F(jω) = a(jω p 1 )(jω p 2 ) Û Ö p i = b± b 2 4ac. ω c = Y X (jω) = 1. 6R 2 C 2 (jω) 2 +7RCjω+1. 1 (6jωRC+1)(jωRC+1) RC, 1. RC = p 1, p
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1 ÓÖ Ò ÊÄ Ò Ò Û Ò Ò Ö Ý ½¾ Ù Ö ÓÖ ÖÓÑ Ö ÓÒ Ò ÄÈ ÐØ Ö ½¾ ½¾ ½» ½½
2 ÓÖ Ò ÊÄ Ò Ò Û Ò Ò Ö Ý ¾ Á b 2 < 4ac Û ÒÒÓØ ÓÖ Þ Û Ö Ð Ó ÒØ Ó Û Ð Ú ÕÙ Ö º ËÓÑ Ñ ÐÐ ÕÙ Ö Ö ÓÒ Ò º Ù Ö ÓÖ ½¾ ÓÖ Ù Ö ÕÙ Ö ÓÖ Ò ØÖ Ò Ö ÙÒØ ÓÒ F(jω) = a(jω) 2 +b(jω)+cº ½ Á b 2 4ac Ø Ò Û Ò ÓÖ Þ F(jω) = a(jω p )(jω p 2 ) ÖÓÑ Ö ÓÒ Ò Û Ö p i = b± b 2 4ac 2a º 0 Y X (jω) = 6R 2 C 2 (jω) 2 +7RCjω+ ÄÈ ÐØ Ö /RC 0.3/RC /RC 3/RC ω = (6jωRC+)(jωRC+) ω c = 0.7 RC, RC = p, p 2 ÔÓÐÝÒÓÑ Ð Û Ö Ð Ó ÒØ Ò ÓÖ ÒØÓ Ð Ò Ö Ò ÒÝ ÓÖ ÕÙ Ö ÓÖ ÓÑÔÐ º ÕÙ Ö ½¾ ¾» ½½
3 Ö ÕÙ Ò Ø ÑÔ Ò Ó Ø Ô ÓÖ ÑÙ Ð Ø Ò 3R Ó Û Ò Ø Ò Ó Ø ÖÙ ØÛÓ ÔÓØ ÒØ Ð Ú Ö ÓÒ Ö Ø ÓØ Ö º º Ø ÙÖÖ ÒØ Ø ÖÓÙ Ø 3R Ò Ð Ð ÓÑÔ Ö Ö Ú ÓÒ Ó ÌÖ Ò Ö ÙÒØ ÓÒ V X 2R +jωcv + V Y 3R = 0 3(V X)+6jωRCV +2(V Y) = 0 (5+6jωRC)V = 3X +2Y º Ã Ä V Ú Y V 3R +jωcy = 0 (+3jωRC)Y = V º Ã Ä Y Ú Ð Ñ Ò Ò V Û Ò Ø ØÛÓ ÕÙ ÓÒ Ú (5+6jωRC)(+3jωRC)Y = 3X +2Y ( ) 5+2jωRC +8(jωRC) 2 2 Y = 3X Y X = 3 3+2jωRC+8(jωRC) 2 = +7jωRC+6(jωRC) 2 = (+6jωRC)(+jωRC) º Ø Ø ÖÓÙ Ø Ö Ø Cµº Ì Ö ÕÙ ÒÝ ÝÑÔØÓØ Ø Ö ÓÖ Ø ÔÖÓ ÙØ Ó Ø ØÓ Y ÙÖÖ ÒØ Ø ØÛÓ ÔÓØ ÒØ Ð Ú Ö Û Ú X 2jωRC 3jωRC = ÝÑÔØÓØ ÓÖ 6(jωRC) 2 º ½¾ ÒÓØ ½ Ó Ð ¾
4 ÊÄ Ò Ò Û Ò Ò Ö Ý ÓÖ Ò ½¾ Ù Ö ÓÖ ÓÖ Ò ËÙÔÔÓ b 2 < 4ac Ò F(jω) = a(jω) 2 +b(jω)+cº»à Ö Õ ÝÑÔØÓØ F LF (jω) = c F HF (jω) = a(jω) 2 ÖÓÑ Ö ÓÒ Ò ÝÑÔØÓØ Ñ ÒÙ ÖÓ Ø ÓÖÒ Ö Ö ÕÙ ÒÝ Ì a(jω c ) 2 = c ωc = c Ò Þ ζ = b Ø ÓÖ ØÓ Ï ( ) 2 ( ) F(jω) = c( j ω ω c +2ζ j ω ω c )+ a º 2aω c = bω c 2c = bsgn(a) 