Republic of the Philippines

Department of Education
PUBLIC TECHNICAL - VOCATIONAL
HIGH SCHOOLS

GRADE 8
 ELECTRICAL INSTALLATION AND MAINTENANCE NC II

UNIT OF COMPETENCY: FUNDAMENTALS OF ELECTRICITY

MODULE TITLE:
LEARNING O
ANALYZES SIGNS, SYMBOLS, AND DATA

7
At the end of

MODULE NO.:
DEFINITION OF TERMS
Ampere (I) - the unit of electrical current (coulombs per second)

Caution - indicates some precautionary measures against potential hazardous situation which, if not
avoided, may result to a minor or moderate injury

Danger - specifies hazardous situation which, if not avoided, will result to a serious injury or even death

Isometric - a kind of drawing which shows the object in 3 dimensional views

Joule (J) - a metric unit of energy: watt per second. 1 Kw hr = 2,655,000 ft-lb = 1.341 hp-hr = 3413 Btu
= 3,600,000 joules

Kilovolt-ampere (KVA) - a measurement of apparent electric power

Kilowatt hour (Kwhr) - a unit of electrical energy or work performed

 Ohm - the unit of electrical resistance (volts/ampere)

Orthographic - a drawing which shows the front top and side view of the object

Volt (E) - the unit of electric pressure or electromotive force which will produce a current of 1
ampere through a resistance of 1 ohm Watts (W) and kilowatts (KW) - are units of electric power.

COMMON ELECTRICAL SYMBOLS

Electrical Symbols are small drawings or pictograms used to represent various electrical devices in a
diagram or plan of an electrical circuit. These symbols are used in sketching schematic diagrams and electrical
plans for numerous types of electrical works. Practically any electrical fixture found in a house has a symbol that
coincides to said fixture on an electrical wiring diagram. These are very useful guide for an electrician or electrical
contractor, thus, making the wiring easier to install as well.

The following are common electrical symbols used in sketching wiring plan and diagram.

Symbol Description Symbol Description

Conductor/Wire Ammeter
Terminal Voltmeter

Switch Galvanometer

Fuse Wattmeter

Connected Wires Wires Not Connected

Circuit Breaker Push Button

Cell Bell

Battery Buzzer

Resistor Speaker

Capacitor Antenna

Diode Male plug

Ground Service Entrance (3 wires)

Lightning Arrester Duplex Convenience Outlet

Kilowatt-Hour Meter Range Outlet

Power Panel Board Special Purpose Outlet

 Lighting Panel Board Weatherproof Outlet

Incandescent Lamp Floor Outlet

Fluorescent Lamp Single Pole Switch

S1

ELECTRICAL SIGNS

Your power tool with its manual may contain "WARNING ICONS" (a picture symbol intended to alert you to,
and/or to instruct you how to avoid a potentially hazardous condition). Knowing and understanding these symbols
will help you operate your tool better and more safely.
 Electrical signs and stickers alert students, workers, and visitors to electrical hazards in the area. Alerting
workers to high voltage areas, electrical hazards, power lines and other electrical equipment in the area, can help
prevent fires and injuries. Proper electrical signs can inform workers of the dangers in the area.

CAUTION indicates some precautionary measures against

potential hazardous situation which, if not avoided, may result to a
minor or moderate injury.

SAFETY ALERT indicates that a person should observe extra

awareness

PROHIBITION means that any activity is not allowed as stated by

the symbol.

DANGER specifies hazardous situation which, if not avoided, will

result to a serious injury or even death.

WARNING specifies a potentially hazardous situation which, if not

avoided, could result to serious injury or even death.
READ AND UNDERSTAND INSTRUCTION MANUAL means that
a person should make some reading before doing any activity.

WEAR EYE PROTECTION indicates that a person should wear

safety goggles or any related protection for the eyes.

ELECTRICAL HAZARD indicates that electrical hazard is present

in the area.

VOLTAGE DANGER indicates high voltage in the area and

surrounding equipment.
 ELECTRICAL WIRING DIAGRAM

The flow of current in a conductor or wire can be represented by diagram. There are two types of
diagram: pictorial diagram and schematic diagram.

A. Pictorial diagram is a sketch of electrical circuit that shows the external appearance of each
component. It is much like a photograph of the circuit and uses simple images of parts.

Sample Pictorial diagram of one bulb controlled by single pole switch using 9 volt battery source.
B. Schematic diagram is a sketch showing the components of the circuit using standard electrical
symbols. It shows the actual number of components and how the wiring is routed but not the actual
location.

Diagram A Diagram B

Sample schematic diagrams of one bulb controlled by single pole switch using direct current (Diagram
A) and alternating current (Diagram B) source.
Schematic diagram of three bulbs connected in parallel circuit controlled
by a single pole switch.
C. Types of Circuit

1. Series Circuit is a circuit in which lamps are arranged in a chain, so that the current has only one
path to take. The current is the same through each load. Example of this is the Christmas lights. It
consists of a number of bulbs that are connected side by side to meet the voltage requirement
which is 220 volts for alternating current.
Pictorial diagram of Christmas light in series circuit
2. Parallel Circuit is a circuit in which lamps are connected across the wires. The voltage across
each load on parallel circuit is the same. The advantage of using parallel circuit is that even if one
of the lamps fails, still the remaining lamps will function.

Pictorial diagram of two bulbs connected in parallel circuit controlled by a single pole switch.
 ELECTRICAL PLAN

Electrical plan is a graphical presentation of electrical wiring connections to install in a particular

house or building. It indicates the position of electrical fixtures such as convenience outlets, switches,
lightings, door bells, and others to be installed.

Sample electrical plan of Single family dwelling

 Number of electrical fixtures found in the electrical plan:
QUANTITY ELECTRICAL FIXTURES

8 pieces Lamp outlets

8 pieces Duplex convenience outlets

4 pieces Single gang switches

1 piece Two gang-switch

1 piece Three gang-switch

ELEMENTS OF ELECTRICITY AND OHM’S LAW

The first, and perhaps most important, the relationship between

current, voltage, and resistance is called Ohm’s Law, discovered by
Georg Simon Ohm and published in his 1827 paper, The Galvanic
Circuit Investigated Mathematically.

