Basic Laws [§â€¸‡®¹ˆ¨Œ‡¼ˆ] - National Chiao Tung Basic Laws ¢â‚¬¢Ohm¢â‚¬â„¢s Law (resistors) ¢â‚¬¢Nodes, Branches,

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  • 2012/9/17

    1

    Basic Laws

    •Ohm’s Law (resistors) •Nodes, Branches, and Loops •Kirchhoff’s Laws •Series Resistors and Voltage Division •Parallel Resistors and Current Division •Wye-Delta Transformations •Applications

    Ohm’s Law •Resistance R is represented by

    •Ohm’s Law:

    A R

     

    Rv +

    _

    i

    1 = 1 V/A

    Cross-section area A

    Meterial resistivity  

    ohm

    Riv 

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    2

    Resistors

    0Riv == R = 0v = 0

    +

    _

    i

    R = v

    +

    _

    i = 0

    0 R v

    limi R

    == ∞→

    Variable resistor Potentiometer (pot)

    Open circuitShort circuit

    Nonlinear Resistors

    i

    v

    Slope = R

    v

    i

    Slope = R(i) or R(v)

    Linear resistor Nonlinear resistor

    •Examples: lightbulb, diodes •All practical resistors may exhibit certain

    nonlinear behavior.

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    3

    Conductance and Power Dissipation

    •Conductance G is represented by

    v i

    R G 

    1 1 S = 1 = 1 A/V

    siemens mho

    G i

    Gvivp

    R v

    Riivp

    vGi

    2 2

    2 2

    ===

    ===

    =

    positive R : power absorption (+)

    negative R: power generation (-)

    Nodes, Branches, & Loops •Branch: a single element (R,

    C, L, v, i)

    •Node: a point of connection between branches (a, b, c)

    •Loop: a closed path in a circuit (abca, bcb, etc) –An independent loop contains

    at least one branch which is not included in other indep. loops.

    –Independent loops result in independent sets of equations.

    +_

    a

    c

    b

    +_

    c

    ba

    redrawn

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    4

    Continued Elements in parallelElements in series

    •Elements in series –(10V, 5)

    • Elements in parallel –(2, 3, 2A)

    •Neither –((5/10V), (2/3/2A))

    10V

    5

    2 3 2A+_

    Kirchhoff’s Laws •Introduced in 1847 by German physicist G. R.

    Kirchhoff (1824-1887).

    •Based on conversation of charge and energy.

    •Two laws are included,Kirchhoff’s current law (KCL) andKirchhoff’s votage law (KVL).

    •Combined withOhm’s law, we have a powerful set of tools for analyzing resistive circuits.

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    5

    KCL

    i1 i2

    in 0211  nn

    N

    n iiii 

    •Assumptions –The law of conservation of charge –The algebraic sum of charges within a system

    cannot change.

    •Statement –The algebraic sum of currents entering a node

    (or a closed boundary) is zero.

    Proof of KCL

    proved)(KCLanyfor0)( )(

    anyfor0)( it.onstoredbetoallowednotisCharge

    object.physicalanotisnodeA

    )()(

    )()( 1

    tti dt

    tdq ttq

    dttitq

    titi

    T T

    T

    TT

    n

    N

    nT

    

     

    

     

    i1 i2

    in

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    Example 1

    i1

    i3i2 i4

    i5

    leaving,entering,

    52431

    54321 0)-()-(

    TT

    T

    ii

    iiiii

    iiiiii

     

    

    Example 2

    321

    312

    IIII

    IIII

    T

    T

     

    I1 I2 I3

    IT IT

    321 IIIIS 

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    7

    Case with A Closed Boundary

    cancelled.arecurrentsbranchInternal

    0

    0

    0

    1111

    

    

    

    

    

    baab

    n bn

    m am

    j bj

    i ai

    iiii

    ii

    ii

    a

    Treat the surface as a big node

    leavingentering ii 

    b

    ia1 ib1

    KVL

    0 1

      m

    M

    m v

    •Assumption –The principle of conservation of energy

    •Statement –The algebraic sum of all voltage drops (or rises)

    around a closed path (or loop) is zero.

    v1+ _ v2+ _ vm+ _

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    8

    Proof of KVL

     

    proved)(KVLanyfor0)( 0)(

    anyfor0)()()()( )(

    anyforconstant)( givesenergyofonconservatitheofprincipleThe

    )()()(

    )()(

    1

    1

    ttv ti

    ttitvtitv dt

    tdw ttw

    dttitvtw

    tvtv

    T

    Tm

    M

    m

    T

    T

    TT

    m

    M

    mT

     

