Transcript ENT163 02-08 - UniMAP Portal

FUNDAMENTALS OF ELECTRICAL
ENGINEERING
[ ENT 163 ]
LECTURE #2
BASIC LAWS
HASIMAH ALI
Programme of Mechatronics,
School of Mechatronics Engineering, UniMAP.
Email: [email protected]
CONTENTS
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Introduction
Ohm’s Law
Nodes, Branches and Loops
Kirchhoff’s Laws
Series Resistors and Voltage Division
Parallel Resistors and current Division
Introduction
• Fundamental laws – govern electric circuits:
1. Ohm’s Law
2. Kirchhoff’s law
• Technique commonly applied:
1.
2.
3.
4.
Resistors in series/ parallel
Voltage division
Current division
Delta-to-wye and wye-to-delta transformations
Ohm’s Law
• Materials in general have a characteristics behavior of resisting the flow of electric
charge.
• Resistance: ability to resist current (R) .
R

A
Material
Resistitivity(   m )
Usage
Silver
1.64 x 10-8
Conductor
Copper
1.72 x 10-8
Conductor
Aluminum
2.8 x 10-8
Conductor
Gold
2.45 x 10-8
Conductor
Carbon
4 x 10-5
Semiconductor
Germanium
47 x 10-2
Semiconductor
Silicon
6.4 x 102
Semiconductor
Paper
1010
Insulator
Mia
5 x 1011
Insulator
Glass
1012
Insulator
Teflon
3 x 1012
Insulator
Ohm’s Law
• Resistor: circuit element that used to model the current – resisting behavior.
•Relationship between current and voltage for a resistor Ohm’s law
Ohm’s law states that the voltage v across a resistor is directly
proportionally 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
v  iR
• Resistance, R: denotes its ability to resist the flow of electric current(Ω)
• The two extreme possible values of R:
• R= 0 is called a short circuit (v= i R=0)
• R=∞ is called open circuit
v
i  lim
R  
R
0
Ohm’s Law
A short circuit is a circuit element with resistance approaching zero
An open circuit is a circuit element with resistance approaching
infinity.
• Resistor is either fixed or variable.
• Fixed resistors have constant resistor.
• The two common types of fixed resistors:
• Wire wound
• Composition
• Variable resistor have adjustable resistance. Example potentiometer.
• Reciprocal of resistance, known as conductance, G:
G
1 i

