R T Y E U A V D A T I T T A N O P T G I H F J E K J Z X C N V R B N L U M T Q W O E J R Y M U B I M I O P A S D F G H J K S A F K O Z X R L A C V H B N M I R O B E R T B O Y L E F G A R C U M Q F M W R O Y U M L I P O T A S K A S D F O I Y E S H V J A I K P N R E T E M T L O V K L Z A C V R G B L E N W M R N Y O O Z J F T W A X Q E W E R T T Y M I A C R A W B A F N U M I P O U M U T C D R E Z X C V E U N S P Q W E D R R X I T K Y S E M I C O N D U C T O R L W E S I R O A A M P S A S D F S T E P H E N G R A Y G H O M E L Z X Y N C V G B W A T T M E T E R B N M Q W E R R A M L U I O K E O R B N E H C S S U M N A V R E T E P B J A P O G A L S D O F G H J K Q K L M N I B J C K X Z S X J C S V B I B N E P M E N O F B C K F X P L M L R G A C X V B Q N N B M G Q D W E R T Y H U I O P A S F E R M Y A G S D F G H M J W I L L I A M G I L B E R T Y T Y O P O O Z I U Y N G T R E W Q J A S D F G H H J K L E M H D H G C H A R L E S A U G U S T I N D E C O U L O M B T A S D F G H J K L A A W R I T Y U I O P P A S D F M G R A N D R E A M P E R E H G H J Z D Y Q W C E D R T A E I B K L C M N B V E K C I R I U G N O V O T T O B V C X S A H E I N R I C H R U D O L P H H E R T Z A Z X C V B
Unit 3
INTRODUCTION TO ALTERNATING CURRENT AND VOLTAGE
1. identify sine waves and measure their characteristics. 2. explain how frequency and period are related.
3. measure the following voltage or current values of a sine wave: instantaneous, peak, peak-to-peak, rms, and average.
4. define a form factor.
5. describe how sine waves are generated.
6. measure points on a sine wave in terms of angular units. 7. determine the phase angle lead and phase lag.
8. express sine waves with a mathematical formula.
9. discuss phasors and how they can be used to represent sine waves.
10. Apply Ohm’s law and Kirchhoff’s laws to ac circuits as well as to dc circuits.
LEARNING OUTCOMES
At the end of this unit, you are expected to:
alternating current peak value
power transmission peak-to-peak value
electricity rms value
sine wave average value
waveform Faraday's law
cycle electromagnetic induction polarity phase alternation phasor period Ohm's law
frequency Kirchhoff's law
instantaneous value
3.1 The Alternating Current
Alternating current circuits improves the versatility and usefulness of electrical power system. Alternating current plays a
vital role in today’s energy generation �
Once a big controversy ensued between the proponents of the of the DC electricity led by Thomas Edison and the advocates of the AC electricity led by George Westinghouse. According to Thomas Edison,
“The AC electricity is dangerous, because it involves high voltage transmission lines.”
The AC advocates on the other hand, countered that:
“The AC alternation is just like a handsaw which cuts on the upstroke and the down stroke. The high voltage in the transmission line could be reduced to the desired voltage as it passes the distribution line.”
3.2 The Power Transmission
Figure 3-1. Electricity leaves the power plant. (2) Its voltage is increased at a step-up transformer. (3) The electricity travels along a transmission line to the area where power is needed. (4) There, in the substation, voltage is decreased with the help of step-down transformer. (5) Again the transmission lines carry the electricity. (6) Electricity reaches the final consumption points.
More than 90 per cent of the electrical energy used for commercial purposes is generated as alternating current. This is not due primarily to any superiority of alternating over direct current so far as applicability to industrial and domestic uses is concerned. In fact, there are many instances
where direct current is absolutely necessary for industrial purposes, such as municipal traction, electrolytic processes, and certain types of arc lamps; also, direct current motors are superior for elevators, printing presses, and many variable-speed drives. However, for these various purposes the energy is generated and transmitted almost always as alternating current and then converted to direct current.
