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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


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


  • With This Analysis Find
    • Time Constant using RTH
    • Final Condition using vTH
the steps 1 4
STEP 1. Assume The Form Of The SolutionThe Steps: 1-4
    • STEP 3: Draw The Circuit At t = 0+
    • 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 constantThe 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 SolutionInductor 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




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