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  • 1

    U1. Transient circuits

    response

    Circuit Analysis, Grado en Ingeniería de Comunicaciones

    Curso 2016-2017

    Philip Siegmann (philip.siegmann@uah.es)

    Departamento de Teoría de la Señal y Comunicaciones

    mailto:philip.siegmann@uah.es

  • Index

    • Recall

    • Goals and motivation

    • Linear differential equation (Recall)

    • The homogeneous solution

    • 1st and 2nd order linear differential equations

    • Examples of 2nd order circuits

    • Initial conditions

    • Transient responses (example on 2nd order circuit)

    • Cause of the transient responses

    • Analysis using Laplace transform

  • Recall

    • Relation between i(t) and v(t) for the

    passive elements R,L,C

    – For R:

    – For C:

    – For L:

    • Energy in these elements:

    – Dissipated in R:

    – Stored in C and L:

    Circuit Analysis / Transient circuits response / Recall

    )()( tRitv 

    dt

    tdv CtitCvtq CCC

    )( )()()( 

    dt

    tdi Ltv LL

    )( )( 

    )(tiL

    L vL(t)

    )(tiC

    C vC(t)

    )(tiR

    R vR(t)

     2)( 2

    1 )( tvCtW CC   

    2 )(

    2

    1 )( tiLtW LL 

     dttiRtW RR )()( 2

    dt

    tdq ti

    dq

    tdW tv

    )( )(,

    )( )( 

  • Goals

    • We want to solve circuits for whatever applied

    source (not only DC and Sinusoidal Steady

    State)

    – Direct resolution in the time domain

    – Resolution using Laplace transforms

    • In particular we want to understand what

    happens when an abrupt change takes place in

    the circuit, which will produce the transient

    response.

    4

    Circuit Analysis / Transient circuits response / Goals

  • Motivation

    • Signal transmission

    Transient response

    Circuit Analysis / Transient circuits response / Motivation

    S.S.S.

  • 6

    Examples of 2nd order circuits

    • RLC-serial (http://en.wikipedia.org/wiki/RLC_circuit )

    • RLC-parallel

    dt

    tde

    L ti

    LCdt

    tdi

    L

    R

    dt

    tid

    tetq Cdt

    tdi LtRi

    tetvtvtv

    g

    g

    gCLR

    )(1 )(

    1)()(

    )()( 1)(

    )(

    )()()()(

    2

    2

    

    

    

     

    dt

    tdi

    C tv

    LCdt

    tdv

    RCdt

    tvd

    dt

    tdi

    dt

    tvd Ctv

    Ldt

    tdv

    R

    titititi dt

    d

    g

    g

    gCLR

    )(1 )(

    1)(1)(

    )()( )(

    1)(1

    )()()()(

    2

    2

    2

    2

    

    

    

    (Energy conservation)

    (Charge conservation)

    Circuit Analysis / Transient circuits response / Example 2nd order circuits

    http://en.wikipedia.org/wiki/RLC_circuit

  • 7

    Transient response

    • Response of a circuit (voltage or current) when

    an abrupt change happens (e.g. switching)

    • Time evolution until achieving a new equilibrium

    • The transition function follows exponential

    variations (decreasing or increasing, fluctuating

    or no fluctuating)

    • They are solutions of linear differential equations

    Circuit Analysis / Transient circuits response / Introduction

  • 8

    General solution

    ),()()(

    ),()(... )()(

    01

    1

    1

    tytyty

    tgtya dt

    tyd a

    dt

    tyd

    ph

    n

    n

    nn

    n

    

     

    yh(t) is the solution for g(t)=0: the Complementary, natural or homogeneous

    solution. Gives the transient behavior of the circuit due to the passive elements.

    It is dependent of the initial conditions.

    yp(t) is a particular solution for the given source or forcing function g(t). The

    particular solution looks like the forcing function, e.g.:

    - If g(t) is constant, then yp(t) is constant

    - If g(t) is sinusoidal, then yp(t) is sinusoidal (i.e. the S.S.S.)

