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 ) 3.. 30. 5. 5 4 3 3 4 5 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.. 4.5 /-30 o or:.. 4.5 30. 3. /5 o or:.. 3. 5. 4.5. cos( 30. 3.897 4.5. sin( 30..5 3.897.5 \ } add 3.. cos( 5. 3.09 3.. sin( 5. 0.88 3.09 +0.88 / Add ral parts: 3.897 3.09 988 3 Add imaginary parts:.5 0.88.4 3 988.4 sum hang 3 back to polar coordinats: 988.4 7.3.4.50 988 O, in Mathcad notation (you'll s ths in futur solutions): 3 7.3 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 8.44. a) using phasor notation, find v 3 v - v.. 5. ( 387... 3.6. ( 8.44. 5. cos( 387. 4 5. sin( 387. 3 4 +3 3. cos( 8.44. 3 3. sin( 8.44. 3 Subtract ral parts: 4. 3. 3 Subtract imaginary parts: 3.. 4 3 +4 v ( t) v ( t ) ( 4. ). Magnitud: (. ) ( 4. ) 4.3 3 4.3 O: 4. Angl: 75.96 arg. 3 75.96 So: v 3 ( t ) v ( t) v ( t ) 4.3. cos( t 75.9. EE 0 / 00 Phasor Exampls p
3. Find,, and in polar phasor form. f. khz EE 0 / 00 Phasor Exampls p ( ω) 0 f. khz 500. ω. f ω.57 0 4 rad sc 80. mh Z.. ω Z.005 k find th currnt: 0.4. µf Z Z q Z 0.99 k Z q 500 +80366 500 806 948.49 806 58.87 Z q 948.5 / 58. o 500 I. 0 magnitud: 36 ma angl: 0. 58.. 58. Z. q 948.5 I 36mA /-58. o find th magnitud find th angl. I 3 500. 3.63 58.. 0. 58. 3.63 /-58. o. I Z 3 005. 358 58.. 90. 3.8 358 / 3.8 o. I Z 3 ma.( 99)..59 58.. ( 90). 3.8 -.59 / 3.8 o O: 3 ma.( 99)..59 58.. ( 90). 48..59 /-48. o O, you can also find ths voltags dirctly, using a voltag dividr. I.E. to find dirctly:.. 6...( ).( ) 6..( ) ω... ω.. 4.053..53 ω.... 4.053.53. ( 4.053.53. ) ( 4.053.53. ).( 4.053.53. ) 4.38 5.078 ( 4.053).53.74 4.38 5.078...069 0.663.74.74 magnitud:.069 0.663.58 angl: 0.663 3.8.069 but this is actually in th third quadrant, so modify your calculator's rsults: 3.8. 80. 48.9.58 /-48. o EE 0 / 00 Phasor Exampls p.( 4.053.53. ) ( 4.053).53
EE 0 / 00 Phasor Exampls p3. mh 4. a) Find Z q. f.5. khz ω. f ω.57 0 4 rad sc 3.46 ft branch Z l ight branch Z l 63.66 Z.. r ω Z r 00 +5.664. µf. 00. 8 mh 5.664 Z q Somtims it's worth simplifying a littl bfor puttiing in numbrs. Z q 3.4. ( 00 5.664. ).. 0.974 3.4. 3.4. 3.4. b).. in 0. 0.974 3.4. 0.974 3.4. 3.4.. ω..... ω ( ( 00.( 0.974 ) 5.664.( 3.4 )) ( 5.664.( 0.974 ) 00. 3.4). ). ( 00.03688 750.79673 ). 0.8084 0.974 3.4 ( 00 5.664. ).( 0.974 3.4. ) 0.974 3.4 3.4. 8.48 69.384. 8.486 37.968 8.49 37.97 4.33 37.97 64.036 Z q 4.4 /-64.04 o 8.49 in. Find I, I 84.09 ma 0. ( 64.04). 84.04 Z. q 4.4 I 84mA / 84.04 o 84. 84.04. I 9.485 +8.569 ma.. I ( 8.486 69.384. ) 84. 8.486 69.384. 0.39 84.04. 69.384 8.486 8.959 0.4 / 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 p3. 0.588 0..039.. 0.5886 / 0 -.039o 0.4 / 8.96 o Sam 0.53 +3.78
EE 0 / 00 Phasor Exampls p4 4. ontinud t's find I. Z.. r ω 00 5.664 30 5.664 Z r 00 +5.664 00 3.4 I 0.4. 8.9 0.4... 30.. 3.4 30. / 8.96-3.4 o 84mA /-3.8 Or, dirctly by urrnt dividr: I. I.. I I ω... dnominator: ω.. (. ) 3.89. 80. 07.4 ω.. I 84. 3.89. 84.04. 07.4.. 84 ma 3.89 / 84.04-07.4 o 84mA /-3.8 I 79.404 34.00 ma How about I? I... ω 0.4 / 8.96 o 0.05708 / 90 o 30mA / 98.96 o Or, dirctly by urrnt dividr: I. I. ω..... I. I ω..... ω. numrator: 80..4 ω.. (. ) 3.7 ω.. Scond quadrant dnominator is th sam as abov..4. 3.7. I. 84. ma.... 3.89. 07.4. 84.04 3.7.. 3.89 84 ma /.4-07.4 + 84.04o 30mA / 98.96 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 9.485 +8.569 ma I 9.485 +8.569 ma 5. a) Find Z. For partial crdit, you must show work and/or intrmdiat rsults. I 5... in 0 Z T in I 0. 0. Z T 344 00 0. 5. ma / -0-0o 400 /-30 o in 0 /-0 o i( t ) 5. cos 377. rad. sc t 0. Z. ( 0 60. ). 50 Z? Z Z T Z ( 344 00. ). 50. ( 0 60. ). 74 40 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
EE 0 / 00 Phasor Exampls p5 a) Find in in polar form. I I. T Z 00 ma Z ( 80 60. ). ω 000. rad sc. in I Z Z in 8 6 50. 8 6 0 6 8 in 0 /-39 o 387 I I Z 00 / 0 o ma Z ( 80 60. ). in b) Find I T in polar form. I. 0. 50 /-39 o 0.. cos(. 0. 39 ). sin( 39. 60 0i ma 50. 50. I T I I Z ( 60 0. ). ma 00. ma 60 0 ma 60 0 8356 0 60 4.78 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 80 60 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 80 60 Must hav a rsistor bcaus thr is a ral part. Must hav a capacitor bcaus th imaginary part is ngativ. Z Z. 80 60. ( 80 60. ). 80 60. Z 8 0 3 +6 0 3 80 60. 80 60.. 80 60 0, 000..008..008 5.00..006 ω 6 µf 5 EE 0 / 00 Phasor Exampls p5
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 83 0 3 rad sc b) Find its valu. o o. S angl is 90. ω 40 o.. ω 60 o. tan( 60. ).73..73 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 83 0 3 rad sc v o ( t) b) Find its valu. o. S angl is 90. ω. 90-60 o... tan( 60. ).73 EE 0 / 00 Phasor Exampls p6..73 0.5 µf