5星型 三角型電阻變換

1、2-2,電阻的,Y,與聯(lián)結(jié)及等效變換,如圖中,I,23,0,時(shí)，對(duì),R,23,支路既不能開路也不能,短路處理，此時(shí)電路無法用電阻串并聯(lián)關(guān)系進(jìn)行,分析，則引出,Y,變換問題,1,2,3,1,2,3,1,2,3,1,2,3,1.Y,形聯(lián)接：三個(gè)電阻的一端,接在同一點(diǎn)上，另一端分別,接到三個(gè)不同端鈕上,2,形聯(lián)接：三個(gè)電阻分,別接到三個(gè)端鈕的每?jī)蓚€(gè),之間,R,ca,R,ab,R,bc,a,b,c,i,1,i,i,2,3,U,ab,U,bc,U,ca,c,a,b,a,R,b,R,c,I,1,I,3,I,2,R,U,ab,U,bc,U,ca,這三個(gè)電阻既非串聯(lián)，又非并聯(lián)，不能用串并聯(lián)化簡(jiǎn)，但可,以通過電

2、阻的,Y,聯(lián)結(jié)的等效變換來簡(jiǎn)化。若上圖兩個(gè)電路,等效，則它們的三個(gè)對(duì)應(yīng)端,a,b,c,的電流,I,a,I,b,I,c,及三個(gè),對(duì)應(yīng)端之間的電壓,U,ab,U,bc,U,ca,應(yīng)相等,對(duì)電阻三角形聯(lián)接的三端網(wǎng)絡(luò)，外加兩個(gè)電流源,i,1,和,i,2,將電流源與電阻的并聯(lián)單口等效變換為一個(gè)電壓源與電阻,的串聯(lián)單口，得到圖,b,電路，由此得到,12,2,23,2,23,12,23,2,12,1,31,12,31,1,31,1,31,23,12,2,23,1,31,12,i,i,R,i,R,i,R,u,i,i,R,i,R,i,R,u,R,R,R,i,R,i,R,i,12,2,2,31,23,12,31,

3、12,23,1,31,23,12,31,23,2,2,31,23,12,31,23,1,31,23,12,23,12,31,1,i,R,R,R,R,R,R,i,R,R,R,R,R,u,i,R,R,R,R,R,i,R,R,R,R,R,R,u,將,i,12,表達(dá)式代入上兩式，得到,式,2,11,和,2,12,分別表示電阻星形聯(lián)接和三角形聯(lián),接網(wǎng)絡(luò)的,VCR,方程,12,2,23,2,23,12,23,2,12,1,31,12,31,1,31,1,31,23,12,2,23,1,31,12,i,i,R,i,R,i,R,u,i,i,R,i,R,i,R,u,R,R,R,i,R,i,R,i,如果要求電阻星

4、形聯(lián)接和三角形聯(lián)接等效，則要,求以上兩,個(gè),VCR,方程的對(duì)應(yīng)系數(shù)分別相等，即,13,2,31,23,12,31,12,23,3,2,31,23,12,31,23,3,31,23,12,23,12,31,3,1,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,由此,解得,14,2,31,23,12,31,23,3,31,23,12,23,12,2,31,23,12,12,31,1,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,12,2,2,31,23,12,31,12,23,1,31,23,12,31,23,2,2,31,23,

5、12,31,23,1,31,23,12,23,12,31,1,i,R,R,R,R,R,R,i,R,R,R,R,R,u,i,R,R,R,R,R,i,R,R,R,R,R,R,u,11,2,2,3,2,1,3,2,2,3,1,3,1,1,i,R,R,i,R,u,i,R,i,R,R,u,電阻三角形聯(lián)接等效變換為電阻星形聯(lián)接的公式為,形三電阻之和,端兩電阻之乘積,接于,i,R,i,當(dāng),R,12,R,23,R,31,R,時(shí)，有,R,R,R,R,R,3,1,3,2,1,14,2,31,23,12,31,23,3,31,23,12,23,12,2,31,23,12,12,31,1,R,R,R,R,R,R,R,

