Basic Laws
Basic Laws
Bởi:
Sy Hien Dinh
INTRODUCTION
Chapter 1 introduced basic concepts such as current, voltage, and power in an electric
circuit. To actually determine the values of this variable in a given circuit requires
that we understand some fundamental laws govern electric circuits. These laws known
as Ohm’s law and Kirchhoff’s laws, from the foundation upon which electric circuit
analysis is build.
In this chapter, in addition to these laws we shall discuss some techniques commonly
applied in circuit design and analysis. These techniques include combining resistors in
series or parallel, voltage division, current division and delta-to-wye and wye-to-delta
transformations
OHM’S LAW
Materials in general have a characteristic behavior of resisting the flow of electric
charge. This physical property or ability to resist current known as resistance and is
represented by the symbol R. the resistance of any material with a uniform crosssectional area A depends on A and its length l, as shown in [link](a). We can represent
resistance (as measured in the laboratory), in mathematical form,
l
R = ρA
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. [link] presents the values of ρ for some common materials
and shows which materials are used for conductors, insulators, and semiconductors.
Resistivities of common materials.
Material
Resistivity ( Ω m) Usage
Silver
1.64x10 − 8
Conductor
Copper
1.72x10 − 8
Conductor
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Aluminum
2.8x10 − 8
Conductor
Gold
2.45x10 − 8
Conductor
Carbon
4x10 − 5
Semiconductor
Germanium 47x10 − 2
Semiconductor
Silicon
6.4x102
Semiconductor
Paper
1010
Insulator
Mica
5x1011
Insulator
Glass
1012
Insulator
Teflon
3x1012
Insulator
a) Resistor b) Circuit symbol for resistance.
The circuit element used to model the current-resisting behavior of a material is the
resistor. For the purpose of constructing circuits, resistors are usually made from
metallic alloys and carbon compounds. The circuit symbol for the resistor is shown in
[link](b), where R stands for the resistance of the resistor. The resistor is the simplest
passive element.
Georg Simon Ohm (1787-1854), a German physicist, is credited with finding the
relationship between current and voltage for a resistor. This relationship is known as
Ohm’s law.
Ohm’s law states that the voltage v across a resistor is directly proportional to the
current i flowing through the resistor.
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That is,
v~i
Ohm defined the constant of proportionality for a resistor to be the resistance, R. Thus,
[link] becomes
v = iR
Which is the mathematical form of Ohm’s law. R in [link] is measured in the unit of
Ohms, designated Ω . Thus,
The resistance R of an element denotes its ability to resist the flow of electric
current; it is measured in ohms ( Ω ).
We may deduce from [link] that
R=
v
i
So that
1 Ω = 1 V/A
To apply Ohm’s law as stated in [link], we must pay careful attention to the current
direction and voltage polarity. The direction of current i and the polarity of voltage v
must conform with the passive sign convention, as shown in [link](b). This implies that
current flows from a higher potential to a lower potential in order for v = iR. If current
flows from a lower potential to a higher potential, v = -iR.
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a) Short cicuit (R=0), b) Open circuit (R= ).
Since the value of R can range from zero to infinity, it is important that we consider the
two extreme possible values of R. An element with R = 0 is called a short circuit, as
shown in [link](a). For a short circuit,
v = iR = 0
Showing that the voltage is zero but the current could be anything. In practice, a short
circuit is usually a connecting wire assumed to be a perfect conductor. Thus,
A short circuit is a circuit element with resistance approaching zero.
Similarly, an element with R = ∞ is known as an open circuit, as shown in [link](b).
For an open circuit,
i = lim
R→ ∞
v
R
=0
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Indicating that the current is zero though the voltage could be anything. Thus,
An open circuit is a circuit element with resistance approaching infinity.
Fixed resistor: a) wire-wound type, b) carbon film thin type.
A resistor is either fixed or variable. Most resistors are of the fixed type, meaning
their resistance remains constant. The two common types of fixed resistors (wirewound
and composition) are shown in [link]. The composition resistors are used when large
resistance is needed. The circuit symbol in [link](b) is for a fixed resistor. Variable
resistors have adjustable resistance. The symbol for a variable resistor is shown in
[link](a). A common variable resistor is known as a potentiometer or pot for short, with
the symbol shown in [link](b). The pot is three-terminal element with the sliding contact
or wiper. By the sliding wiper, the resistances between the wiper terminal and the fixed
terminals vary. Like fixed resistors, variable resistors can either be of wirewound or
composition type, as shown in [link]. Although resistors like those in [link] and [link]
are used in circuit designs, today most circuit components including resistors are either
surface mounted or integrated, as typically shown in [link].
