Sinusoidal Response Notes

Transcription

1 ECE 30 Sinusoidal Rspons Nots For BIBO Systms AStolp /29/3 Th sinusoidal rspons of a systm is th output whn th input is a sinusoidal (which starts at tim 0) Systm Sinusoidal Rspons stp input H( s) output stp rspons X m x( y( y ss ( t + t t y tr ( t Sinusoidal Input Complt stp rspons stady-stat rspons + cos( ω u( t ) <> transint rspons Y( s ) X( s) H( s ) X m H( u( t ) + Y tr ( s) H( ω ) s sin( ω u( t ) <> Gnral sinusoidal input: X mc cos( ω t ) X ms ( sin( ω u( ) u( X( s ) Stady-Stat Rspons & H( X Y( s ) X( s) mc s X ms ω H( s ) H( s ) X mc s X ms ω partial fraction xpansion: Y( s ) H( s ) Complt sinusoidal rspons A s B ω + phasor-typ transfr function C ( ) b X mc s X ms ω D ( ) E ( ) + A s B ω C D E + s ( ) ( ) ( ) stady-stat rspons + transint rspons Y ss ( s ) + Y tr ( s) multiply both sids by: X mc s X ms ω H( s ) A s B C D E ω + ( ) ( ) ( ) st s ω X mc ω X ms ω H( ω ) A ω B C D E ω + 0 ( ) ( ) ( ) X( H( ω ) A ω B ω Y ss ( s ) stady-stat rspons in phasor form X( ω ) th input xprssd in phasor form H( ω ) th stady-stat sinusoidal transfr function phasor-typ transfr function Th transint part would b found by finishing th partial-fraction xpansion Sinusoidal Rspons Nots p

2 Stady-Stat Rspons by Phasors Exprssion of signals as phasors T Priod Sinusoidal Rspons Nots p2 f frquncy, cycls / scond f T ω 2 π ω radian frquncy, radians/sc ω 2 π f A amplitud t Phas: φ T 360 or: φ t T 2 π rad y( t ) A cos( ω t φ) Phasor voltag: v( t ) V p cos( ω t φ) V( ω ) V p φ Phas: currnt: i( t ) I p cos( ω t φ) I( ω ) I p φ Ex Lt's assum th input to your systm is v ( t ) 32 cos( ω t V ( ω ) 32 or: V ( ω ) 32V / o In rctangular form: or: 32 cos( 309 V 32 sin( 0828 V V ( ω ) ( ) V Ex2 What if a signal is th sum of two sinusoids v ( t ) 32 cos( ω t v ( v 2 ( t ) 4 sin( ω t 60 v 3 ( t ) v ( v 2 ( I'm going to drop th ( notation from th phasor notation, it gts cumbrsom, but rmmbr that phasors ar in th frquncy domain From Ex: V 32 V 32V / o V V v 2 ( tim Phasors ar basd on cosins, so xprss v 2 ( as a cosin Rmmbr: sin( ωt ) cos( ω t 90 So: v 2 ( t ) 4 cos( ω t ) 4 cos( ω t 30 V 2 4V /-30 o or: V 2 4 V 30 4 cos( V 4 sin( V V V \ } V V / Add ral parts: Add imaginary parts: Chang V 3 back to polar coordinats: atan OR, in Mathcad notation (you'll s ths in futur solutions): V 3 V 3 73 V arg V 3 V 3 ( ω ) 73V /- o or: V 3 ( ω ) 73 may also b convrtd back to th tim domain: v 3 ( t ) v ( v 2 ( t ) 73 cos( ω t V Sinusoidal Rspons Nots p2 V 3 V V 2 add V V sum drawing of th phasor diagram

3 Magnitud and Phas of transfr functions With stady-stat sinusoidal inputs Ex3 a) Find th magnitud and phas of th following transfr function at this frquncy: ω 2 rad sc H( s ) s ω H( ω ) 2 s 2 s 20 s 2 s 0 2 ( 2 ( 20 ( 2 sc ( ω ) 2 s 2 sc s 20 sc 2 s 2 sc s 0 sc 2 0 sc 2 Exprssd with propr units 20 2 ω 2 ( 0 ω 2 ( without units ( 2) ( 2) H( ω ) M / H( atan 0 atan b) Find th stady-stat sinusoidal output if th input is: 32 cos( 2 t V in 32 V 32V / o V outss ( ω ) V in ( H( ω ) 32 V v outss ( t ) 74 cos( 2 t 3637 ( V Ex4 a) Find th magnitud and phas of th following transfr function at this frquncy: f Hz ω 2 π f H( s ) s 2 20 sc s 000 sc 2 s 2 0 sc s 800 sc 2 s ω 342 rad sc H( ω ) ( 2 20 sc ( ω ) ( 2 0 sc ( ω ) 000 sc sc 2 ( 342 ) 20 ( 342) 000 ( 342) 2 0 ( 342) ω 342 rad sc without units H( ω ) M / H( atan b) Find th stady-stat sinusoidal output if th input is: x( t ) 4 cos( 2 π Hz X( ω ) 4 0 and thn Y( ω ) 4 ( ) Not that you can us th rctangular form of H( y( t ) 836 cos 342 rad sc t 364 sin 342 rad sc t not that th sin carris th opposit sign as th imaginary part atan y( t ) 688 cos 342 rad sc t 32 Sinusoidal Rspons Nots p3

