ECE 2210 / 00 Phasor Examples

Transcription

1 EE 0 / 00 Phasor Exampls. Add th sinusoidal voltags v ( t ) 4.5. cos( t 30. and v ( t ) 3.. cos( t 5. v ( t) using phasor notation, draw a phasor diagram of th thr phasors, thn convrt back to tim domain form. v ( t ) 4.5. cos( t 30. ( ω ) 4.5 /-30 o or: ( ω ) 4.5. and v ( t ) 3.. cos( t 5. ( ω ) 3. /5 o or: ( ω ) v ( t) A.Stolp 3//0 rv 0/7/06 I'm going to drop th (ω) notation from th phasor notation, it gts cumbrsom, but rmmbr that phasors ar in th frquncy domain /-30 o or: /5 o or: cos( sin( \ } add 3.. cos( sin( / Add ral parts: Add imaginary parts: sum hang 3 back to polar coordinats: O, in Mathcad notation (you'll s ths in futur solutions): arg 3.5 hang 3 back to th tim domain: v 3 ( t ) v ( t) v ( t ) 7.3. cos( t.5.. drawing of th phasor diagram tim. Two sinusoidal voltags: v ( t ) 5. cos( t 387. and v ( t ) 3. cos( t a) using phasor notation, find v 3 v - v.. 5. ( ( cos( sin( cos( sin( Subtract ral parts: Subtract imaginary parts: v ( t) v ( t ) ( 4. ). Magnitud: (. ) ( 4. ) O: 4. Angl: arg So: v 3 ( t ) v ( t) v ( t ) 4.3. cos( t EE 0 / 00 Phasor Exampls p

2 3. Find,, and in polar phasor form. f. khz EE 0 / 00 Phasor Exampls p ( ω) 0 f. khz 500. ω. f ω rad sc 80. mh Z.. ω Z.005 k find th currnt: 0.4. µf Z Z q Z 0.99 k Z q Z q / 58. o 500 I. 0 magnitud: 36 ma angl: Z. q I 36mA /-58. o find th magnitud find th angl. I /-58. o. I Z / 3.8 o. I Z 3 ma.( 99) ( 90) / 3.8 o O: 3 ma.( 99) ( 90) /-48. o O, you can also find ths voltags dirctly, using a voltag dividr. I.E. to find dirctly: ( ).( ) 6..( ) ω... ω ω ( ) ( ).( ) ( 4.053) magnitud: angl: but this is actually in th third quadrant, so modify your calculator's rsults: /-48. o EE 0 / 00 Phasor Exampls p.( ) ( 4.053).53

3 EE 0 / 00 Phasor Exampls p3. mh 4. a) Find Z q. f.5. khz ω. f ω rad sc 3.46 ft branch Z l ight branch Z l Z.. r ω Z r µf mh Z q Somtims it's worth simplifying a littl bfor puttiing in numbrs. Z q 3.4. ( ) b).. in ω..... ω ( ( 00.( ) ( 3.4 )) ( ( ) ). ). ( ) ( ).( ) Z q 4.4 / o 8.49 in. Find I, I ma 0. ( 64.04) Z. q 4.4 I 84mA / o I ma.. I ( ) / 8.96 o To find You could thn us anothr voltag dividr to find or. dirctly:. in.... in ω.. ω ω.... ω. in. ω..... in ω ω... in ω.. ω... EE 0 / 00 Phasor Exampls p / o 0.4 / 8.96 o Sam

4 EE 0 / 00 Phasor Exampls p4 4. ontinud t's find I. Z.. r ω Z r I / o 84mA /-3.8 Or, dirctly by urrnt dividr: I. I.. I I ω... dnominator: ω.. (. ) ω.. I ma 3.89 / o 84mA /-3.8 I ma How about I? I... ω 0.4 / 8.96 o / 90 o 30mA / o Or, dirctly by urrnt dividr: I. I. ω..... I. I ω..... ω. numrator: ω.. (. ) 3.7 ω.. Scond quadrant dnominator is th sam as abov I. 84. ma ma / o 30mA / o This currnt is gratr than th input currnt. What's going on? Th angl btwn I & I is big nough that thy somwhat cancl ach othr out. hck Kirchoff's urrnt aw: I I ma I ma 5. a) Find Z. For partial crdit, you must show work and/or intrmdiat rsults. I 5... in 0 Z T in I Z T ma / -0-0o 400 /-30 o in 0 /-0 o i( t ) 5. cos 377. rad. sc t 0. Z. ( ). 50 Z? Z Z T Z ( ). 50. ( ) b) ircl : i) Th sourc currnt lads th sourc voltag <--- answr, bcaus 0 o > -0 o. ii) Th sourc voltag lads th sourc currnt EE 0 / 00 Phasor Exampls p4

5 EE 0 / 00 Phasor Exampls p5 a) Find in in polar form. I I. T Z 00 ma Z ( ). ω 000. rad sc. in I Z Z in in 0 /-39 o 387 I I Z 00 / 0 o ma Z ( ). in b) Find I T in polar form. I /-39 o 0.. cos( ). sin( i ma I T I I Z ( ). ma 00. ma 60 0 ma I T 86mA /-4.8 o c) ircl : i) Th sourc currnt lads th sourc voltag ii) Th sourc voltag lads th sourc currnt answr i), -4.8 o > -39 o 7. d) Th impdanc Z is mad of two componnts in sris. What ar thy and what ar thir valus? Z Must hav a rsistor bcaus thr is a ral part. ( Z ) 80 Must hav a capacitor bcaus th imaginary part is ngativ. Im( Z) 60. ω Im( Z) 667 µf ) Th impdanc Z is mad of two componnts in paralll. What ar thy and what ar thir valus? Z Must hav a rsistor bcaus thr is a ral part. Must hav a capacitor bcaus th imaginary part is ngativ. Z Z ( ) Z , ω 6 µf 5 EE 0 / 00 Phasor Exampls p5

6 EE 0 / 00 Phasor Exampls p6 8. You nd to dsign a circuit in which th th "output" voltag lads th input voltag (v S (t)) by 40 o of phas. a) What should go in th box:,,? v S ( t) o. S v o ( t) f. khz angl of is 40 o. This can only happn if th angl of is positiv, so is a inductor ω. f ω rad sc b) Find its valu. o o. S angl is 90. ω 40 o.. ω 60 o. tan( 60. ) ω 34.5 mh 9. You nd to dsign a circuit in which th th "output" voltag lags th input voltag (v S (t)) by 40 o of phas.. 80 a) What should go in th box:,,? o. S v S ( t) angl of is -40 o. This can only happn if th angl of is ngativ, so is a capacitor f. khz ω. f ω rad sc v o ( t) b) Find its valu. o. S angl is 90. ω o... tan( 60. ).73 EE 0 / 00 Phasor Exampls p µf