Transcript Introduction to Technical Mathematics

01 – DC Theory
01 – DC Theory
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The intent of this presentation is to present enough information to provide the reader with a
fundamental knowledge of Direct Current fundamentals used within Michelin and to better
understand basic system and equipment operations.
By enrolling in this self-study course, you have demonstrated a desire to improve yourself and
Michelin Manufacturing. However, this self-study course is only one part of the total Michelin
training program. Practical experience, IMS (AP) school, selected reading, and your desire to
succeed are also necessary to successfully round out a fully meaningful training program.
Although the words “he,” “him,” and “his” are used sparingly in this course to
enhance communication, they are not intended to be gender driven or to affront or discriminate
against anyone.
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01 - DC Theory
Basic Electrical Theory
Simply put, electricity is nothing more than the flow of electrons through a conductor.
Electric Charge
Since some atoms can lose electrons and other atoms can gain electrons, it is possible to cause a
transfer of electrons from one object to another. When this takes place, the equal distribution of the
positive and negative charges in each object no longer exists. Therefore, one object will contain an
excess number of electrons and its charge must have a negative (-) electric polarity. The other object
will contain an excess number of protons and its charge must have a positive (+) polarity.
When a pair of objects contains the same charge (both are positive or both are negative), the objects are
said to have like charges. When a pair of bodies contains different charges, they are said to have unlike
or opposite charges. The law of electric charges may be stated as follows.
Like charges repel each other and unlike charges attract each other.
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01 - DC Theory
The Coulomb
The magnitude of electric charge a body possesses is determined by the number of electrons compared
with the number of protons within the body. The symbol for the magnitude of the electric charge is Q,
expressed in units of coulomb (C). The charge of one coulomb means a body contains a charge of 6.28
x 1018 more electrons than protons.
The Electrostatic Field
The fundamental characteristic of an electric charge is its ability to exert a force. This force is present
within the electrostatic field surrounding every charged object. When two objects of opposite polarity are
brought near each other, the electrostatic field is concentrated in the area between them. Lines of force
drawn between the two objects indicate the electric field. If an electron is released at point A in this field,
it will be repelled by the negative charge and will be attracted to the positive one. Thus both charges will
tend to move the electron in the direction of the lines of force between the two objects. The arrowheads
in this drawing indicate the direction of motion that would be taken by the electron if it were in different
areas of the electrostatic field.
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01 - DC Theory
Potential Difference
Because of the force of its electrostatic field, an electric charge has the ability to do the work of moving
another charge by attraction and repulsion. The ability of a charge to do work is called its potential.
When one charge is different from the other, there must be a difference in potential between them.
The sum of the differences of potential of all the charges in the electrostatic field is referred to as
electromotive force (emf).
The basic unit of potential difference is the volt (V). The symbol for potential difference is also V,
indicating the ability to do the work of forcing electrons to move. Because the volt unit is used, potential
difference is called voltage.
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01 - DC Theory
Current
The movement or the flow of electrons is called current. To produce current, the electrons must be
moved by a potential difference. Current is represented by the letter symbol I. The basic unit in which
current is measured is the ampere (A). One ampere of current is defined as the movement of one
coulomb (6.28 x 1018 electrons) past any point of a conductor during one second of time.
Current Flow
In a conductor, such as copper wire, the free electrons are charges that can be forced to move with
relative ease by a potential difference. If a potential difference is connected across two ends of a copper
wire, the applied voltage forces the free electrons to move. This current is a drift of electrons from the
point of negative charge moving through the wire, and returning to the positive charge at the other end.
The direction of the electron drift is from the negative side of the source, through the wire, and back to
the positive side of the source. The direction of electron flow is from a point of negative potential to a
point of positive potential.
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01 - DC Theory
Current Summary
Static electricity is dealing with electrons at rest, whereas electrical current deals with electric charges in
motion. As shown before, an electrical current will flow when a conductor joins two points. One point is
more negative than the other point. Hence, as long as there is a conductor to carry them, electrons must
move from negative regions to more positive regions (or less negative regions). As long as there is
some relative difference in the states of the charges and a conduction path, electrons will flow.
Definition:
An electric current is a movement of electrons in a general direction through a conductor.
