Introduction to Circuit Theory
Basic Laws
2012-09-25
Jieh-Tsorng Wu
National Chiao-Tung University
Department of Electronics Engineering
Outline
1.
2.
3.
4.
5.
Ohm’s Law
Nodes, Branches, and Loops
Kirchhoff’s Laws
Resistor Networks
Applications
2. Basic Laws
2
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Ohm’s Law
Ohm’s law states that the voltage across a resistor is directly proportional to
the current flowing through the resistor.
v
v2
2
Resistance  R 
Power  p  vi  i R 
i
R
1   1 V/A
i
Conductance  G 
v
1 S  1   1 A/V
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3
i2
Power  p  vi  v G 
G
2
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Short Circuit and Open Circuit
2. Basic Laws
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Resistor
l
R  
A
2. Basic Laws
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Resistivity
2. Basic Laws
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Fixed Resistors
2. Basic Laws
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Variable Resistors
2. Basic Laws
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Linear and Nonlinear Resistors
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Nodes, Branches, and Loops
 A branch represents a single element such as a voltage source or a
resistor.
 A node is the point of connection between two or more branches.
 A loop is any closed path in a circuit. A loop is independent if it contains
at least one branch which is not a part of any other independent loop.
 A network with b branches, n nodes, and l independent loops will satisfy
the fundamental theorem of network topology:
b  l  n 1
 Two or more elements are in series if they exclusively share a single node
and consequently carry the same current.
 Two or more elements are in parallel if they are connected to the same
two nodes and consequently have the same voltage across them.
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Example 1
How many branches, nodes and loops are there?
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Example 2
Should we consider it as one
branch or two branches?
How many branches, nodes and loops are there?
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Kirchhoff’s Current Law (KCL)
Kirchhoff’s current law (KCL) states that the algebraic sum of currents
entering a node (or a closed boundary) is zero.
i
n
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0
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Kirchhoff’s Voltage Law (KVL)
Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages
around a closed path (or loop) is zero.
v
m
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0
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KCL Example 1
Find current I for the circuit shown below.
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KCL Example 2
Find current io and vo for the circuit shown below.
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KVL Example
Find current I for the circuit shown below.
I
2. Basic Laws
va  vb
R1  R2  R3
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KCL+KVL Example
Find currents and voltages for the circuit shown below.
KCL
i1  i2  i3
KVL
30  v1  v2  0   30  8i1  3i2  0
v2  v3  0   3i2  6i3  0
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Current Sources in Parallel
2. Basic Laws
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Voltage Sources in Series
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Series Resistors and Voltage Division
 Series: Two or more elements are in series if they are cascaded or connected
sequentially and consequently carry the same current.
 The equivalent resistance of any number of resistors connected in a series is the
sum of the individual resistances.
N
Req  R1  R2    RN  Rn
n 1
 The voltage divider can be expressed as
vn  v 
2. Basic Laws
Rn
R1  R2    RN
v  i  Req
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Series Resistor Example
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Parallel Resistors and Current Division
 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.
 The equivalent resistance of a circuit with N resistors in parallel is:
N
1
1
1
1
1
 
 

Req R1 R2
RN n 1 Rn
N
Geq  G1  G2    Gn  Gn
n 1
 The current divider can be expressed as:
in  i 
2. Basic Laws
Gn
G1  G2    GN
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i  v  Geq
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Parallel Resistors Example
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Resistor Example 1
Find Req.
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Resistor Example 2
Find Rab.
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Resistor Example 3
Find io and vo. Calculate the power dissipated in the 3 resistor.
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Resistor Example 4
Find vo. Find the power supplied by the current source. Find the
power dissipated by each resistor.
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Y (T) Network and  () Network
Y Network
R12  R1  R3 
Rb ( Ra  Rc )
Ra  Rb  Rc
R13  R1  R2 
Rc ( Ra  Rb )
Ra  Rb  Rc
R34  R2  R3 
Ra ( Rb  Rc )
Ra  Rb  Rc
 Network
 Network
T Network
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Y- Transformations
-to-Y
Y-to-
R1 
Rb Rc
( Ra  Rb  Rc )
Ra 
R1 R2  R2 R3  R3 R1
R1
R2 
Rc R a
( Ra  Rb  Rc )
Rb 
R1 R2  R2 R3  R3 R1
R2
R3 
Ra Rb
( Ra  Rb  Rc )
Rc 
R1 R2  R2 R3  R3 R1
R3
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Resistor Example
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Applications: Lighting Systems
v1  v2  ...  vN 
v1  v2  ...  vN  V0
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V0
N
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Applications: DC Meters
Parameters:
IFS: full-scale current
Rm: meter resistance
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Voltmeter and Ammeter
RA  R
RV  R
2. Basic Laws
R
R
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Voltmeter
Single Range
VFS  I M   Rn  Rm 
Multiple Range
VFS 1  I M   R1  Rm 
VFS 2  I M   R2  Rm 
VFS 3  I M   R3  Rm 
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Ammeter
Single Range
Rn
I M  I FS 
Rn  Rm
Multiple Range
I FS 1  I M   R1  Rm  R1
I FS 2  I M   R2  Rm  R2
I FS 3  I M   R3  Rm  R3
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Ohmmeter
Rx 
V
I
Rx 
E
 ( R  Rm )
Im
Let I m  I M  I FS when Rx  0, then
E  ( R  Rm ) I FS
I

Rx   FS  1 ( R  Rm )
 Im

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Voltmeter and Ammeter Combination
I meter  I element
I meter  I element
Vmeter  Velement
Vmeter  Velement
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