Voltage, Current, and Resistance

An electric circuit is formed when a conductive path is created to
allow electric charge to continuously move. This continuous
movement of electric charge through the conductors of a circuit is
called a current, and it is often referred to in terms of “flow,” just like
the flow of a liquid through a hollow pipe.
The force motivating charge carriers to “flow” in a circuit is
called voltage. Voltage is a specific measure of potential energy that
is always relative between two points.
When we speak of a certain amount of voltage being present in a
circuit, we are referring to the measurement of how
much potential energy exists to move charge carriers from one
particular point in that circuit to another particular point. Without
reference to two particular points, the term “voltage” has no
meaning.
Current tends to move through the conductors with some degree
of friction, or opposition to motion. This opposition to motion is
more properly called resistance. The amount of current in a circuit
depends on the amount of voltage and the amount of resistance in
the circuit to oppose current flow.
Just like voltage, resistance is a quantity relative between two
points. For this reason, the quantities of voltage and resistance are
often stated as being “between” or “across” two points in a circuit.
Units of Measurement: Volt, Amp,
and Ohm
To be able to make meaningful statements about these quantities
in circuits, we need to be able to describe their quantities in the
same way that we might quantify mass, temperature, volume,
length, or any other kind of physical quantity. For mass we might
use the units of “kilogram” or “gram.”
For temperature, we might use degrees Fahrenheit or degrees
Celsius. Here are the standard units of measurement for electrical
current, voltage, and resistance:
The “symbol” given for each quantity is the standard alphabetical
letter used to represent that quantity in an algebraic equation.
Standardized letters like these are common in the disciplines of
physics and engineering and are internationally recognized.
The “unit abbreviation” for each quantity represents the
alphabetical symbol used as a shorthand notation for its particular
unit of measurement. And, yes, that strange-looking “horseshoe”
symbol is the capital Greek letter Ω, just a character in
a foreign alphabet (apologies to any Greek readers here).

Each unit of measurement is named after a famous experimenter in

electricity: The amp after the Frenchman Andre M. Ampere,
the volt after the Italian Alessandro Volta, and the ohm after the
German Georg Simon Ohm.
The mathematical symbol for each quantity is meaningful as well.
The “R” for resistance and the “V” for voltage are both self-
explanatory, whereas “I” for current seems a bit weird. The “I” is
thought to have been meant to represent “Intensity” (of charge
flow), and the other symbol for voltage, “E,” stands for
“Electromotive force.” From what research I’ve been able to do,
there seems to be some dispute over the meaning of “I.”
The symbols “E” and “V” are interchangeable for the most part,
although some texts reserve “E” to represent voltage across a
source (such as a battery or generator) and “V” to represent voltage
across anything else.
All of these symbols are expressed using capital letters, except in
cases where a quantity (especially voltage or current) is described in
terms of a brief period of time (called an “instantaneous” value). For
example, the voltage of a battery, which is stable over a long period
of time, will be symbolized with a capital letter “E,” while the voltage
peak of a lightning strike at the very instant it hits a power line
would most likely be symbolized with a lower-case letter “e” (or
lower-case “v”) to designate that value as being at a single moment
in time.
This same lower-case convention holds true for current as well, the
lower-case letter “i” representing current at some instant in time.
Most direct-current (DC) measurements, however, being stable over
time, will be symbolized with capital letters.

Coulomb and Electric Charge

One foundational unit of electrical measurement often taught in the
beginnings of electronics courses but used infrequently afterward, is
the unit of the coulomb, which is a measure of electric charge
proportional to the number of electrons in an imbalanced state.
One coulomb of charge is equal to 6,250,000,000,000,000,000
electrons.
The symbol for electric charge quantity is the capital letter “Q,” with
the unit of coulombs abbreviated by the capital letter “C.” It so
happens that the unit for current flow, the amp, is equal to 1
coulomb of charge passing by a given point in a circuit in 1 second
of time. Cast in these terms, current is the rate of electric charge
motion through a conductor.

As stated before, voltage is the measure of potential energy per unit

charge available to motivate current flow from one point to
another. Before we can precisely define what a “volt” is, we must
understand how to measure this quantity we call “potential energy.”
The general metric unit for energy of any kind is the joule, equal to
the amount of work performed by a force of 1 newton exerted
through a motion of 1 meter (in the same direction).
In British units, this is slightly less than 3/4 pound of force exerted
over a distance of 1 foot. Put in common terms, it takes about 1
joule of energy to lift a 3/4 pound weight 1 foot off the ground, or
to drag something a distance of 1 foot using a parallel pulling force
of 3/4 pound. Defined in these scientific terms, 1 volt is equal to 1
joule of electric potential energy per (divided by) 1 coulomb of
charge. Thus, a 9-volt battery releases 9 joules of energy for every
coulomb of charge moved through a circuit.

These units and symbols for electrical quantities will become very
important to know as we begin to explore the relationships
between them in circuits.
The Ohm’s Law Equation
Ohm’s principal discovery was that the amount of electric current
through a metal conductor in a circuit is directly proportional to the
voltage impressed across it, for any given temperature. Ohm
expressed his discovery in the form of a simple equation, describing
how voltage, current, and resistance interrelate:
In this algebraic expression, voltage (E) is equal to current (I)
multiplied by resistance (R). Using algebra techniques, we can
manipulate this equation into two variations, solving for I and for R,
respectively:
Analyzing Simple Circuits with Ohm’s
Law
Let’s see how these equations might work to help us analyze simple
circuits:
In the above circuit, there is only one source of voltage (the battery,
on the left) and only one source of resistance to current (the lamp,
on the right). This makes it very easy to apply Ohm’s Law. If we
know the values of any two of the three quantities (voltage, current,
and resistance) in this circuit, we can use Ohm’s Law to determine
the third.
In this first example, we will calculate the amount of current (I) in a
circuit, given values of voltage (E) and resistance (R):
What is the amount of current (I) in this circuit?

In this second example, we will calculate the amount of resistance

(R) in a circuit, given values of voltage (E) and current (I):
What is the amount of resistance (R) offered by the lamp?

In the last example, we will calculate the amount of voltage

supplied by a battery, given values of current (I) and resistance (R):
What is the amount of voltage provided by the battery?