4ac ØÓ ÒÓØ Ò Ø ÜÔÖ ÓÒ ÈÖÓÔ ÖØ c Ù Ø Ò ÓÚ Ö ÐÐ Ð ÓÖº µ ω c Ð Ø Ö ÕÙ ÒÝ Ü Ò F(jω) ÙÒØ ÓÒ Ó Ù Ø ω ω º c µ Ì Ô Ó Ø F(jω) Ö Ô ÖÑ Ò ÒØ Ö ÐÝ Ý ζº µ Ì ÕÙ Ö ÒÒÓØ ÓÖ Þ b 2 < 4ac ζ < º µ ω = ω c ÝÑÔØÓØ Ò = c ÙØ F(jω) = c 2jζº µ Û ÓÑ Ñ Ù Ø ÕÙ ÐÝ ÓÖ Q 2ζ = aω c b º ÐØ ÖÒ Ú ÐÝ ½¾» ½½
5 Ò Ò Ö Ý ÊÄ ½¾ Ù Ö ÓÖ ÓÖ Ò ÊÄ Y I = = R + jωl +jωc jωl LC(jω) 2 + L R jω+ ω c = c a = 000, ζ = b 2aω c = ÝÑÔØÓØ jωl Ò jωc º ÖÓÑ Ö ÓÒ Ò Ò Û Ò ÓÖ Ý Ö ØÓÖ Y 2 º ÁØ Ô ÕÙ ω = 000º Ì Ö ÓÒ ÒØ Ö ÕÙ ÒÝ ω r ÖÔÐÝ Û Ò Ø ÑÔ Ò ÔÙÖ ÐÝ Ö Ð ω r = 000, Z RLC = Y I = Rº Ý Ø Ñ Û ØÖÓÒ Ô Ò ÔÓÛ Ö ÓÖÔØ ÓÒ Ö ÓÒ ÒØ Ý Ø Ñº ½¾» ½½
6 ÓÖ Ò Ò Ò Ö Ý ½¾ Ù Ö ÓÖ ÊÄ ÖÓÑ Ö ÓÒ Ò ω = 000 Z L = 00j, Z C = 00jº Z L = Z C I L = I C I = I R +I L +I C = I R = Y = I R R = = 56dBV I L = Y Z L = j = 6j Ò Û Ò Ä Ö ÙÖÖ ÒØ Ò L Ò C ÜÐÝ Ò Ð ÓÙØ I R = I Ò Z = R Ö Ðµ ½¾» ½½
7 ÓÖ Ò ÊÄ Ò Ò Û Ò Ò Ö Ý ÖÓÑ Ö ÓÒ Ò ½¾ Ù Ö ÓÖ ÖÓÑ Ö ÓÒ Ò ÄÈ ÐØ Ö ω = 2000 Z L = 200j, Z C = 50j ( ) Z = R + Z L + Z C = Y = I Z = = 36dBV I R = Y R = I L = Y Z L = , I C = ÅÓ Ø ÙÖÖ ÒØ ÒÓÛ ÓÛ Ø ÖÓÙ C ÓÒÐÝ 0. Ø ÖÓÙ Rº ½¾» ½½
8 ÊÄ ÐÐ Ð ¹ÔÓÛ Ö Ò Û ÓÖ Ð Ó Ò Û º Ò Û Ò ½¾ Y I = /R+j(ωC /ωl) Ù Ö ÓÖ ÓÖ Ò ÖÓÑ Ö ÓÒ Ò Ò Û Ò Ò Ò Ö Ý Ø Ö Ò Ó Ö ÕÙ Ò ÓÖ Ò Û Y 2 Ö Ö Ø Ò Ð Ô º I Û Y I 2 = ( /R) 2 +(ωc /ωl) 2 Y I (ω 0) 2 = R 2 ω È 0 = 000 ω Y 3dB I (ω 3dB) 2 = Y 2 I (ω 0) 2 ( ω 3dB RC R ω 3dB L) = 2 ( /R) 2 +(ω 3dB C /ω 3dB L) 2 = R2 ω 3dB RC R /ω 3dB L = ± ω 2 3dB RLC ±ω 3dBL R = 0 ÈÓ Ú ÖÓÓØ ω 3dB = ±L+ L 2 +4R 2 LC 2RLC = {920, 086} Ö» Q ÓÖ ω 0 B = 2ζ = 6º Q Ù ÐÝ µ Ò Û B = = 67rad/sº ½¾» ½½