Some of the reasons for generating electrical energy as alternating current are the following: Alternating current can be generated at comparatively high voltages, and these voltages can
be raised and lowered readily by means of static transformers.
For constant-speed work, the alternating-current induction motor is more efficient than the
direct-current motor and is loess in first cost and in maintenance, owing in part to the fact that the induction motor has no commutator.
A circuit operating at increased voltage, has a lower power loss, power voltage drop, and economically constructed for using smaller copper wires. On transmission and distribution line, power loss is the most important problem to resolve. This is the main reason why Alternating Current (AC) gained more favor and acceptance during the middle part of the 19th century. In the USA, an ordinary house current is described as 120 volts 60 Hertz.
3.3 The Sine Wave
Waveforms are a graphical representation of how voltage or current varies with time. The most common type of a waveform is the sine wave. The sine wave is one very common type of alternating (ac) and alternating voltage. It is also referred to as a sinusoidal wave or, simply, sinusoid. The electrical service provided by the power companies is in the form of sinusoidal voltage and current.
Sine waves are produced by two types of sources: rotating electrical machines (ac generators) and electronic oscillator circuits, which are in instruments known as electronic signal generators.
Figure 3.2. The Sine Wave
-15 -10 -5 0 5 10 15 0 1 2 3 4 5 6 7 v o l t a g e o r c u r r e n t time (sec)
Figure 3-3. Parts of a Sine Wave
3.3.1 Cycle
A cycle is a complete set of positive and negative values.
3.3.2 The Polarity of a Sine Wave
(a) Positive voltage: current direction as shown.
(b) Negative voltage: current reverses direction
Figure 3-4. Alternating current and voltage.
Positive alternation Vs 0 t + Vs _ + Vs _ _ Vs + VS Negative alternation Voltage (V) or
Current (I) Peak (maximum) value
Negative-going zero-crossing Positive-going
3.3.3 The Period of a Sine Wave
The period (T) of a sine wave is the time required to complete one cycle. 3.3.4 The Frequency of a Sine Wave
Frequency is the number of cycles that a sine wave completes in 1 sec. The Unit of Frequency
Frequency (f) is measured in units of hertz, abbreviated Hz. One hertz is equivalent to one cycle per second; 60 Hz is 60 cycles per second; and so on.
3.3.5 Relationship of Frequency and Period
The relationship between frequency and period is very important. The formulas for this relationship are as follows:
f = 1𝑇 ; T = 𝑓1
Example 3.1. The period of a certain sine wave is 10 ms. What is the frequency?
Solution:
f = 1𝑇 = 10 ms 1 = 100 Hz
Example 3.2. The frequency of a sine wave is 60 Hz. What is the period?
Solution:
T = 1𝑓 = 160 Hz 1 = 16.67 ms
Problem Set No. 4
SINE WAVES
1. Describe one cycle of a sine wave.
2. At what point does a sine wave change polarity?
3. How many maximum points does a sine wave have during one cycle?
4. How is the period of a sine wave measured?
5. Define frequency, and state its unit.
6. Determine f when T = 5 s. a. 200 kHz b. 20 kHz c. 2 kHz d. 200 Hz 7. Determine T when f = 120 Hz. a. 8.33 ms b. 6.33 ms c. 4.25 ms d. 2.45 ms
8. A sine wave goes through 5 cycles in 10 s. What is its period? a. 1 s
b. 2 s c. 2 s d. 2 s
9. A sine wave has a frequency of 50 kHz. How many cycles does it complete in 10 ms? a. 200 cycles
b. 400 cycles c. 500 cycles d. 600 cycles
3.4 THE 60-Hz AC POWER LINE
Practically all homes in the Unites States are supplied alternating voltage between 115 and 125 V rms, at a frequency of 60 Hz. This is a sine-wave voltage produced by a rotary generator. The electricity is distributed by high voltage power lines from the generating station and reduced to the lower voltages used in the home. Here the incoming voltage is wired to all the wall outlets and electrical equipment in parallel. The 120-V source of commercial electricity is the 60-Hz power line or the mains, indicating it is the main line for all the parallel branches.