Ohm’s Law Triangle Technique

Ohm’s Law is a very simple and useful tool for analyzing electric
circuits. It is used so often in the study of electricity and electronics
that it needs to be committed to memory by the serious student.
For those who are not yet comfortable with algebra, there’s a trick
to remembering how to solve for anyone quantity, given the other
two.
First, arrange the letters E, I, and R in a triangle like this:
If you know E and I, and wish to determine R, just eliminate R from
the picture and see what’s left:
If you know E and R, and wish to determine I, eliminate I and see
what’s left:
Lastly, if you know I and R, and wish to determine E, eliminate E and
see what’s left:
Eventually, you’ll have to be familiar with algebra to seriously study
electricity and electronics, but this tip can make your first
calculations a little easier to remember. If you are comfortable with
algebra, all you need to do is commit E=IR to memory and derive
the other two formulae from that when you need them!

REVIEW:

 Voltage is measured in volts, symbolized by the letters “E”

or “V”.
 Current is measured in amps, symbolized by the letter “I”.
  Resistance is measured in ohms, symbolized by the letter
“R”.
 Ohm’s Law: E = IR ; I = E/R ; R = E/I

ELECTRON THEORY AND OHM’S

LAW
OBJECTIVES

After studying this unit, the student should be able to

• list the fundamental properties of matter.
• describe the structure of an atom.
 • explain the basic electrical concepts of current, voltage, resistance, and electrical
polarity.
• define Ohm’s law.

MATTER

Anything that occupies space and has weight is called matter. All liquids, gases, and
solids are examples of matter in different forms. Matter is made up of smaller units called
atoms.

ATOMS
An atom resembles the solar system with the sun as the center around which a
series of planets revolve, as shown in Figure 2-1. In the atom, there is a relatively large
mass at the center called the nucleus. Electrons revolve in orbital patterns around the
nucleus.
Figure 2-1 Atomic structure
of Helium.
ELECTRONS

ELECTRON
ORBITS PROTONS
AND NUCLEUS
NEUTRONS

7
 Unit 2 Electron Theory and Ohm’s Law 50

ELECTRICAL CHARGE
A material is said to have an electrical charge when it attracts or repels another
charged material. A material may have either a positive or a negative electrical charge.
Two objects with positive charges repel each other. Two objects with negative charges
also repel each other. Two objects with unlike charges attract each other.

PROTONS AND NEUTRONS

Part of the nucleus of an atom is made up of protons. Each proton has a positive
electrical charge and attracts electrons; neutrons form the remainder of the nucleus.
Neutrons are electrically neutral. They can neither attract nor repel other electrical
charges.
 Unit 2 Electron Theory and Ohm’s Law 51

ELECTRONS

One or more electrons revolve continuously around the nucleus of an atom (just as
the planets revolve about the sun). Electrons possess a negative electrical charge and are
very much lighter in weight than protons. All electrons are alike regardless of the atoms of
which they are a part. An atom contains the same number of electrons as protons. For
example, the aluminum atom has thirteen electrons and thirteen protons.

CURRENT

Electrons in motion result in an electrical current. Copper wire is often used to carry
electrical current (moving electrons). For each atom of copper in the wire, electrons are
revolving around the nucleus. When electrical pressure (voltage) from a battery or
generator is applied, it is possible to force these electrons out of their circular paths and
cause them to pass from atom to atom along the length of the wire (conductor).
 Unit 2 Electron Theory and Ohm’s Law 52

The greater the number of electrons passing a given point in a circuit, the greater the
intensity of the current. The intensity of an electrical current is measured in amperes (A).
The instrument used to measure current is called an ammeter as shown in Figure 2-2. An
ammeter must be connected in series with other devices in a circuit. The letter “I” is used
to represent the amount of current in a circuit.

Current Types

The following three types of current are shown in Figure 2-3:

• Direct current (DC) is the movement of electrons in one direction in a conductor.

• Pulsating direct current is a current in one direction that varies in intensity at a
regular interval of time.
• Alternating current (AC) is a current that changes in direction and intensity at a
regular interval of time.
 Unit 2 Electron Theory and Ohm’s Law 53

Figure 2-2 In-line ammeter.

 Unit 2 Electron Theory and Ohm’s Law 54

Figure 2-3 Types of

electrical current.
CURRENT

0 0 0
TIME TIME TIME

DIRECT CURRENT PULSATING ALTERNATING

DIRECT CURRENT CURRENT

VOLTAGE

Aclosed circuit and a source of electrical pressure are necessary to produce an

electrical current. Electrical pressure, known as voltage, or potential difference, is
 Unit 2 Electron Theory and Ohm’s Law 55

obtained from many sources. Generators are widely used for high-powered AC and DC
installations. Storage batteries are used extensively for DC power in automobiles and
aircraft. Photoelectric cells convert light energy into electrical energy. These cells are used
as voltage sources in light-operated devices. A thermocouple, which consists of a junction
of two unlike metals, generates a low voltage when heated. Of all the voltage sources
mentioned, the generator is most commonly used because of its suitability for commercial
and residential applications.

The letter “E” is used to represent a voltage. The volt (V) is the unit used to express
the quantity of electrical pressure. The instrument used to measure voltage is the
voltmeter. The voltmeter must be connected in parallel with the load to be measured.
 Unit 2 Electron Theory and Ohm’s Law 56

ELECTRICAL POLARITY

All DC sources of electrical pressure have two terminals to which electrical devices
are connected. These terminals have electrical polarity. One terminal is the positive
terminal, whereas the other is the negative terminal. Electrons flow through the device
from the negative terminal of the source to the positive terminal of the source. The
source maintains a supply of electrons on its negative terminal.

RESISTANCE

The property of a material that causes it to oppose the movement of electrons is

called resistance. All materials have some resistance. Materials that offer little resistance
to electron movement are called conductors. Those that offer high resistance are called
nonconductors or insulators.
 Unit 2 Electron Theory and Ohm’s Law 57

Resistance is measured in ohms. The symbol for ohms is the Greek letter omega, Ω.
This symbol, representing ohms, and the letter “R,” representing resistance, are used in
formulas. The instrument used to measure resistance is called an ohmmeter. Electrical
power must be disconnected in a circuit when using an ohmmeter. The meter shown in
Figure 2-4 is commonly used to measure resistance, voltage, and current.
Unit 2 Electron Theory and Ohm’s Law 58
 Unit 2 Electron Theory and Ohm’s Law 59

Figure 2-4 Volt-ohm-milliampere meter. (Courtesy

of Triplett Corp.)
OHM’S LAW

It is extremely important to understand the methods used to control the amount of

current in a circuit. Asimple formula, Ohm’s law, is used to show the relationship of
current, voltage, and resistance. Ohm’s law states that in any electrical circuit, the current
is directly proportional to the voltage applied to the circuit and is inversely proportional to
the resistance in the circuit. Note that both resistance and voltage affect the current.

