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# Three Phase Power

In this article, we will begin a discussion of three phase power. The name “three phase” comes from the fact that there are three distinct line-to-neutral voltages: $$V_a = V_{LN}cos(\omega_0 t + 0) \\ V_b = V_{LN}cos(\omega_0t + \frac{2\pi}{3}) \\ V_c = V_{LN}cos(\omega_0t - \frac{2\pi}{3})$$

It is also possible to measure the voltage difference across two lines, for example: $$V_{ab} = V_a - V_b = \sqrt{3}V_{LN}cos(\omega_0t + \frac{\pi}{6})$$

There are several benefits to using three phase power in comparison to single phase power. Firstly, power dissipation in transmission lines is proportional to $I_L^2R$. For a three phase system, there are three conductors, each carrying $\frac{1}{3}$ of the current required, therefore reducing power loss and improving efficiency. Additionally, the use of three phases causes the torque of rotating electrical machinery to be smoother.

## Common Connections in Three Phase Circuits

The two common configurations in three phase circuits are delta (Also written as $\Delta$): and wye (also known as star):

## Relations between Line and Phase Voltage and Currents

For Wye Circuits: $$V_{line} = \sqrt{3} V_{phase} \\ I_{line} = I_{phase}$$

For Delta Circuits: $$V_{line} = V_{phase}\\ I_{line} = \sqrt{3}I_{phase}$$

Where $V_{line}$ is the line to line voltage, $V_{phase}$ is the voltage across one phase of the load, $I_{line}$ is the current conducted by one line, and $I_{phase}$ is the current through one phase of the load.