6、R,R,R,R,R,R,R,R,R,R,R,17,2,2,1,3,3,2,2,1,31,1,1,3,3,2,2,1,23,3,1,3,3,2,2,1,12,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,電阻星形聯(lián)接等效變換為電阻三角形聯(lián)接的公式為,18,2,端相連的電阻,不與,形電阻兩兩乘積之和,mn,R,mn,由式,2,14,可解得,c,a,b,a,R,b,R,c,R,a,R,ca,R,ab,R,bc,a,b,c,b,對(duì)星形聯(lián)結(jié)的電阻，根據(jù)等效電阻的應(yīng)用，轉(zhuǎn)變?yōu)槿切温?lián)結(jié)的電阻，則,有,b,a,c,c,b,b,a,ca,a,a,c,c,b,b

7、,a,bc,c,a,c,c,b,b,a,ab,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,三角形聯(lián)結(jié)的電阻,星形聯(lián)結(jié)電阻中各電阻兩兩相乘積之和,星形聯(lián)結(jié)中另一端鈕所連電阻,若,R,a,R,b,R,c,R,Y,Y,ca,bc,ab,3,R,R,R,R,R,則有,對(duì)三角形聯(lián)結(jié)的電阻，根據(jù)等效電阻的應(yīng)用，轉(zhuǎn)變?yōu)樾切温?lián)結(jié)的電阻，則,有,ca,bc,ab,ca,bc,c,ca,bc,ab,bc,ab,b,ca,bc,ab,ab,ca,a,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,c,a,b,a,R,b,R,c,R,b,R,

8、ca,R,ab,R,bc,a,b,c,a,若,R,ab,R,bc,R,ca,R,則有,R,R,R,R,R,3,1,Y,c,b,a,星形聯(lián)結(jié)的電阻,三角形聯(lián)結(jié)電阻中兩相鄰電阻之積,三角形聯(lián)結(jié)電阻之和,注：電阻星形聯(lián)結(jié)有時(shí)又稱為,T,形電阻，電阻三角形聯(lián)結(jié)也稱為形電路,例,2-6,圖示電路，已知,U,s,100V,R,1,100,R,2,20,R,3,80,R,4,R,5,40，求電流,I,I,R,5,R,4,R,3,R,2,R,1,d,a,c,b,U,s,R,c,U,s,d,a,c,b,R,4,R,5,I,R,b,R,a,Y,40,80,20,100,100,80,8,80,20,100,80,

9、20,10,80,20,100,20,100,3,2,1,1,3,c,3,2,1,3,2,b,3,2,1,2,1,a,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,A,5,2,A,40,100,40,40,8,40,40,40,8,40,40,10,ad,s,5,b,4,c,5,b,4,c,a,ad,R,U,I,R,R,R,R,R,R,R,R,R,R,解,2-2-1,將圖,2-15(a,的,T,形電路等效變換為圖,2-15(b,的,形電路。已知,R,a,3,R,b,R,c,6,試求,R,ab,R,bc,R,ca,12,6,72,24,3,72,12,6,72,b,a,

10、c,c,b,b,a,ca,a,a,c,c,b,b,a,bc,c,a,c,c,b,b,a,ab,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,R,圖,2-15,2-2-2,將圖,2-17(a,的電路等效,變換為圖,2-17(b,的,T,形電路,已知,R,1,4,R,2,8,R,3,12,R,4,4,試求,R,a,R,b,R,c,解,R,1,R,2,R,3,為形聯(lián)結(jié)，變換成,T,形電路如圖,c,所示,d,b,a,c,R,a,R,b,R,d,R,4,c,a,圖中，有,R,ab,R,3,12,R,bd,R,2,8,R,da,R,1,4,則,2,da,bd,ab,da,ab,a,R,R,R,R,R,R,4,da,bd,ab,bd,ab,b,R,R,R,R,