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Circuit symbol for: a) a variable resistor in general, b) a potentiometer.
Variable resistor: a) composition type, b) slider pot.
It should be pointed out that not all resistors obey Ohm’s law. A resistor that obeys
Ohm’s law is known as a linear resistor. It has a constant resistance and thus its currentvoltage characteristic is as illustrated in [link](a): its i-v graph is a strait passing through
the origin. A nonlinear resistor does not obey Ohm’s law. Its resistance varies with
current and its i-v characteristic is typically shown in [link](b). 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 book
that all elements actually designated as resistors are linear.
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Resistors in a thin-film circuit.
The i-v charcteristic of: a) a linear resistor, b) a nonlinear resistor.
A useful quantity in circuit analysis is the reciprocal of resistance R, known as
conductance and denoted by G:
G=
1
R
=
i
v
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The conductance is a measure of how well an element will conduct electric current. The
unit of conductance is the siemens (S), the SI unit of conductance:
1S = 1A / V
Thus,
Conductance is the ability of an element to conduct electric current; it is measured
in siemens (S).
The same resistance can be expressed in ohms or siemens. For example, 10 Ω is the
same as 0.1 S. from [link], we may write
i = Gv
The power dissipated by a resistor can be expressed in term of R. using [link] and [link],
p = vi = i2R =
v2
R
The power dissipated by a resistor may also be expressed in terms of G as
p = vi = v2G =
i2
G
We should note two things from [link] and [link]:
1. The power dissipated in a resistor is a nonlinear function of either current or voltage.
2. Since R and G are positive quantities, the power dissipated in a resistor is always
positive. Thus, a resistor always absorbs power from the circuit. This confirms the idea
that a resistor is a passive element, incapable of generating energy.
NODES, BRANCHES AND LOOPS
Since the elements of an electric circuit can be interconnected in several ways, we
used to understand some basic concepts of network topology. To differentiate between
a circuit and a network, we may regard a network as an interconnection of elements
or devices, whereas a circuit is a network providing one or more closed paths. The
convention, when addressing network topology, is to use the word network and circuit
mean the same thing when used in this context. Such elements include branches, nodes,
and loops.
A branch represents a single element such as a voltage source or a resistor.
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In the other words, a branch represents any two-terminal element. The circuit in [link]
has five branches, namely, the 10 V voltage source, the 2 A current source, and the three
resistors.
A node is the point of connection between two or more branches.
Nodes, branches, and loops.
A node is usually indicated by a dot in a circuit. If a short circuit (a connecting wire)
connects two nodes, the two nodes constitute a single node. The circuit in [link] has
three nodes a, b, and c. notice that the three points that form node b are connected
by perfectly conducting wires and therefore constitute a single point. The same is true
of the four points forming nodes c. we demonstrate that the circuit in [link] has only
three nodes by redrawing the circuit in [link]. The two circuits in [link] and [link] are
identical. However, for the sake of the clarity, nodes b and c are spread out with perfect
conductors as in [link].
A loop is any closed path in a circuit.
A loop is a closed path formed by starting at a node, passing through any node more
than one. A loop is said to be independent if it contains at least one branch which is not
a part of any other independent loop. Independent loops or paths result in independent
sets of equations.
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Three node circuit of Figure 2.8 is redrawn.
For example, the closed path abca containing the 2- Ω resistor in [link] is a loop.
Another loop is the closed path bcb containing the 3- Ω resistor and current source.
Although one can identify six loops in [link], only three of them are independent.
A network with b branches, n nodes, and l independent loops will satisfy the
fundamental theorem of network topology:
b=l+n-1
As the next two definitions show, circuit topology is of great value to the study of
voltages and currents in an electric circuit.
Two or more elements in series if they exclusively share a single node and
consequently carry the same current.
Two more elements are in parallel if they are connected to the same two nodes and
consequently have the same voltage across them.
