CHAPTER 2
Basic Laws
Here we explore two fundamental laws that govern electric circuits
(Ohm’s law and Kirchhoff’s laws) and discuss some techniques commonly
applied in circuit design and analysis.
2.1. Ohm’s Law
Ohm’s law shows a relationship between voltage and current of a resistive element such as conducting wire or light bulb.
2.1.1. Ohm’s Law: The voltage v across a resistor is directly proportional to the current i flowing through the resistor.
v = iR,
where R = resistance of the resistor, denoting its ability to resist the flow
of electric current. The resistance is measured in ohms (Ω).
• To apply Ohm’s law, the direction of current i and the polarity of
voltage v must conform with the passive sign convention. This implies that current flows from a higher potential to a lower potential
15
16
2. BASIC LAWS
in order for v = iR. If current flows from a lower potential to a
higher potential, v = −iR.
i
l
Material with
Cross-sectional resistivity r
area A
+
v
–
R
2.1.2. The resistance R of a cylindrical conductor of cross-sectional area
A, length L, and conductivity σ is given by
L
R=
.
σA
Alternatively,
L
R=ρ
A
where ρ is known as the resistivity of the material in ohm-meters. Good
conductors, such as copper and aluminum, have low resistivities, while
insulators, such as mica and paper, have high resistivities.
2.1.3. Remarks:
(a) R = v/i
(b) Conductance :
1
i
=
R v
1
The unit of G is the mho (f) or siemens2 (S)
G=
1Yes, this is NOT a typo! It was derived from spelling ohm backwards.
2In English, the term siemens is used both for the singular and plural.
2.1. OHM’S LAW
17
(c) The two extreme possible values of R.
(i) When R = 0, we have a short circuit and
v = iR = 0
showing that v = 0 for any i.
+
i
v=0 R=0
–
(ii) When R = ∞, we have an open circuit and
v
=0
i = lim
R→∞ R
indicating that i = 0 for any v.
+
v
i=0
R=∞
–
2.1.4. A resistor is either fixed or variable. Most resistors are of the
fixed type, meaning their resistance remains constant.
18
2. BASIC LAWS
A common variable resistor is known as a potentiometer or pot for
short
2.1.5. Not all resistors obey Ohms law. A resistor that obeys Ohms
law is known as a linear resistor.
• A nonlinear resistor does not obey Ohms law.
• Examples of devices with nonlinear resistance are the lightbulb and
the diode.
• Although all practical resistors may exhibit nonlinear behavior under certain conditions, we will assume in this class that
all elements actually designated as resistors are linear.
2.1. OHM’S LAW
19
2.1.6. Using Ohm’s law, the power p dissipated by a resistor R is
v2
p = vi = i2 R = .
R
Example 2.1.7. In the circuit below, calculate the current i, and the
power p.
i
30 V
DC
5 kΩ
+
v
–
Definition 2.1.8. The power rating is the maximum allowable power
dissipation in the resistor. Exceeding this power rating leads to overheating
and can cause the resistor to burn up.
Example 2.1.9. Determine the minimum resistor size that can be connected to a 1.5V battery without exceeding the resistor’s 41 -W power rating.
illustrative circuit. We will then apply the same systematic method to solve
more complicated examples, including the one shown in Figure 2.1.
20
2.1 T E R M I N O L O G Y2.
BASIC LAWS
Lumped circuit elements are the fundamental building blocks of electronic circuits. Virtually2.2.
all of Node,
our analyses
will be conducted
on circuits containing
Branches
and Loops
two-terminal elements; multi-terminal elements will be modeled using combinations of two-terminal
elements.
We have already
several two-terminal
Definition
2.2.1. Since
the elements
of anseen
electric
circuit can be inelements such as resistors, voltage sources, and current sources. Electronic
terconnected in several ways, we need to understand some basic concept
access to an element is made through its terminals.
of network An
topology.
electronic circuit is constructed by connecting together a collection of
separate elements
at their terminals, as
in Figure
The junction points
• Network
= interconnection
ofshown
elements
or2.2.
devices
at
which
the
terminals
of
two
or
more
elements
are
connected
are referred to as
• Circuit = a network with closed paths
the nodes of a circuit. Similarly, the connections between the nodes are referred
to as the edges
or branches
of a A
circuit.
Note represents
that each element
in Figure
2.2
Definition
2.2.2.
Branch:
branch
a single
element
such
forms a single branch. Thus an element and a branch are the same for circuits
as a voltage
source
or a resistor.
A branch
represents
any two-terminal
comprising
only two-terminal
elements.
Finally, circuit
loops are defined
to be
element.closed paths through a circuit along its branches.
Several nodes, branches, and loops are identified in Figure 2.2. In the circuit
Definition
2.2.3.
A node
the10“point”
between
in Figure 2.2,
there Node:
are 10 branches
(andis
thus,
elements) of
andconnection
6 nodes.
another example, a is a node in the circuit depicted in Figure 2.1 at
two or moreAsbranches.
which three branches meet. Similarly, b is a node at which two branches meet.