R v
The conductance is the ability of an element to conduct electric current ,
it is measured in mhos or Siemens (S).
Ohm’s Law
• The power is dissipated by a resistor is expressed by:
v2
p  vi  i R 
R
2
or
i2
p  vi  v G 
G
2
Note:
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. Thus
resistor is a passive element, incapable of generating energy.
Nodes, Branches and Loops
• A network – interconnection of elements or devices.
• A circuit – network providing one/ more closed paths.
• Branch represents a single elements such as a voltage source or
resistor. A branch represents any two-terminal element.
• Node is the point of connection between two or more branches.
• It usually indicated by a dot in a circuit.
• Loop is 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 ant node more than once.
5Ω
b
a
10 V +
-
2Ω
3Ω
c
2A
Nodes, Branches and Loops
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.
Nodes, Branches and Loops
Example:
Determine the number of branches and nodes in the circuit shown
in Figure 2.12. Identify which elements are in series and which are
in parallel.
5Ω
10 V +
-
6Ω
Figure 2.12
2A
Kirchhoff’s Law
•
First introduced in 1874 by the German physicist Gustav Robert
Kirchhoff (1824-1887)
•
Kirchhoff’s current law (KCL):
1. The algebraic sum of currents entering a node (or close boundary) is zero.
2. The sum of the currents entering a node is equal to the sum of the currents
leaving the node.
N
i
n 1
n
0
i1  ()i2  i3  i4  ()i5  0
i5
i1
i3
i2
i4
Fig: Currents at a node illustrating KCL
i1  i3  i4  i2  i5
Kirchhoff’s Law
IT
IT
a
a
I1
I2
I3
I S  I1  I 2  I 3
=
b
b
IT  I 2  I1  I 3
IT  I1  I 2  I 3
Kirchhoff’s Law
•
Kirchhoff’s voltage law (KVL):
1. The algebraic sum of all voltages around a closed path (or loop) is zero.
M
v
m 1
m
0
2. Sum of voltage drops is equal to sum of voltage rises.
+ v2 -
v1
+ v3 -
+
-
 v1  v2  v3  v4  v5  0
+
v4
v2  v3  v5  v1  v4
- v5 +
Sum of voltage drop = Sum of voltage rises
Kirchhoff’s Law
•
Kirchhoff’s voltage law (KVL):
a
+
Vab
+
V1
+
V2
a+
=
b
+
b -
+
Vab
VS  V1  V2  V3
-
V3
v2  v3  v5  v1  v4
Kirchhoff’s Law
Example:
For the circuit in Fig. 2.21 find voltages v1 and v2.
2Ω
+ v1 20V
+
-
v2
+
3Ω
Kirchhoff’s Law
Problem:
Find v1and v2 in the circuit of the Fig.
4Ω
+ v1 10V
+
-
8V
+ v2 2Ω
+
Kirchhoff’s Law
Problem:
Determine voand i in the circuit of the Fig.
i
4Ω
2vo
+ -
+ v1 12V
+
-
4V
+ vo -
6Ω
+
Kirchhoff’s Law
Problem:
Find current io and voltage vo in the circuit shown in Fig. below.
a
io
+ vo -
0.5io
4Ω
3A
Series Resistors and Voltage Division
Consider a single-loop circuit with two resistors in series
i
v
•a
+ v1 -
+ v2 -
R1
R2
Applying Ohm’s law of each resistor
v1  iR
+
-
(1)
If apply KVL to the loop (CW), we have
b
i
•a
+
-
+ v
Req
-
Combing (1) and (2):
v  v1  v2  i( R1  R2 )
v  iReq
b
•
(2)
 v  v1  v2  0
•
v
v2  iR2
or
i
Req  R1  R2
v
R1  R2
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  R3 ...  RN   RN
n 1
To determine the voltage across each resistor shown in Fig,
v1 
R1
v
R1  R2
, v2 
Principle of voltage division:
vn 
RN
v
R1  R2  ...  RN
R2
v
R1  R2
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,
Req 
R1 R2
R1  R2
• For a circuit with N resistors in parallel,
1
1
1
1
 
 .. 
Req R1 R2
RN
Parallel Resistors And Current Division
Consider the circuit below where two resistors are connected in parallel and
therefore have the same voltage across them. From Ohm’s law
Node a
i
i2
i1
v
+
-
R2
R1
i1 
Node b
Substitute (3) into (4), we get
i
v
v
1
1
v

 v(  ) 
,
R1 R2
R1 R2
Req
1
1
1
 
Req R1 R2
v1  i1R1  i2 R2
Req 
R1 R2
R1  R2
v
,
R1
i2 
v
,
R2
(3)
Applying KCL at node a gives
i  i1  i2
(4),
• For a circuit with N resistors in parallel,
1
1
1
1
 
 .. 
Req R1 R2
RN
Parallel Resistors And Current Division
• Principle of current division:
i1 
•
R2i
R1  R2
i2 
R1i
R1  R2
Extreme cases:
1. R2=0,-Req=0 ; entire current flows through the short circuit.
2. R2= ∞ , Req=R1; current flows through the path of least resistance.
Wye – Delta Transformation
•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 – Delta Transformations
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.
Ra 
R1 R2  R2 R3  R3 R1
R1
Rb 
R1 R2  R2 R3  R3 R1
R2
Rc 
R1 R2  R2 R3  R3 R1
R3
Wye – Delta Transformations
Y and ∆ networks are said to be balanced when
R1  R2  R3  RY
Ra  Rb  Rc  R
Therefore conversion formulas:
R1 
R
3
R  3RY