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Sahithyan's S1
Sahithyan's S1 — Electrical Fundamentals

3-Phased System

The phases are denoted by R,Y,B\text{R},\text{Y},\text{B} in that order.

Why 3-phase?

Why not 1-phase?

  • The current can be distributed into 3 wires instead of just 1.
    There is a maximum limit of how much current a wire can carry.
  • Economical as less amount of wires.
    3-phase system requires 44 wires (33 if balanced) while single phase system requires 66.

Why not 2 or 4 or 6?

  • Generators are volume constrained.
    3-phase system has the most optimum volume utilization in a generator.
  • 3-wired 3-phase system carry 3 times the power of a 2-wired 1-phase system.
  • As number of phases increases, the power advantage diminishes.
  • As number of phases increases, the complexity of construction and maintanance increases.
  • 3-phase motors have the most optimum volume utilization for a given output.

Balanced 3-phase

A 3-phase system is said to be balanced iff:

  • Supply is balanced
  • Loads are the same in each phase

Power source

A 3-phase power source which produces 3 phase voltages of equal rms value, but with 120°120° phase difference.

Phasor diagram

A graphical representation of the magnitude and phase relationship between different waveforms.

In a balanced 3-phase system, the phasor diagram shows three phasors (vectors) of equal length, separated by 120° angles. Each phasor represents the voltage of one phase. The diagram helps in visualizing the phase differences and the symmetry of the system.

Phase voltage

Voltage between a phase wire and the neutral wire.

VRNV_{\text{RN}}, VYNV_{\text{YN}}, VBNV_{\text{BN}} are the phase voltages.

Line-to-line voltage

Voltage between any 2 phase wires. Line-to-line voltages also have a 120°120° phase difference.

VRYV_{\text{RY}}, VYBV_{\text{YB}}, VBRV_{\text{BR}} are the line-to-line voltages or line voltages.

VBR=2×VBNcos(30°)=3VBN\big| V_{\text{BR}} \big| = 2 \times \big| V_{\text{BN}} \big| \cos(30°) = \sqrt{3} \big| V_{\text{BN}} \big|

Analysis

Analysis of 3-phase circuit

IN=E[10°zR+1120°zY+1120°zB]I_N=E \bigg[ \frac{1\angle 0°}{z_R} + \frac{1\angle -120°}{z_Y} + \frac{1\angle 120°}{z_B} \bigg]

When the loads are balanced: zR=zY=zB=zz_R=z_Y=z_B=z, IN=0I_N = 0
In this case, neutral wire is optional and can be eliminated. IN=0I_N=0 have to be maintained so that the voltage is equal to ground voltage in neutral wire. This makes sure there are no power losses in neutral wire.

Real-life Usage

Most domestic loads are single-phase. In case of 3-phase domestic wiring, the single-phase loads are distributed among the 3 phases at the main distribution board.

Devices that have a 3-phase power input, doesn’t require a neutral line.

Per-phase Equivalent Circuit

Power, voltage, current, power factor are same for all 33 phases.

When a 3-phase system is balanced, it is sufficient to consider only a single phase. The diagram showing the single-phase equivalent of the power system using standard symbols.

Per-phase equivalent circuit

Here:

  • EE - voltage across the source
  • VV - voltage across the load
Per-phase power=VpIlcosθ=13×3-phase power        Vl=3Vp\text{Per-phase power} = |V_p||I_l|\cos\theta = \frac{1}{3}\times\text{3-phase power} \;\; \land \;\; |V_l|= \sqrt 3 |V_p|     3-phase power=3VlIlcosθ\implies \text{3-phase power}= \sqrt{3}|V_l||I_l|\cos\theta

Here:

  • VpV_p - phase voltage
  • VlV_l - line voltage
  • IlI_l - line current
  • cosθ\cos\theta - power factor
  • The power can either be source power, load power, or transmission power losses.

Unbalanced 3-phase system

A 3-phase system becomes unbalanced, when load distribution is not equal among the phases. IN0I_N\neq 0. Highly undesirable. Neutral wire is the return path for the line currents and is compulsory.

Large currents in the neutral wire could cause:

  • If neutral wire have significant impedance, different points of the neutral wire will have different voltage
  • Series voltage unbalances can happen if the neutral wire is broken

Each phase will be different. Complete system has to be considered when analyzing the circuit.