Delta Connection Style

In the delta connection style, the 3 phases of the source or load are connected in series, and the three connection points are used as terminals for connection to the rest of the circuit.

The phasors of phase voltages, which are also line voltages, can be written as:

$$\begin{array}{rl}{\vec{V}}_{AB}& =V_{\text{phase}}\angle 0°=V_{LL}\angle 0°\\ {\vec{V}}_{BC}& =V_{\text{phase}}\angle -120°=V_{LL}\angle -120°\\ {\vec{V}}_{CA}& =V_{\text{phase}}\angle 120°=V_{LL}\angle 120°\end{array}$$

where $V_{\text{phase}}=V_{LL}$ is the rms value of the phase or line voltage.

Voltage and Current Relationships

Relationship	Value
Phase-to-neutral and line-to-line voltage magnitudes	$V_{LL}=V_{\text{phase}}$
Phase and line current magnitudes	$I_{LL}=\sqrt{3}{I}_{\text{phase}}$
Phase and line current phase angles	${\vec{I}}_x$ lags ${\vec{I}}_{xy}$ by $30°$ e.g. ${\vec{I}}_a$ lags ${\vec{I}}_{ab}$ by $30°$

Loads and Currents

Below a 3-phase $\Delta$-connected balanced load is shown.

The relationships between the phasors of phase voltages and currents are:

$$\begin{array}{rl}{\vec{I}}_{ab}& =\frac{{\vec{V}}_{ab}}{Z_{\text{phase}}}=\frac{V_{LL}\angle 0°}{|Z_{\text{phase}}\angle \theta |}=\frac{V_{LL}}{|Z_{\text{phase}}|}\angle -\theta =I_{\text{phase}}\angle -\theta \\ {\vec{I}}_{bc}& =\frac{{\vec{V}}_{bc}}{Z_{\text{phase}}}={\vec{I}}_{\text{phase}}\angle -120°-\theta \\ {\vec{I}}_{ca}& =\frac{{\vec{V}}_{ca}}{Z_{\text{phase}}}={\vec{I}}_{\text{phase}}\angle +120°-\theta \end{array}$$

The phasors of line currents can be found with KCL:

$$\begin{array}{rl}{\vec{I}}_a& ={\vec{I}}_{ab}-{\vec{I}}_{ca}=\sqrt{3}{I}_{\text{phase}}\angle -\theta -30°\\ {\vec{I}}_b& ={\vec{I}}_c-{\vec{I}}_{ab}=\sqrt{3}{I}_{\text{phase}}\angle -120°-\theta -30°=\sqrt{3}{I}_{\text{phase}}\angle -150°-\theta \\ {\vec{I}}_b& ={\vec{I}}_c-{\vec{I}}_{ab}=\sqrt{3}{I}_{\text{phase}}\angle +120°-\theta -30°=\sqrt{3}{I}_{\text{phase}}\angle 90°-\theta \end{array}$$

These current phasors are shown below

Note that, in a $\Delta$-connected source, we have

$$\begin{array}{rl}{\vec{V}}_{AB}+{\vec{V}}_{BC}+{\vec{V}}_{CA}=0& \text{or}{\vec{V}}_{ab}+{\vec{V}}_{bc}+{\vec{V}}_{ca}=0\\ {\vec{I}}_{AB}+{\vec{I}}_{BC}+{\vec{I}}_{CA}=0& \text{or}{\vec{I}}_{ab}+{\vec{I}}_{bc}+{\vec{I}}_{ca}=0\\ {\vec{I}}_A+{\vec{I}}_B+{\vec{I}}_C=0& \text{or}{\vec{I}}_a+{\vec{I}}_b+{\vec{I}}_c=0\end{array}$$