• Itabisand
usually
indicated by a dot in a circuit.
bc are examples of branches in the circuit. The circuit has five branches
• Ifand
a four
short
circuit (a connecting wire) connects two nodes, the two
nodes.
Since
we assume that
the interconnections
between the elements in a circuit
nodes
constitute
a single
node.
are perfect (i.e., the wires are ideal), then it is not necessary for a set of elements
to be joined2.2.4.
togetherLoop:
at a single
space
for closed
their interconnection
be
Definition
Apoint
loopin is
any
path in atocircuit.
A
considered
a singleby
node.
An example
this ispassing
shown inthrough
Figure 2.3.aWhile
closed path
is formed
starting
at a of
node,
set of nodes
the four elements in the figure are connected together, their connection does
and returning
to the starting node without passing through any node more
not occur at a single point in space. Rather, it is a distributed connection.
than once.
Nodes
Loop
Elements
E 2.2 An arbitrary circuit.
Branch
Definition 2.2.5. Series: Two or more elements are in series if they
are cascaded or connected sequentially and consequently carry the same
current.
Definition 2.2.6. Parallel: Two or more elements are in parallel if
they are connected to the same two nodes and consequently have the same
voltage across them.
2.2. NODE, BRANCHES AND LOOPS
21
2.2.7. Elements may be connected in a way that they are neither in
series nor in parallel.
Example 2.2.8. How many branches and nodes does the circuit in the
following figure have? Identify the elements that are in series and in parallel.
5Ω
1Ω
2Ω
DC
10 V
4Ω
2.2.9. A loop is said to be independent if it contains
a branch which
2.2 Kirchhoff’s Laws
is not in any other loop. Independent loops or paths result in independent
sets of equations. A network with b branches, n nodes, and ` independent
loops will satisfy the fundamental theorem of network topology:
B
Elements
b = ` + n − 1.
C
B
withinAa
circuitCare
Definition 2.2.10. The primary signals
its currents
A
D
and voltages, which we denote by the symbols i and v, respectively. We
Distributed node
D
define a branch current
as the current along a branch of the circuit, and
Ideal wires
a branch voltage as the potential difference measured across a branch.
CHAPTER T
F I G U R E 2.3
interconnectio
elements that
at a single nod
i
Branch
current
v
+
Branch
voltage
Nonetheless, because the interconnections are perfect, the connection can be
considered to be a single node, as indicated in the figure.
The primary signals within a circuit are its currents and voltages, which we
denote by the symbols i and v, respectively. We define a branch current as the
current along a branch of the circuit (see Figure 2.4), and a branch voltage as the
potential difference measured across a branch. Since elements and branches are
the same for circuits formed of two-terminal elements, the branch voltages and
currents are the same as the corresponding terminal variables for the elements
forming the branches. Recall, as defined in Chapter 1, the terminal variables for
F I G U R E 2.4
definitions illu
a circuit.
22
2. BASIC LAWS
2.3. Kirchhoff ’s Laws
Ohm’s law coupled with Kirchhoff’s two laws gives a sufficient, powerful
set of tools for analyzing a large variety of electric circuits.
2.3.1. Kirchhoff ’s current law (KCL): the algebraic sum of currents
departing a node (or a closed boundary) is zero. Mathematically,
X
in = 0
n
KCL is based on the law of conservation of charge. An alternative
form of KCL is
Sum of currents (or charges) drawn as entering a node
= Sum of the currents (charges) drawn as leaving the node.
i5
i1
i4
i2
i3
Note that KCL also applies to a closed boundary. This may be regarded
as a generalized case, because a node may be regarded as a closed surface
shrunk to a point. In two dimensions, a closed boundary is the same as a
closed path. The total current entering the closed surface is equal to the
total current leaving the surface.
Closed boundary
2.3. KIRCHHOFF’S LAWS
23
Example 2.3.2. A simple application of KCL is combining current
sources in parallel.
IT
a
I1
b
I2
I3
(a)
IT
a
IT = I 1 – I2 + I 3
b
(b)
A Kirchhoff ’s voltage law (KVL): the algebraic sum of all voltages
around a closed path (or loop) is zero. Mathematically,
M
X
vm = 0
m=1
KVL is based on the law of conservation of energy. An alternative
form of KVL is
Sum of voltage drops = Sum of voltage rises.
+
v2 –
+
v1
v3 –
v4
– v5 +
24
2. BASIC LAWS
Example 2.3.3. When voltage sources are connected in series, KVL
can be applied to obtain the total voltage.
a
+
V1
Vab
V2
a
Vab
V3
b
–
+
b
VS = V1 + V2 – V3
–
Example 2.3.4. Find v1 and v2 in the following circuit.
4Ω
+ v1 –
10 V
8V
+
v2 –
2Ω
Example 2.3.5.
(a)
1
(b)
Fig. 27-20
(c)
Question 5.