Elements are in series when they are chain-connected or connected sequentially, end to
end. For example, two elements are in series if they share one common node and no
other element is connected to that common node. Elements in parallel are connected to
the same pair of terminals. Elements may be connected in a way that they are neither
in series nor in parallel. In the circuit show in [link], the voltage source and the 5Ω resistor are in series because the same current will flow through them. The 2- Ω
resistor, the 3- Ω resistor, and the current source are in parallel because they are
connected to the same two nodes (b and c) resistors and neither in series nor in parallel
with each other.
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KIRCHHOFF’S LAWS
Ohm’s law is not sufficient to analyze circuits. However, when it is coupled with
Kirchhoff’s two laws, we have a sufficient, powerful set of tools for analyzing a large
variety of electric circuits. Kirchhoff’s laws were first introduced in 1847 by the German
physicist Gustav Robet Kirchhoff (1824-1887). These laws are formally known as
Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL).
Kirchhoff’s first law is based on the law of conservation of charge, which requires that
the algebraic sum of charges within a system cannot change.
Kirchhoff’s current law (KCL) states that the algebraic sum of currents entering a
node (or a closed boundary) is zero.
Mathematically, KCL implies that
∑Nn = 1 in = 0
Where N is number of branches connected to the node and in is nth current entering
(or leaving) the node. By the law, current entering a node may be regarded as positive,
while currents leaving the node may be taken as negative or vice versa.
To prove KCL assume a set of currents ik(t), k = 1, 2 … flow into a node. The algebraic
sum of currents at the node is
iT(t) = i1(t) + i2(t) + i3(t) + ...
Integrating both sides of [link] gives
qT(t) = q1(t) + q2(t) + q3(t) + ...
Where qk(t) = ∫ik(t)dtand qT(t) = ∫iT(t)dt but the law of conservation of electric charge
requires that the algebraic sum of electric charges at the note must not change; that is,
the node stores no net charge. Thus qT(t) = 0 → i(t) = 0, confirming the validity of KCL.
Consider the node in [link]. Applying KCL gives
i1 + ( − i2) + i3 + i4 + ( − i5) = 0
Since current i1, i2 and i4 are entering the node, while current i2and i5 are leaving it. By
rearranging the terms, we get
i1 + i3 + i4 = i2 + i5
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[link] is an alternative form of KCL:
The sum of the current entering a node is equal to the sum of the currents leaving
the node.
Currents at a node illustrating KCL.
Note that KCL also applies to a closed boundary. This may be regarded as 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. As typically illustrated in the
circuit of [link], the total current entering the closed surface is equal to the total current
leaving the surface.
Appling KCL to a closed boundary.
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A simple application of KCL is combining current sources in parallel. The combined
current is the algebraic sum of the current supplied by the individual sources. For
examples, the current sources shown in [link](a) can be combined as in [link](b). The
combined or equivalent current source can be found by applying KCL to node a.
IT + I2 = I1 + I3
or
IT = I1 − I2 + I3
A circuit cannot contain two different currents, I1 and I2, in series, unless I1 = I2;
otherwise KCL will be violated.
Current sources in parallel: a) origin circuit; b) equivalent circuit.
Kirchhoff’s second law is based on the principle of conservation of energy:
Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around
a closed path (or loop) is zero.
Expressed mathematically, KVL states that
∑M
m = 1 vm = 0
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Where M is the number of voltages in the loop (or the number of branches in the loop)
and vm is the mth voltage.
To illustrate KVL, consider the circuit in [link]. The sign on each voltage is the polarity
of the terminal encountered first as we travel around the loop. We can start with any
branch and go around the loop either clockwise or counterclockwise. Suppose we start
with the voltage source and go clockwise around the loop as shown; then voltages would
be − v1, +v2, +v3, − v4, and +v5, in that order. For example, as we reach branch 3, the
positive terminal is met first; hence we have +v3. For branch 4, we reach the negative
terminal first; hence, − v4. Thus, KVL yields
− v1 + v2 + v3 − v4 + v5 = 0
Rearranging terms gives
v2 + v3 + v5 = v1 + v4
Which may be interpreted as
Sum of voltage drops = sum of voltage rises
This is an alternative form of KVL. Notice that if we had traveled counterclockwise, the
result would have been +v1, − v5, +v4, − v3, and − v2, which is the same as before except
that the sign are reversed. Hence, [link] and [link] remain the same.
A single loop circuit illustrating KVL.