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01 - DC Theory
The strength or magnitude of the current will be a measure of the number of electrons that pass through
any part of the conductor in one second. The magnitude of a current is measured in AMPERES. When
a conductor is carrying a current of 1 ampere, it means that 6.28 x 10 18 electrons are passing any point
of the conductor in one second. This difference of 6.28 x 10 18 electrons is represented by a coulomb.
Hence, if 1 Ampere flows for 1 second, a quantity of 1 Coulomb of electricity has been used. 10
Coulombs/second is a current of 10 amperes. A current of 30 amperes is equal to: 30coulombs/second.
To find the current flowing in a circuit, the quantity in coulombs is divided by the time in seconds.
1 Ampere = 1 Coulomb / 1 SECOND
Current in AMPERES
=I
Quantity in COULOMBS = Q
Time in SECONDS
= t Therefore,
Q = I x t coulombs
I = Q/t amperes
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01 - DC Theory
The Electric Circuit Concept
1. One of the first requirements for an electrical circuit is a source of electromotive force (emf). Emf is a
force that causes electrons in a circuit to flow and is measured in volts. The Volt may be defined as that
difference in potential which, when applied across a resistance of 1 OHM it will produce a current flow of
1 AMPERE.
2. The second requirement for an electrical circuit is a complete path from one side of the emf source
to the other side of the emf source for current flow. Look at the following circuit:
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01 - DC Theory
The Electric Circuit Concept
The battery is the emf source. The switch and copper wire will provide a complete path for current flow
when the switch is closed. When the switch is closed, the battery supplies energy to electrons in the
copper wire and causes them to flow from the negative side to the positive side of the battery.
S1
V
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01 - DC Theory
Resistance
Resistance is the opposition to current flow. In the same way that friction resists the motion of
mechanical devices and the flow of fluids through pipes, resistance opposes the flow of electrons
through a conductor. In order to maintain a constant flow in a fluid system, pressure must be maintained
to overcome the friction in the system.
COMPARISON OF FLUID AND ELECTRICAL SYSTEMS
FLUID
Pressure (PSI or BARS)
Flow (gallons/min. or liters/min.)
Resistance/Restriction
(Dynamic Pressure Drop)
ELECTRICAL
Potential Difference - (Volts)
Current (Amperes)
Resistance (Ohms)
Resistance is always present in an electrical circuit. To add resistance to a circuit, electrical components
called resistors are used. A resistor is a device whose resistance to current flow is a known or specified
value.
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01 - DC Theory
Resistance
Resistance is measured in ohms () and is represented by the symbol R in equations. One ohm can be
defined, as that amount of resistance that will limit the current in a conductor to one ampere when the
voltage applied to the conductor is one volt. Resistors are common components of many electrical and
electronic circuits. Some frequent uses for resistors are to establish the proper value of circuit voltage,
to limit current, and to provide a load.
Ohm's Law
In the early part of the 19th century Georg Simon Ohm proved by experiment that a precise relationship
exists between current, voltage, and resistance. This relationship is called Ohm's Law and is stated as
follows:
The current in a circuit is DIRECTLY proportional to the applied voltage and
INVERSELY proportional to the circuit resistance.
Ohm's Law written as a mathematical relationship:
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01 - DC Theory
Look at the following circuit:
The current flowing through the circuit is 1 Ampere. Let's see what the current is if we use different
voltages for the battery.
01 – DC Theory
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R
BATTERY
CORRESPONDING CURRENT
10 
5v
0.5 A
10 
20 v
2.0 A
10 
100 v
10.0 A
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01 - DC Theory
Example 1:
For the circuit shown, find the Voltage:
Given: I = 1.5A
R = 10
Ohm’s Law:
Solution: Transpose the equation; V = I * R
(1.5) (10) = 15 Volts
Example 2:
For the circuit shown, find the Resistance:
I = 28 mA
V = 56v
Solution:
I = 56 / 0.028 = 2000  = 2K 
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01 - DC Theory
Example 3:
For the circuit shown, find the Current:
Given: V = 24Volts
R = 10
Ohm’s Law:
Solution:
24 / 100 = .24Amps = 240mA
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01 - DC Theory
Series Circuits
The above circuit is classified as a series circuit.
In this circuit there is only one current path. This distinguishes the series circuit from the parallel circuit.
The battery is the voltage source and is usually referred to as the applied or total voltage of the circuit.