According to Ohm’s law, when the resistance of a circuit is constant, the current can
be changed by changing the voltage: current will increase when the voltage is increased,
and current will decrease when the voltage is decreased. Similarly, when the voltage is
constant, current will increase when the resistance is decreased, and current will decrease
when resistance is increased.
 Unit 2 Electron Theory and Ohm’s Law 60

The exact relationship of voltage, current, and resistance is expressed by the

equation for Ohm’s law:

I=E
R

Where= I intensity of current in amperes

= quantity of E electrical pressure in volts
= R amount of resistance in ohms Two other
forms of Ohm’s law follow:

E
E =IRand R =
 Unit 2 Electron Theory and Ohm’s Law 61

I
Example: If a voltage of 24 volts appears across a resistance of 4 ohms, find the
current through the resistance.

I = E = 24 volts = 6 amperes
R 4Ω

Example: Find the voltage that appears across an 8-ohm resistance if the current
through it is 10 amperes.

E = IR = (10 amperes) (8 Ω) = 80 volts

 Unit 2 Electron Theory and Ohm’s Law 62

SUMMARY

Ohm’s law is the basic formula for understanding electrical fundamentals. The
relationships among current, voltage, and resistance provide a foundation for
understanding various types of electrical circuits and systems. Current is the movement of
electrons.

Voltage is the electrical pressure that causes the electrons to move. Resistance is a
property of all materials that tends to prevent electrons from moving. The lower the
resistance, the greater the current.

ACHIEVEMENT REVIEW

1. Name the particles that revolve in orbital patterns around the nucleus of an atom.
 Unit 2 Electron Theory and Ohm’s Law 63

___________________________________________________________________
___________________________________________________________________

2. Will a proton attract or repel an electron? _______________________________

3. A current that changes direction and intensity at a regular interval of time is called:
___________________________________________________________________

4. Explain the meaning of voltage, current, and resistance. ____________________

___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
 Unit 2 Electron Theory and Ohm’s Law 64

___________________________________________________________________

5. State Ohm’s law, and write three forms of Ohm’s law using equations.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________

6. What instruments are used to measure voltage, current, and resistance? ________
___________________________________________________________________
___________________________________________________________________
 Unit 2 Electron Theory and Ohm’s Law 65

___________________________________________________________________
___________________________________________________________________
7. What units of measure are used for voltage, current, and resistance? __________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________

8. A trouble light has a resistance of 12 ohms and is rated at 1/2 ampere.

What voltage must be applied to obtain the rated current? ___________________
___________________________________________________________________
___________________________________________________________________

9. What current is taken by a heater with a resistance of 24 ohms when connected to a

120-volt supply? __________________________________________________
 Unit 2 Electron Theory and Ohm’s Law 66

___________________________________________________________________
10. Determine the resistance of a lamp that draws 3 amperes when connected to a
120-volt supply. __________________________________________________
___________________________________________________________________

11. If the lamp in problem 10 is connected to a 240-volt supply, what is the new value of
current? (Assume there is no change in resistance as the temperature of the lamp
changes.) __________________________________________________
___________________________________________________________________
___________________________________________________________________

12. An 8-ohm resistor is connected to a 120-volt circuit. What current will it draw?
___________________________________________________________________
 Unit 2 Electron Theory and Ohm’s Law 67

13. If 60 volts are applied to an 8-ohm resistor, what is the value of current through the
resistor? _______________________________________________________
___________________________________________________________________

14. A toaster is connected to a 120-volt supply and it draws 8 amperes. Find the
resistance. ________________________________________________________
___________________________________________________________________

15. A 5-ohm heater draws 9 amperes from a power supply. What is the voltage of the
power supply? _______________________________________________________
___________________________________________________________________
___________________________________________________________________

16. If the 5-ohm heater in problem 15 is replaced with a 15-ohm heater, what current
will the 15-ohm heater draw from the same power supply? _________________
 Unit 2 Electron Theory and Ohm’s Law 68

___________________________________________________________________
17. What voltage must be applied to a 6.4-ohm lamp filament to develop 20 amperes of
current? ________________________________________________________
___________________________________________________________________
___________________________________________________________________
18. An ammeter placed in a lighting circuit registers a current of 3 amperes. If a
24-volt source has been applied, what is the circuit resistance? ______________
___________________________________________________________________
19. If an ohmmeter measures the resistance of a load as 7 ohms, and a source of
28 volts is applied, what is the current? _________________________________
___________________________________________________________________
 Unit 2 Electron Theory and Ohm’s Law 69

20. If the resistance in a circuit remains constant, what will happen to the current if the
voltage increases? _______________________________________________
___________________________________________________________________
21. If the voltage of a circuit remains constant, what will happen to the current if the
resistance increases? ________________________________________________
___________________________________________________________________
22. What is the term given to anything that has weight and occupies space? _______
___________________________________________________________________
 U•N•I•T 3
SERIES CIRCUITS
OBJECTIVES

After studying this unit, the student should be able to

• describe the basic relationships of voltage, current, and resistance in a series circuit.
• apply Ohm’s law to determine unknown quantities.
Knowing certain basic rules in the operation of series, parallel, and series-parallel
circuits is important in developing a facility for locating faults in electrical equipment.
 Unit 3 Series Circuits 71
Understanding electrical problems is, in VOLTAGE
fact, impossible without this knowledge.
The total voltage applied to a series
A series circuit is one in which devices circuit is distributed across the various
are connected so that there is only one components of the circuit in a series of
path for current. The direction of the voltage drops.
current in the wire is the same as the
The three equal resistors shown in
direction of electron movement. Figure 3-1
Figure 3-2 are connected in series. The
illustrates three lamps connected in a
voltage across each component is equal to
series with a voltage source.
onethird of the total voltage. In Figure 3-3,
the voltage across each resistor is
proportional to the resistance. The higher
the resistance, the greater the voltage drop
in a series circuit.
 Unit 3 Series Circuits 72
L1 L2 L3
1 2 3

1 2 3
I

E
– +
A B

Figure 3-1 Three lamps connected in series.

