EGR 2201
Circuit Analysis
Professor Nick Reeder
Reminders



Please turn off cell phones.
No food or soft drinks in the
classroom.
Stow water bottles at floor level.
EGR 2201 Unit 1
Basic Concepts



Read Alexander & Sadiku, Chapter 1.
Homework #1 and Lab #1 due next
week.
Quiz next week.
What This Course Is About
In this course you’ll learn
mathematical techniques for
studying electric circuits.
Our focus is not on practical circuits
that do interesting things.



You’ll study those in later courses,
using the techniques that you learn in
this course.
The Math That We’ll Use
Calculator or Math Software
Some of the math we’ll do is timeconsuming with a basic calculator.


It’s faster with a powerful calculator that
can solve systems of linear equations
and manipulate complex numbers.





Examples: TI-30 or Casio fx-115
Examples: TI-86 or TI-89
You can use any calculator on exams, but
no cell phones.
Recommendation: Learn one calculator and
use it for all homework and exams.
Another option: Use MATLAB software,
which you may have used in other courses.
Calculator or Math Software (2)


Be aware of the calculator policy for the
Fundamentals of Engineering exam and the
Principles and Practice of Engineering exam,
administered by the National Council of
Examiners for Engineering and Surveying
(NCEES).
In a few years you may decide to take
these exams for professional advancement.
What is a Circuit?
Our book’s definition (page 4):

An electric circuit is an interconnection of
electrical elements.
Five electrical elements that we’ll
focus on:






Resistors
Capacitors
Inductors
Voltage Sources
Current Sources
Example Circuit: A Power Supply
from a Flat-Screen Television
Resistor
Inductor
Capacitors
Schematic Diagrams


To discuss circuits, we draw
schematic diagrams that represent
those circuits.
Schematic diagrams do not show the
parts of the circuit as they actually
look. Instead, they contain standard
symbols that represent electrical
elements.
Example Schematic Diagram: A
Radio Transmitter (from book’s page 4)
Resistor Symbol
Inductor Symbol
Capacitor Symbol
A Simpler Example Schematic
Diagram: Flashlight
Switch
Light Bulb
Battery (Voltage Source)


When the switch is open (as drawn), no current
flows, so the bulb is dark.
When the switch is closed, current flows, and the
bulb lights.
Another Simple Example: A
Voltage Source And Two Resistors
Polarity of a Battery

Note that the symbol for a battery is
asymmetric. The end with the longer line
represents the battery’s positive terminal,
and the other end represents its negative
terminal.
Positive terminal
+

Negative terminal
Direction of Current Flow


For historical reasons, we say that in our
simple circuit current flows out of the
battery’s positive terminal and into its
negative terminal.
Modern science tells us that electrons actually
move in the opposite direction, but we’ll
follow the standard convention shown above.
Element Ratings

The schematic diagrams so far have
been incomplete.


They show what kind of elements are in
the circuit and how those elements are
connected to each other.
But they do not show numerical ratings
that let us quantify the circuit’s behavior.
Every voltage source has a numerical rating
in volts (V).
 Every resistor has a numerical rating in
ohms ().

Examples of Voltage Sources

What is the rating of these sources?

Flashlight battery ____ V

Wall outlet ____ V

But the battery is a DC voltage source,
while the outlet is an AC voltage source.
DC Versus AC

In a direct-current (DC) circuit,
current flows in one direction only.


The textbook’s Chapters 1 through 8
cover DC circuits.
In an alternating-current (AC)
circuit, current periodically reverses
direction.

The book’s Chapters 9 through 11
cover AC circuits.
Schematic Symbols for
Independent Voltage Sources

Several different symbols are
commonly used for voltage sources:
Type of Voltage
Source
Generic voltage source
(may be DC or AC)
DC voltage source
AC voltage source
Symbol Used in Our Symbol Used in
Textbook
Multisim Software
V or v?



