Presentation W12D2 Answers

Embed Size (px)

Citation preview

  • 8/9/2019 Presentation W12D2 Answers

    1/37

    1

    W12D2RC, LR, and

    Undriven RLC Circuits;Experiment 4

    Today’s Reading Course Notes: Sections 11.7-11.9, 11.10,

    11.13.6; !"t. #: $ndri%en R&C Circuits

  • 8/9/2019 Presentation W12D2 Answers

    2/37

    Announcements

    'at( Re%ie) *ee+ 13 Tuesday 9"-11 " in 6-1

    /S 9 due *ee+ 13 Tuesday at 9 " in o!es outside 3-0 or 6-1

    Ne!t Reading 2ssignent *13 Course Notes: Sections 11.-9, 11.1-11.13

    2

  • 8/9/2019 Presentation W12D2 Answers

    3/37

    3

    Outline

    !"erient #: /art 1 RC and &R Circuits

    Si"4e 5aronic sci44ator 

    $ndri%en R&C Circuits

    !"erient #: /art $ndri%en R&C Circuits

  • 8/9/2019 Presentation W12D2 Answers

    4/37

    4

    RC Circuit Car!in!

    So4ution to t(is euation )(en

    s)itc( is c4osed at t 8 0: dQ

    dt 

     = −1

    RC

    Q− Cε ( )

     

  • 8/9/2019 Presentation W12D2 Answers

    5/37

    5

    RC Circuit" Discar!in!

    So4ution to t(is euation )(en

    s)itc( is c4osed at t  8 0

    tie constant:

     

    dQ

    dt 

     = −1

    RC

    Q

     Q(t ) =Qoe−t /RC

  • 8/9/2019 Presentation W12D2 Answers

    6/37

    6

    RL Circuit" #ncreasin! Current

    So4ution to t(is euation )(ens)itc( is c4osed at t 8 0:

     =

    t /(

     units: seconds

     

    I (t ) =  ε 

    R

    (1− e−t /τ )

  • 8/9/2019 Presentation W12D2 Answers

    7/377

    RL Circuit" Decreasin! Current

    So4ution to t(is euation )(ens)itc( is o"ened at t  8 0:

     0− 

    units: seconds

     

    I (t ) =  ε 

    R

    (1− e−t /τ )

  • 8/9/2019 Presentation W12D2 Answers

    8/378

    $easurin! %ime Constant

      y 

    2(t 

    2) = y 

    2(t 

    1 + τ ) = y 

    0e

    −(t 1 +τ ) /τ  = y 

    0e

    −(t 1

    /τ )e−1 = y 

    1(t 

    1)e−1

    /ic+ a "oint 1 )it(

    ind "oint suc( t(at

    n t(e 4a you )i44 "4ot sei-4og and =it cur%e a+e sureyou e!c4ude data at ot( ends

      y 1(t 1) = y 0e− (t 

    1/τ )

     τ 

     ≡t 

    2 −t 1

      y 

    2(t 

    2) = y 

    1(t 

    1)e−1

     

    ln( y (t ) / y 0) = lne− (t /τ ) = −(t / τ ) ⇒

    ln y (t ) = ln y 0 − (t / τ ) ⇒ τ  = −1/slope  y (t ) = y 

    0e− (t /τ ) ⇒

  • 8/9/2019 Presentation W12D2 Answers

    9/379

    Experiment 4"

    RC and RL Circuits

  • 8/9/2019 Presentation W12D2 Answers

    10/3710

    $ass on a &prin!"

    &imple 'armonic $otion

  • 8/9/2019 Presentation W12D2 Answers

    11/3711

    Demonstration

    $ass on a &prin!"

    &imple 'armonic $otion$ass on a &prin! (C 2)

    (tt":??scri"ts.it.edu?@tsg?)))?deo."("A4etnu8CB0s(o)80

    http://scripts.mit.edu/~tsg/www/demo.php?letnum=C%202&show=0http://scripts.mit.edu/~tsg/www/demo.php?letnum=C%202&show=0

  • 8/9/2019 Presentation W12D2 Answers

    12/3712

    $ass on a &prin!

