Voltage regulation across power factors

Reference notes

Use Next → on the narrator above to walk through six steps that build the secondary-side phasor construction, derive the approximate regulation formula, and sweep the load power factor from heavy lagging through unity to leading.

What "voltage regulation" actually means

For a transformer, the no-load secondary voltage V_NL is what V₂ would be if you suddenly opened the load — basically V₁/a, the source's view of the secondary. When you apply rated load, the actual terminal voltage V₂ is lower by an amount that depends on the load current and the equivalent series impedance R_eq + jX_eq. Voltage regulation is the percent drop:

Regulation scales with load — at half load the drops are half, so the regulation is roughly half. At constant load, it depends on the load power factor.

The phasor construction

Place V₂ along the reference and let the load current I₂ sit at angle θ behind V₂ (lagging) or ahead (leading). The two equivalent series-impedance drops are:

• I₂·R_eq — in phase with I₂ (resistive drop)
• I₂·X_eq — 90° leading I₂ (inductive drop)

Adding them tip-to-tail to V₂ gives V_NL:

The magnitude of V_NL minus V₂ is the regulation — a small number, but on a power system carrying tens of MW, very much worth getting right.

The approximate regulation formula

For small drops (a few per cent of V₂), the bend in the phasor triangle is tiny and we can just project the drops onto the V₂ direction:

where ε_r = I₂·R_eq / V₂ and ε_x = I₂·X_eq / V₂ are the per-unit resistive and reactive drops at full load. The sign of the second term depends on the power factor:

• Lagging PF uses +ε_x·sin θ — both terms are positive, regulation is at its largest.
• Unity PF kills the sin θ term — regulation reduces to about ε_r alone (a small positive number).
• Leading PF uses −ε_x·sin θ — the two terms can subtract enough to make the whole thing negative, meaning the no-load voltage is smaller than the full-load voltage. In other words, the secondary terminal voltage rises when you connect the load.

Maximum regulation

Differentiating the approximate formula with respect to θ shows that regulation is maximum when the load PF angle equals the equivalent-impedance angle:

and the maximum value equals the per-unit magnitude of the equivalent impedance, ε_z = √(ε_r² + ε_x²).

Why this matters in practice

• Distribution transformers feeding industrial loads (motors, fluorescent lights) see lagging PF → larger regulation, so terminal voltage sags. PF-correction capacitor banks pull the load PF closer to unity to limit the sag.
• Long, lightly-loaded transmission lines feeding into a transformer can present a leading load (line-charging capacitance dominates) → negative regulation, with terminal voltage rising at full load. Utilities have to actively manage this.
• The same equivalent-circuit parameters (R_eq, X_eq) we extracted from the SC test in the previous lesson are exactly what we need here. The two lessons fit together: SC test measures the parameters; regulation uses them to predict behaviour.

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