Some authors use uppercase letters
for constant quantities, such as V for
the voltage of a constant DC voltage
source.
And they use lowercase letters for
time-varying quantities, such as v for
the voltage of an AC voltage source.
Our textbook mentions this
convention on pages 7 and 10, but
usually uses lowercase letters for both
constant and time-varying quantities.
DC Voltage Sources on Our
Trainer
Fixed +5 V voltage source
No matter which red
socket you use, you
must also use the
GROUND socket.
Fixed -5 V voltage source
Variable positive
voltage source,
controlled by the lefthand knob. We’ll
usually use this one.
Variable negative voltage source, controlled by the right-hand knob.
Using a Digital Multimeter to
Measure Voltage


We’ll use a digital multimeter, like the
Fluke 45 shown, to measure voltage.
Note that the meter has a red lead
and a black lead. See next slide ….
Meter’s Red and Black Leads

When you measure a voltage, the
order of the red and black leads
determines whether the value is
displayed as positive or negative.
Meter will display 5.00 V
Meter will display 5.00 V
Resistance




Resistance is opposition to the flow of
electrons.
Resistance’s unit of measure is the
ohm ().
A perfect conductor would have zero
resistance and a perfect insulator
would have infinite resistance.
A resistor is a device manufactured
to have a specific amount of
resistance.
Resistor Ratings


The resistors in our labs range in
value from 10  to 10,000,000 .
Instead of having the value printed in
numbers on the case, our resistors are
marked with a four-band color code to
indicate the value.
Resistor Color Code

The first three color bands specify
the resistance’s nominal value.
Digit
Color
0
Black
1
Brown
2
Red
3
Orange
4
Yellow
5
Green
6
Blue
7
Violet
8
Gray
9
White
Resistor Color Code (2)



The fourth band (“tolerance band”)
gives the percent variation from the
nominal value that the actual
resistance may have.
Tolerance
Color
5%
Gold
10%
Silver
20%
None
Many websites have color-code charts
and calculators, such as this one.
Tolerance Calculations

To find a resistor’s tolerance in
ohms, multiply its nominal value by
the percentage tolerance.
Example: For a 220  resistor with
5% tolerance, the tolerance in ohms
is
220   0.05 = 11 .

Then…

Tolerance Calculations (2)




To find the minimum value that a resistor
can have, subtract its tolerance in ohms
from its nominal value.
In example above, the nominal value was
220  and the tolerance was 11 . So the
minimum value is
220   11  = 209 .
To find the maximum value that a resistor
can have, add its tolerance in ohms to its
nominal value.
In example above, the maximum value is
220  + 11  = 231 .
Using a Digital Multimeter to
Measure Resistance


Digital multimeters can measure
resistance as well as voltage.
When measuring a resistor’s
resistance, the resistor must be out of
circuit: definitely no power applied
and disconnected from other
elements.
Selecting the Measurement Type
on the Digital Multimeter
DC Voltage
AC Voltage
DC Current
AC Current
Resistance
Plugging the Meter’s Leads into
the Jacks
Red lead here
to measure
voltage or
resistance.
Black lead
always in this
jack.
Red lead here to
measure current.
Same Circuit Layout, but Different
Element Ratings

These two circuits will perform
differently. In particular, the different
element ratings will result in:


Different current values
Different voltage values
Current




Current is the flow of electric charge
through a circuit.
We use the symbol I or i to represent
current.
Current’s unit of measure is the
ampere, or amp (A).
For example,

To say that a current is 2.5 amperes, we
write
I = 2.5 A
or
i = 2.5 A
Voltage




Voltage is a measure of how
forcefully charge is being pushed
through a circuit.
We use the symbol V or v to represent
voltage.
Voltage’s unit of measure is the volt
(V).
For example,

To say that a voltage is 5 volts, we write
V=5V
or
v=5V
Summary of Some Electrical
Quantities, Units, and Symbols

Quantity
Symbol
SI Unit
Symbol for
the Unit
Current
I or i
ampere
A
Voltage
V or v
volt
V
Resistance
R
ohm

Plumbing Analogy




It may help to think of a circuit as
being like a plumbing system, with
water flowing through pipes.
On this analogy, voltage is like the
water pressure in a pipe. Its value will
be different at different points in the
circuit.
Current is like the volumetric flow rate
through a pipe.
See Wikipedia article on Hydraulic analogy.
Plumbing Analogy in Our Simple
Circuit
A wire is like a water pipe. The amount of
electricity per second flowing through a wire is the
current, which is measured in amperes.
The voltage
(pressure)
at this point
is greater than
the voltage
at this point.
A voltage source is like
a water pump. Its
voltage rating (in volts)
tells you how strong it is.
Resistors are like partial blockages
in the pipe. They restrict the amount
of current that flows through the circuit.
The Goal of Circuit Analysis



