Appendix (1) - Proof that the fault current in a capacitor is limited to three times the line current for an ungrounded wye capacitor bank.

Appendix (2) - Unbalance voltage formula for ungrounded wye connected capacitor banks.

APPENDIX (1)

Proof that the fault current in a capacitor is limited to three times the line current for an ungrounded wye capacitor bank.

First we will look at what happens to the voltage when a capacitor fails in one phase just before the fuse clears the faulted capacitor.

Figure 1 shows a balanced floating wye capacitor connection. Let 1 PU (per unit) equal VaN, VbN and VcN.

Assume a capacitor is failing in leg (a). Just before the fuse attached to this capacitor clears, the neutral (N) shifts to Va or, N’. See figure 2.

There is an increase in voltage across the capacitors in lines VbN’ and VcN’. As the capacitor fails and prior to the capacitor fuse clearing the neutral will shift from N to N’. Calculating the legs of the triangle created by sides b =(VbN), a=(VaN) and b’= (VbN’) we can use the law of Cosines:

(b’)² = a² + b² –2ab(CosN )

(b’)² = (VbN’)

But a=VaN =b=VbN=NN’=1PU, then a² = b² = 1

N = 120⁰

CosN = (-0.5)

Then (VbN’)² = 1² + 1² – 2(1)(1)(-0.5) = 3

(VbN’) = sqrt(3) or, approximately 1.73205

Using this knowledge for our vector analysis we will let Side a = Side b = sqrt(3). Therefore, the voltage on these legs will be at line-to-line potential.

The shift in the capacitor current will have the same effect as the voltage shift. Under normal conditions the current leads the voltage by 90^o . See figure 3.

Using Kirchhoff’s law, under balance conditions, we have:

I_N = IØa + IØb + IØc = 0

When the failure in leg (a) occurs as described above, the current will change accordingly.

See figure 4.

In figure 3, I_N = 0. However, I_N shifts to IØa position and is now considered I_N’. Again using Kirchhoff’s law we have the following:

I_N’ = IØb + IØc

Under normal conditions IØa = IØb = IØc, but as shown in the voltage calculation when unit in leg (a) begins to fail, Vb = Vc = (sqrt(3) * 1pu.) The same calculations will apply in the current calculations and IØb = IØc, = (sqrt(3) * normal line current.)

Using Polar and Complex number calculations from figure 4, we have the following:

I_N’ = (sqrt(3) * Ð330^O) + (sqrt(3) * Ð30^O)

= (1.5 - j0.866) + (1.5 + j0.866)

= (3 ± j0)

Therefore, I_N’ = 3 times normal line current.

See figure 5 for a vectorial solution. You will note that the figure is a Rhombus. The diagonals of a Rhombus intersect at right angles. Also, the diagonals divide the Rhombus into four equal and congruent triangles. Then A = B, let A and B equal (sqrt(3)). Then we have the following:

Vector A + Vector B = 2C (Resultant)

C = (Cos30⁰)(sqrt(3))

C = (0.866)(sqrt(3))

C = 1.5

\ 2C=3

APPENDIX (2)

Unbalance voltage formula for ungrounded wye connected capacitor banks.

The neutral shift due to removal of "F" units in one phase (with one series group) will be as follows:

%V_NS = (100F)/(3S(N-F)+2F

and V_NS =( % V_NS/100)*V_T (Volts)

The voltage on the remaining units with "F" units removed

%V_R = (100NS)/(S(N-F)+F)*(1+(F/(3S(N-F0+2F)*V_T/SV

Where:

See Table 1 for the affects of an unbalance on capacitor banks with four units in parallel, to meet the standard requirement, on 12,470, 13,200 and 13,800 voltage systems.