Important Properties
The amount of current flowing in one part of the circuit is equal to the amount flowing in any other part of
the circuit. In other words, the same current flows through R1, R2, and R3.
IT = IR1 = IR2 = IR3
As seen from Ohm's Law, a current flowing through a resistor produces a potential difference across the
terminals of the resistor. (V = I x R) This potential difference is called a Voltage drop.
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01 - DC Theory
Series Circuits
Kirchhoff's Voltage Law
VT = VR1 + VR2 + VR3
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01 - DC Theory
VT = VR1 + VR2 + VR3 (Kirchhoff's Voltage Law for Series Circuits)
IT = IR1 = IR2 = IR3
(Current common in series circuits)
VR1 = R1 IT
VR2 = R2 IT
(Ohm's Law)
VR3 = R3 IT
VT = RT IT
RT IT = (R1 IT) + (R2 IT) + (R3 IT) (Substitute Ohm’s Law into Kirchhoff’s Law)
Divide both sides of the equation by IT
IT cancels and the following equation is the result:
Rule for Series Circuits: Total resistance for any series resistive circuit is the sum of the
individual resistances.
RT = R1 + R2 + R3
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01 - DC Theory
Unit of Power
Power is the rate of doing work, and the unit of electrical power is the watt.
Power is expressed as VOLTS X AMPS or as an equation.
Note that:
P=VxI
I = V  R and V = I X R (Ohm's Law).
By substituting Ohm's Law into the expression P = V x I we get:
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01 - DC Theory
Unit of Power
Example:
A device is connected across a supply of 230V. There is a current flow of 5 AMPS. What is the power
dissipated by the circuit?
P=VxI
P = 230V x 5A = 1150W or 1.15KW
Power for electric motors is sometimes rated in watts or kilowatts, but more usually in Horsepower (hp).
1 hp = 746 watts
Energy and work are essentially the same and are expressed in identical units. Power is different,
however, because it is the time rate of doing work. The Kilowatt-hour (KWh) is the practical commercial
unit of electrical energy or work performed. If a device, as in the above example, were connected to the
supply for a period of ten (10) hours, then 11.5 KWh would be used (1.15 KW x 10 Hrs.).
Work = Power x time
01 – DC Theory
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01 - DC Theory
Practice Exercises
Find the Power PT:
1.
VT = 48v
IT = 2A
PT =
2.
R1 = 10
IT = 8A
PT =
3.
VT = 40v
R1 = 100
PT =
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01 - DC Theory
Practice Exercises
Given:
R1 = 20
R2 = 30
IT = 2A
Find:
RT =
PT =
VT =
VR1 =
PR1 =
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01 - DC Theory
Practice Exercises
Given:
IT = 60 A
R1 = 5 
R2 = 3 
R3 = 9 
R4 = 5 
R5 = 8 
Find:
VT =
VR1 =
VR2 =
VR3 =
VR4 =
PT =
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01 - DC Theory
Practice Exercises
Given:
VT = 210 V
R1 = 6 
R2 = 19 
R3 = 5 
Find:
IT =
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01 - DC Theory
Practice Exercises
Given:
VT = 250 V
IT = 25 A
R1 = 5 
R2 = 2.5 
Find:
RT =
R3 =
State Ohm’s Law: __________
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01 - DC Theory
Resistor Color Code
COLOR
BAND 1, 1st #
BAND 2, 2nd #
BAND 3
MULTIPLIER
BLACK
0
0
BROWN
1
1
1
RED
2
2
ORANGE
3
3
YELLOW
4
4
10,000
GREEN
5
5
100,000
BLUE
6
6
VIOLET
7
7
GRAY
8
8
WHITE
9
9
10
100
1,000
1,000,000
10,000,000
100,000,000
1,000,000,000
0.1
GOLD
0.01
SILVER
NONE
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BAND 4
TOLERANCE
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+
-
5%
+
-
10%
+
-
20%
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01 - DC Theory
The Digital Multi-Meter (DMM)
Digital multi-meters (DMM) have generally replaced the analog-type multi-meter (VOM) as the test
device of choice for maintainers because they are easier to read, are often more compact and have
greater accuracy. The DMM performs all standard VOM measurement functions of AC and DC. Some
offer frequency and temperature measurement. Many have such features as peak-hold display that
provides short-term memory for capturing the peak value of transient signals as well as audible and
visual indications for continuity testing and level detection.