This course’s main goal: to learn how,
given the schematic diagram of a
circuit, to compute the voltages and
currents in the circuit.
For some circuits, such
as this one, the math is
simple (basic algebra).
More complicated circuits require more
powerful math (trig, complex numbers,
calculus, differential equations…).
Large and Small Numbers



We must often deal with very large or
very small numbers.
Example: a resistor might have a
resistance of 680,000  and a current
of 0.000145 A.
It’s not convenient to use so many
zeroes when writing or discussing
numbers. Instead we use SI prefixes
(or engineering prefixes), which are
abbreviations for certain powers of 10.
Table 1.2

1,000,000,000,000
1,000,000,000
1,000,000
1,000
We rarely
use these.
1 / 1,000
1 / 1,000,000
1 / 1,000,000,000
1 / 1,000,000,000,000

Engineering Prefix Game

You must memorize these prefixes.

To practice, play my Metric Prefix
matching game at
http://people.sinclair.edu/nickreeder/flashgames.htm.

You must also be able to convert
between numbers written with
engineering prefixes and numbers
written in everyday (floating-point)
notation.

To practice, play my EngineeringNotation game.
Using Engineering Prefixes


Whenever you have a number that’s
greater than 1000 or less than 1,
you should use these prefixes.
Examples:


Instead of writing 680,000 ,
write 680 k
(pronounced “680 kilohms”).
Instead of writing 0.000145 A,
write 145 A
(pronounced “145 microamps”).
Calculator’s Exponent Key


Scientific calculators have an
exponent key (usually labeled EE,
EXP, or E) that lets you easily enter
numbers with engineering prefixes.
Examples:


To enter 680 k, press 680 EE 3.
To enter 145 , press 145 EE −6.
Calculator’s Engineering Mode


Most scientific calculators also have
an engineering mode, which forces
the answer always to be displayed
with one of the engineering powers
of 10.
Learn how to use this feature of
your calculator. It will save you
from making mistakes.
Measuring Voltage



A voltmeter is an instrument designed to
measure voltage (also called potential
difference).
Voltage measurements
are always made
across elements.
To measure a
voltage in a circuit,
you don’t need to
disconnect any
Measuring the voltage
across R1.
elements.
Positive or Negative Voltage?




When you measure a voltage, the displayed
value may be positive or negative.
In the drawing, the meter’s
+ lead is connected
to point a and its
a
– lead to point b.
To indicate this, we
would say that we’re
b
measuring vab.
If we swapped the leads, we’d be measuring vba.
These two voltages, vab and vba, have the same
magnitude but different signs.

Example: If vab = 1.60 V, then vba must be 1.60 V.
Voltage Drops and Rises



If vab = 1.60 V, we
say that there’s a
voltage drop of
1.60 V from
point a to point b.
Equivalently, we say
that there’s a
voltage rise of
1.60 V from point b to point a.
a
b
Though it may seem confusing, we could also say
that there’s a voltage rise of 1.60 V from point a
to point b, or that there’s a voltage drop of 1.60 V
from point b to point a.
Measuring Current


An ammeter is an instrument designed to
measure current.
To measure the current
at a point, you must
break the circuit at
that point and
insert the ammeter
in such a way that
the current flows
through the ammeter.
Measuring current.
Positive or Negative Current?



When you measure a current, the displayed
value may be positive or negative.
Note that in the
drawing, the meter’s
+ lead is connected to
the battery and its –
lead to R1.
The displayed
value is the current
flowing into the + lead
and out of the – lead.
Positive or Negative Current? (2)

As with voltage measurements, swapping
the leads would give the same magnitude
but opposite sign.


Example: If the meter displays 34.0 mA when
connected as shown, then it would
display 34.0 mA if you swapped
the leads.
We could express this by
saying either that
a current of 34.0 mA flows
from V1 to R1 (clockwise),
or that a current of
34.0 mA flows from
R1 to V1 (counter-clockwise).
Measuring Resistance


An ohmmeter is an instrument designed to
measure resistance.
To measure
an element’s
resistance, you
must remove
the element
from the
circuit.
Measuring R1’s resistance.