    The homogeneous solution decreases exponentially so that

    y(t)  yp(t)

    Linear differential equation of order n:

    Circuit Analysis / Transient circuits response / Linear differential equation

  • 9

    The homogeneous solution

    yh(t) is the solution for g(t)=0 (external energy

    supply =0)

    The solution has the form: yh(t)=A·exp(st) (“Ansatz”)

    Circuit Analysis / Transient circuits response / The homogeneous solution

      stkst k

    k

    AsA dt

    d ee :since 

     

    

    

    

    n

    k

    ts

    k

    ts

    n

    tsts

    h

    n

    n

    n

    n

    kn AAAAty

    ssss

    asas

    1

    21

    21

    0

    1

    1

    ee...ee)(

    },...,,{

    equation polynomial sticcharacteri ,0...

    21

    A1, A2, … are obtained with the initial (and/or boundary) conditions

    characteristic polynomial equation

    (“Eigenwerte”)

  • 10

    1st and 2nd order linear

    differential equations

    2

    0

    1

    012

    2

    2

    ),()( )()(

    n

    n

    a

    a

    tgtya dt

    tdy a

    dt

    tyd

    

    

     is called the damping ratio (accounts for the energy loss) n is called the natural frequency (maximum energy storage) (http://en.wikipedia.org/wiki/Damping)

    1

    ),()( )(

    0

    0

    

    a

    tgtya dt

    tdy

     is a time constant (how fast yh(t) decreases)

    Circuit Analysis / Transient circuits response / 1st nd 2nd order differential equations

    1st order

    2nd order

    For circuits containing

    one energy storage

    element (C or L)

    For circuits containing

    two independent energy

    storage element

    http://en.wikipedia.org/wiki/Damping

  • 11

    Example of 2º orden circuits

    • RLC-serial (http://en.wikipedia.org/wiki/RLC_circuit )

    • RLC-parallel

    L

    R

    LC

    dt

    tde

    L ti

    LCdt

    tdi

    L

    R

    dt

    tid

    n

    n

    g

     

    2 ,

    1

    )(1 )(

    1)()( 2

    2

    

    

    CRLC

    dt

    tdi

    C tv

    LCdt

    tdv

    RCdt

    tvd

    n

    n

    g

     

    2

    1 ,

    1

    )(1 )(

    1)(1)( 2

    2

    

    

    Circuit Analysis / Transient circuits response / Ejemplos circuitos 2º orden

    http://en.wikipedia.org/wiki/RLC_circuit

  • 12

    Initial conditions

    • For each energy storage element we need an

    initial condition (at t=t0):

    – For C: vC(t=t0) = V0

    – For L: iL(t=t0) = I0

    Circuit Analysis / Transient circuits response / Initial conditions

     2)( 2

    1 )( tiLtW LL 

     2)( 2

    1 )( tvCtW CL 

    vC(t) iL(t)

    (t)

    E(t)

  • 13

    Condiciones iniciales

    • When an abrupt change happens at t=t0 in a

    circuit, there is always continuity in the

    variation of the energies in C and L 

    – There is continuity in the voltage at C:

    – There is continuity in the current trough L:

    Circuit Analysis / Transient circuits response / Initial conditions

    )()( 00 

     tvtv CC

    )()( 00 

     titi LL

    Justo después de t0 Justo antes de t0

    (not so for the current iC(t))

    (not so for the voltage vL(t))

  • 14

    Example of transient response

    0 1

    :equationsticCharacteri

    ,e)(

    form theof solutionsth equaion wi free) (source Homogeneus

    ,0)( 1)()(

    ,0 )(

    )( 1

    )(

    0For

    )0(

    0)0(

    :0atconditionsInitial

    2

    2

    

    

    

      

    LC s

    L

    R s

    Ati

    ti LCdt

    tdi

    L

    R

    dt

    tid

    dt

    tdi Ltq

    C tRi

    t

    Ev

    i

    t

    2

    st

    gC

    Discharge of the capacitor

    Characteristic equation:

    Circuit Analysis / Transient circuits response / Transient responses

    dt

    tdi LtRitvtvtv LRC

    )( )()()()( 

    Once i(t) is known we can obtain vC(t):

    For t ≥0

  • 15

    Type of solutions of

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