balanced/unbalanced supply and load concepts

balanced/unbalanced supply and load concepts

1. What Balanced Means

Balanced system: magnitudes equal and phases 120° apart (both supply and load). Unbalanced: either magnitudes unequal or phases not 120°, or loads unequal.

• Balanced supply: $V_a=V_{ph}∠0°$, $V_b=V_{ph}∠-120°$, $V_c=V_{ph}∠+120°$, $|V_a|=|V_b|=|V_c|$.
• Balanced load: impedances of three phases equal:

(same $|Z|$ and $pf$). Could be star (Y) or delta (Δ) connected.

Unbalanced: either supply unbalanced (fault, utility rarely) or load unbalanced (e.g., single-phase loads on different phases drawing different $I$).

2. Consequences

Balanced	Unbalanced
Line currents: equal magnitude, 120° apart, neutral current $I_n=I_a+I_b+I_c=0$ → neutral not needed (3-wire system)	Neutral current ≠0, must carry return, neutral wire needed (4-wire)
Power:

constant, easy calculation | Need per-phase calculation

, neutral carries |
| Voltage: phase voltages remain balanced | Neutral shift: star point not at zero ⇒ phase voltages unbalanced, some load over-voltage |
| Analysis: one phase suffices (single-phase equivalent) | Need three-phase analysis, symmetrical components (for faults) |

3. Star vs Delta Relation (balanced)

• Star (Y):

($V_{ph}$ phase-neutral, $V_L$ line-line)

• Delta (Δ):

Power both

4. Worked Checks

(a) Balanced star load Z=10∠30° Ω, V_ph=230V ⇒ I_ph=23∠-30° A, I_L=23A, I_n=0 ✓

(b) Unbalanced star: Z_a=10Ω, Z_b=20Ω, Z_c=30Ω at same V_ph ⇒ I_a=23A, I_b=11.5A, I_c=7.67A ⇒ I_n=23∠0 +11.5∠-120 +7.67∠120 ≠0 ⇒ neutral carries ~12A ⇒ neutral wire essential.

(c) Why 4-wire for distribution: residential single-phase loads on each phase are inherently unbalanced ⇒ neutral carries imbalance.

(d) Unbalanced supply causes negative-sequence currents in motors ⇒ heating.

5. Exam Points

• Define balanced (equal magnitude, 120°) vs unbalanced — 5 marks, very frequent (almost certain to ask).
• Show balanced $\Rightarrow I_n=0$, neutral not needed / 3-wire; unbalanced ⇒ $I_n≠0$ neutral needed, neutral shift — 5-7 marks, very frequent.
• Give $V_L$ vs $V_{ph}$ for Y ($√3$) and Δ — 5 marks, very frequent.
• Power formulas $P=√3 V_L I_L cosφ$ vs per-phase sum — 3-5 marks, frequent.