When voltage sources are connected in series, KVL can be applied to obtain the total
voltage. The combined voltage is the algebraic sum of the voltage of the individual
sources. For example, for the voltage shown in [link](a), the combined or equivalent
voltage source in [link](b) is obtained by applying KVL.
− Vab + V1 + V2 − V3 = 0
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or
Vab = V1 + V2 − V3
To avoid violating KVL, a circuit cannot contain two different voltages V1 and V2 in
parallel unless V1 = V2.
Voltage source in series: a) original circuit, b) equivalent circuit.
SERIES RESISTORS AND VOLTAGE DIVISION
The need to combine resistors in series or in parallel occurs so frequently that it warrants
special attention. The process of combining the resistors is facilitated by combining two
of them at a time. With this in mind, consider the single-loop circuit of [link] the two
resistors are in series, since the same current i flows in both of them. Applying Ohm’s
law to each of the resistors, we obtain
v1 = iR1
and
v2 = iR2
If we apply KVL to the loop (moving in the ckockwise direction), we have
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− v + v1 + v2 = 0
Combining [link] and [link], we get
v = v1 + v2 = i(R1 + R2)
or
i=
v
(R1 + R2)
A single loop circuit with two resistor in series.
Notice that [link] can be written as
v = iReq
implying that the two resistors can be replaced by an equivalent resistor Req; that is
Req = (R1 + R2)
Thus, [link] can be replaced by the equivalent circuit in [link]. Two circuits in [link]
and [link] are equivalent because they exhibit the same voltage-current relationship at
the terminals a-b. An equivalent circuit such as the in [link] is useful in simplying the
analysis of a circuit. In general,
The equivalent resistance of any number of resistors connected in series is the sum
of the individual resistance.
For N resistors in series then,
R = R1 + R2 + ⋯ + RN = ∑Nn = 1 Rn
eq
To determine the voltage across each resistor in [link], we substitute [link] into [link]
and obtain
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v1 =
R1
R1 + R2 v,
v2 =
R2
R1 + R2 v
Equivalent circuit of the Figure 2.15 circuit.
Note that the voltage v is divided among the resistor in direct proportion to their
resistances; the larger the resistance, the larger the voltage drop. This is called the
principle of voltage division, and the circuit in [link] is called a voltage divider. In
general, if a voltage divider has N resistors ( R1, R2, RN) in series with the source voltage
v, the nth resistor ( Rn) will have a voltage drop of
vn =
Rn
R1 + R2 + ⋯ + RN v,
PARALLEL RESISTORS AND CURRENT DIVISION
Consider the circuit in [link], where two resistors are connected in parallel and therefore
have the same voltage across them. From Ohm’s law,
v = i1R1 = i2R2
or
i1 =
v
R1 , i 2
=
v
R2
Applying KCL at node a gives the total current I as
i = i1 + i2
Substituting [link] into Equation (34), we get
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i=
v
R1
+
v
R2
1
= v( R +
1
1
R2 )
=
v
Req
where Req is the equivalent resistance of the resistors in parallel:
1
Req
=
1
R1
+
1
R
2
Or
=
R1 + R2
R 1R 2
Req =
R1R2
R1 + R2
1
Req
Or
Thus,
The equivalent resistance of two parallel resistors is equal to the product of their
resistance divided by their sum.
Two resistor in parallel.
It must be emphasized that this applies only to two resistors in parallel. From [link], if
R1 = R2, then Req = R1 / 2.
We can extend the result in [link] to the general case of a circuit with N resistors in
parallel. The equivalent resistance is
1
R
eq
=
1
R1
+
1
R2
+ ... +
1
RN
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Note that Req is always smaller than the resistance of the smallest resistor in the parallel
combination. If R1 = R2 = ... = RN = R, then
Req =
R
N
For example, if four 100 Ω resistors are connected in parallel, their equivalent
resistance is 25 Ω .
It is often more convenient to use conductance rather than resistance when dealing with
resistors in parallel. From [link], the equivalent conductance for N resistors in parallel is
Geq = G1 + G2 + ...+GN
Where Geq = 1 / Req, G1 = 1 / R1, G2 = 1 / R2, G3 = 1 / R3 … GN = 1 / RN. [link] states:
The equivalent conductance of resistors connected in parallel is the sum of their
individual conductances.