Advantages. The incoming electric service to residences is normally given as 120 V rms. With an rms value of 120 V, the ac power is equivalent to 120-V dc power in heating effect. If the value were higher, there would be more danger of a fatal electric shock. Lower voltages would be less efficient in supplying power.
Higher voltage can supply electric power with less I2R; sine the same power is produced with less
I. Note that the I2R power loss increases as the square of the current. For applications where large
amounts of power are used such as central air-conditioners and clothes dryers, a line voltage of 240 V is often used.
The advantage of ac over dc power is greater efficiency in distribution from the generating station. Alternating voltages can easily be stepped up by means of a transformer, with very little loss, but a transformer cannot operate on direct current. The reason is that a transformer needs the varying magnetic field produced by an ac voltage.
Using a transformer, the alternating voltage at the generating station can be stepped up to values as high as 500 kV for high-voltage distribution lines. These high-voltage lines supply large amounts of power with much less current and less I2R loss, compared with a 120-V line. At the
home, the lower voltage required is supplied by a down transformer. The up and step-down characteristics of a transformer refer to the ration of voltages across the input and output connections.
The frequency of 60 Hz is convenient for commercial ac power. Much lower frequencies would require much bigger transformers because larger windings would be necessary. Also, too low a frequency for alternating current in a lamp would cause the light to flicker. For the opposite case, too high a frequency results in excessive iron-core heating in the transformers because of eddy currents and hysteresis losses. Based on these factors, 60 Hz is the frequency of the ac power line in
the United States. It should be noted that the frequency of the ac power mains in England and most European counters is 50 Hz.
3.5 Voltage and Current Values of a Sine Wave
3.5.1 Instantaneous Value
Figure 3-5 illustrates that any point in time on a sine wave, the voltage (or current) has an instantaneous value. This instantaneous value is different at different point in the curve.
Instantaneous values are positive during the positive alternation and negative during the negative alternation. Instantaneous values of voltage or current are symbolized by lower case v and i, respectively.
Figure 3-5. Example of instantaneous value of a sine voltage.
3.5.2 Peak Value
The peak value of a sine wave is the value of voltage (or current) at the positive or the negative maximum (peaks) with respect to zero. Since the peaks are equal in magnitude, a sine wave is characterized by a single peak value, as is illustrated in Figure 3-6. For a given sine wave, the peak value is constant and is represented by VP or Vm for voltage and
I
P orI
m for current.-10 -8 -6 -4 -2 0 2 4 6 8 10 0 1 2 3 4 5 6 7 V v o l t s time (ms) v2 v1 t1 t2
Figure 3-6. Example of a peak value of a sine wave.
3.5.3 Peak-to-Peak Value
The peak-to-peak value of a sine, as illustrated in Figure 3-6, is the voltage (or current) form the positive peak to negative peak. Of course it is always twice the peak value as expressed in the following equations:
VPP = 2VP
I
PP = 2IP3.5.4 rms Value
The term rms stands for root mean square. It refers to the mathematical procedure which this value is derived. The rms value is also referred to as the effective value. Most ac voltmeters display rms voltage. The 220 V at your wall outlet is an rms value.
The rms value of a sine wave is actually a measure of the heating effect of a sine wave. For an example, when a resistor is connected across an ac (sine wave) source, as shown in Figure 3-7(a), a certain amount of heat is generated by the resistor. Part (b) shows the same resistor connected across a dc voltage source.
-15 -10 -5 0 5 10 15 0 1 2 3 4 5 6 7 v o l t a g e o r c u r r e n t time (sec) Vp Vp-p
Figure 3-7. When the same amount of heat is being produced in both cases, the sine wave has an rms value equal to the DC voltage.
The value of the AC voltage can be adjusted so that resistor gives off the same amount of heat as it does when connected to the DC source.
