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1. INTRODUCTION


Fundamental laws – govern electric circuits:
a) Ohm’s law
b) Kirchhoff’s law
Techniques commonly applied:
a) resistors in series / parallel
b) voltage division
c) current division
d) delta-to-wye and wye-to-delta transformations
2. OHM’S LAW



Materials – have a characteristic behavior of resisting the flow of electric
charge
Resistance: ability to resist current (R)
Mathematically,
R

A
Fig. 2.1: a) Resistor b) Circuit symbol for resistance
Material
Silver
Copper
Aluminum
Gold
Carbon
Germanium
Silicon
Paper
Mica
Glass
Teflon
Resistivity (.m)
1.64 x 10-8
1.72 x 10-8
2.8 x 10-8
2.45 x 10-8
4 x 10-5
47 x 10-2
6.4 x 102
1010
5 x 1011
1012
3 x 1012
Usage
Conductor
Conductor
Conductor
Conductor
Semiconductor
Semiconductor
Semiconductor
Insulator
Insulator
Insulator
Insulator
Resistivities of common materials





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Resistor: circuit element – used to model the current-resisting behavior
Relationship between current and voltage for a resistor – Ohm’s Law
Ohm’s Law: the voltage v across a resistor is directly proportional to the
current I flowing through the resistor
Georg Simon Ohm (1787-1854), a German physicist, is credited with finding
the relationship between current and voltage for a resistor
Ohm – defined the constant of proportionality for a resistor – resistance, R
Resistance, R: denotes its ability to resist the flow of electric current ()
Two extreme possible values of R:
a) R = 0
b) R = 


short circuit (v = i R = 0)
lim


v
 0 
open circuit  i 
 R R

Short circuit: circuit element with resistance
approaching zero
Open circuit: circuit element with resistance
approaching infinity
Fig. 2.2: a) Short circuit b) Open
Circuit
Fixed
Resistor
Variable
Fig. 2.3: Symbol for a) variable
resistor, b) potentiometer

Conductance, G – how well an element
will conduct electric current
G


1 i

R v
Note:
Conductance: the ability of an element to
conduct electric current; measured in
mho or siemens (S)
Power dissipated by a resistor:
v2
p  vi  i R 
R
2
OR
p  vi  v 2 G 
i2
G
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 (resistor –
absorbs power from circuit,
resistor – passive element,
incapable in generating
energy)
3. NODES, BRANCHES AND LOOPS





Network – interconnection of elements
or devices
Circuit – network providing one / more
closed paths
Branch: represents a single element (e.g.
voltage source, resistor); i.e. represents
any two-terminal element
Node: point of connection between two
or more branches; usually indicated by a
dot in a circuit
Loop: any closed path in a circuit;
formed by starting at a node, passing
through a set of nodes, and returning to
the starting node without passing
through any node more than once.
Note:
1. Two or more elements are
in series if they
exclusively share a single
node and consequently
carry the same current.
2. Two or more elements are
in parallel if they are
connected to the same two
nodes and consequently
have the same voltage
across them.
Fig. 2.4: Nodes, branches and loops
4. KIRCHHOFF’S LAWS


First introduced in 1874 by the German physicist Gustav Robert Kirchhoff
(1824-1887)
Kirchhoff’s current law (KCL):
a) The algebraic sum of currents entering a node (or closed boundary) is
zero.
b) The sum of the currents entering a node is equal to the sum of the currents
leaving the node.
c) Mathematically,
N
i
n 1
n
0

Kirchhoff’s voltage law (KVL):
a) The algebraic sum of all voltages around a closed path (or loop) is zero.
b) Sum of voltage drops = sum of voltage rises
c) Mathematically,
M
v
m 1
Fig. 2.5: KCL
m
0
Fig. 2.6: KVL
5. SERIES RESISTORS AND VOLTAGE DIVISION


The equivalent resistance of any number of resistors connected in series is the
sum of the individual resistances.
Mathematically,
N
Req  R1  R2    R N   Rn
n 1

Principle of voltage division:
vn 
Rn
v
R1  R2    RN
Fig. 2.7: a) A single-loop circuit with two resistors in series, b) equivalent circuit
6. PARALLEL RESISTORS AND CURRENT DIVISION


The equivalent resistance of two parallel resistors is equal to the product of
their resistances divided by their sum.
Mathematically,
RR
Req  1 2
R1  R2
For a circuit with N resistors in parallel,
1
1
1
1



Req R1 R2
RN

Principle of current division:
i1 

R2 i
R1 i
, i2 
R1  R2
R1  R2
Extreme cases:
a) R2 = 0,
Req = 0 (refer equation); entire current flows
through the short circuit
b) R2 = ,
Req = R1; current flows through the path of least
resistance
Fig. 2.8: a) Two resistors in parallel, b) equivalent circuit
Fig. 2.9: a) A shorted circuit, b) an open circuit
7. WYE – DELTA TRANSFORMATIONS



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Implementation – three-phase networks, electrical filters, etc.
Delta to wye conversion: each resistor in the Y network is the product of the
resistors in the two adjacent  branches, divided by the sum of the three 
resistors.
R1 
Rb Rc
Ra  Rb  Rc
R2 
Rc Ra
Ra  Rb  Rc
R3 
Ra Rb
Ra  Rb  Rc
Wye to delta conversion: each resistor in the  network is the sum of all
possible products of Y resistors taken two at a time, divided by the opposite Y
resistor.
R R  R2 R3  R3 R1
Ra  1 2
R1
Rb 
R1 R2  R2 R3  R3 R1
R2
Rc 
R1 R2  R2 R3  R3 R1
R3
Y and  networks are said to be
balanced when
R1 = R2 = R3 = RY,
R
Ra = Rb = Rc =
Therefore conversion formulas:
RY 
R
3
or
R  3 RY
Fig. 2.10: Wye-delta transformations