When measuring resistance, the meter will
never display a negative value.
Multimeter


A multimeter can measure voltage,
current, or resistance, depending on
the setting of a selector switch.
A multimeter must not be set to
measure current when it is connected
as a voltmeter, or set to measure
voltage when it is connected as an
ammeter.
Multimeter Challenge Game


You must learn how to use a
multimeter.
To learn the basics, play my
Multimeter Challenge game at
http://people.sinclair.edu/nickreeder/flashgames.htm.
Some Quantities and Their Units



Three that we have discussed:
Quantity
Symbol
SI Unit
Symbol for
the Unit
Current
I or i
ampere
A
Voltage
V or v
volt
V
Resistance
R
ohm

Four new ones:
Quantity
Symbol
SI Unit
Symbol for
the Unit
Charge
Q or q
coulomb
C
Time
t
second
s
Energy
W or w
joule
J
Power
P or p
watt
W
Charge





All electrical phenomena are based on the
movement or separation of electric charge.
We don’t often measure charge directly,
but sometimes we need to calculate it.
The symbol for charge is Q or q.
Charge’s unit of measure is the coulomb
(C).
For example,
 To indicate a charge of 450
microcoulombs, we write
Q = 450 µC
or
q = 450 µC
Basic Facts About Charge




There are two kinds of charge, which
we call positive and negative.
Opposite charges attract.
Like charges repel.
The smallest known charge is the
charge on a proton or an electron,
1.602 × 10-19 C. Most practical
charges that we deal with are much
larger than this—for example,
nanocoulombs (nC) or
microcoulombs (µC).
Formal Definition of Current


We’ve seen that current can
informally be thought of as being like
the flow rate of water through a
plumbing system.
More formally, current is defined as
the rate of change of charge per
time:
dq
i
dt

One ampere is equal to one coulomb
per second (1 A = 1 C/s).
Differentiation and Integration


Recall that differentiation and
integration are inverse operations.
Therefore, any relationship between
two quantities that can be expressed
in terms of derivatives can also be
expressed in terms of integrals.
Charge and Current

We saw above that current is
the derivative with respect to
time of charge:

Therefore charge is the
integral with respect to time
of current:

In typical problems, we know
the initial charge at time t0
and wish to find the charge q (t )
at later time t. In such cases
we use the definite integral:
dq
i
dt
q   i dt
t
  i dt  q (t0 )
t0
Calculus or Algebra?

As we’ve seen, the equations relating charge and
current contain derivatives and integrals:
dq
i
dt


q   i dt
Some problems involving current and charge
therefore require calculus. (For example,
Problems 1.2 and 1.3 in the textbook.)
But for many problems—in particular, problems
in which current is constant—these equations
simplify to algebraic equations:
q
i
t
q  it
Energy





Energy is perhaps the most fundamental
physical concept, underlying all areas of
physics.
We don’t often measure energy directly,
but sometimes we need to calculate it.
The symbol for energy is W or w.
Energy’s unit of measure is the joule (J).
For example,
 To indicate an energy of 780 nanojoules,
we write
W = 780 nJ
or
w = 780 nJ
Formal Definition of Voltage


We’ve seen that voltage can informally be
thought of as being like water pressure in a
plumbing system.
More formally, the voltage between two
points is defined as the amount of energy
needed to move a unit charge from one
point to the other:
dw
v
dq

One volt is equal to one joule per coulomb
(1 V = 1 J/C).
Supplies energy
Power



Absorb energy
At any time, some elements in a circuit
supply energy, and some elements absorb
energy.
An element’s power is the rate at which
that element supplies or absorbs energy.
The symbol for power is P or p:
dw
p
dt

Power’s unit of measure is the watt (W).
One watt is equal to one joule per second
(1 W = 1 J/s).
Energy and Power

We saw above that power is
the derivative with respect to
time of energy:

Therefore energy is the
integral with respect to time
of power:

In typical problems, we know
the initial energy at time t0
and wish to find the energy
at later time t. In such cases
we use the definite integral:
dw
p
dt
w   p dt
t
w(t )   p dt  w(t0 )
t0
Calculus or Algebra?

As we’ve seen, the equations relating energy and
power contain derivatives and integrals:
dw
p
dt


w   p dt
Some problems involving power and energy
therefore require calculus.
But for many problems—in particular, problems
in which power is constant—these equations
simplify to algebraic equations:
w
p
t
w  pt
Positive or Negative Power?


