basic laws (a)web.nchu.edu.tw/~ycchiang/circuits_i/ckt1_ch2a.pdfbasic laws (a) 2 ohm’s law •...

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Basic Laws (a)

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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

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Resistivity

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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

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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

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Resisters

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Resisters

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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

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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: