Delta Connections  Calculation of Capacitor Unbalance
The industrial market has become aware of the money that can be saved with the use of powerfactor correction capacitors and has given rise to more deltaconnected 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
X_{c} = 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 = approximately 192.6 amps. This confirms the fact that in a balanced deltaconnected 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 cofactor 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 AB
Let
V = 7200 V
AB = 600 KVAR
BC = 800 KVAR
CA = 800 KVAR
Again, using or C (in mfd) = and
We find
(See Figure 4)
Then
Figure 5
So We Have
But, as before, we can use for the leg currents.
You will notice I_{c} remained at in both cases. This is because I_{ca }+ I_{cb} didn't change in either case.
Again, mathematics can be used to solve the line currents. Ignoring the displacement again, we can easily see I_{a} and I_{b} will be equal as far as the integer cofactor is concerned.
Using The Law of Cosines, We Can Obtain the Formula
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
