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# Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege - PowerPoint PPT Presentation

Engineering 43. Chp 7 Step-by-Step Pulse Response. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. 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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Chp 7

Step-by-StepPulse Response

Bruce Mayer, PE

Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

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

• 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

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

• 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-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

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

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

Find RTH With Respect to the L Terminals

Inductor Example cont.3

• Then The Time Constant, 

• RTH by Series Calc

Inductor Example cont.4

• Then The Solution

• Alternatively use x = v in:

• Let’s Work This 1st Order Cap Problem

• Power Source DISengaged

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

VOLTAGE STEP

CURRENT STEP

TIME SHIFTED STEP

Pulse Construction

• Examples

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

• 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

• Let’s Work This 1st Order Cap Problem

• R1→4 = 2 kΩ

• Power Source ENgaged

• IF we Have Time