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Engineering 43. Chp 7 Step-by-Step Pulse Response. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] 1 st Order Ckts: Step-by-Step. This Approach Relies On The Known Form Of The Solution But Finds The ODE Parameters Using Basic Circuit Analysis Tools

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Engineering 43

Chp 7

Step-by-StepPulse Response

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]


1 st order ckts step by step
1st Order Ckts: Step-by-Step

  • This Approach Relies On The Known Form Of The Solution But Finds The ODE Parameters Using Basic Circuit Analysis Tools

  • This Method Eliminates the Need For The Determination Of The Differential Equation Model

  • Most Useful When Variable of Interest is NOT vC or iL


Basic concept
Basic Concept

  • Recall The form of the ODE Solution for a Ckt w/ One E-Storage Element and a Constant Driving Ckt

  • Where

    • K1 The final Condition for the Variable of Interest

      • Can Be determined by Analyzing the Ciruit in Steady State; i.e., t→

    • x(0+)  The Initial Condition for the Variable

      • Provides the Second Eqn for Calculating K2

    •   Ckt Time Constant

      • Determine By Finding RTH Across the Storage Element


The general approach

FC

The General Approach

  • Obtain The Voltage Across The Capacitor or The Current Through The Inductor

Thevenin

  • With This Analysis Find

    • Time Constant using RTH

    • Final Condition using vTH


The steps 1 4

STEP 1. Assume The Form Of The Solution

The Steps: 1-4

  • STEP 3: Draw The Circuit At t = 0+

  • The CAPACITOR Acts As a VOLTAGE SOURCE

  • The INDUCTOR Acts As a CURRENT SOURCE

  • Determine The VARIABLE of INTEREST At t=0+

  • Determine x()

    • STEP 4: Draw The Circuit a Loooong Time After Switching to Determine The Variable In Steady State

    • Determine x(0+)

      • STEP 2: Draw The Circuit In Steady State just PRIOR To Switching And Determine Capacitor-Voltage Or Inductor-Current


    The steps 5 6

    STEP 5: determine the time constant

    The Steps: 5-6

    • With These 3-Parameters Write the Solution For the Variable of interest using The Assumed Solution

    • RTH Determined at Cap/Ind Connection Terminals

    • Step-By-Step DOWNside

      • Do NOT have ODE So Can NOT easily Check Solution

        • Can usually chk the FINAL Condition

    • STEP 6: Determine The Constants K1 & K2,


    Step by step inductor example
    Step-By-Step: Inductor Example

    • STEP-1: The Form of the Soln

    • For the Circuit Below Find vO for t>0

    • STEP-2: Initial inductor current (L is Short to DC)

    • Note That vO is NOT Directly Related to The Storage Element

      • → Use Step-by-Step


    Inductor example cont

    STEP 3: Determine output at 0+

    By Inductor Physics

    Inductor Example cont.

    • Note That at t=0+

      • The 6V Source is DISCONNECTED from the Ckt Elements

        • No Connection on Supply Side

      • Single Loop Ckt

    • At t=0+, Replace The L with a 3A Current Src


    Inductor example cont 2

    STEP 4: Find Output In Steady State After The Switching

    By Inductor Physics In Steady State

    L is SHORT to DC

    Inductor Example cont.2

    • Recall at t=0+The 6V Source is DISconnected from the Ckt Elements

      • The Ckt Has NO Power Source

      • Over A long Time All the Energy Stored by The Inductor Will be Dissipated as HEAT by The Resistors, Hence


    Inductor example cont 3

    STEP 5: Find Time Constant After Switch

    Find RTH With Respect to the L Terminals

    Inductor Example cont.3

    • Then The Time Constant, 

    • RTH by Series Calc


    Inductor example cont 4

    STEP 6: Find The Solution

    Inductor Example cont.4

    • Then The Solution

    • Alternatively use x = v in:


    Whiteboard work
    WhiteBoard Work

    • Let’s Work This 1st Order Cap Problem

      • Power Source DISengaged


    Pulse response
    Pulse Response

    • Consider The Response Of Circuits To A Special Class Of SINGULARITY functions

    VOLTAGE STEP

    CURRENT STEP

    TIME SHIFTED STEP


    Pulse construction

    Pulse = Sum of Steps

    Pulse Construction

    • Examples


    Piecewise transient repsonse

    Non-Zero Initial Condition (std ODE)

    PieceWise Transient Repsonse

    • This expression will hold on ANY interval where the sources are CONSTANT

    • The values of the constants may be different and must be evaluated for each interval

    • The values at the END of one interval will serve as INITIAL conditions for the NEXT interval

    • The Response is Shifted From the Time Origin by an Amount t0

    • For CONSTANT fTH, The Time-Shifted Exponential Solution


    Piecewise example

    Piecewise constant source

    PieceWise Example

    • The Switch is Initially At a. At Time t=0 It Moves To b, and At t=0.5 it moves back to a.

    • Find vO(t) for t>0

    • On Each Interval Where The Source is Constant The Response Will Be of the Form


    Piecewise example cont

    For 0t<0.5 (Switch at b)

    t0 = 0

    Assume Solution

    PieceWise Example cont

    • Now Piece-2 (Switch at a)

      • t0=0.5S

    • Find Parameters And Piece-1 Solution



    Whiteboard work1
    WhiteBoard Work

    • Let’s Work This 1st Order Cap Problem

      • R1→4 = 2 kΩ

      • Power Source ENgaged

        • IF we Have Time


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