2.3. KIRCHHOFF’S LAWS
25
6 Res-monster maze. In Fig. 27-21, all the resistors have a
resistance of 4.0 and all the (ideal) batteries have an emf of 4.0
V. What is2.3.6
the current
(If below,
you canallfind
Example
(HRW through
p. 725).resistor
In the R?
figure
thethe
resistors
loop through
maze,
answer
the question
a
haveproper
a resistance
of 4.0 this
Ω and
all you
the can
(ideal)
batteries
are 4.0with
V. What
is
few seconds
of mental
calculation.)
the current
through
resistor
R?
ies? (b) Are
t resistances
R2, to a batrallel. Rank
ent through
R
–
R2
x
d
e
Fig. 27-21
Question 6.
Question 4.
that R1 R2
7 A resistor R1 is wired to a battery, then resistor R2 is added in
series. Are (a) the potential difference across R1 and (b) the cur-
26
2. BASIC LAWS
2.4. Series Resistors and Voltage Division
2.4.1. When two resistors R1 and R2 ohms are connected in series, they
can be replaced by an equivalent resistor Req where
Req = R1 + R2
.
In particular, the two resistors in series shown in the following circuit
i
R1
R2
+ v1 –
+ v2 –
a
v
b
can be replaced by an equivalent resistor Req where Req = R1 +R2 as shown
below.
i
a
Req
+ v –
v
b
The two circuits above are equivalent in the sense that they exhibit the
same voltage-current relationships at the terminals a-b.
Voltage Divider: If R1 and R2 are connected in series with a voltage
source v volts, the voltage drops across R1 and R2 are
R1
R2
v
and
v2 =
v
v1 =
R1 + R2
R1 + R2
2.5. PARALLEL RESISTORS AND CURRENT DIVISION
27
Note: The source voltage v is divided among the resistors in direct proportion to their resistances.
2.4.2. In general, for N resistors whose values are R1 , R2 , . . . , RN ohms
connected in series, they can be replaced by an equivalent resistor Req
where
N
X
Req = R1 + R2 + · · · + RN =
Rj
j=1
If a circuit has N resistors in series with a voltage source v, the jth
resistor Rj has a voltage drop of
Rj
v
vj =
R1 + R2 + · · · + RN
2.5. Parallel Resistors and Current Division
When two resistors R1 and R2 ohms are connected in parallel, they can
be replaced by an equivalent resistor Req where
1
1
1
=
+
Req
R1 R2
or
R1 R2
Req = R1 kR2 =
R1 + R2
i
Node a
i1
v
i2
R1
R2
Node b
Current Divider: If R1 and R2 are connected in parallel with a current
source i, the current passing through R1 and R2 are
R2
R1
i1 =
i
and
i2 =
i
R1 + R2
R1 + R2
Note: The source current i is divided among the resistors in inverse proportion to their resistances.
28
2. BASIC LAWS
Example 2.5.1.
Example 2.5.2. 6k3 =
Example 2.5.3. (a)k(na) =
Example 2.5.4. (ma)k(na) =
In general, for N resistors connected in parallel, the equivalent resistor
Req = R1 kR2 k · · · kRN is
1
1
1
1
=
+
+ ··· +
Req
R1 R2
RN
Example 2.5.5. Find Req for the following circuit.
4Ω
1Ω
2Ω
Req
5Ω
8Ω
6Ω
3Ω
Example 2.5.6. Find the equivalent resistance of the following circuit.
2.5. PARALLEL RESISTORS AND CURRENT DIVISION
i
4Ω
i0
a
6Ω
12 V
29
+
v0
–
3Ω
b
Example 2.5.7. Find io , vo , po (power dissipated in the 3Ω resistor).
Example 2.5.8. Three light bulbs are connected to a 9V battery as
shown below. Calculate: (a) the total current supplied by the battery, (b)
the current through each bulb, (c) the resistance of each bulb.
I
9V
15 W
20 W
9V
10 W
I1
I2
+
V2
–
R2
+
V3
–
R3
+
V1
–
R1
30
2. BASIC LAWS
2.6. Practical Voltage and Current Sources
An ideal voltage source is assumed to supply a constant voltage. This
implies that it can supply very large current even when the load resistance
is very small.
However, a practical voltage source can supply only a finite amount of
current. To reflect this limitation, we model a practical voltage source as
an ideal voltage source connected in series with an internal resistance rs ,
as follows:
Similarly, a practical current source can be modeled as an ideal current
source connected in parallel with an internal resistance rs .
2.7. Measuring Devices
Ohmmeter: measures the resistance of the element.
Important rule: Measure the resistance only when the element is disconnected from circuits.
Ammeter: connected in series with an element to measure current
flowing through that element. Since an ideal ammeter should not restrict
the flow of current, (i.e., cause a voltage drop), an ideal ammeter has zero
internal resistance.
Voltmeter:connected in parallel with an element to measure voltage
across that element. Since an ideal voltmeter should not draw current away
from the element, an ideal voltmeter has infinite internal resistance.