Therefore,
I = 0.5 ampere

Figure 3-2 Voltage and current distribution:

resistors of equal value in series.

15
 Unit 3 Series Circuits 73

3V 9V

1 2

R1 R2

4 12

12 16
Therefore,
I = 0.75 ampere

Figure 3-3 Voltage and current distribution: Figure 3-4 Digital multimeter. resistors of
unequal value in series. (Courtesy of Advanced Test Products)
 Unit 3 Series Circuits 74
As shown in the previous figures, the sum of the voltages across the individual
devices is equal to the total applied voltage. This leads to the following important rule for
a series circuit:

The sum of the voltage drops across individual resistors in

a series circuit is equal to the total applied voltage. In
other words,
E T = El + E2 + E3 + . . . + En

CURRENT

Because only one path for current exists, the current through all components in the
circuit is the same. This statement can be expressed as

I T = Il = I2 = I3 = In

Where= total current IT

Il
I2
I3
In
 Unit 3 Series Circuits 75
= current through component 1
= current through component 2
= current through component 3
= current through nth component

Figure 3-5 Wire-wound resistor. (Courtesy of PowerRohm Resistors, Inc.)

 Unit 3 Series Circuits 76
RESISTANCE

The total resistance of a series circuit is equal to the sum of resistances of all
resistors in the circuit. The total resistance in Figure 3-1 is the resistance from terminal A
to terminal B with the voltage source disconnected.

In equation form,

R T = Rl + R2 + R3 + . . . + Rn

Where= total circuit RT resistance

= resistance of resistor 1 Rl
R2 = resistance of resistor 2

R3 = resistance of resistor 3
= resistance of nth
Rn
resistor
 Unit 3 Series Circuits 77
Example: The total resistance for Figure 3-3 is RT = R1 + R2.

RT = 4 Ω + 12 Ω = 16 Ω

An alternate path of very low resistance in a circuit is called a short circuit (Figure 3-
6). For example, if the two wires leading to a lamp come in contact with each

other, a path of practically zero open, such as a switch, or malfunctioning,

resistance is formed. When this such as a burned-out fuse or a broken
happens, there is a very large
Lamp Lamp
current in the wires leading to the
place of contact, and the wires will
overheat. Short Open

An open circuit occurs when

some part of a circuit is either
E E
 Unit 3 Series Circuits 78
Short Circuit Open Circuit Figure 3-6 Short circuit and open circuit.
wire. There is no current anywhere in the circuit. However, the source voltage must be
accounted for. If a voltmeter is used at an open point in a circuit, it will indicate the
source voltage.
Example: Find the total resistance, total current, and voltage drops for the circuit
shown in Figure 3-7.

R T = R1 + R2 + R3 2

R2
= 2 Ω + 3 Ω + 7 Ω = 12 Ω
3
R1 R3
ET 240 V 1 3
IT = = = 20 amperes 2 7

RT 12 Ω
ET = 240 V
I T = I1 = I2 = I3
 Unit 3 Series Circuits 79
El = ITR1 = (20)(2) = 40 volts

E2 = ITR2 = (20)(3) = 60 volts E3 = ITR3 Figure 3-7 Sample problem.

= (20)(7) = 140 volts
Note that the sum of the voltage drops is equal to the total voltage.

E l + E2 + E3 = ET
40 + 60 + 140 = 240 volts

Example: Find the total current for the circuit shown in Figure 3-8.

R T = R1 + R2 + R3 I 1
=2+6+2 2

2
= 10 Ω 120E V 6

R3

2
 Unit 3 Series Circuits 80
IT = ET = 120 V = 12 amperes
RT 10 Ω

Figure 3-8 Sample problem.

SUMMARY
A series circuit means that the resistive loads are connected one after another. In
this type of circuit, the current is the same in all parts of the circuit. To determine the
current, the total resistance must first be calculated. The total resistance is the sum of all
the resistances in the circuit. The current is then the supplied voltage divided by the total
resistance.

Rules for a series circuit:

E T = El + E2 + E3 + . . . + En

I T = Il = I2 = I3 = In
 Unit 3 Series Circuits 81
R T = Rl + R2 + R3 + . . . + Rn

ACHIEVEMENT REVIEW

1. Four loads are connected in series across 110 volts DC. The loads fail to operate. A
voltmeter connected in succession across each device reads 0 across the first three
loads and 110 volts across the fourth load. What circuit fault is indicated at the
fourth load?_____________________________________________________
___________________________________________________________________

2. Four loads are connected in series across 120 volts and a 3-ampere current exists.
One load fails to operate. The voltage across each of the other devices is 40 volts.

What circuit fault is indicated? ________________________________________

3. State three characteristics of a series circuit. ______________________________

___________________________________________________________________
 Unit 3 Series Circuits 82
___________________________________________________________________

4. Find the voltage drop across a 10-ohm resistor, if the current through the resistor is
1.7 amperes. _____________________________________________________
___________________________________________________________________

5. Find the resistance of a resistor if the voltage drop across it is 51 volts, and the
current through it is 3 amperes. ________________________________________
___________________________________________________________________

6. Solve for the unknown values in the circuit in Figure 3-9.

R1 = 5 R2 = 10 R3 = 15
RT = ________________
E1 = ___ E2 = ___ E3 = ___
IT = ________________

E1 = ________________
 Unit 3 Series Circuits 83
E2 = ________________ 150 V

E3 = ________________ Figure 3-9 Series circuit.

7. Solve for the unknown values if IT = 10 amperes in Figure 3-10.
E= E = 80 volts
1 2

R1 = 4 R 2 = ___
E1 = ________________

EG = ________________
EG

RT = ________________ Total Voltage

R2 = ________________

Figure 3-10 Series circuit.

8. Find E1 and E2 in the circuit in Figure 3-11.

 Unit 3 Series Circuits 84
Figure 3-11
2
Finding voltages.
2
30
1 E = 128 V
1
10

9. Find ET in Figure 3-12.

Figure 3-12 Finding total voltage. 1
8

E 2
6
I = 3A R3

12
 Unit 3 Series Circuits 85
10. If E2 = 54 volts, find E1 in Figure 3-13.

R2

9
R1
1 R3
3

ET
Figure 3-13 Finding voltage.
11. Using the circuit in problem 10, find E2 if E1 = 6 V.
12. Find E1 and E3 in Figure 3-14.
 Unit 3 Series Circuits 86
R2

I = 4A
R2

E1 R1 R3
E3
2 5

ET

Figure 3-14 Finding voltages.

