Delta Connections
Calculation of Capacitor Unbalance
(Article # GES4233)
By Neal S. Ciurro
The industrial market has become aware of the money that can be saved with the use of
power-factor correction capacitors and has given rise to more delta-connected capacitor
banks. Customers showing concern for identifying failures within the banks find that the
biggest problem is in finding an easy method of determining the current differences in
unbalanced conditions so that relay and/or alarms can be used. This problem can be
reduced considerably with mathematics. First, consider the balanced delta connection of
Figure 1. Using a simple and standard connection, let the system be 7200 volts, delta, with
2400 kvar. This means 800 kvar per phase.
Figure 1
Converting kvar into microfarads, we have:
Using the parameters set up:
From here, the capacitive reactance can be determined.
Where:
V = phase to phase voltage
C = capacitance in microfarads
Xc = capacitive reactance
Figure 2. Hypothetical circuit
Based on the balanced load, we know the phase voltages are 120° apart. We also know that
current leads the voltage by 90° degrees. This is shown in Figure 3.
Figure 3. Phasor diagram of the circuit.
From Figures 2 and 3, we have:
The leg currents will then become:
Using Kirchoff's Law:
or
Note that 111.3 amps times
= approximately192.6 amps. This confirms the fact that in
a balanced delta-connected load, the line voltage and phase voltage are equal, and the line
current is
times larger than the phase current.
This arduous method is simplified by mathematics if the displacement of the currents is not
considered and only the integer co-factor is considered. This is simply:
In our case, it would be:
Line current is
Leg current is
But the main concern is the unbalanced delta. What happens to the line and leg currents?
The above calculation will not apply. Again, in order to appreciate the mathematics, we will
go through the pains of the total calculations.
The phase currents need to be computed and then Kirchoff's current law applied at the
junctions to obtain the three line currents. The line currents will not be equal nor will they
have a 120° difference as in a balanced load.
Figure 4.
Assume we lose a unit in leg A-B:
Let
V = 7200V
AB = 600 kvar
BC = 800 kvar
CA = 800 kvar
Again, using
or C (in mfd) =
We find
(See Figure 4)
Then:
and
Figure 5
So, we have:
But, as before, we can use
for the leg currents.
You will notice Ic remained at
change in either case.
in both cases. This is because Ica + Icb didn't
Again, mathematics can be used to solve the line currents. Ignoring the displacement again,
we can easily see Ia and Ib will be equal as far as the integer co-factor is concerned.
Using the law of cosines, we can obtain the formula:
or
In our case:
Summary
The balanced delta capacitor circuit and calculations are basic but still prove to be time
consuming. For the most part, angular displacement is not significant and there is no reason
to go through the long arduous task of finding the displacement just to obtain the resultant,
or current. The formula
provides the resultant we seek.
The same holds true on the unbalanced delta condition and can be easily handled with the
formula