The rms value of a sine wave is equal to the dc voltage that produces the same amount of heat in a resistance as the sinusoidal voltage.
The peak value of a sine wave can be converted to the corresponding rms value using the following relationships for either voltage or current:
Vrms = Vp 2 Irms = Ip 2
Using these formulas, we can also determine the peak value knowing the rms value, as follows: VP = 2 Vrms
I
p = 2I
rms V 12 V R 30Ω XWM1 V I R 30Ω XWM1 V I E 12 Vrms 60 Hz 0° (a) (b)3.5.5 Average Value
The average value of a sine wave when taken over one complete cycle is always zero, because the positive values (above the zero crossing) offset the negative values (below the zero crossing).
To be useful for comparisons, the average value of a sine wave is defined over a half-cycle rather than over a full cycle. The average value is the total area under the half-cycle divided by the distance of the curve along the horizontal axis. The result is expressed in terms of the peak value as follows for both voltage and current sine waves:
Vavg = 2 V𝑃
I
avg = 2 𝐼
𝑃Problem Set No. 5
VALUES OF SINE WAVE VOLTAGE AND CURRENT 1. Derive the formula for the rms value of current.
2. Derive the formula for the average value of a sine wave in half-cycle. 3. A sine wave has a peak value of 12 V. Determine the following values:
(a) rms (b) peak-to-peak (c) half-cycle average
4. A sinusoidal current has an rms value of 5 mA. Determine the following value: (a) peak (b) half-cycle average (c) peak-to-peak
5. A direct current of 12.5 A flows in a 25-ohm noninductive resistance. Determine maximum value of an alternating current that will produce heat at the same rate in this resistance. a. 2 A
b. 12.5 A c. 17.68 A d. 31.25 A
6. Number 6 AWG underground cable, which supplies a series incandescent lamp system with alternating current is guaranteed to operate safely with 5,000 volts (rms) alternating. If the system were changed to direct current, at what voltage would it be safe to operate the system?
a. 3535.53 V b. 5,000 V c. 7071.07 V d. 10,000 V
3.6 Sine Wave Voltage Sources
A. An AC Generator
An AC generator is a rotating electrical machine that uses the principle of electromagnetic induction. It converts mechanical energy into electrical energy.
Figure 3-8. An elementary generator.
3.6.1 Parts of an Elementary Generator
1. Magnetic Poles – provide the magnetic field
2. Loop of wire – cuts the magnetic field so that emf will be induced. 3. Slip rings – connected to the loop of wire.
4. Brushes – slide with the slip rings to collect the current from the loop of wire. 5. Terminals - connected to the load
3.6.2 Generation of Alternating EMFs.
1
2
3
4
The generation of emf started with the discovery of electromagnetism by Hans Christian Oersted in 1820. He found that when current flows through a coil a magnetic field is established around it as shown in Figure 3-9.
Figure 3-9. Magnetic lines of force around a current-carrying conductor.
After the discovery (by Oersted) that electric current produces a magnetic field, scientists began to search for the converse phenomenon from about 1821 onwards. The problem they put to themselves was how to ‘convert’ magnetism into electricity. It is recorded that Michael Faraday was in the habit of walking about with magnets in his pockets so as to constantly remind him of the problem. After nine years of continuous research and experimentation, he succeeded in producing electricity by ‘converting magnetism’. In 1831, he formulated basic laws underlying the phenomenon of electromagnetic induction (known after his name), upon which is based the operation of most of the commercial apparatus like motors, generators and transformers etc.
3.6.3 Faraday’s Laws of Electromagnetic Induction
Faraday summed up the above facts into two laws known as Faraday’s Laws of Electromagnetic Induction.
First Law. It states that whenever the magnetic flux linked with a circuit changes, an e.m.f. is always induced in it.
or
whenever a conductor cuts magnetic flux, an e.m.f. is induced in that conductor.
Second Law. It states that the magnitude of the induced e.m.f. is equal to the rate of change of flux-linkages.
