rasm. Karrali argumentli trigonometrik funksiyalardan

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rasm. Karrali argumentli trigonometrik funksiyalardan
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bet | 2/14 | Sana | 03.12.2023 | Hajmi | 395,87 Kb. | | #110584 |
Bog'liq 610-211.1 rasm. Karrali argumentli trigonometrik funksiyalardan foydalanish usuli.
NE VATi uchinchi darajali polinom bilan approksimatsiyalangan bo‘lsin,
i=a0+a1U+a2U2+a3U3. (1.5)
Uning kirishiga bitta garmonik tebranish ta’sir etsin,
Uk(t)=Ukcos(ω0t+ φ0). (1.6)
(1.7) ni (1.8) ifodaga qo‘yib, hamda
cos2α=0,5(1+cos2α) (1.7)
cos3α=3/4 cosα+1/4 cos3α (1.8)
trigonometrik formulalardan foydalanib, NE dan o‘tayotgan tok spektral tashkil etuvchilar yig‘indisi shaklida ifodalaymiz
i=a0+a1Ukcos(ω0t+φ0)+a2U2kcos2(ω0t+φ0)+a3U3kcos3(ω0t+φ0)=
=a0+ a1Ukcos(ω0t+φ0)+0,5a2U2k+0,5a2U2kcos(2ω0t+2φ0)+
+0,75a3U3kcos(ω0t+φ0)+0,25a3U3kcos(3ω0t+3φ0). (1.9)
Ushbu tok ω1 chastotali tashkil etuvchidan tashqari, tok doimiy tashkil etuvchisi (ω0=0), ikkinchi garmonika (2ω0) va uchinchi garmonika (3ω0)tashkil etuvchilardan iborat. Bu tashkil etuvchilar quyidagi qiymatlarga ega:
I0=a0+0,5a2U2k ;
I1=a1Uk+0,75a3U3k ;
I2=0,5a2U2k ;
I2=0,25a3U3k . (1.10)
Bunda tokning doimiy tashkil etuvchisi va juft garmonikalari approksimatsiyalovchi polinomning juft darajali tashkil etuvchilari va toq garmonikalari toq darajali tashkil etuvchilari hisobiga paydo bo‘ladi, shu bilan birga aniqlanadigan tokning eng yuqori garmonikasi approksimatsiyalovchi polinom darajasiga teng bo‘ladi. Ko‘p hollarda aniqlanadigan garmonika soni oshgan sari uni qiymati avvalgilariga nisbatan kamayib boradi. 1.2-rasmda tok aniqlangan spektral tashkil etuvchilari keltirilgan.
1.2-rasm.
VAXi uchinchi darajali polinom bilan ifodalangan NE kirishiga ikkita tebranish ta’sir etgan holatni ko‘rib chiqamiz. Bunda
U1=U1cos(ω1t+ φ1) va U2=U2cos(ω2t+ φ2) (1.11)
va ularning chastotasi ω2 >ω1 bo‘lsin.
1.11 yig‘indisini 1.5 yig‘indiga qo‘yamiz va NE dan o‘tayotgan tok ifodasini olamiz
i=a0+a1U1cos(ω1t+φ1)+a1U2cos(ω2t+φ2)+a2[U1cos(ω1t+φ1)+U2cos(ω2t+φ2)]2++a3[U1cos(ω1t+φ1)+U2cos(ω2t+φ2)]3. (1.12)
(a+b)2 , (a+b)3 ni yoyish va
cosα∙cosβ=0,5cos(α+β)+0,5cos(α-β) ;
cosα∙cos2β=0,5cosα+0,25cos(2α+β)+0,25cos(2α-β) ;
cos2α∙cosβ=0,5cosβ+0,25cos(α+2β)+0,25cos(α-2β) ;
tirgonometrik formulalardan foydalanib (1.12) ni quyidagi ko‘rinishga keltiramiz
i=a0+a1U1cos(ω1t+φ1)+a1U2cos(ω2t+φ2)+0,5a2U21+0,5a2U22+ +0,5a2U1cos(2ω1t+2φ1)+0,5a2U22cos(2ω2t+2φ2)+a2U1U2cos[(ω1+ ω2)t+(φ1+φ2)]+ +a2U1U2cos[(ω1-ω2)t+(φ1-φ2)]+ 0,75a3U31cos(ω1t+φ1)+0,75a3U32cos(ω2t+φ2)+ +0,25a3U31cos(3ω1t+3φ1)+0,25a3U32cos(3ω2t+3φ2)+1,5a3U21U2cos(ω2t+φ2)+ +0,75a3U21U2cos[(ω1-2ω2)t +(φ1-2φ2)]+ 0,75a3U21U2cos[(ω1+2ω2)t+(φ1+2φ2)]+ +1,5a3U1U22cos(ω1t+φ1)+0,75a3U1U22cos[(2ω1+ω2)t+(2φ1+φ2)]+ +0,75a3U1U22cos[(2ω1-ω2)t +(2φ1-φ2)] . (1.13)
1.13 ifodadagi NE orqali o‘tgan tok spektral tashkil etuvchilari spektrini chizamiz (1.3-rasm).
1.3-rasm.
Nochiziqli element orqali umumiy holda: birinchi signal va uning garmonikalari (nω1+nφ1); ikkinchi signal va uning garmonikalari (mω2+mφ2) va kombinatsion chastotalar [(nω1+nφ1)∙(mω2+mφ2)] paydo bo‘ladi. Kombinatsion chastotalar murakkabligi ularning tartibi N=|n|+|m| orqali aniqlanadi (n va m butun natural sonlar). Masalan ω1+2ω2 – uchinchi tartibli, 2ω1+2ω2 – to‘rtinchi tartibli kombinatsion tashkil etuvchilar hisoblanadilar.
1.13 ifodadagi tok har bir spektral tashkil etuvchilari qiymati (amplitudasi) mos chastotali spektral tashkil etuvchilar yig‘indisi bilan aniqlanadi.
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