     

    F  = −kx = ma= md2 x 

    dt 2

    md2 x 

    dt 2  + kx = 0

      x (t ) = x 

    0

    cos(ω 0

    t + φ )

    1

    3 #

    *(at is 'otionA

     

    ω 0 =

    m= Angular freuenc!

    Si"4e 5aronic 'otion

     x 0: 2"4itude o= 'otion

    φ : /(ase tie o==set

  • 8/9/2019 Presentation W12D2 Answers

    13/3713

    &imple 'armonic $otion

     2"4itude !0

      x (t ) = x 

    0cos(ω 

    0t + φ )

     

    "erio# =

    1

    freuenc! → T  =

    1

    "erio# =2π 

    angular freuenc!→ T  =

    2π 

    ω  

    "$ase %$ift (φ ) = −π 

    2

  • 8/9/2019 Presentation W12D2 Answers

    14/37

    Concept *uestion" &imple

    'armonic Oscillator 

    *(ic( o= t(e =o44o)ing =unctions x t  (as a second

    deri%ati%e )(ic( is "ro"ortiona4 to t(e negati%e o= t(e

    =unction

     x (t ) = Acos2π 

    T  t 

     

     

     

     ÷

      x (t ) = Ae−t /T 

      x (t ) = Aet /T  

     x (t ) =1

    2at 2

    2

    2&d x   x 

    dt µ −

    1.

    .

    3.

    #.

  • 8/9/2019 Presentation W12D2 Answers

    15/37

    Concept *uestion Ans+er" &imple

    'armonic Oscillator 

    Ans+er 4 

  • 8/9/2019 Presentation W12D2 Answers

    16/3716

    $ass on a &prin!" Ener!-

     x (t ) = x 0 cos(ω 0t + φ )

    1 S"ring 'ass 3 S"ring # 'ass

    nergy (as "arts: 'ass Dinetic and S"ring /otentia4

     

    K  = 12m dx 

    dt   

      ÷

    2

    = 12kx 

    0

    2 sin2 (ω 0t + φ )

    Us =

    1

    2

    kx 2 =1

    2

    kx 0

    2cos2 (ω 0t + φ )

     

    dx 

    dt  = v  x (t ) = −ω 0 x 0 sin(ω 0t + φ )

    nergy

    s4os(es ac+

    and =ort(

  • 8/9/2019 Presentation W12D2 Answers

    17/37

    17

    LC Circuit

    1. Set u" t(e circuit ao%e)it( ca"acitor, inductor,

    resistor, and attery.

    . &et t(e ca"acitor ecoe

    =u44y c(arged.

    1. T(ro) t(e s)itc( =ro a to b.

    1. *(at (a""ensA

  • 8/9/2019 Presentation W12D2 Answers

    18/37

    18

    LC Circuit

    >t undergoes si"4e (aronic otion, Eust 4i+e aass on a s"ring, )it( trade-o== et)een c(arge on

    ca"acitor S"ring and current in inductor 'ass.

    ui%a4ent4y: trade-o== et)een energy stored in

    e4ectric =ie4d and energy stored in agnetic =ie4d.

  • 8/9/2019 Presentation W12D2 Answers

    19/37

  • 8/9/2019 Presentation W12D2 Answers

    20/37

  • 8/9/2019 Presentation W12D2 Answers

    21/37

    21

    Concept * Ans+er" LC Circuit

     2ns)er: #. T(e current is

    a!iu )(en t(e c(arge on

    t(e ca"acitor is Fero

    Current and c(arge are e!act4y 90 degrees out o=

    "(ase in an idea4 &C circuit no resistance, so )(en

    t(e current is a!iu t(e c(arge ust e

    identica44y Fero.