13. For the circuit in problem 12, find ET if R2 = 4 Ω.

14. For the circuit in problem 12, if IR changes to 6A and R2 is unknown, find E3.
2
 Unit 3 Series Circuits 87
15. In Figure 3-14, find the current through R3 if E1 = 18 volts and IR is unknown.
2

16. In Figure 3-14, if IR is unknown and E3 is 15 volts, find E1.

2
This page intentionally left blank
 U•N•I•T 4
PARALLEL CIRCUITS
OBJECTIVES

After studying this unit, the student should be able to

• describe the characteristics of parallel circuits.

• demonstrate a procedure for solving parallel circuit problems.
Because of their unique characteristics, parallel circuits are more widely used than
any other type of circuit. The distribution of power in a large city is accomplished by a
maze of feeder lines all connected in parallel. A parallel circuit has more than one path for
current.

VOLTAGE

The circuit shown in Figure 4-1 is an

example of a simple parallel circuit. Note that R1 R2 R3
ET
each resistor is placed directly across the main 3 6 8

source of voltage. This causes each resistor to

operate at the same voltage as the source. An
electrical component should never be placed in Figure 4-1 Unequal resistors connected a
parallel circuit if it has a voltage rating less in parallel.

than the source voltage.

 The fact that all components in a parallel circuit operate at the same voltage is
expressed by the following equation:

E T = E1 = E2 = E3 = En

Where= total voltage ET

= voltage across component 1 E1

E2 = voltage across
component 2
E3
= voltage across
En component 3

= voltage across nth component

 23
CURRENT

The components in a parallel circuit operate independently of one another. Each

component takes current in accordance with its resistance. The number of separate paths
for current is equal to the number of components in parallel. The total current in a parallel
circuit is I T = I 1 + I 2 + I 3 . . . + In equal to the sum
of the currents in the
separate components. The equation that expresses
this I T statement follows:
I1
Where= total current
I2
= current through component
I3
1
In

RT will always be less than the smallest R in the

circuit when two or more resistors are present.
 = current through component 2

= current through component 3

= current through nth component

RESISTANCE

It is apparent from studying the previous equation that adding more parallel
branches to the circuit will increase the total current. Ohm’s law (R T = ET/IT) shows that the
total circuit resistance decreases as current increases in parallel circuits. Therefore, adding
parallel branches results in a decrease in total resistance.

Equal Resistors

As seen in Figure 4-3, in a parallel circuit that con-

 Figure 4-2 DC-AC clamp-on
sists of devices with equal resistance, the total circuit ammeter. (Courtesy of Advanced
resistance is numerically equal to the resistance value of Test Products)

ET 15 15 15

Figure 4-3 Equal resistors connected in parallel.

 Unit 4 Parallel Circuits 95

one device divided by the number of devices connected in parallel. Expressed as an

equation, this statement becomes

R 15
RT = = = 5 ohms
N 3
Where= total RT resistance in ohms
= resistance R of one of the equal valued resistors in ohms
N = number of parallel resistors

Unequal Resistance

In practice, parallel circuits with resistors that have unequal values are more
frequently used than parallel circuits with resistors that have equal values. No simple rule
 Unit 4 Parallel Circuits 96

applies in this case because each resistor takes a different value of current for the same
applied voltage.

To find the total resistance of a parallel circuit, apply a known source voltage to the
circuit and determine the total current. Ohm’s law is then used to find the total
resistance.

RT = ET
IT
Where= total circuit resistance RT in ohms
= total voltage in volts ET

IT = total current in
amperes
 Unit 4 Parallel Circuits 97

The total circuit resistance also can be found by using the following formula. This
formula may be applied to any parallel circuit with any number of parallel branches.

Known as the “reciprocal” formula, it is expressed as

1 = 1 + 1 + 1 . . . + 1 Rn
RT R1 R2 R3
 Unit 4 Parallel Circuits 98

Where= total resistance RT

= resistance of resistor 1 R1
R2 = resistance of
resistor 2
R3
= resistance of
Rn resistor 3
= resistance of nth resistance
Example: Find the total resistance of the circuit in Figure 4-1.

1=1+1+1
RT 3 6 8
Lowest common 1=8+4+3
denominator is 24 RT 24 24 24
 Unit 4 Parallel Circuits 99

1 = 8 + 4 +3 = 15
RT 24 24

1 15
= (cross multiply)
RT 24
Solving for RT 15RT = 24
 Unit 4 Parallel Circuits 100

24
RT = == 1.6 ohms
15
An alternate solution to this problem is as follows:
1
RT =
1/3 + 1/6 + 1/8

RT = 1
0.333 + 0.167 + 0.125
1
RT = 1
0.625
RT =1.6 ohms
 Unit 4 Parallel Circuits 101

A simple method of solving circuits consisting of only two resistors in parallel (with
either equal or unequal values) is called the “product over the sum” method.
Example: A 3-ohm resistor and a 6-ohm resistor are connected in parallel.
Determine their combined resistance.

RT = R1 × R2 = 3 × 6 = 18 = 2 ohms
R 1 + R2 3+ 6 9
Example: For the circuit in Figure 4-4, find the total current and the current in R 2.

E R1
I1 I2 R2
24 V 6Ω
4Ω
IT
 Unit 4 Parallel Circuits 102

Figure 4-4 Sample problem.

RT = R1 × R2 = 6 × 4 = 24 = 2.4 Ω
R 1 + R2 6+ 4 10

IT = ET = 24 = 10 amperes
RT 2.4

E T = E1 = E2

E2 24
I2 = = = 6 amperes
R2 4
Note: IT may also be found by adding the currents I1 and I2.
 Unit 4 Parallel Circuits 103

E1 24
Find I1: I1 = = = 4 amperes
R2 6

Therefore, IT = I1 + I2 = 4 + 6 = 10 amperes.

Example: Find IT in the circuit shown in Figure 4-5.

I

E
R1 R2 R3
120 V
6Ω 12 Ω 16 Ω

Figure 4-5 Sample problem.

 Unit 4 Parallel Circuits 104

1=1+1+1
RT R1 R2 R3

1
1 = ++
RT 6

1=1× 8+1× 4+1× 3

RT 6 8 12 4 16 3
Lowest common

=
denominator is 48 ++

Cross 1 = 15
 Unit 4 Parallel Circuits 105

RT 48
multiply

15RT = 48
 Unit 4 Parallel Circuits 106

RT = = 3.2 Ω

IT = ET = 120 = 37.5 A
RT 3.2
SUMMARY
A parallel circuit has branches of resistance. The voltage is the same across each
branch, but the current may not be the same in each branch. The current is determined
by the amount of resistance in the branch. If the branch currents are added together, the
sum is the total current.

Rules for parallel circuits:

ET = E1 = E2 = E3 . . . = En

I T = I 1 + I 2 + I 3 . . . + In

1=1+1+1...+1
RT R1 R2 R3 Rn

ACHIEVEMENT REVIEW

1. Four 12-ohm resistors are connected in parallel. Calculate the total circuit resistance.
_______________________________________________________
___________________________________________________________________

2. Four resistors are connected in parallel. The resistance values are 4 ohms, 8 ohms, 12
ohms, and 16 ohms. Calculate the total circuit resistance.
___________________________________________________________________
___________________________________________________________________

3. The resistors mentioned in problem 2 are connected in parallel across a 120-volt DC

supply.
a. Calculate the current through each resistor.
b. Find the total current.

___________________________________________________________________
 Unit 4 Parallel Circuits 107

c. Find the total circuit resistance.

___________________________________________________________________
___________________________________________________________________
___________________________________________________________________

4. Determine the total resistance of a 10-ohm resistor and a 30-ohm resistor connected
in parallel. ________________________________________________________
___________________________________________________________________
5. If the circuit in problem 4 is connected to a 150-volt supply, find the current through
each resistor. ________________________________________________________
___________________________________________________________________
6. Find the total voltage, ET, for the circuit shown in Figure 4-6.

Figure 4-6 Finding 60

total voltages.

40

3A

___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
7. Find the current through R3 in the circuit shown in Figure 4-7.

___________________________________________________________________
 Unit 4 Parallel Circuits 108

15 A
Figure 4-7 Finding current.
3A 6A ?

R1 R2 R3
E
10 Ω 5Ω

___________________________________________________________________
___________________________________________________________________
8. For the circuit in problem 7, what is the value of R 3? _______________________
___________________________________________________________________
9. Find the value of R2 for the circuit shown in Figure 4-8 if the total circuit resistance is
7.5 ohms.

Figure 4-8
Finding resistance. E
75 volts
R1 R2 = _
10

___________________________________________________________________
___________________________________________________________________
___________________________________________________________________

10. What is the total current in problem 9? __________________________________

___________________________________________________________________

11. The ammeters in the circuit in Figure 4-9 are indicating 4 amperes and 9 amperes as
shown. Find the values of R3 and RT.

___________________________________________________________________
 Unit 4 Parallel Circuits 109

Figure 4-9 9A
Finding resistance.
E 4A
R = Ω R2 = 60 Ω R3 = Ω

R1 = 30 Ω

___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________

12. For problem 11, what is the total voltage, E? _____________________________

___________________________________________________________________
 Unit 4 Parallel Circuits 110
13. Find IT for the circuit shown in Figure 4-10.

10
Figure 4-10 Ω

Finding current.

100 V ET 20 Ω 20 Ω

IT

_____________________________________________________________________________
_
_____________________________________________________________________________
_
 _____________________________________________________________________________
_
_____________________________________________________________________________
_ 14. Using the circuit in Figure 4-10, what is the current through the 10-ohm resistor?

_____________________________________________________________________________
_
_____________________________________________________________________________
_

15. In Figure 4-10, if the 10-ohm resistor is changed to 20-ohms, and E T is changed to 120
volts, find IT.
_____________________________________________________________________________
_
_____________________________________________________________________________
_
16. In Figure 4-10, if there is a break in the 10-ohm resistor causing an “open circuit”to
occur in the 10-ohm branch, what will be the total current, IT?
_____________________________________________________________________________
_
_____________________________________________________________________________
_
This page intentionally left blank
 U•N•I•T 5
SERIES-PARALLEL CIRCUITS
OBJECTIVES

After studying this unit, the student should be able to

• explain the characteristics of series-parallel circuits.

• demonstrate a procedure for solving problems involving series-parallel circuits.
Combining series and parallel circuits is often necessary to meet electrical
requirements and to group devices in a load circuit to obtain a particular value of
resistance.
Grouping devices in series-parallel Figure 5-2 illustrates another
circuits is also necessary in control series-parallel circuit. Resistors R1 and R2
circuits for auditorium and stage are in parallel with respect to each
lighting as well as for motor control. In other. Resistors R3 and R4 constitute
many instances, it is desirable to group another parallel combination. The
voltage sources, particularly batteries, parallel combination of R1 and R2 is in
to obtain the correct voltage and series with the parallel combination of
current capacity. R3 and R4.

The circuit shown in Figure 5-1 is In Figure 5-3, the resistors are
an example of a series-parallel circuit. In grouped in another type of series-
this circuit, lamps L1 and L2 constitute a parallel
parallel circuit. The rheostat R, used to
control the current in this circuit, is in
series with L1 and L2 as a group.
 R1 R3
R
3 6

E
L1 L2 R2 R4
9 12

Figure 5-1 A series-parallel circuit.

R1 R3 E

Figure 5-3 A series-parallel circuit.

R2 R4
33

Figure 5-2 A series-parallel circuit.

 Unit 5 Series-Parallel Circuits 116

circuit. In this circuit, R1 and R3 are in series, and R2 and R4 are in series. The two series
branches are then in parallel.

EQUIVALENT CIRCUITS

The methods used to determine current, voltage, and resistance for series and
parallel circuits apply to combination circuits as well. Solving problems in series-parallel
circuits is made easier by resolving these circuits into equivalent circuits.