The display shows other functions as well.
• Low battery indicator
• Annunciators show what is being measured (volts, ohms, amps, etc.)
• Autopolarity indicates negative readings with a minus sign when the
• leads are connected incorrectly without any damage
• Autoranging automatically selects proper measurement range
• One selector switch makes it easy to select measurement functions
• Overload protection prevents damage to the meter and the circuiit, and protects the user
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01 - DC Theory
Measuring Resistance
The figure below shows the steps that should be followed when measuring resistance. Remember that
resistance measurements are carried out without the power being applied to the component under test,
and resistance values can vary by as much as 20% due to the tolerance of certain resistors. Do not be
misled if your meter reading is slightly different from the color band on the resistor. If a resistor’s value is
off and exceeds the tolerance, the resistor should be replaced. A resistor will rarely short, but typically
will open. If a resistor does open, the DMM display will flash on and off or display OL (open line)
because the resistor has an infinite resistance.
• Turn off power to the circuit
• Select resistance 
• Plug the black test lead into the COM jack and the red test lead into the  jack
• Connect the probe tips across the component or portion of the circuit for which
you want to determine the resistance
• View the reading and be sure to note the unit of measure: , K, M, etc.
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01 - DC Theory
Measuring Voltage
The figure below shows the steps that should be followed when measuring voltage. The measurement of
both voltage and resistance is where the DMM finds its greatest utilization. For voltage and resistance
measurement, the red lead is inserted into the V – Ω (volt or ohm) meter jack.
Select volts AC (V~), volts DC (V---), milli-volts (V---) as desired
Plug the black test lead into the COM jack and the red test lead into the V jack
Touch the probe tips to the circuit across a load or power source as shown
(parallel to the circuit to be tested) View the reading being sure to note the
unit of measure
Note: For DC readings of the correct polarity (+ or -), touch the red test probe
to the positive side of the circuit, and the black test probe to the negative side
of the circuit ground. If you reverse the connections, a DMM with auto-polarity
will merely display a minus sign indicating negative polarity. With an analog
meter you risk damaging the meter.
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01 - DC Theory
Measuring Current
The figure below shows the steps that should be followed when measuring current. The measurement of
current is rarely performed when troubleshooting, as the circuit path has to be opened to insert the DMM
in series with the current flow. However, if current is to be measured, the red lead is inserted into one of
the ampere jacks, 10 amp (10A) or 300 milliamp (300 mA) input jack depending on the expected value of
the reading.
• Turn off the power to the circuit
• Disconnect, cut or unsolder the circuit, creating a place where the meter probes
can be inserted
• Select amps AC (A~), or amps DC (A---) as desired
• Plug the black test lead into the COM jack and the red test lead into 10 amp
(10A) or 300 milliamp (300mA) jack depending on the expected value of the reading
• Connect the probe tips to the circuit across the break as shown so that all current
will flow through the meter ( a series connection)
• Turn the circuit power back on
• View the reading being sure to note the unit of measure
Note: If test leads are reversed a negative (-) sign will be displayed
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01 - DC Theory
Parallel Circuits
The above circuit is classified as a parallel circuit. In this circuit a total current (IT) leaves the
negative terminal of the battery. When the current reaches the junction labeled (A), it splits into two
branch currents (IR1) and (IR2).
When (IR1) and (IR2) reach the junction labeled B, they join back together as IT and continue on to the
positive side of the battery. Since no current can be lost, the sum of IR1 and IR2 must be equal to IT.
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01 - DC Theory
Parallel Circuits
Important Properties: The voltage across one branch of a parallel circuit is equal to the voltage
across any other parallel branch of the circuit.
VT = VR1 = VR2
The sum of the currents entering a junction must equal the sum of currents leaving the
junction.
Kirchhoff's Current Law
IT = IR1 + IR2
The sum of the currents entering a junction must
equal the sum of currents leaving the junction.
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01 - DC Theory
Parallel Circuits
When two resistors are connected in parallel, the electrons can move more easily. In a parallel circuit
the electrons can move through one branch of the circuit without having to go through the other
branch. The total current flowing through a parallel circuit splits up, some going one way and some
going the other. The branch currents will sum together.
