Bruce Mayer, PE Licensed Electrical &amp; Mechanical Engineer [email protected]

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Engineering 43. Chp 7 Step-by-Step Pulse Response. Bruce Mayer, PE Licensed Electrical &amp; 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]

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
• 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

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
STEP 1. Assume The Form Of The SolutionThe 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
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-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
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
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
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
STEP 6: Find The SolutionInductor Example cont.4
• Then The Solution
• Alternatively use x = v in:
WhiteBoard Work
• Let’s Work This 1st Order Cap Problem
• Power Source DISengaged
Pulse Response
• Consider The Response Of Circuits To A Special Class Of SINGULARITY functions

VOLTAGE STEP

CURRENT STEP

TIME SHIFTED STEP

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 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
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 Work
• Let’s Work This 1st Order Cap Problem
• R1→4 = 2 kΩ
• Power Source ENgaged
• IF we Have Time