Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11EE201 Lecture 30 P. 1Complex Forcing FunctionsWhat are these curves?EE201 Lecture 30 P. 2Now what are they?t t t 0 0 0 sin (t)cos (t)cos (t) + sin (t)A cos (t) + B cos (t) = (A2 + B2)1/2 cos (t + ) = tan-1 (-B/A) = phase angleWouldn’t it be convenient to have a single function that contained ALL of the information above for ac circuits?ttsin(t + )cos(t + )jyxt1jEE201 Lecture 30 P. 3Complex number representationComplex number z = x + jyx = cos(t+); y = sin(t+)z = cos(t+) + j sin(t+)From Euler’s equation:z = ejt+ = ejt ejForm of complex excitations)()(tjstjseIeVForm of complex responses)()(tjmtjmeIeVEE201 Lecture 30 P. 4Now on to Sinusoidal Steady State Response (SSS). These are circuits that are driven by an ac sources. Premise for computing SSS response: The sum of complex exponentials, Anejωt, their derivatives, or their indefinite integrals of any order, is a complex exponential of the same frequency ω.Assume frequency (ω) is constant (for now).EE201 Lecture 30 P. 5Example: Find the SSS response for iL(t) given is(t) = Iscos(ωt)is(t)ir(t)R LiL(t)sinusoidal forcing functionStep 1 : Find differential equation for circuitKCL: iL(t) + ir(t) = is(t))()()(tiLRtiLRttisLL(1)EE201 Lecture 30 P. 6Step 2: Determine form of response )sin()cos()( tBtAtiLNote: Since input is sinusoidal (cosine wave), the response is sinusoidal(2))]sin()cos([)]sin()cos([ tBtALRtBtAt)cos( tILRs0)sin()cos( tALRBtILRLRABsEvaluate at t = 0 and t = 20sILRLRABAt t = 0:EE201 Lecture 30 P. 720 ALRBAt t = :Solving simultaneous equations for A , B2222LRIRAs;222LRRLIBsStep 3: Express the steady state response)sin()cos()(2222222tLRRLItLRIRtissLPut into form )cos()( tItimL222BAIm22222LRRIBAIsmEE201 Lecture 30 P. 8RLIRRLIss121tantanRLtLRRItisL1222tancos)(ANow do same example using complex forcing functions.Note minus sign!!Basic concept: exchange real inputs and responses with complex ones, e.g.)()cos(tjsseVtVEE201 Lecture 30 P. 9Example: Find the SSS response)(tiLgiventjsseIti)(is(t)ir(t)R LiL(t)Step 1: Write differential equation)()()(tiLRtiLRttisLLEE201 Lecture 30 P. 10Step 2: Determine form of response22)()()()()()()()(LRLjRRILjRRIeIILReILRejIeILReeILReejIeILReILRejIejIttieItissjmsjmjmtjsjtjmjtjmtjstjmtjmtjmLtjmLEE201 Lecture 30 P. 11RLLRRIIsm122tan)(Magnitude of responsePhase of responseStep 3: Write responseRLtjseLeLRRIRti1tan(22)()(RLtLRRItisL1222tancos)(ANote: this is identical to previous
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