V-V Connection Power Factor Analysis

The V-V connection, also known as the Open Delta connection, is used to supply a three-phase load using only two single-phase transformers. This configuration is employed when one transformer in a Delta-Delta bank fails or when the initial three-phase load does not warrant a full three-transformer bank.

When two single phase transformers are connected in V-V fashion to supply a balanced three phase load with power factor $\cos \theta$, the voltage across each transformer is the line-to-line voltage, and the current through each transformer is the line current.

Phase Relationships in V-V Connection

In a balanced three-phase system, the line voltages are displaced by 120 degrees from each other. Similarly, the line currents are also displaced by 120 degrees from each other.

For a balanced load with a power factor $\cos \theta$ (where $\theta$ is the angle between the phase voltage and phase current in the equivalent three-phase load), the line currents lag the corresponding phase voltages by the angle $\theta$.

In the V-V configuration, the voltages across the two transformers can be considered as two of the three line voltages, say $V_{AB}$ and $V_{BC}$. The currents through the transformers are the corresponding line currents, $I_A$ and $I_C$.

Transformer Power Factors

The power factor of an individual transformer is the cosine of the angle between the voltage across it and the current through it. Due to the specific phasor relationships in the V-V connection supplying a balanced load, the phase angles between the transformer voltage and current are not simply $\theta$.

Considering the standard phasor diagram for a V-V connection supplying a balanced load with power factor $\cos \theta$ lagging:

• The voltage $V_{AB}$ (across the first transformer) leads the phase voltage $V_{AN}$ by $30^\circ$.
• The current $I_A$ lags the phase voltage $V_{AN}$ by $\theta$.
• Therefore, the angle between $V_{AB}$ and $I_A$ is $30^\circ - (-\theta) = 30^\circ + \theta$. The power factor of the first transformer is $\cos (30^\circ + \theta)$.

• The voltage $V_{BC}$ (across the second transformer) leads the phase voltage $V_{BN}$ by $30^\circ$.
• The current $I_C$ leads the line voltage $V_{BC}$ by $(30^\circ - \theta)$. This is another way to look at the phase relationships in V-V connection.
• Therefore, the angle between $V_{BC}$ and $I_C$ is $30^\circ - \theta$. The power factor of the second transformer is $\cos (30^\circ - \theta)$.

The two transformers operate at different power factors, even when supplying a balanced load. These power factors are $\cos (30^\circ - \theta)$ and $\cos (30^\circ + \theta)$.

Summary of Power Factors

The power factors of the two single-phase transformers connected in V-V configuration supplying a balanced three-phase load with power factor $\cos \theta$ are $\cos (30^\circ - \theta)$ and $\cos (30^\circ + \theta)$.

Comparing this result with the given options, the correct power factors are \(\cos (30^\circ - \theta)\) and \(\cos (30^\circ + \theta)\).