1
Basic Laws (a)
2
Ohm’s Law
• Resistance (R)– The ability of an element to resist the flow of electric
current, in ohms ().
• Resistivity ()– A general property of
materials: the ability to resist current measuredin ohm-meters (-m).
(2.1)RA
A
3
Resistivity
4
Ohm’s Law
• Resistor– The circuit element used to model the current-resisting
behavior of a material.• Ohm’s law
– The voltage v across a resistor is directly proportional to the current i flowing through the resistor.
(2.2)v i
(2.3)v iR
(2.4)vRi
1 = 1 V/A
5
Ohm’s Law
• Short circuit– A circuit element with resistance approaching zero.
• Open circuit– A circuit element with resistance approaching infinity.
0R 0 (2.5)v iR
R
lim 0 (2.6)R
viR
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Ohm’s Law
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Resisters
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Resisters
9
Resisters
10
Resisters
11
Resistors
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Conductance
• Conductance (G)– The ability of an element to conduct electric current, in
mhos ( ) or siemens (S).
– For a resistor,
1 (2.7)iGR v
1 S 1 1 A/V (2.8) (2.9)i Gv
10 0.1 S
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Power Dissipated
• Using Eqs. (1.7) and (2.3):2
2 (2.10)vp vi i RR
22 (2.11)ip vi v G
G
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Example 2.1
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Practice Problem 2.1
• The essential component of a toaster is an electrical element (aresistor) that converts electrical energy to heat energy. How much current is drawn by a toaster with resistance 10 at 110 V?
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Example 2.2
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Example 2.2 (cont.)
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Practice Problem 2.2
• For the circuit shown in Fig. 2.9, calculate the voltage v, the conductance G, and the power p.
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Example 2.3
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Practice Problem 2.3
• A resistor absorbs an instantaneous power of 20cos2t mWwhen connected to a voltage source v = 10 cost V. Find i and R.
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Nodes, Branches, and Loops
• Branch – A branch represents a single element such as a voltage
source or a resistor…• Node
– A node is the point of connection between two or more branches
• Loop – A loop is any closed path in a circuit
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Nodes, Branches, and Loops
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Nodes, Branches, and Loops
• The fundamental theorem of network topology – A network with b branches, n nodes, and l independent
loops will satisfy:
• Series – 2 or more elements exclusively share a single node and
consequently carry the same current.• Parallel
– 2 or more elements are connected to the same 2 nodes and consequently have the same voltage across them.
1 (2.12)b l n
24
Example 2.4
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Example 2.4 (cont.)
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Practice Problem 2.4
• How many branches and nodes does the circuit in Fig. 2.14 have? Identify the elements that are in series and in parallel.
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Kirchhoff’s Laws
• Kirchhoff’s current law (KCL)– The algebraic sum of currents entering a node (or a closed
boundary) is zero.
10 (2.13)
N
nn
i
Law of conservation of electric charge
1 2 3 4 5( ) ( ) 0 (2.16)i i i i i
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Kirchhoff’s Laws
• KCL– The sum of the currents
entering a node is equalto the sum of thecurrents leaving thenode.
1 3 4 2 5 (2.17)i i i i i
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Kirchhoff’s Laws
• A current can not contain 2 different currents, I1 and I2, in series, unless I1 = I2; otherwise KCL will be violated.
(2.18)
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Kirchhoff’s Laws
• Kirchhoff’s voltage law (KVL)– The algebraic sum of all voltages around a closed path (or
loop) is zero.
10 (2.19)
M
mm
v
1 2 3 4 5 0 (2.20)v v v v v
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Kirchhoff’s Laws
• KVL: Sum of voltage drops = Sum of voltage rises (2.22)
2 3 5 1 4 (2.21)v v v v v
1 2 3 (2.23)abV V V V
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Example 2.5
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Example 2.5 (cont.)
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Practice Problem 2.5
• Find v1 and v2 in the circuit of Fig. 2.22.
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Example 2.6
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Example 2.6 (cont.)
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Practice Problem 2.6
• Find vx and vo in the circuit of Fig. 2.24.
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Example 2.7
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Example 2.7 (cont.)
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Practice Problem 2.7
• Find vo and io in the circuit of Fig. 2.26.
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Example 2.8
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Example 2.8 (cont.)
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Example 2.8 (cont.)
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Example 2.8 (cont.)
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Practice Problem 2.8
• Find the currents and voltages in the circuit shown in Fig. 2.28.
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Series Resistors and Voltage Division
• Applying Ohm’s law
• Applying KVL1 1 2 2, (2.24)v iR v iR
1 2 0 (2.25)v v v 1 2 1 2( ) (2.26)v v v i R R
1 2
(2.27)( )
viR R
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Series Resistors and Voltage Division
• Equivalent resistance of series resistors– Sum of individual resistances– For N resistors in series,
• Principle of voltage division
– For N resistors in series, the nth resistor have a voltage drop:
eq (2.28)v iR eq 1 2 (2.29)R R R
eq 1 21
(2.30)N
N nn
R R R R R
1 21 2
1 2 1 2
, (2.31)R Rv v v vR R R R
1 2
(2.32)nn
N
Rv vR R R
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Parallel Resistors and Current Division
• From Ohm’s law:
• Applying KCL:
1 1 2 2v i R i R 1 21 2
, (2.33)v vi iR R
1 2 (2.34)i i i
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Parallel Resistors and Current Division
• Equivalent resistance of 2 parallel resistors– Product of their resistance divided by their sum
• Equivalent resistance of N parallel resistors
– For N equal resistors in parallel:
1 2 1 2 eq
1 1 (2.35)v v vi vR R R R R
eq 1 2
1 1 1 (2.36)R R R
1 2eq
1 2
(2.37)R RRR R
eq 1 2
1 1 1 1 (2.38)NR R R R
eq (2.39)RRN
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Parallel Resistors and Current Division
• Equivalent conductance of N parallel resistors– Sum of individual conductances
• Principle of current divisioneq 1 2 3 (2.40)NG G G G G
2 11 2
1 2 1 2
, (2.43)R i R ii iR R R R
1 2eq
1 2
(2.42)iR Rv iRR R
1 21 2
1 2 1 2
, (2.44)G i G ii iG G G G
1 2
(2.45)nn
N
Gi iG G G
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Parallel Resistors and Current Division
• 2 extreme cases: