Chapter 5: Energy Storage and Dynamic...

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Chapter 5: Energy Storage and Dynamic Circuits

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  • Chapter 5: Energy Storage and Dynamic Circuits

  • Chapter 5: Outline

  • Capacitor (Brief)

    dtdvCvvip == :power ousinstantane

    221 :energy stored ousinstantane Cvw =

    Electrical memory

    Branch equation

    vc is the preferred variable.

    +==t

    tdi

    C

    ttvdi

    Ctv

    0

    )(1)0()(1)(

    +

    +=+

    jt

    jtdi

    Cjtvjtv )(

    1)()(

    Voltage continuity when the current is finite.

  • Inductor (Brief)

    dtdiLivip == :power ousinstantane

    221 :energy stored ousinstantane Liw =

    Branch equation

    iL is the preferred variable.

    Electrical memory

    Current continuity when the voltage is finite.

    +==t

    tdv

    L

    ttidv

    Lti

    0

    )(1)0()(1)(

  • Dynamic Circuits

  • Dynamic Circuits

    l A circuit is dynamic when currents or voltages are time-varying.

    l Dynamic circuits are described by differential equations.

    l Order of the circuit is determined by order of the differential equation.

    l The differential equations are derived based on Kirchhoffs laws and device (branch) equations.

  • First-Order Dynamic Circuits

    RidtdiLv += :formdirect Ri

    dtdiLv += :formindirect

    )(01 :ial)(different form Generic tfyadtdy

    a =+

    Forcing function

  • First-Order Circuits

    ivRdt

    dvc =+

    1 =++=

    t

    dtdv

    iCdt

    diRid

    CRiv

    1

    1

    KCL KVL

    or vvdt

    dvRC c

    c =+Generally, use vC or iL.

  • Second-Order Circuits

    l Second-order circuits: circuits described by a second-order differential equation.

    )(012

    2

    2 :form Generic tfyadtdya

    dt

    yda =++

  • Second-Order Circuits

    CRL vvvv ++= :KVL

    iCdt

    diR

    dtid

    Ldtdv 1

    :form aldifferentiIn 22

    ++=

  • Example 5.8: Second-Order Amplifier Circuit

    inoutout v

    dtdv

    RCdtvd

    LC =+22

    dtdvC

    i out

    =Ridtdi

    Lvin +=

  • Response: - Natural, Forced- Transient, Steady state- many more,

  • Natural Response

    l Natural response yN(t) is the solution of the circuit equation with the forcing function set to zero. It is also known as the complementary solution.

    l With the forcing function set to zero, the differential equation becomes homogeneous.

    l A homogeneous differential equation can be solved using the characteristic equation along with the initial condition.

    l First-order circuits require one initial condition. Second-order circuits require two initial conditions.

  • First-Order Circuits

    001 =+ NyadtNdya

    0

    1

    0

    01

    )0(condition initialby determined is

    )(

    0 :equation sticCharacteri

    1

    0

    YAyA

    AeAety

    aa

    s

    asa

    N

    taa

    stN

    ==

    ==

    =

    =+

    +

  • Second-Order Circuits

    0012

    2

    2 =++ NyadtNdya

    dtNyda

    dtdy

    yAA

    eAeAty

    aaaaa

    s

    asasa

    NN

    ttN

    )0( and )0( conditions initial by two determined are ,

    )(

    24

    ,

    0 :equation sticCharacteri

    21

    21

    2

    02211

    21

    012

    2

    21

    ++

    +=

    =

    =++

  • Stable Circuit

    l A circuit is stable if the circuit variable yN(t) 0, as t.

    l A circuit is exponentially stable if the circuit variable yN(t) 0, as t in an exponential form.

  • Example 5.9: Capacitor Discharge

    C=300FR=2Mv(0-)=1000V 0,1000)(

    y)(continuit 1000)0()0(

    )(60011

    01

    0

    600

    600/

    >=

    ===

    =

    ===+

    =+

    +

    tVetv

    vvA

    AetvRC

    sRCs

    vdt

    dvRC

    t

    N

    NN

    tN

    NN

  • Forced Response

    l Forced response yF(t) is the solution of the inhomogeneous differential equation (i.e., the forcing function is not zero), independent of any initial conditions. The solution is also known as the particular solution.

    l Please refer to Table 5.3 for method of undetermined coefficients.

  • Table 5.3

  • Example 5.10 Sinusoidal Forced Response

    R=4L=0.1HV(t)=25sin30t

  • Example 5.10 Sinusoidal Forced Response

    ))30cos()((or 30sin430cos3)(

    4,3 have weon,substitutiafter 30sin30cos)(

    30sin2541.0

    43

    43

    +=+=

    ==+=

    =+

    tAtittti

    kktktkti

    tidtdi

    FF

    F

    FF

  • If the forcing function contains any term proportional to a component of the natural response (i.e., excitation of a natural frequency), then the term must be multiplied t.

  • Example 5.11: Exponential Forced Response

    btev

    ss

    vidtdi

    =

    ==+

    =+

    10

    40041.0 :solution shomogeneou

    41.0

    Aeti

    k

    ekti

    b

    tF

    tF

    20

    2

    202

    5)(

    5

    )(

    20 if

    =

    ==

    =

    Ateti

    k

    tekti

    b

    tF

    tF

    40

    2

    402

    100)(

    100

    )(

    40 if

    =

    =

    =

    =

  • Complete Response

    l Complete response is the sum of the natural response and the forced response, i.e., y(t) = yF(t) + yN(t) .

    l The constants in yN(t) are evaluated from the initial conditions on with the complete response.

    l For a stable circuit, y(t)= yF(t), as t, since yN(t)0, as t.

    l The circuit is in the steady state if yN(t) is negligible compared to yF(t).

    l Before arriving the steady state, the circuit is in the transient state.

  • Example 5.12: Complete Response Calculation (RL)

    14140)0(

    280sin2280cos14)()()( :response complete

    280sin2280cos14)( :solution particular

    )( :solution shomogeneou

    0for ,280sin400)(

    0for 0)( ,41.0

    40

    40

    =+==

    +=+=

    +==

    >=

  • Example 5.12: Complete Response Calculation (RL)

  • Chapter 5: Problem Set

    l 42, 44, 45, 48, 54, 61, 64