K. N. Toosi University of Technology
Electric Circuits
Chapter 1. Circuit Variables
By:
B
y: FARHAD
FARHAD FARADJI,
FARA
ADJI, Ph.D.
Ph..D.
Assistant
A
ssistant Professor,
Professor,
Electrical and Computer Engineering,
K. N. Toosi University of Technology
Reference:
ELECTRIC CIRCUITS, 9th edition, 2011,
James W. Nilsson, Susan A. Riedel
1
1.1. Introduction
KNTU
• An electric circuit is a mathematical model that approximates
the behavior of an actual electrical system.
• The term electric circuit is commonly used to refer to an
actual electrical system as well as to the model that
represents it.
• When we talk about an electric circuit, we always mean a
model, unless otherwise stated.
• Circuit theory is a special case of electromagnetic field
theory: the study of static and moving electric charges.
Electric Circuits
Chapter 1. Circuit Variables
2
1.1. Introduction
KNTU
Three basic assumptions to use circuit theory, rather than electromagnetic
field theory:
1. Electrical effects happen instantaneously throughout a system.
– Electric signals travel at or near the speed of light.
em is physically small, electric signals move tthrough it so
– If the system
quickly.
gn
nals aaffect
ffect eevery
verry point
point in the
the system
systtem
m simultaneously.
simulta
– Electric signals
– Such a small
system.
malll system
system is
is called
called a lumped-parameter
lumped-parametter syst
2. The net charge on every component in the system is always zero.
– No component can collect a net excess of charge.
– Some components can hold equal but opposite separated charges.
3. There is no magnetic coupling between the components in a system.
– Magnetic coupling can occur within a component.
Electric Circuits
Chapter 1. Circuit Variables
3
1.1. Introduction
KNTU
How small does a physical system have to be to qualify as a lumped-parameter system?
9 Electric signals propagate by wave phenomena.
9 If the physical dimensions of the system is small compared to the wavelength of the signal, we have a
lumped-parameter system.
9 ʄ= c/f , c = 3 X 108 m/s.
How do we define smaller?
er? A good rule is the rule of 1/10th.
The dimension of the system
tem has to
o be 1/10th
1/1
10th (or
(o
or smaller)
sm
malleer) of
of the
the dimension
dimeensio
on of
of ʄ.
ʄ.
Example 1. Power systemss in
operate
50
6000
n IIran
ran o
perate att 5
0 Hz.
Hz. Thus,
Thus, ʄ= 6
000 km..
The dimension of the power
must
bee lless
lumped-parameter system.
werr system
system m
ust b
ess than
than 600
600 km
km to
to be
be treated aass a lump
Example 2. The propagation frequency of radio signals is on the order of 109 Hz. Thus, ʄ = 0.3 m.
The dimension of a communication system must be less than 3 cm to qualify as a lumped-parameter system.
¾
Whenever any of the pertinent physical dimensions of a system under study approaches the wavelength
of its signals, we must use electromagnetic field theory to analyze that system.
¾
We study circuits derived from lumped-parameter systems.
Electric Circuits
Chapter 1. Circuit Variables
4
1.2. The International System of Units
KNTU
¾ Engineers compare theoretical results to experimental results and compare competing engineering
designs using quantitative measures.
¾ Modern engineering is a multidisciplinary profession in which teams of engineers work together on
projects, and they can communicate their results in a meaningful way only if they all use the same
units of measure.
¾ The International System
used
societies
ystem off Units (abbreviated
(abbreeviateed SI)
SI) iss u
sed by all tthe
he major
major engineering
e
and most engineers thr
throughout
roughout the
the world.
world.
Electric Circuits
Chapter 1. Circuit Variables
5
1.2. The International System of Units
Electric Circuits
Chapter 1. Circuit Variables
KNTU
6
1.2. The International System of Units
KNTU
• In many cases, the SI unit is either
too small or too large to use
conveniently.
• Engineers often use only the ones
for powers divisible
sible by 3.
• Engineers often select
seelect the
the prefix
prefix that
thatt
places the base nu
number
umber iin
n the
the range
range
between 1 and 1000.
• For example:
10 ʅs = 0.00001 s = 0.01 ms =
10,000,000 ps.
Electric Circuits
Chapter 1. Circuit Variables
7
1.3. Circuit Analysis: An Overview
KNTU
¾ Need may come from the desire to improve on
an existing design, or it may be something
brand-new.
¾ Design specifications are measurable
characteristics of a proposed design.
¾ Insight comes from
m education and experience.
exxperiencee.
¾ Concept derives fro
from
om ccomplete
omplete u
understanding
nderstandingg
of the design specifications
cifications coup
coupled
pled with th
the
he
insight.
¾ A circuit model is a commonly used
mathematical model for electrical systems.
¾ An ideal circuit component is a mathematical
model of an actual electrical component,
representing the behavior of actual component
to an acceptable degree of accuracy.
Electric Circuits
Chapter 1. Circuit Variables
8
1.3. Circuit Analysis: An Overview
KNTU
¾ The tools of circuit analysis, the focus of this
course, are then applied to the circuit model.
¾ A comparison between the desired behavior,
from the design specifications, and the
predicted behavior, from circuit analysis, may
lead to refinements
nts in the circuit model and its
ideal circuit elements.
ents.
•
The physical prototype
ottype iss an
an actual
acctual electrical
eleectrical
system, constructed
ed
d ffrom
rom aactual
ctual eelectrical
lectriccal
components.
•
Measurements determine the actual
quantitative behavior of the physical system.
•
The iterative process, in which models and
systems are continually refined, may produce a
design that accurately matches the design
specifications and thus meets the need.