    LC Ci it &i l ' i

  • 8/9/2019 Presentation W12D2 Answers

    22/37

    22

    LC Circuit" &imple 'armonic

    Oscillator 

     

    d2Q

    dt 2  +

    1

    LCQ = 0 ⇒

     

    Q(t ) = Q0cos(ω 

    0t + φ )

     ω 

    0 = 1 / LC

    C(arge:

     2ngu4ar =reuency:

     2"4itude o= c(arge osci44ation:

     

    /(ase tie o==set:

     

    Q

    C − LdI

    dt  = 0 ' I = −dQ

    dt 

    φ 

    Si"4e (aronic osci44ator:

     Q

    0

  • 8/9/2019 Presentation W12D2 Answers

    23/37

    23

    LC Oscillations" Ener!-

     U =U

    E +U

    B =

    Q2

    2C +

    1

    2LI 2 =

    Q0

    2

    2C

     

    UE =

    Q2

    2C

     =Q

    0

    2

    2C

     

     

     

     ÷cos2ω 

    0t 

     

    UB =

    1

    2

    LI 2 =1

    2

    LI0

    2 sin2ω 0t =

    Q0

    2

    2C

     

     

     

     ÷sin2ω 

    0t 

    Tota4 energy is conser%ed GG

    Notice re4ati%e "(ases

  • 8/9/2019 Presentation W12D2 Answers

    24/37

    LC CircuitOscillation

    &ummar-

  • 8/9/2019 Presentation W12D2 Answers

    25/37

    25

    Addin! Dampin!"

     RLC Circuits

  • 8/9/2019 Presentation W12D2 Answers

    26/37

    26

    Demonstration

    Undriven RLC Circuits (. 1/0)

    http://scripts.mit.edu/~tsg/www/demo.php?letnum=Y%20190&show=0http://scripts.mit.edu/~tsg/www/demo.php?letnum=Y%20190&show=0

  • 8/9/2019 Presentation W12D2 Answers

    27/37

    RLC Circuit" Ener!- Can!es

     >nc4ude =inite resistance:

    'u4ti"4y y

     I =

    dQ

    dt  ⇒

    ecrease in stored

    energy is eua4 to Hou4e

    (eating in resistor 

  • 8/9/2019 Presentation W12D2 Answers

    28/37

    28

    Damped LC Oscillations

    Resistor dissi"ates energy

    and syste rings do)n o%er

    tie. 24so, =reuencydecreases:

     Q(t ) =Q

    0e−(R/ 2L)t cos(ω t )

     ω  =   ω 02

    − (R / 2L)2

     ω 

    0 = 1 / LC > R / 2L

  • 8/9/2019 Presentation W12D2 Answers

    29/37

    29

    Experiment 4" art 2

    Undriven RLC Circuits

  • 8/9/2019 Presentation W12D2 Answers

    30/37

    30

    Appendix" Experiment 4"

    art 2Undriven RLC Circuits

    roup ro3lem

    and Concept *uestions

  • 8/9/2019 Presentation W12D2 Answers

    31/37

    31

    ro3lem" LC Circuit

    Consider t(e circuit s(o)n in t(e =igure. Su""ose t(e s)itc(

    t(at (as een connected to "oint a =or a 4ong tie is sudden4y

    t(ro)n to b  at t  8 0. ind t(e =o44o)ing uantities:

    a t(e =reuency o= osci44ation o= t(e circuit.

    t(e a!iu c(arge t(at a""ears on t(e ca"acitor.

    c t(e a!iu current in t(e inductor.

    d t(e tota4 energy t(e circuit "ossesses as a =unction o= tie t .

     

  • 8/9/2019 Presentation W12D2 Answers

    32/37

    1. 1

    .

    3. 3

    #. #

    Concept *uestion" Expt 4>n today’s 4a t(e attery turns on

    and o==. *(ic( circuit diagra isost re"resentati%e o= our circuitA

    1. .

    3. #.

    &oad 4a )(i4e )aitingI

  • 8/9/2019 Presentation W12D2 Answers

    33/37

    Concept *uestion Ans+er" Expt 4

     2ns)er: 1.

    T(ere is resistance in t(e circuitin our non-idea4 inductor.

