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    ENAD Formula Sheet

    Uploaded by

    Rachel Tan
    This document contains a formula sheet for electrical network analysis and design. It includes formulas and equations for second order transient analysis of RLC circuits, characteristic equations, damping factors, resonant frequencies, general solutions to natural and step responses, distinct real and complex roots, useful Laplace transform pairs, convolution integrals, frequency selective circuits, transfer functions for different filter types, magnitudes of transfer functions at cutoff frequencies, two-port networks, and transformers.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    ENAD Formula Sheet

    Uploaded by

    Rachel Tan
    101 views7 pages
    This document contains a formula sheet for electrical network analysis and design. It includes formulas and equations for second order transient analysis of RLC circuits, characteristic equations, damping factors, resonant frequencies, general solutions to natural and step responses, distinct real and complex roots, useful Laplace transform pairs, convolution integrals, frequency selective circuits, transfer functions for different filter types, magnitudes of transfer functions at cutoff frequencies, two-port networks, and transformers.

    Original Description:

    electrical networks formula sheet

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    This document contains a formula sheet for electrical network analysis and design. It includes formulas and equations for second order transient analysis of RLC circuits, characteristic equations, damping factors, resonant frequencies, general solutions to natural and step responses, distinct real and complex roots, useful Laplace transform pairs, convolution integrals, frequency selective circuits, transfer functions for different filter types, magnitudes of transfer functions at cutoff frequencies, two-port networks, and transformers.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    101 views7 pages

    ENAD Formula Sheet

    Uploaded by

    Rachel Tan
    This document contains a formula sheet for electrical network analysis and design. It includes formulas and equations for second order transient analysis of RLC circuits, characteristic equations, damping factors, resonant frequencies, general solutions to natural and step responses, distinct real and complex roots, useful Laplace transform pairs, convolution integrals, frequency selective circuits, transfer functions for different filter types, magnitudes of transfer functions at cutoff frequencies, two-port networks, and transformers.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 7
    At a glance
    Powered by AI
    The key takeaways are about second order circuit analysis including characteristic equations, solutions to natural and step responses, and damping factors. It also discusses useful Laplace transform pairs and frequency selective circuits.

    The different types of transient responses for second order circuits are overdamped response, underdamped response, and critically damped response. The document provides the general solutions for the natural response and step response of each case.

    Some common frequency selective circuit transfer functions discussed are the general form for low-pass filters, high-pass filters, band-pass filters, and band-reject filters. Each is expressed as a function of s in the Laplace domain.

    Formula

    Sheet for ELEN30009 Electrical Network Analysis and Design




    Second Order Transient Analysis of RLC Circuits




    Characteristic Equation
    Neper frequency or damping factor

    (rad/s)
    2
    2
    s + 2 s + 0 = 0

    1

    Parallel RLC, =
    2RC



    R

    Series RLC, =
    2L



    Resonant radiant frequency (rad/s)
    Damped radiant frequency (rad/s)

    1
    0 =
    d = 0 2 2

    LC


    General solutions to natural response of
    second-order circuits

    2
    2
    Overdamped response, 0 <

    x t = A1e s1t + A2e s2t

    s1 , s2 = 2 02


    02 > 2
    Underdamped response,

    x ( t ) = B1e t cos d t + B2e t sin d t
    s1 , s2 = jd




    02 = 2
    Critically damped response,

    t
    + D2e t
    x t = D1te

    s1 = s2 =

    ()

    ()


    General solutions to step response of
    second-order circuits

    2
    2
    Overdamped response, 0 <
    x t = X f + A '1 e s1t + A '2 e s2t

    s1 , s2 = 2 02



    2
    2
    Underdamped response, 0 >
    x t = X f + B '1 e t cos d t
    + B '2 e t sin d t
    s , s = j
    1 2
    d


    02 = 2
    Critically damped response,
    t
    t

    ()

    ()

    x ( t ) = X f + D '1 te

    s1 = s2 =

    22

    + D '2 e







    Distinct Real Roots


    Repeated Real Roots


    Distinct Complex Roots


    Repeated Complex Roots


    Useful Transform Pairs



    F(s)

    K
    s+a

    Ke at u ( t )

    (s + a)
    K
    s + j
    K

    ( s + j )

    f(t)

    Kte at u ( t )

    2 K e t cos ( t + ) u ( t )

    s + + j
    K

    ( s + + j )

    2t K e t cos ( t + ) u ( t )



    Convolution Integral

    y ( t ) = h ( t ) x ( t ) = x ( t ) h (t )


    y ( t ) = h ( t ) x ( t ) = h ( ) x ( t ) d

    y (t ) = x (t ) h (t ) =

    x ( ) h (t ) d


    Frequency selective circuits


    General form of transfer function for
    low-pass filters
    K

    H (s) =


    General form of transfer function for
    high-pass filter
    Ks

    H (s) =

    s + C


    General form of transfer function for
    band-pass filter
    s
    H ( s ) = 2
    2


    General form of transfer function for
    band-reject filter

    s 2 + 02

    H (s) = K

    s + s + 0

    s + C

    23

    s 2 + s + 02


    Magnitude of transfer function, defined at
    at cutoff frequency, C

    H ( jC ) =

    1
    H max
    2


    Cutoff frequencies of bandpass and
    bandreject filters

    2

    C1 ,C1 = + 2 + o


    Two Port Terminal Equations

    V1 = z11 I1 + z12 I 2
    V2 = b11V1 b12 I1


    V2 = z21 I1 + z22 I 2
    I 2 = b21V1 b22 I1



    I1 = y11V1 + y12V2
    V1 = h11 I1 + h12V2


    I 2 = y21V1 + y22V2
    I 2 = h21 I1 + h22V2



    V1 = a11V2 a12 I 2
    I1 = g11V1 + g12 I 2

    I1 = a21V2 a22 I 2

    V2 = g 21V1 + g 22 I 2

    24

    25

    26

    27




    Transformers (positive dot convention)

    M = k L1L2
    di
    di
    v1 = L1 1 + M 2
    dt
    dt

    N2
    v1
    N1
    N
    i2 = 1 i1
    N2
    v2 =

    v2 = M

    di1
    di
    + L2 2
    dt
    dt

    N2
    v1
    N1
    N
    i2 = ! 1 i1
    N2
    v2 = !

    28

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