By convention, we assign a positive sign to
a power value if the element is absorbing
energy, and we assign a negative sign if
the element is supplying energy.
For example,
 To say that an element is absorbing 50
milliwatts, we would write
p = 50 mW

To say that an element is supplying 250
milliwatts, we would write
p = 250 mW
Kilowatt-hours


We’ve seen that in the SI system of
units, energy is measured in joules (J)
and power is measured in watts (W),
with
1J=1W1s
But the electrical power industry uses
different units: the kilowatt (kW) for
power and the kilowatt-hour (kWh)
for energy.
1 kWh = 1 kW  1 hour
The Power Law

We now have the following definitions:
dw
p
dt


dq
i
dt
The chain rule of calculus tells us that :
dw dw dq

dt dq dt

dw
v
dq
Therefore we can write: p  vi
In words, an element’s power is equal to
its voltage times its current.
The Passive Sign Convention


To get the correct sign (+ or ) on the
power value when we use the power law
(p=vi), we must be careful
with the signs of v and i.
We’ll always follow the
passive sign convention,
which says that we regard
the positive direction for
current as being current
into an element’s positive
terminal.
Conservation
Supplies energy
of Energy


Any circuit must obey the law of
conservation of energy.
Therefore the algebraic sum of the
powers in a circuit must equal 0.


Absorb energy
Recall that an energy supplier’s power is
negative, while an energy absorber’s power
is positive.
Example: In the circuit shown, if we know
that the voltage source’s power is 100 mW,
and R1’s power is 75 mW, then what must
R2’s power be?
Review: Some Quantities and Their
Units
Quantity
Symbol
SI Unit
Symbol for
the Unit
Current
I or i
ampere
A
Voltage
V or v
volt
V
Resistance
R
ohm

Charge
Q or q
coulomb
C
Time
t
second
s
Energy
W or w
joule
J
Power
P or p
watt
W
Active Elements


Circuit elements can be classified as
active or passive, depending on
whether they are capable of
generating electric energy.
Active elements can generate
electric energy.

Examples:
Voltage sources
 Current sources

Passive Elements

Passive elements cannot generate
electric energy.

Examples:
Resistors
 Capacitors
 Inductors



An important difference among these is
that capacitors and inductors can
store energy for later use.
Resistors cannot store energy: they
always dissipate energy as heat.
Ideal Sources


The most important active elements are
voltage sources and current sources.
In each case the word “ideal” means that
these are simplified models that ignore some
of the effects present in real sources.
Ideal Independent Voltage Source



An ideal independent voltage source
maintains a specified terminal
voltage no matter what the rest of
the circuit looks like.
We’ve discussed
these previously.
The book’s Figure
1.11 shows two
symbols for ideal
independent
voltage sources.
Ideal Independent Current Source


An ideal independent current source
supplies a specified current no
matter what the rest of the circuit
looks like.
The arrow identifies it
as a current source
and shows the
direction of
positive current
flow.
Ideal Dependent Voltage Source



An ideal dependent voltage source
maintains a terminal voltage whose
value depends on a voltage or
current somewhere else in the
circuit.
The diamond-shaped
body tells us that it’s a
dependent source.
The +/- inside tells us
that it’s a voltage
source, and shows the
voltage polarity.
Ideal Dependent Current Source



An ideal dependent current source
supplies a current whose value
depends on a voltage or current
somewhere else in the circuit.
The diamond-shaped
body tells us that it’s a
dependent source.
The arrow inside tells us
that it’s a current source
and shows the direction
of current flow.
Summary of Symbols for Ideal
Sources
Ideal
independent
voltage
source
Ideal
independent
current
source
Ideal
dependent
voltage
source
Ideal
dependent
current
source
Four Kinds of Dependent Sources

A dependent source’s value depends
on a voltage or current somewhere
else in the circuit, giving rise to four
kinds:





A
A
A
A
voltage-controlled
current-controlled
voltage-controlled
current-controlled
voltage source.
voltage source.
current source.
current source.
Text next to the symbol will let you
tell exactly which kind it is….
Examples of Symbols for
Controlled (Dependent) Sources
5v
Voltagecontrolled
voltage
source
5i
Currentcontrolled
voltage
source
5v
Voltagecontrolled
current
source
5i
Currentcontrolled
current
source
Example of a Controlled Source in
a Schematic Diagram

If i in this circuit is equal to 2.5 A,
then the dependent voltage source’s
value is 25 V.