    T(e attery s)itc(ing o== doesn’trea+ t(e circuit ut a44o)s it to

    ring do)n

  • 8/9/2019 Presentation W12D2 Answers

    34/37

    Concept *uestion" LC Circuit

    T(e "4ot s(o)s t(e c(argeon a ca"acitor 4ac+ cur%e

    and t(e current t(roug( it

    red cur%e a=ter you turn

    o== t(e "o)er su""4y. >= you"ut a core into t(e inductor

    )(at )i44 (a""en to t(e

    tie T &ag

    A 0 #0 0 10-1.0J

    0

    -0.J0

    0.0J0

    0.J0

    1.0J0

    -1.0>0

    -0.>0

    0.0>0

    0.>0

    1.0>0

     

    T4ag

       C   (  a  r  g  e  o  n   C  a  "  a  c   i   t  o  r

    Tie S

     C(arge

        C  u  r  r  e  n   t   t   (  r  o  u  g   (   C  a  "  a  c   i   t  o  r

     Current

    1. >t )i44 increase. >t )i44 decrease3. >t )i44 stay t(e sae

    #. > don’t +no) 34

    Concept *uestion Ans+er" LC

  • 8/9/2019 Presentation W12D2 Answers

    35/37

    35

    Concept *uestion Ans+er" LC

    Circuit

      2ns)er 1.

    T &ag )i44 increase.

    utting in a core increases

    (e inductor’s inductance and

    ence decreases t(e natura4reuency o= t(e circuit.

    o)er =reuency eans

    onger "eriod. T(e "(ase )i44

    eain at 90K a uartereriod so T 

    &ag )i44 increase.

    0 #0 0 10

    -1.0J0

    -0.J0

    0.0J0

    0.J0

    1.0J0

    -1.0>0

    -0.>0

    0.0>0

    0.>0

    1.0>0

     

    T4ag

       C   (  a  r  g  e  o  n   C  a  "  a  c   i   t  o  r

    Tie S

     C(arge

        C  u  r  r  e  n   t   t   (  r  o  u  g   (   C

      a  "  a  c   i   t  o  r

     Current

  • 8/9/2019 Presentation W12D2 Answers

    36/37

    36

    Concept *uestion" LC Circuit

    >= you increase t(eresistance in t(e circuit

    )(at )i44 (a""en to rate

    o= decay o= t(e "ictured

    a"4itudesA0 #0 0 10

    -1.0J0

    -0.J0

    0.0J0

    0.J0

    1.0J0

    -1.0>0

    -0.>0

    0.0>0

    0.>0

    1.0>0

     

    T4ag

       C

       (  a  r  g  e  o  n   C  a  "  a  c   i   t  o  r

    Tie S

     C(arge

        C  u  r

      r  e  n   t   t   (  r  o  u  g   (   C  a  "  a  c   i   t  o

      r

     Current

    1. >t )i44 increase decay ore ra"id4y

    . >t )i44 decrease decay 4ess ra"id4y3. >t )i44 stay t(e sae#. > don’t +no)

    Concept *uestion Ans+er" LC

  • 8/9/2019 Presentation W12D2 Answers

    37/37

    Concept *uestion Ans+er" LC

    Circuit2ns)er: 1. >t )i44 increase

    decay ore ra"id4y

    Resistance is )(at dissi"ates "o)er in t(e circuit

    and causes t(e a"4itude o= osci44ations todecrease. >ncreasing t(e resistance a+es t(e

    energy and (ence a"4itude decay ore ra"id4y.

    0 #0 0 10

    -1.0J0

    -0.J0

    0.0J0

    0.J0

    1.0J0

    -1.0>0

    -0.>0

    0.0>0

    0.>0

    1.0>0

     

    T4ag

       C   (  a

      r  g  e  o  n   C  a  "  a  c   i   t  o  r

    Tie S

     C(arge

        C  u  r  r  e  n

       t   t   (  r  o  u  g   (   C  a  "  a  c   i   t  o  r

     Current