Power in Three-Phase Circuits

Lecture-3: Overview

• \(1-\phi\) and \(3-\phi\) Circuits & Quantities

• Voltages and Currents in \(Y\) and \(\Delta\) Connections

• \(Y \leftrightarrow \Delta\) conversion

• Power in \(3-\phi\) circuits

\(3-\phi\) Circuits

• Mostly electricity is generated by \(3-\phi\) AC generators

• \(3-\phi\) system

  • has three phases, i.e., the current will pass through the three wires

  • there will be one neutral wire for passing the fault current to the earth ( \(3-\phi\) 4-wire system)

• \(3-\phi\) system can be used as a \(1-\phi\) if one of their phase and the neutral wire is taken out from it

• C.S.A of the neutral conductor is half of the live wire.

\(1-\phi\) System

\(3-\phi\) System

Why \(3-\phi\) is Preferred Over \(1-\phi\)?

Advantages of \(3-\phi\) over \(1-\phi\) system:

• can be used as 3 \(\times\) \(1-\phi\) system

• conductor needed is 75% of \(1-\phi\) circuit

• instantaneous power in \(1-\phi\) system falls down to zero as can seen from the sinusoidal curve but in \(3-\phi\) the net power from all the phases gives a continuous power to the load.

• higher efficiency and minimum losses

Phase Sequence, Line and Phase Quantities

• Sinusoidal steady state and balanced condition \(\rightarrow\) 3-voltages are equal in magnitude and displaced by \(120^{\circ}\) w.r.t each other

• Phase sequence: of voltage \(a-b-c\), where \(a\)-phase leads \(b\) by \(120^{\circ}\), \(b\) leads \(c\) by \(120^{\circ}\) and so on

• Phase sequence applies to both time-domain and phasor-domain

• Line Voltage: measured between any two line conductors

• Phase Voltage: measured across any one component (source winding or load impedance)

• Line current through any one line between a three-phase source and load

• Phase current through any one component comprising a three-phase source or load.

• \(3-\phi\) AC circuits are either \(Y\) or \(\Delta\) connected

Line/Phase and Load in \(3-\phi\) System

STAR (Y): Current & Voltage Relations

\[ \begin{aligned} &\mathrm{E}_{\mathrm{NR}}=\mathrm{E}_{\mathrm{NY}}=\mathrm{E}_{\mathrm{NB}}=\mathrm{E}_{\mathrm{ph}}(\text{ in mag.})\\ &\overline{\mathrm{E}_{\mathrm{NR}}}+\overline{\mathrm{E}_{\mathrm{RY}}}-\overline{\mathrm{E}_{\mathrm{NY}}}=0\\ \Rightarrow&\overline{\mathrm{E}_{\mathrm{RY}}}=\overline{\mathrm{E}_{\mathrm{NY}}}-\overline{\mathrm{E}_{\mathrm{NR}}}(\text { vector diff.}) \\ \mathrm{E}_{\mathrm{RY}}& =\sqrt{\mathrm{E}_{\mathrm{NY}}^{2}+\mathrm{E}_{\mathrm{NR}}^{2}+2 \mathrm{E}_{\mathrm{NY}} \mathrm{E}_{\mathrm{NR}} \cos 60^{\circ}}\\ \Rightarrow & E_{L}=\sqrt{E_{p h}^{2}+E_{p h}^{2}+2 E_{p h} E_{p h} \times 0.5 }\\ \Rightarrow&\mathrm{E}_{\mathrm{L}}=\sqrt{3 \mathrm{E}_{\mathrm{ph}}^{2}}=\sqrt{3} \mathrm{E}_{\mathrm{ph}} \text { (in mag.) } \\ &\mathrm{I}_{\mathrm{NR}}=\mathrm{I}_{\mathrm{NY}}=\mathrm{I}_{\mathrm{NB}}=\mathrm{I}_{\mathrm{ph}} \\ &\mathrm{I}_{\mathrm{R}}=\mathrm{I}_{\mathrm{Y}}=\mathrm{I}_{\mathrm{B}}=\mathrm{I}_{\mathrm{L}} \end{aligned} \]

Delta(\(\Delta\)): Current & Voltage Relations

\[ \begin{aligned} &\mathrm{E}_{12}=\mathrm{E}_{\mathrm{RY}};~\mathrm{E}_{23}=\mathrm{E}_{\mathrm{YB}};~ \mathrm{E}_{31}=\mathrm{E}_{\mathrm{BR}} \\ &\mathrm{E}_{12}=\mathrm{E}_{23}=\mathrm{E}_{31}=\mathrm{E}_{\mathrm{ph}} \\ &\mathrm{E}_{\mathrm{RY}}=\mathrm{E}_{\mathrm{YB}}=\mathrm{E}_{\mathrm{BR}}=\mathrm{E}_{\mathrm{L}} \\ &\mathrm{I}_{12}=\mathrm{I}_{23}=\mathrm{I}_{31}=\mathrm{I}_{\mathrm{ph}} \\ & \overline{\mathrm{I}_{31}}=\overline{\mathrm{I}_{\mathrm{R}}}+\overrightarrow{\mathrm{I}_{12}} \Rightarrow \overline{\mathrm{I}_{\mathrm{R}}}=\overline{\mathrm{I}_{31}}-\overline{\mathrm{I}_{12}} \\ & \mathrm{I}_{\mathrm{R}}=\sqrt{\mathrm{I}_{31}^{2}+\mathrm{I}_{12}^{2}+2 \mathrm{I}_{31} \mathrm{I}_{12} \cos 60^{\circ}} \\ & \mathrm{I}_{\mathrm{L}}=\sqrt{\mathrm{I}_{\mathrm{ph}}^{2}+\mathrm{I}_{\mathrm{ph}}^{2}+2 \mathrm{I}_{\mathrm{ph}} \mathrm{I}_{\mathrm{ph}} \times 0.5} \\ & \mathrm{I}_{\mathrm{L}}=\sqrt{3 \mathrm{I}_{\mathrm{ph}}^{2}}=\sqrt{3} \mathrm{I}_{\mathrm{ph}} \end{aligned} \]

\(Y \leftrightarrow \Delta\) Transformation

\[\begin{aligned} R_{a b} &=\frac{R_{a} R_{b}+R_{a} R_{c}+R_{b} R_{c}}{R_{c}} \\ R_{a c} &=\frac{R_{a} R_{b}+R_{a} R_{c}+R_{b} R_{c}}{R_{b}} \\ R_{b c} &=\frac{R_{a} R_{b}+R_{a} R_{c}+R_{b} R_{c}}{R_{a}} \end{aligned}\]