EE201 Lecture 30 Complex Forcing Functions What are these curves P 1 EE201 Lecture 30 P 2 Now what are they sin t 0 t cos t t 0 0 t cos t sin t A cos t B cos t A2 B2 1 2 cos t tan 1 B A phase angle Wouldn t it be convenient to have a single function that contained ALL of the information above for ac circuits EE201 Lecture 30 P 3 Complex number representation sin t 1 jy j t x t cos t Complex number z x jy x cos t y sin t t z cos t j sin t From Euler s equation EE201 Lecture 30 P 4 Form of complex excitations Vs e j t I s e j t Form of complex responses Vm e j t I m e j t Now 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 Anej t their derivatives or their indefinite integrals of any order is a complex exponential of the same frequency Assume frequency is constant for now EE201 Lecture 30 P 5 Example Find the SSS response for iL t given is t Iscos t is t ir t R iL t L sinusoidal forcing function Step 1 Find differential equation for circuit KCL iL t ir t is t iL t R R iL t is t t L L 1 EE201 Lecture 30 P 6 Step 2 Determine form of response Note Since input is sinusoidal cosine wave the response is sinusoidal iL t A cos t B sin t 2 R A cos t B sin t A cos t B sin t t L R I s cos t L RA R RB B I cos t A sin t 0 s L L L Evaluate at t 0 and t RA 2 R At t 0 B L L I s 0 EE201 Lecture 30 At t 2 P 7 RB A 0 L Solving simultaneous equations for A B R2Is A 2 R L2 2 B RLI s R 2 L2 2 Step 3 Express the steady state response R2Is RLI s iL t 2 2 2 cos t 2 2 2 sin t R L R L Put into form iL t I m cos t 2 I m A2 B 2 2 2 Im A B RI s R 2 L2 2 EE201 Lecture 30 P 8 RLI s 1 L tan 2 tan R R Is 1 Note minus sign 1 L iL t cos t tan 2 2 2 R R L RI s Now do same example using complex forcing functions Basic concept exchange real inputs and responses with complex ones e g Vs cos t Vs e j t A EE201 Lecture 30 Example Find the SSS response iL t given is t I s e j t is t iL t ir t R L Step 1 Write differential equation iL t R R iL t is t t L L P 9 EE201 Lecture 30 P 10 Step 2 Determine form of response iL t I m e j t iL t I m j e j t t R R j t j t I m j e I me I s e j t L L R R j t j j t j I m j e e I m e e I s e j t L L R R j j I m j e I m e I s L L RI s RI s R j L j I me 2 R j L R L 2 EE201 Im Lecture 30 P 11 RI s R 2 L 2 L tan 1 R Magnitude of response Phase of response Step 3 Write response L j t tan 1 RI s R iL t Re e 2 2 R L 1 L iL t cos t tan A R R 2 L2 2 RI s Note this is identical to previous solution
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