    

    

    v1+ _ v2+ _ vM+ _

    i

    Example 1

    41532

    54321 0

    vvvvv

    vvvvvvRT 

    

    v4v1

    v5

    +_ +_

    +_

    v2+ _ v3+ _

    Sum of voltage drops = Sum of voltage rises

  • 2012/9/17

    9

    Example 2

    321

    321 0

    VVVV

    VVVV

    ab

    ab

     

    V3

    V2

    V1

    Vab

    +_

    +_

    +_

    +

    _

    a

    b

    Vab +_

    +

    _

    a

    b 321 VVVVS 

    Example 3 Q: Find v1 and v2. Sol:

    V12,V8 A4 205

    03220 (2),Eq.into(1)Eq.ngSubstituti

    (2)020 givesKVLApplying

    (1)3,2 ,lawsOhm'From

    21

    21

    21

      

    

    

    

    vv i i

    ii

    vv

    iviv

    v1+ _

    v2 +

    _

    20V

    2

    3+_ i

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    10

    Example 4 Q: Find currents and voltages. Sol:

      (3)8330 03830

    030 1,looptoKVLAppying

    (2)0 givesKCL,nodeAt

    (1)6,3,8 ,lawsOhm'By

    21

    21

    21

    321

    332211

    ii

    ii

    vv

    iii a

    iviviv

     

    

    

    

    V6V,6V,24

    A1A,3A2 gives(2)Eq.(5),Eq.&(3)Eq.By

    (5)236 (1),Eq.By

    (4)0 2,looptoKVLAppying

    321

    312

    2323

    2332

    

    

    

    

    vvv

    iii

    iiii

    vvvv

    v1+ _

    30V

    8

    3+_

    i1

    6 +

    _ v3

    i3

    i2

    Loop 1 Loop 2

    a

    +

    _ v2

    b

    Example 5

    Q: Find vo. Sol:

    V5,A1 053035

    (2),Eq.into(1)Eq.ngSubstituti

    (2)0235 givesKVLApplying

    (1)5,10 ,lawsOhm'From

     

    

    

    o

    oxx

    ox

    vi ii

    vvv

    iviv

    i

  • 2012/9/17

    11

    Series Resistors

     

    (5)

    ,Let

    (4)or

    (3) (2),Eq.&(1)Eq.By

    (2)0 KVL,Applying

    (1), ,lawsOhm'By

    21eq

    eq

    21

    2121

    21

    2211

    RRR

    iRv RR

    v i

    RRivvv

    vvv

    iRviRv

    

     

    

    

     v1+ _

    v

    R1

    +_

    i

    v2+ _

    R2a

    b

    v +_

    i

    v+ _

    Reqa

    b

    Voltage Division

    v R R

    v RR

    R iRv

    v R R

    v RR

    R iRv

    eq

    2

    21

    2 22

    eq

    1

    21

    1 11

     

    

     

     v1+

    _

    v

    R1

    +_

    i

    v2+ _

    R2a

    b

    v +_

    i

    v+ _

    Reqa

    b

  • 2012/9/17

    12

    Continued

    v R R

    v RRR

    R v

    GGGG

    RRRR

    eq

    n

    N21

    n n

    N21eq

    N21eq

    1111

     

    

    

    v +_

    i

    v+ _

    Reqa

    b

    v1+ _

    v

    R1

    +_

    i

    v2+ _

    R2a

    b

    vN+ _

    RN

    Parallel Resistors

    (5)or

    (4) 111

    (3) 11

    (2),Eq.&(1)Eq.By (2)

    ,nodeatKCLApplying

    (1),or

    ,lawsOhm'By

    21

    21

    21eq

    eq2121

    21

    2 2

    1 1

    2211

    RR RR

    R

    RRR

    R v

    RR v

    R v

    R v

    i

    iii a

    R v

    i R v

    i

    RiRiv

    eq  

    

     

      

     

    

    

     i a

    b

    R1+_ R2v

    i1 i2

    i a

    b

    Req or Geq+_v v

  • 2012/9/17

    13

    Current Division

    i GG

    G RR iR

    R v

    i

    i GG

    G RR iR

    R v

    i

    RR RiR

    iRv

    21

    2