IT = IR1 + IR2
(Kirchhoff's Current Law for Parallel Circuits
VT = VR1 = V R2
(Voltage common in parallel circuits)
IR1 = VT  R1
IR2 = VT  R2
(Ohm's Law)
IT = VT  RT
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01 - DC Theory
Parallel Circuits
(Substitute Ohm’s Law into Kirchhoff’s Current Law)
(Divide both sides of the equation by VT)
And then,
1
1
VT
V
V
(
)= ( T + T ) 
RT V T
R1 R 2 V T
VT now cancels out and the following equation is the result:
Use the reciprocal to get
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01 - DC Theory
Parallel Circuits
Given:
VT = 120 V
R1 = 15 
R2 = 10 
Find:
RT =
IT =
IR1 =
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01 - DC Theory
Parallel Circuits
Given:
IT = 3 A
R1 = 12 
R2 = 15 
R3 = 10 
IT
VT
+
IR1
R1
IR2
R2
IR3
R3
Find:
RT =
VT =
IR1 =
IR2 =
IR3 =
01 – DC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
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01 - DC Theory
Parallel Circuits
Given:
VT = 12 V
R3 = 1 
IT
VT
Find:
PR3 =
01 – DC Theory
Presentation : IMS – Tech Managers Conference
+
Author : IMS Stafff
Author : IMS Staff
IR1
R1
Creation date : 31 Oct 2012
Creation date : 08 March 2012
IR2
R2
Classification : D3
Classification : D3
IR3
R3
Conservation :
Page : ‹#›
01 - DC Theory
Parallel Circuits
Given:
IT = 6 A
R1 = 4 
R2 = 4 
R3 = 4 
IT
VT
+
IR1
R1
IR2
R2
IR3
R3
Find:
RT =
VT =
IR1 =
IR3 =
PR2 =
01 – DC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›
01 - DC Theory
Series and Parallel Circuits
If resistors are arranged in both series and parallel in the same circuit, then solving the parallel
section first and then the entire circuit is the simplest solution. In the circuit shown, R2 and R3 are in
parallel and this combination is in series with R1. We may call the resistance of the parallel section
REQ and find it by the reciprocal method:
01 – DC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
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Classification : D3
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01 - DC Theory
Series and Parallel Circuits
We may then redraw the circuit as shown below, where the branch with resistance REQ has
replaced the parallel section R2 and R3.
Now the resistance of the entire circuit is simply:
RT = R1 + REQ.
Note: REQ = Equivalent resistance for R2 in parallel with R3 and is shown as a broken line.
01 – DC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
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01 - DC Theory
Series and Parallel Circuits
Given:
VT = 150 V
R1 = 6 
R2 = 4 
R3 = 14 
R4 = 16 
Find:
IT =
IR1 =
IR3 =
VR4 =
PR1 =
PR2 =
01 – DC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
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01 - DC Theory
Series and Parallel Circuits
Given:
VT = 30 V
R1 = 10 
R2 = 15 
R3 = 20 
Find:
RT =
IT =
VR1 =
IR3 =
01 – DC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›
01 - DC Theory
Series and Parallel Circuits
Given:
VT = 60 V
R1 = 30 
R2 = 20 
R3 = 10 
R4 = 5 
Find:
RT =
IT =
PR2 =
01 – DC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›
01 - DC Theory
Series and Parallel Circuits
Given:
IT = 1 A
R1 = 2 
R2 = 10 
R4 = 4 
VR2 = 6 V
Find:
R3 =
RT =
VT =
VR4 =
PT =
01 – DC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
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01 - DC Theory
Supplemental Problems
Known: Pt = 3600 W
V1 = 60V
R1
V2 = 20V
R2
V3 = 20 V
R3
P2 = 400W
VT
Calculate :
Rt = ______
01 – DC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
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01 - DC Theory
Supplemental Problems
Known: R1 = 6Ω
I2 = 3A
P3 = 432W
Calculate :
Rt = ______
01 – DC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
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01 - DC Theory
Supplemental Problems
Calculate :
IR1
IR5
IR2
VR4
01 – DC Theory
Presentation : IMS – Tech Managers Conference
Author : IMS Stafff
Author : IMS Staff
Creation date : 31 Oct 2012
Creation date : 08 March 2012
Classification : D3
Classification : D3
Conservation :
Page : ‹#›