Three-Phase Circuits

Y-Connection

Source

3 Phase Y Source

\begin{matrix} V_{l i n e} = V_{p h a s e} * \sqrt{3} ∠ 30 ° \\ I_{l i n e} = I_{p h a s e} \end{matrix}

Load

3 Phase Y Load

\begin{matrix} V_{l i n e} = V_{p h a s e} * \sqrt{3} ∠ 30 ° \\ I_{l i n e} = I_{p h a s e} \\ I_{A} = \frac{V_{A}}{Z}, I_{B} = \frac{V_{B}}{Z}, I_{C} = \frac{V_{C}}{Z} \end{matrix}

Δ-Connection

Source

3 Phase Delta Source

\begin{matrix} V_{l i n e} = V_{p h a s e} \\ I_{l i n e} = I_{p h a s e} * \sqrt{3} ∠ - 30 ° \end{matrix}

Load

3 Phase Delta Load

\begin{matrix} V_{l i n e} = V_{p h a s e} \\ I_{l i n e} = I_{p h a s e} * \sqrt{3} ∠ - 30 ° \\ I_{A B} = \frac{V_{A B}}{Z}, I_{B C} = \frac{V_{B C}}{Z}, I_{C A} = \frac{V_{C A}}{Z} \end{matrix}

Summary

$\large V_{phase} = V_{rms} \angle{0\degree}$ $\large I_{phase} = I_{rms} \angle{0\degree}$

Y-Connection	Δ-Connection
$\Large V_{phase} = V_A = V_{rms} \angle{0\degree}$	$\Large V_{phase} = V_{AB} = V_{rms} \angle{0\degree}$
$\Large V_{line} = V_{AB} = \sqrt{3} V_{rms} \angle{30\degree}$	$\Large V_{line} = V_{phase} = V_{AB} = V_{rms} \angle{0\degree}$
$\Large I_{phase} = I_A = \frac{V_A}{Z} = I_{rms} \angle{\phi}$	$\Large I_{phase} = I_{AB} = \frac{V_{AB}}{Z} = I_{rms} \angle{\phi}$
$\Large I_{line} = I_A = I_{phase} = I_{rms} \angle{\phi}$	$\Large I_{line} = I_A = \sqrt{3} I_{phase} \angle{-30\degree} = \sqrt{3} I_{rms} \angle{\phi - 30\degree}$

Y-Δ Conversion

Y-Delta Conversion

To convert from Y to Δ, multiply the impedances by 3, and to convert from Δ to Y, divide the impedances by 3.

Power

active power $\large P_{3\phi}$ reactive power $\large Q_{3\phi}$ apparent power $\large S_{3\phi}$ power factor $\large pf_{3\phi}$ for three-phase circuits are given by:

\begin{matrix} P_{3 ϕ} = 3 | V_{p h a s e} | | I_{p h a s e} | c o s ϕ \\ P_{3 ϕ} = \sqrt{3} | V_{l i n e} | | I_{l i n e} | c o s ϕ \\ Q_{3 ϕ} = 3 | V_{p h a s e} | | I_{p h a s e} | s i n ϕ \\ Q_{3 ϕ} = \sqrt{3} | V_{l i n e} | | I_{l i n e} | s i n ϕ \\ S_{3 ϕ} = 3 V_{p h a s e} I_{p h a s e}^{*} \\ | S_{3 ϕ} | = 3 | V_{p h a s e} | | I_{p h a s e} | \\ | S_{3 ϕ} | = \sqrt{3} | V_{l i n e} | | I_{l i n e} | \\ p f_{3 ϕ} = c o s ϕ = c o s (θ_{v_{p h a s e}} - θ_{i_{p h a s e}}) \end{matrix}