9 ÓÖ Ò ÊÄ ÖÓÑ Ö ÓÒ Ò Ò Ò Ö Ý ½¾ Ù Ö ÓÖ v(t)i(t) ÓÖ P L P Ò C Ò P ÓÔÔÓ R º 2 Li2 L + 2 Cy2 ËØÓÖ Ò Ö Ý L Ò Cº Û Ò ÐÓ Ò Û Ò Ò Ò Ö Ý Q ω W ØÓÖ P R = ω 2 C IR 2 2 I 2 R= ωrc ω 000 = Y 600 = I R = I L 6j = I C = +6j Q ω Ô ØÓÖ Ò Ö Ý Ú Ö ÔÓÛ Ö ÐÓ º ½¾» ½½
10 ÚÓÐØ Ô ÓÖ ÖÓ Ø Ø Ö Ô Ú ÓÑÔÓÒ ÒØ V = IZ = 600 = 600Vº Ì Û Ú ÓÖÑ Ì ØÓ Ø Ô ÓÖ v(t) = 600cosωt Ò ÔÐÓØØ Ò Ø ÙÔÔ Ö Ö Ö Ô º ÖÓÑ ÒÓÛ Ò ÓÖÖ ÔÓÒ Ò I Ò L +6j Ó = p R (t) = 600cosωt 6sinωt = 3600sinωtcosωt 800sin2ωtº Ì Ö = Ò Ø ÐÓÛ Ö Ð Ö Ô º ÔÐÓØØ Ö Ú ÓÒ Ó Ò Ò Ö Ý Ï Ú ÓÖÑ ÒÔÙØ ÙÖÖ ÒØ Ô ÓÖ I = º º i(t) = cosωt Û Ö ω = 000rad/sµº Ì ÓÑÔÐ Ü ÑÔ Ò Ö Z Ì L = jωl = Ò 00jΩ Z C = 00jΩº Í Ò Ø ÓÖÑÙÐ ÓÖ = Ô Ö ÐÐ Ð ÑÔ Ò Ø ØÓØ Ð ÑÔ Ò Z = j + 00j = 600 jωc º ËÓ Ø Ö ÓÒ ÒØ Ö ÕÙ ÒÝ Ø ÑÔ Ò Ó L Ò C Ò Ð ÓÙØ Ò Ø ØÓØ Ð ÑÔ Ò Ù Ø Z = 600Ωº Ò Ç Ñ³ Ð Û ØÓ ÛÓÖ ÓÙØ Ø Ò Ú Ù Ð ÙÖÖ ÒØ Ô ÓÖ Ò Ø Ø Ö ÓÑÔÓÒ ÒØ V Û Ù I R = V R = = I C = V = 600 Z C 00j = 6j I L = V = 600 Z L 00j = 6jº Ò Ì Û Ú ÓÖÑ ÓÖÖ ÔÓÒ Ò ØÓ Ø Ø Ö Ô ÓÖ Ö ÔÐÓØØ Ò Ø ÙÔÔ Ö Ð Ö Ô º Ô ÓÖ ØÓ Ö Ó Ò³Ø ÖÐÝ Ú Ø ÓÖÖ Ö ÙÐØ Ò Ó Û ÐÙÐ Ø ÔÓÛ Ö ÅÙÐØ ÔÐÝ Ò ÖÐÝ Ý ÑÙÐØ ÔÐÝ Ò v(t) i(t)º ÓÖ Ø Ö ØÓÖ V = 600 Ò I Û Ú ÓÖÑ R Ó = p R (t) = 600cosωt cosωt = 600cos 2 ωt = cos2ωtº ÓÖ Ø Ò ÙØÓÖ V = 600 Ò I L = 6j Ó p R (t) = 600cosωt 6sinωt = 3600sinωtcosωt = 800sin2ωtº Ò ÐÐÝ ÓÖ Ø Ô ÓÖ V = 600 Ì Ò Ö Ý ØÓÖ Ò Ò Ò ÙØÓÖ w L (t) = 2 Li2 (t) = sin2 ωt =.8sin 2 ωt = 0.9( cos2ωt)º Ì Ò Ö Ý ØÓÖ Ò Ô ÓÖ w C (t) = 2 Cv2 (t) = cos 2 ωt =.8cos 2 ωt = 0.9(+cos2ωt)º Ì Ö ÔÐÓØØ Ò Ø ÐÓÛ Ö Ö Ö Ô º Ì ØÓØ Ð ØÓÖ Ò Ö Ý Ò Ø ÖÙ w L (t)+w C (t) =.8J Û Ó ÒÓØ Ú ÖÝ Û Ø Ñ º ½¾ ÒÓØ ½ Ó Ð