Electric Circuits
Chapter 1. Circuit Variables
9
1.4. Voltage and Current
KNTU
¾ The concept of electric charge is the basis for describing all electrical
phenomena.
¾ Important characteristics of electric charge:
9 The charge is bipolar, meaning that electrical effects are described in
terms of positive
negative
sitive and negativ
ve charges.
9 The electric charge
ch
harge exists
exists in
in discrete
discrete quantities,
quantitiees,, which
whicch are
ar integral
C.
multiples off th
the
electronic
charge,
1.6022
he ele
ectronicc ch
hargge, 1.6
6022 X 10-199 C.
feccts aare
re aattributed
ttributed tto
ob
oth tthe
he separation
separatiion of charge and
9 Electrical effects
both
charges in motion.
¾ In circuit theory, the separation of charge creates an electric force (voltage),
and the motion of charge creates an electric fluid (current).
Electric Circuits
Chapter 1. Circuit Variables
10
1.4. Voltage and Current
KNTU
9 Whenever positive and negative charges are separated, energy is expended.
9 Voltage is the energy per unit charge created by the separation.
9 We express this ratio in differential form as:
Electric Circuits
Chapter 1. Circuit Variables
11
1.4. Voltage and Current
KNTU
9 The electrical effects caused by charges in motion depend on the rate of
charge flow.
9 The rate of charge flow is known as the electric current,
which is expressed as:
Electric Circuits
Chapter 1. Circuit Variables
12
1.5. The Ideal Basic Circuit Element
KNTU
™ An ideal basic circuit element has three attributes:
1. it has only two terminals, which are points of connection to other circuit
components;
2. it is described mathematically in terms of current and/or voltage; and
3. It cannot be subdivided into other elements.
™ Algebraically the notion of positive charge flowing in one direction is
equivalent to the notion of negative charge flowing in the opposite direction.
Electric Circuits
Chapter 1. Circuit Variables
13
1.5. The Ideal Basic Circuit Element
Electric Circuits
Chapter 1. Circuit Variables
KNTU
14
1.5. The Ideal Basic Circuit Element
KNTU
¾ The assignmentss o
off tthe
he reference
reference polarity
polarityy for
for voltage
volttagge and
and the
the reference
direction for current
arbitrary.
rrent are entirelyy arbi
itrraryy.
¾ However, once you have assigned the references, you must write all
subsequent equations to agree with the chosen references.
¾ The most widely used sign convention applied to these references is called the
passive sign convention, which we use throughout this course.
Electric Circuits
Chapter 1. Circuit Variables
15
1.5. The Ideal Basic Circuit Element
KNTU
¾ The passive sign
n cconvention
onvention can
can be
be stated
stated
d as
as follows:
follo
ows:
Whenever the reference
rence direction
directtion for
for the
the current
currrent iin
n an element
ellementt is in the direction
of the reference voltage drop across the element (as in Fig. 1.5), use a positive
sign in any expression that relates the voltage to the current. Otherwise, use a
negative sign.
Electric Circuits
Chapter 1. Circuit Variables
16
1.6. Power and Energy
KNTU
‰ Power is the time rate of expending or absorbing energy.
‰ Mathematically, energy per unit time is expressed in the form of a derivative,
or:
Electric Circuits
Chapter 1. Circuit Variables
17
1.6. Power and Energy
KNTU
¾ The power associated with the flow of charge follows directly from the
definition of voltage and current in Eqs. 1.1 and 1.2, or:
Electric Circuits
Chapter 1. Circuit Variables
18
1.6. Power and Energy
KNTU
™ If we use the passive sign convention, Eq. 1.4 is correct if the reference
direction for the current is in the direction of the reference voltage drop
across the terminals.
with
inals. Otherwise, EEq.
q. 1.4 must
mu
ust be written
wriitten w
it a minus sign.
™ In other words, if the
the current
currrent reference
referen
nce is in the direction
direection of a reference
voltage rise across
osss the
the terminals,
terminals, tthe
he expression
expression for the power
pow is:
Electric Circuits
Chapter 1. Circuit Variables
19
1.6. Power and Energy
Electric Circuits
Chapter 1. Circuit Variables
KNTU
20
1.6. Power and Energy
KNTU
¾ Interpreting algebraic sign of power:
9 If the power is positive (p > 0), power is being delivered to the circuit
inside the box.
9 If the powerr is negative (p < 0), power is being extracted from the circuit
inside the box.
ox.
Electric Circuits
Chapter 1. Circuit Variables
21
1.6. Power and Energy
KNTU
¾ Examples:
Fig. 1.6(b): i= 4A , v= -ϭϬsїƉс-(-10)(4)= 40W
Thus the circuit inside the box is absorbing 40 W.
Fig. 1.6(c): i= -4A , v= 10V їƉс-(10)(-4)= 40W
Thus the circuit inside the box is absorbing 40 W.
Electric Circuits
Chapter 1. Circuit Variables
22
1.6. Power and Energy
Electric Circuits
Chapter 1. Circuit Variables
KNTU
23
1.6. Power and Energy
KNTU
• Something is wrong!
rong!
• If the values for vo
voltage
oltage aand
nd ccurrent
urrent in
in this
thiis circuit
circuit are
are correct,
correct the total power
should be zero!
• There is an errorr in
powers if
in the
the data
data and
and we
we can
can find
find it
it from
from tthe
he ccalculated
alc
the error exists in the sign of a single component.
• If we divide the total power by 2, we get -10 W, which is the power calculated
for component d.
• If the power for component d was +10 W, the total power would be 0.
• Circuit analysis techniques from upcoming chapters can be used to show that
the current through component d should be -1 A, not +1 A given in Table 1.4.
Electric Circuits
Chapter 1. Circuit Variables
24