bus. The matrix [ZTOT] represents the Thevenin equivalent impedance matrix at the faulted bus. The
fault impedance is represented byZfin Fig. 21.34.
Kirchhoff’s voltage law in matrix form can be applied to the circuit of Fig. 21.33.
Ea
Eb
Ec2
43(^5) ¼
Zaa Zab Zac
Zba Zbb Zbc
Zca Zcb Zcc
2
4
3
5
Ifa
Ifb
Ifc
2
4
3
(^5) þ
Zf 00
0 Zf 0
00 Zf
2
4
3
5
Ifa
Ifb
Ifc
2
4
3
(^5) þ
Vax
Vbx
Vcx
2
4
3
(^5) þ
Vxg
Vxg
Vxg
2
4
3
(^5) (21:200)
Equation (21.188)can be written in compressed form as
½¼Eabc ½ZTOT½þIfabc ½ZF½þIfabc ½þVabcx Vxg
(21:201)
Combine terms in Eq. (21.201).
½¼Eabc ½ZEQ½þIfabc ½þVabcx Vxg
(21:202)
where½ZEQ¼½ZTOTþ½ZF (21:203)
Solve Eq. (21.202) for the fault currents:
½¼Ifabc ½YEQ½Eabc ½YEQ½Vabcx ½YEQ Vxg
(21:204)
where½¼YEQ ½ZEQ^1 (21:205)
Since the matrices [YEQ] and [Eabc] are known, define
½¼IPabc ½YEQ½Eabc (21:206)
Substituting Eq. (21.206) into Eq. (21.204) results in the following expanded equation:
Ifa
Ifb
Ifc
2
4
3
(^5) ¼
IPa
IPb
IPc
2
4
3
(^5)
Yaa Yab Yac
Yba Ybb Ybc
Yca Ycb Ycc
2
4
3
5
Vax
Vbx
Vcx
2
4
3
(^5)
Yaa Yab Yac
Yba Ybb Ybc
Yca Ycb Ycc
2
4
3
5
Vxg
Vxg
Vxg
2
4
3
(^5) (21:207)
[ZTOT]
Z (^) f a
Ifa Vax
- Z (^) f b
Ifb
+Vbx x
−
−
− −
−
Z (^) f c
Ifc
Faulted
Bus
Vcx
Vxg
−
Ec
−
Eb
−
Ea
FIGURE 21.34 Unbalanced feeder short-circuit analysis model.