Phasor Diagram and Voltage Regulation

Definition

Phasor diagram is a vector diagram used in AC circuit analysis to represent sinusoidal quantities such as voltage and current by arrows (phasors) showing their magnitude and phase angle.

Voltage regulation is the measure of change in terminal voltage when the load changes from no-load to full-load, with all other conditions kept constant, usually expressed as a percentage of full-load voltage.

For a transformer, voltage regulation is given by:

$$\text{Voltage Regulation} = \frac{V_{no\ load} - V_{full\ load}}{V_{full\ load}} \times 100\%$$

A low voltage regulation means the equipment maintains a nearly constant output voltage under varying load, which is desirable in power systems.

Main Content

1. Phasor Diagram in AC Circuits and Transformers

• A phasor diagram is used to show the relationship between induced emf, applied voltage, current, and impedance drops in an AC circuit.
• In transformers, phasor diagrams help explain how primary and secondary voltages behave under different load power factors, making them very useful for performance analysis.

In a transformer:

• The primary voltage $V_1$ is balanced by the back emf $E_1$, the resistive drop $I_1R_1$, and the reactive drop $I_1X_1$.
• The secondary terminal voltage $V_2$ depends on the secondary induced emf $E_2$, copper loss drops, and the load current angle.

Phasor diagrams are drawn by taking the flux $\phi$ as the reference. The induced emfs $E_1$ and $E_2$ lag the flux by 90 degrees. The current phasor depends on the load:

• For lagging power factor, current lags the voltage.
• For leading power factor, current leads the voltage.
• For unity power factor, current is in phase with the voltage.

These diagrams are important because they show why the terminal voltage may drop or rise depending on the nature of the load.

2. Voltage Regulation and Its Dependence on Load Power Factor

• Voltage regulation tells us how much the output voltage changes when a device supplies load current.
• The amount of voltage drop is strongly affected by the load power factor and internal impedance of the transformer or alternator.

For a transformer, the approximate change in voltage is influenced by:

Resistance drop

• : $IR$, which is in phase with current.

Reactance drop

• : $IX$, which is 90 degrees ahead of current.

The sign and magnitude of these drops vary with the power factor:

Lagging power factor load

• : both resistance and reactance drops usually reduce the secondary voltage, so regulation becomes positive and comparatively high.

Unity power factor load

• : voltage drop is mainly due to resistance, so regulation is moderate.

Leading power factor load

• : reactance drop may oppose the resistive drop, and the terminal voltage may rise, producing very low or even negative regulation.

This is why capacitive or leading loads can sometimes improve the terminal voltage of a transformer. In practical systems, good regulation is important because sensitive equipment requires stable voltage for proper operation.

The approximate voltage regulation of a transformer can be written as:

$$\%VR \approx \frac{I_2R_{eq}\cos\phi \pm I_2X_{eq}\sin\phi}{V_2} \times 100$$

where:

• $R_{eq}$ = equivalent resistance
• $X_{eq}$ = equivalent reactance
• $\phi$ = load power factor angle

The plus sign is used for lagging power factor and the minus sign for leading power factor.

3. Construction and Interpretation of Phasor Diagrams for Regulation Analysis

• Phasor diagrams are constructed by selecting one phasor as a reference and drawing all other voltages and currents according to their phase relationships.
• They help determine whether the terminal voltage increases or decreases under load and therefore help calculate voltage regulation accurately.

Typical construction steps include:

• Draw the load current $I_2$ as reference, or in some cases the terminal voltage $V_2$ as the base reference.
• Add the internal drops $I_2R_{eq}$ and $I_2X_{eq}$ vectorially to obtain the induced emf $E_2$.
• Compare $E_2$ and $V_2$ to determine voltage regulation.

Interpretation:

• If $E_2 > V_2$, the transformer has positive regulation, meaning voltage falls under load.
• If $E_2 \approx V_2$, regulation is small and performance is good.
• If $E_2 < V_2$, regulation is negative, usually due to leading power factor load.

Example: A transformer supplying an inductive load will show a larger voltage drop because the reactive component of current causes a larger internal drop. This can be visualized directly in the phasor diagram, where the reactance drop adds substantially to the induced emf required to maintain the terminal voltage.

In alternators, phasor diagrams similarly show the relation: $$E = V + I_aR_a + jI_aX_s$$ where $E$ is generated emf, $V$ is terminal voltage, $I_a$ is armature current, $R_a$ is armature resistance, and $X_s$ is synchronous reactance. The same idea of voltage regulation applies to machines delivering different load types.

Working / Process

1. Draw the reference phasor, usually terminal voltage or flux, depending on the machine being analyzed.
2. Draw the current phasor according to the load power factor: lagging, leading, or unity.
3. Add the internal voltage drops $IR$ and $IX$ vectorially to obtain the induced emf, then compare it with terminal voltage to find voltage regulation.

Advantages / Applications

• Phasor diagrams make AC circuit analysis simple, visual, and easy to understand.
• Voltage regulation helps evaluate the performance of transformers, alternators, and other power equipment under load.
• These concepts are widely used in power transmission, distribution systems, and electrical machine design to ensure stable voltage supply.
• They are useful in estimating whether a device is suitable for sensitive industrial or domestic loads.
• They help engineers improve system efficiency by minimizing unwanted voltage drops and selecting proper compensation methods.

Summary

• Phasor diagrams show the phase and magnitude relationships between AC quantities.
• Voltage regulation measures the change in terminal voltage from no-load to full-load.
• Load power factor strongly affects voltage regulation, with lagging loads causing greater drop and leading loads sometimes causing voltage rise.
• Together, these concepts are essential for analyzing and improving transformer and alternator performance.