This means that we may replace the circuit in [link] with that in [link]. Notice the
similarity between [link] and [link]. The equivalent conductance of parallel resistors is
obtained the same way as the equivalent resistance of resistors in series is obtained just
the same way as the resistance of resistors in parallel. Thus the equivalent conductance
Geq of N resistors in series (such as shown in [link]) is
1
Geq
=
1
G1
1
G2
+
+
1
G3
+ ... +
1
GN
Given the total current i entering node a in [link], how do we obtain current i1 and i2?
We know that the equivalent resistor has the same voltage, or
v = iReq =
iR1R2
R1 + R2
Combining [link] and [link] results in
i1 =
iR2
R1 + R2 ,
and
i2 =
iR1
R1 + R2
Which shows that the total current i is shared by the resistor in inverse proportion to their
resistances. This is known as the principle of current division, and the circuit in [link]
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known as a current divider. Notice that the larger current flows through the smaller
resistance.
As an extreme case, suppose one of the resistors in [link] is zero, say R2 = 0; that is, R2
is a short circuit, as shown in [link](a). From [link], R2 = 0 implies that i1 = 0, i2 = i. this
means that the entire current i bypasses R1 and flows through the short current R2 = 0,
the path of least resistance. Thus when a circuit is short circuited, as shown in [link](a),
two things should be kept in mind:
1. The equivalent resistance Req = 0. [See what happens when R2 = 0 in [link].}
2. The entire current flows through the short circuit.
As another extreme case, suppose R2 = ∞ , that is R2 is an open circuit, as shown in
[link](b). The current still flows through the path of least resistance, R1. By taking the
limit of [link] as R2 → ∞ , we obtain Req = R1 in this case.
Equivalent circuit to Figure 2.17.
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a) A shorted circuit, b) an open circuit.
If we divide both the numerator and denominator by R1R2, [link] becomes
i1 =
i2 =
G1
G1 + G2 i
G2
G1 + G2 i
Thus, in general, if a current divider has N conductors ( G1, G2, …, GN) in parallel with
the source current i, the nth conductor ( GN) will have current
in =
Gn
G1 + G2 + ... + GN i
In general, it is often convenient and possible to combine resistors in series and parallel
and reduce a resistive network to a single equivalent resistance Req. Such a single
equivalent resistance is the resistance between the designated terminals of the network
and must exhibit the same i-v characteristics as the original network at the terminals.
WYE-DELTA TRANSFORMATIONS
Situations often arise to circuit analysis when the resistors are neither in parallel nor in
series. For example, consider the bridge circuit in [link]. How do we combine resistors
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R1 through R6 when the resistors are neither in series nor in parallel? Many circuits of
the types shown in [link] can be simplified by using three terminal equivalent networks.
There are the wye (Y) or tee (T) network shown in [link] and the delta ( Δ) or pi ( Π
) network shown in [link]. These networks occur by themselves or as part of a larger
network. They are used in three phase networks, electrical filters, and matching network.
Our main interest here is in how to identify them when they occur as part of a network
and how to apply wye-delta transformation in the analysis of that network.
The bridge network.
Two forms the same network; a)Y, b)T.
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Two forms of the same network: a)⌂ b)п.
Delta to wye conversion
Suppose it is more convenient to work with a wye network in a place where the circuit
contains a delta configuration. We superimpose a wye network on the existing delta
network and find the equivalent resistances in the wye network. To obtain the equivalent
resistances in the wye network, we compare the two networks and make sure that
the resistance between each pair of nodes in the Δ (or Π) network is the same as the
resistance between the same pair of nodes in the Y (or T) network. For terminals 1 and
2 in [link] and [link], for example,
R12(Y) = R1 + R3
R12(Δ) = Rb // (Ra + Rc)
Setting R12(Y) = R12(Δ) gives
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R = R1 + R3 =
12
Rb(Ra + Rc)
Ra + Rb + Rc
Similarly,
R13 = R1 + R2 =
R34 = R2 + R3 =
Rc(Ra + Rb)
Ra + Rb + Rc
Ra(Rb + Rc)
Ra + Rb + Rc
Subtracting [link] from [link]. We get
R1 − R2 =
Rc(Rb − Ra)
Ra + Rb + Rc
Adding [link] and [link] gives
R1 =
RbRc
Ra + Rb + Rc
and subtracting [link] from [link] yields
R2 =
RcRa
Ra + Rb + Rc
Subtracting [link] from [link]. We obtain
R3 =
RaRb
Ra + Rb + Rc
Superposition of Y and ⌂ networks as an aid in transforming one to the other.