4 Impdancs sris: Z q Z Z 2 Z 3 + Exampl: Z q R C s L s Voltag dividr: V Zn V total Z n Z Z 2 Z 3 + paralll: Exampl: Z q + Z Z 2 Z 3 Currnt dividr: Z q R C s L s I Zn I total R C s Z n + Z Z 2 Z 3 L s Ex4 a) Find th stady-stat V R and v R ( givn v S ( is a 2 Vpp cosin wav at: f 2 khz b) Find th currnt: I 6 R L ( R 00 Ω V S ( ω ) 6 0 ω 2 π f ω L 80 mh Transfr function for V R as th output: H( s ) C 04 µf 0 I( s) V( s) C ( V R ( s) V S ( s) 266 rad sc R R L s V R ( ω ) R 6 V V V R 363V /-82 o R L ( C ( v R ( t ) 363 cos 266 rad sc t 82 Z( s) I( s ) V( s) Z( s) 6 V ( 266) s ω 266 rad sc ( 266 ) 6 V magnitud: 6324 ma 9488 Ω angl: I 6324mA /-82 o 6 V C s 6 V atan c) Draw a phasor diagram of all th voltags V L I Z L 6324 ma 00 Ω 636 V V L 636V / 38 o V C I Z C 6324 ma ( 99) Ω 28 V 82 ( 90) 38 V C -28V / 38 o 28V /-482 o Sinusoidal Rspons Nots p4

5 Sinusoidal Rspons Nots p L 2 mh Ex a) Find th stady-stat V C and v C ( givn v in ( is a 2 Vp cosin wav at: with a 20 o lading phas angl Transfr function for V C as th output: H( s ) H( s ) L s R L 2 s R L 2 s C s C s R L 2 s R L 2 s f 2 khz V C ( s) C s C s /\ V in C µf R 200 Ω L 2 8 mh L s R L 2 s C s V in 2 V 20 ω 2 π f ω 708 rad sc H( ω ) L ( R L 2 ( C ( 0002 ( 708 ) ( 708) 0 6 ( 708) ( 00708) V C V in ( H( ω ) 20 2 V / o 204V / 896 o v c ( t ) 204 cos 708 rad sc t 896 a) Find th stady-stat I L2 and i L2 ( V C ( s) Transfr function for I L2 as th output: H( s ) I L2 ( s) R L 2 s V C ( s) R L 2 s H( s ) L s C s R L 2 s R L 2 s L s L s C s R L 2 s R L 2 s H( ω ) L ( ω ) L C ( 2 R L 2 ( R L 2 ( Ω 798 kω 438 I L2 V in ( H( ω ) kω kω / o 8638mA / -238 o i L2 ( t ) 8638 ma cos 708 rad sc t 238 Sinusoidal Rspons Nots p

6 Sinusoidal Rspons Nots p6 Ex6 This systm: H( s ) s 20 s Has this input: x( t ) 4 sin( 2 t 40 u( a) Us stady-stat AC analysis to find th stady-stat output y ss ( t )? AC stady-stat H( H( 2 ) atan 2 20 atan / 3646 X( ω ) 4 30 Not 90 o phas-lag bcaus it's givn as a sin wav Y ss ( ω ) / / 6646 y ss ( t ) 776 cos( 2 t 6646 b) Exprss th output, and sparat into 3 partial fractions that you can find in th laplac transform tabl without using complx numbrs Show what thy ar, but don't find th cofficints Find th input as a sum of a pur sin and cosin 4 sin( cos( so: 4 sin( 2 t 40 u( t ) ( 3064 sin( 2 t ) 27 cos( 2 ) u( Y( s ) s s 2 44 ( s 20) ( s ) A s + B s s C 2 s 2 44 c) Continu with th partial fraction xpansion ust far nough to find th transint cofficint as a numbr ( s )( s 20 ) A s B s ( s ) + C 2 ( s ) lt s ( ( ) )( 20 ) A A ( s 20 )( s) 2 44 A 4404 d) Exprss th complt (both transint and stady-sta output as a function of tim y( t )? 776 cos( sin( y( t ) 4404 t 776 cos( 2 t 6646 u( Eithr answr y( t ) 4404 t 697 cos( 2 t ) 68 sin( 2 u( ) What is th tim constant of th transint part this xprssion? τ? Sinusoidal Rspons Nots p6