Why Square root of Three enters all Three phase calculations

Have you ever thought why there is always a √3 present in every 3 phase calculations, let it be a delta or star transformer.

Actually this was not derived with heavy thinking or with supercomputers; rather we have to apply the simple basics of transformers which we have studied primarily.

And keep in mind, one main fact is that the differentiation between phase voltage and line voltage. Let me explore that first.

Line voltage is nothing but the output voltage measured at transformer externally i.e. Voltage available between any 2 wires out of 3 wires. We call it as V_L.

But Phase voltage is the output voltage available internally i.e. Voltage available between single phase wire and neutral. We call it as V_ph.

And the same definition holds good for current available which shall be called as I_L & I_ph.

Our basic rule is to measure power available at secondary which is nothing but the product of Phase Voltage x Phase Current and summation of all three phases,

                                              P             =             3 x V_ph x I_ph                                          …. (1)

But as you see it is very difficult to measure the phase voltages from a transformer which may involve complex connections and may be impossible rather it’s easy to measure from the lines present outside.

So here we go with a bit trigonometric work to derive the phase data from available line data.

As we know in a star connected transformer, the line voltage is nothing but the combination of two phase voltages which are out of phase by 120°. But current will be same for both phase and line. And in delta connected transformer, two currents will be out of phase by 120° and voltages will be equal.

Here we take the case of star connected transformer,

Since the line voltage is combination of two phase voltages we cannot just add together to make V_L  = 2 x V_ph which doesn’t make sense since there is a phase shift of 120°.

So,                                        V_L  = 2 x V_ph x sin(120)

                                              V_L  = 2 x V_ph x √3/2

                                              V_L  = √3 x V_ph

                       V_ph  = V_L  /√3                                                      …. (2)

Substituting (2) in (1),

We get                                 P = 3 x V_L  /√3 x I_L               since I_ph  = I_L  

                                              P = √3 .V_L .I_L

This holds good for transformers and for other loads like motors, generators etc another factor called power factor gets introduced in the equation and converted as,

                                    P = √3 .V_L .I_L.cosθ

Open Delta Transformers

Open Delta Connection:

An open-delta connection is a method of providing a three-phase supply, using two single-phase transformers. It is particularly useful if , say, one single-phase transformer, part of a three single-phase transformers forming a three-phase transformer bank, becomes damaged -allowing the two remaining transformers to provide a temporary three-phase supply to the load.
The drawback with this connection is that the capacity of the transformer bank is reduced, and it can only provide a lower load current.

It should be noted that the output power of an open delta connection is only 87% of the rated power of the two transformers. For example, assume two transformers, each having a capacity of 25 kVA, are connected in an open delta connection. The total output power of this connection is 43.5 kVA (50 kVA x 0.87 = 43.5 kVA).

Another figure given for this calculation is 58%. This percentage assumes a closed delta bank containing 3 transformers. If three 25 kVA transformers were connected to form a closed delta connection, the total output would be 75 kVA (3 x 25 = 75 kVA). If one of these transformers were removed and the transformer bank operated as an open delta connection, the output power would be reduced to 58% of its original capacity of 75 kVA. The output capacity of the open delta bank is 43.5 kVA (75 kVA x .58% = 43.5 kVA).

The voltage and current values of an open delta connection are computed in the same manner as a standard delta-delta connection when three transformers are employed. The voltage and current rules for a delta connection must be used when determining line and phase values of voltage current.

Closing a Delta:

When closing a delta system, connections should be checked for proper polarity before making the final connection and applying power. If the phase winding of one transformer is reversed, an extremely high current will flow when power is applied. Proper phasing can be checked with a voltmeter at delta opening. If power is applied to the transformer bank before the delta connection is closed, the voltmeter should indicate 0 volts. If one phase winding has been reversed, however, the voltmeter will indicate double the amount of voltage. It should be noted that a voltmeter is a high impedance device. It is not unusual for a voltmeter to indicate some amount of voltage before the delta is closed, especially if the primary has been connected as a wye and the secondary as a delta. When this is the case, the voltmeter will generally indicate close to the normal output voltage if the connection is correct and double the output voltage if the connection is incorrect

Calculation on how 58% arrived for open delta transformers:

There are two biggest advantages:

1) In an unearthed system like capacitor banks, it is possible to detect ground faults though earth path is not available.

2) In case of bulk power transmission where transformer banks are used, their advantage is that if one leg is out for maintenance/replacement than also approx. 57% power can be transmitted.

Diff between Earthed & Unearthed cable

In 3phase earthed system, phase to earth voltage is 1.732 times less than phase to phase 

voltage. Therefore voltage stress on cable to armor is 1.732 times less than voltage stress

between conductor to conductor. Whereas in unearthed system, (if system  neutral is not

grounded) phase to ground voltage can be equal to phase to phase voltage. In such case the

insulation level of conductor to armor should be equal to insulation level of conductor to 

conductor.

Also can be detailed as:

For cables to be used in solidly earthed systems, the phase-to-armour insulation has to be rated for U/root(3) only which is the phase-to-ground voltage when the system-neutral is  solidly earthed with no intentional resistance in the neutral grounding circuit. But in the case of system-neutral being resistance-earthed, then the phase-to-ground voltage of the two healthy phases rise up when an earth fault occurs on the third phase. When the system-neutral is high-resistance-earthed or left unearthed, the phase-to-ground voltage of healthy phases come close to or attain phase-to-phase values depending on the degree of effectiveness of system-neutral earthing. Therefore the phase-to-armour insulation of cables used in ungrounded systems could be rated for the full phase-to-phase voltage U instead of for U/root(3). The cables to be used in solidly earthed systems can have the phase-to-armour insulation rated for U/root(3). The U/Uo rating of the cable indicates the voltage rating of the core-to-core insulation and the core-to-armour insulation. For example for a 6.6kV ungrounded system, 6.6kV/6.6kV (UE) class cable has to be used while 6.6kV/3.8kV (E) class cables are adequate for solidly earthed systems. The UE-class cable is
naturally costlier than the earthed class of cable.