Figure 5-4 is equivalent to Figure 5-3. In this case, R1 and R3 are combined as a single
resistance RA, equal in value to the sum of R1 and R3. Similarly, RB replaces R2 and R4. RA and
RB then may be combined into one resistor, RC, to result in the final equivalent circuit of
Figure 5-5. The total current in the original series parallel circuit, Figure 5-3, is equal to the
current in the simple series circuit of Figure 5-5.
 Unit 5 Series-Parallel Circuits 117

RA
R
9 C
6.3

RB
21

E E

Figure 5-4 Equivalent circuit. Figure 5-5 Equivalent circuit.

CIRCUIT SOLUTION

After the total resistance of a circuit is found, the total current, as well as the current
in other parts of the circuit, can be determined according to Ohm’s law. In Figure 5-6, the
equivalent resistance of the parallel resistors R2 and R3 is 12 ohms.
 Unit 5 Series-Parallel Circuits 118

Therefore, Figure 5-7 is the series circuit equivalent of Figure 5-6, and the total
R1 R1
8 8

E
E 120 V
R2 R3 R 2, 3
120 V
20 30 I 12
T
6A

Figure 5-6 A series-parallel circuit. Figure 5-7 Equivalent circuit.

resistance is as follows:

R T = R 1 + R 2× R 3
R 2+ R 3
 Unit 5 Series-Parallel Circuits 119

RT = 8+ 20 × 30
20 + 30

RT = 8+ = 8 12+

RT = 20 Ω

120 volts
The total current is IT = = 6 amperes 20 ohms

The voltage across R2,3 is IT × R2,3 = 6 amperes × 12 ohms = 72 volts. Because R2,3 is the
equivalent resistance of the parallel combination of R2 and R3, the voltage across these
resistors is 72 volts, as shown in Figure 5-8. Finally, the current through
R2 is R1
8 I3

I
T
E 6A
120 V I2 R2 R3
72 V
20 30
 Unit 5 Series-Parallel Circuits 120

I2 = 72 volts = 3.6 amperes 20 ohms

and the current through R3 is

I3 = 72 volts = 2.4 amperes

30 ohms
Figure 5-8 Circuit problem.

Example: Find the total current (IT) in the circuit shown in Figure 5-9.
Figure 5-9 Sample problem. 8 12

8 12

120 V
2
 Unit 5 Series-Parallel Circuits 121

RT = 2+ 8×8 + 12 ×12
8+8 12+12

RT = 2 + 4 + 6 = 12 Ω

IT = ET = 120 = 10 A
RT 12
SUMMARY

In a simple series-parallel circuit, the total currrent is equal to the sum of the branch
currents. This current passes through the resistances that are in series with the voltage
source. The total current may also be computed by changing the series-parallel circuit into
a series circuit. The resistances of the branches may be converted into a single resistance.
 Unit 5 Series-Parallel Circuits 122

This resistance is then in series with the other resistances in the circuit, and the total
resistance is the sum. By using Ohm’s law, the total current can be calculated.

ACHIEVEMENT REVIEW
1. a. In Figure 5-6, what circuit components are connected in series?
____________________________________________________________________
____________________________________________________________________
b.What circuit components are in parallel with each other?

____________________________________________________________________
____________________________________________________________________
 Unit 5 Series-Parallel Circuits 123

2. Assume that each resistor shown in Figure 5-2 has a resistance of 100 ohms. Findthe
total circuit resistance. _____________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________ 3.
a. In Figure 5-10, what circuit components are connected in series?

____________________________________________________________________
____________________________________________________________________
Figure 5-10 R1 R2
 Unit 5 Series-Parallel Circuits 124

R3 R4

EG
Series and parallel.
 Unit 5 Series-Parallel Circuits 125

b.What components are connected in parallel?

____________________________________________________________________
____________________________________________________________________ 4.
Determine the total current in the circuit in Figure 5-11.

____________________________________________________________________
____________________________________________________________________
____________________________________________________________________

Figure 5-11
8 3
Finding total current.
120 V
6

5. Find the current through the 6-ohm resistor for the circuit used in problem 4.
____________________________________________________________________
____________________________________________________________________
6. Determine the total resistance of the circuit in Figure 5-12 between points A and B.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________

Figure 5-12 B
10
Finding total resistance.
5 7

4 12

____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
 Unit 5 Series-Parallel Circuits 126

7. If 120 volts are connected across points A and B in the circuit shown in problem
6,what is the current through the 4-ohm resistor?
8. Five 4-ohm resistors are connected so that their combined resistance will equal5
ohms. Draw the circuit diagram.

9. The two resistors in branch A-B of the circuit in Figure 5-13 are of equal value.
Figure 5-13A 8A
Finding resistance.
R

240 V 120

What is the value of each resistor if the ammeter indicates 8 amperes?

____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
10. For the circuit in problem 9 (Figure 5-13), what is the voltage across one of theR
resistors? _______________________________________________________
____________________________________________________________________
____________________________________________________________________

11. If IT = 8 amperes in the circuit in Figure 5-14, find Eg and I3.

____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
 Unit 5 Series-Parallel Circuits 127

Figure 5-14 Finding 10

voltage and current. I2 I3

IT

Eg 45 V 15 9

____________________________________________________________________

____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
 Unit 5 Series-Parallel Circuits 128

12. For the circuit in problem 11 (Figure 5-14), if the voltage across the parallelbranches
is changed from 45 volts to 90 volts, find the total current, IT.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
13. Draw the series equivalent circuit diagram for the circuit in problem 11 (Figure 5-14).

14. Find the total current for the circuit shown in Figure 5-15.

Figure 5-15 6 8
Finding total current.

4 12

72 V

IT

____________________________________________________________________
____________________________________________________________________
____________________________________________________________________

____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
 Unit 5 Series-Parallel Circuits 129

15. Using the circuit in problem 14 (Figure 5-15), find the voltage across the 4-
ohmresistor. ___________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
16. What is the value of the voltage across the 8-ohm resistor in problem 14?

____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
 Unit 5 Series-Parallel Circuits
130
17. Find the current through the 6-ohm resistor in the circuit in problem 14.
__________________________________________________________________
__
__________________________________________________________________
__
__________________________________________________________________
__
18. What is the value of current through the 12-ohm resistor in the problem 14
circuit diagram?
__________________________________________________________________
__
__________________________________________________________________
__
__________________________________________________________________
__

____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
 Unit 5 Series-Parallel Circuits
131

____________________________________________________________________
____________________________________________________________________
____________________________________________________________________