11 ÊÄ ÖÓÑ Ö ÓÒ Ò ½¾ Ù Ö ÓÖ Y X = /jωc R+jωL+ jωc = LC(jω) 2 +RCjω+ ÓÖ Ò Ò Û Ò Ò Ò Ö Ý LC (jω) 2 º ÝÑÔØÓØ Ò ω c = c a = 000, ζ = b 2aω c = R 200 ω c : Z L = Z C = 00j I = X R, Y X = RCω = 2ζ, Y X = π 2 ÈÐÓØ Å ÒØÙ ζ Ð ÐÓ Ö Ô Ñ ÐÐ Ö Ò Û º ËÑ ÐÐ Ä Ö ζ ÑÓÖ ÐÓ Ñ ÐÐ Ö Ô ÐÓÛ Ö ω Ð Ö Ö Ò Û º ÈÐÓØ È ζ Ø Ô Ò ÓÚ Ö π º ËÑ ÐÐ ) 2ζ Y X π 2 (+ ζ log 0 ωωc 0 ζ < ω ÓÖ ω c < 0 +ζ C L R=5, ζ= R=20, ζ=0. R=60, ζ=0.3 0 R=20, ζ= π R=5, ζ=0.03 R=20, ζ=0. R=60, ζ=0.3 R=20, ζ=0.6 R k 0k ω (rad/s) k 3.98k 0k ω (rad/s) ½¾» ½½
12 ÊÄ ÖÓÑ Ö ÓÒ Ò Ò Ò Ö Ý ÄÈ ÐØ Ö ½¾ Ù Ö ÓÖ Y X = LC(jω) 2 +RCjω+ = (j ω ωc )2 +2ζj ω ωc + ÓÖ Ò ω c = c a = 000, ζ = b 2aω c = bω c 2c = R 200 Y ω X ω Ó Ó c ω c Ù Ø Ð Ö ÕÙ ÒÝ Ü ÓÒ ÐÓ Ü µº ÙÒØ ÓÒ Ò Û Ò Ì ÓÖ ζ Þ µ ÖÑ Ò Ø Ô Ó Ø Ô º È ÄÈ ÐØ Ö ÓÖ Ö ÕÙ ÒÝ È ÙÒ Ö ÓÖÒ Ö Ô Ô ÒÓ ÓÖ Þ Ò ω p = ω c 2ζ , 4dB R=20, ζ=0. 906, 5dB R=60, ζ=0.3 ζ , 4dB ζ ζ Ò Ö Ð Ú ØÓ ÝÑÔØÓØ ω p 30 2ζ ζ 2 999, 26dB ω c ω (krad/s) R=5, ζ=0.03 R=20, ζ=0.6 2ζ Q Ö ÕÙ Ò ω Ì Ö p Ô ω c ÝÑÔØÓØ ÖÓ ω r Ö Ð ÑÔ Ò ζ < 0.3 ω ÓÖ p ω c ω r ÐÐ ÐÐ Ø Ö ÓÒ ÒØ Ö ÕÙ Òݺ º Ü Ö Ð ÓÒ Ô Û Ò ω p ω Ì c ω Ò r Ø Ò Ø Ò Ý ÒÝ ÓØ Ö ÓÖÒ Ö Ö ÕÙ Ò Ò Ø Ö ÔÓÒ º Ö ÕÙ Ò ½¾ ½¼» ½½
13 ÊÄ ÖÓÑ Ö ÓÒ Ò ÓÖ Ò Ô Ò Ò Ö Ý ÓÖÔØ ÓÒ ÓÖ Ö ÖÙ Ö Ð ÑÔ Ò ω r Ô Ö ÔÓÒ Ñ Ý Ð ÐÝ Ö ÒØ Ö ÕÙ ÒÝ ½¾ Ù Ö ÓÖ Ò Û Ò Ò Ò Ö Ý Ì ÕÙ ÐÝ ÓÖ Q Ó Ø Ö ÓÒ Ò Q ω Ò Ö Ý 0 ØÓÖ Ò Û ÔÓÛ Ö Ò R Ò Û Û Ö ÔÓÛ Ö ÐÐ Ý 3 ÚÓÐØ Ý 2 ÓÖ 2 Ò Ö Ý ÐÓ Û Ò Ò L ØÓÖ Ì C ω 0 2ζ Ù Ö ÓÖ ( jω ω c ) 2 +2ζ ( jω ω c )+ a(jω) 2 +b(jω)+c ω c = c a Ò ζ = b 2aω c = bsgn(a) 4ac ±40» ÐÓÔ Ò Ò Ñ ÒÙ Ö ÔÓÒ Ô Ò Ö Ô ÐÝ Ý 80 ÓÚ Ö ω = 0 ζ ω c Ò ÖÖÓÖ Ò ÝÑÔØÓØ 2ζ Q ω 0 ÓÖ ÙÖØ Ö Ð À ÝØ ½ ÓÖ ÁÖÛ Ò ½¾º ½¾ ½½» ½½
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