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We do not need to memorize [link] to [link]. To transform a Δ network to Y, we create
extra node n as shown in [link] and follow this conversion rule:
Each resistor in the Y network is the product of the resistors in the two adjacent Δ
branches, divided by the sum of the three Δ resistors.
One can follow this rule and obtain [link] to [link] from [link].
Wye to delta conversion
To obtain the conversion formulas for transforming a wye network to an equivalent delta
network, we note from [link] to [link] that
R1R2 + R R3 + R3R1 =
2
RaRbRc(Ra + Rb + Rc)
(Ra + Rb + Rc)
2
=
RaRbRc
Ra + R + Rc
b
Dividing [link] by each of [link] to [link] leads to the following equations:
Ra =
Rb =
Rc =
R1R2 + R2R3 + R R1
3
R1
R1R2 + R2R3 + R R1
3
R2
R1R2 + R2R3 + R R1
3
R3
From [link] to [link] and [link], the conversion rule for Y to Δ is as follows:
Each resistor in the Δ network is the sum of all possible products of Y resistors taken
two at a time, divided by the opposite Y resistor.
The Y and Δ networks are said to the balanced when
R1 = R2 = R3 = RY
and
Ra = Rb = Rc = RΔ
Under these conditions, conversion formulas become
RY =
RΔ
3
or
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RΔ = 3RY
One may wonder why RY is less than RΔ. Well, we notice that the Y-connection is like a
“series” connection while the Δ-connection is like a “parallel” connection.
Note that in making the transformation, we do not take anything out of the circuit or
put in anything new. We are merely substituting different but mathematically equivalent
three-terminals network patterns to create a circuit in which resistors are either in series
or in parallel, allowing us to calculate Req if necessary.
SUMMARY
1. A resistor is a passive element in which the voltage v across it is directly
proportional to the current I through it. That is, a resistor is a device that obeys
Ohm’s law,
V = iR
where R is resistance of the resistor.
1. A short circuit is a resistor (a perfectly conducting wire) with zero resistance (R
= 0). An open circuit is a resistor with infinite resistance (R = ∞ ).
2. The conductance G of a resistor is the reciprocal of its resistance:
G=
1
R
1. A branch is a single two-terminal element in electric circuit. A node is the point
of connection between two or more branches. A loop is closed path in a circuit.
The number of branches b, the number of the node n, and the number of
independent loops l in a network are related as
b=l+n-1
1. Kirchhoff’s current law (KCL) states that the currents at any node algebraically
sum to zero. In other words, the sum of the currents entering a node equals the
sum of currents leaving the node.
2. Kirchhoff’s voltage law (KVL) states that the voltages around closed path
algebraically sum to zero. In other words, the sum of the voltage rises equals
the sum of the voltage drops.
3. Two elements are in series when they are connected sequentially, end to end.
When elements are in series, the same current flows through them ( i1 = i2).
They are in parallel if they are connected to the same to nodes. Elements in
parallel always have the same voltage across them ( v1 = v2).
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4. When two resistors R1( = 1 / G1) and R2( = 1 / G2) are in series, their equivalent
resistance Req and equivalent conductance Geq are
Req = R1 + R2, Geq =
G1G2
G1 + G2
1. When two resistors R1( = 1 / G1) and R2( = 1 / G2) are in parallel, their equivalent
resistance Req and equivalent conductance Geq are
Req =
R1R2
R1 + R2 ,
Geq = G1 + G2
1. The voltage division principle for two resistors in series is
v1 =
R1
R1 + R
2
v, v2 =
R2
R1 + R
v
2
11. The current division principle for two resistors in parallel is
i1 =
R2
R1 + R
2
i, i2 =
R1
R1 + R
i
2
1. The formulas for a delta-to-wye transformation are
R1 =
RbRc
Ra + Rb + Rc ,
R2 =
RcRa
Ra + Rb + Rc ,
R3 =
RaRb
Ra + Rb + Rc
1. The formulas for a wye-to-delta transformation are
Ra =
R1R2 + R2R3 + R3R1
,
R1
Rb =
R1R2 + R2R3 + R3R1
,
R2
Rc =
R1R2 + R2R3 + R3R1
R3
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