Harmonics and Parallel Operation of Transformers

This module delves into two critical aspects of transformer operation relevant to competitive exams like GATE: harmonics and the principles of parallel operation. Understanding these concepts is vital for analyzing power system behavior and ensuring efficient transformer utilization.

Understanding Harmonics in Transformers

Transformers, due to their non-linear magnetic characteristics, can introduce harmonic currents and voltages into the power system. These harmonics are sinusoidal components of voltage or current that have frequencies that are integer multiples of the fundamental frequency.

Non-linear magnetisation curves cause harmonic distortion.

The flux in a transformer core is proportional to the applied voltage. However, due to the non-linear B-H curve of the core material, the magnetizing current required to produce this flux is not purely sinusoidal. This non-sinusoidal magnetizing current is the primary source of harmonics.

The magnetic flux ( $\phi$ ) in a transformer core is directly related to the applied voltage ( $V$ ) and inversely related to the frequency ( $f$ ) by the equation $V = 4.44 f N \phi_{max}$ . However, the relationship between magnetic flux density ( $B$ ) and magnetic field strength ( $H$ ) in ferromagnetic materials is non-linear, described by the B-H curve. The magnetizing current ( $I_m$ ) is proportional to $H$ . As the flux ( $\phi$ ) varies sinusoidally, the required $H$ (and thus $I_m$ ) deviates from a sinusoidal waveform. This deviation is due to the saturation of the magnetic core. The non-sinusoidal magnetizing current can be represented by a Fourier series, which includes the fundamental component and its odd harmonics (primarily the 3rd, 5th, and 7th).

The most significant harmonics generated by transformers are the odd harmonics, particularly the third harmonic, due to the symmetrical nature of the B-H curve.

Effects of Harmonics

Harmonics can have several detrimental effects on power systems and connected equipment:

Increased Losses: Harmonics increase RMS current and voltage, leading to higher I²R losses in windings and increased core losses (hysteresis and eddy current losses).

Overheating: Increased losses result in overheating of transformers, motors, and cables, potentially leading to premature failure.

Malfunctioning of Electronic Equipment: Sensitive electronic devices, communication systems, and control circuits can be disrupted or malfunction due to harmonic distortion.

Resonance: Harmonics can cause resonance issues when combined with system capacitances and inductances, leading to dangerously high voltages and currents.

Parallel Operation of Transformers

Transformers are often operated in parallel to meet load demands that exceed the capacity of a single unit, to improve reliability, and for maintenance purposes. For successful parallel operation, certain conditions must be met.

Condition	Requirement	Reason
Same Voltage Ratio	Primary and secondary voltage ratings must be identical.	Ensures equal voltage across windings, preventing circulating currents.
Same Per-Unit Impedance	Per-unit impedance values must be equal (or very close).	Ensures proper load sharing. If impedances differ, the transformer with lower impedance will be overloaded.
Same Polarity	Transformers must be connected with correct polarity.	Incorrect polarity leads to a short circuit across the secondary windings.
Same Phase Displacement (for three-phase)	Vector groups must be the same.	Ensures that the secondary voltages are in phase, preventing circulating currents.
Same Frequency	Operating frequency must be the same.	Essential for proper synchronization and load sharing.

What is the most critical condition for parallel operation of transformers to ensure proper load sharing?

Having the same per-unit impedance.

When transformers with identical per-unit impedances are connected in parallel, they share the total load in proportion to their individual ratings. If the per-unit impedances are slightly different, the load sharing will be approximately proportional to the inverse of their per-unit impedances.

Let $S_{total}$ be the total load, $S_1$ and $S_2$ be the loads carried by transformer 1 and 2 respectively, and $Z_{pu1}$ and $Z_{pu2}$ be their per-unit impedances. For successful parallel operation, $S_1 + S_2 = S_{total}$ . If $Z_{pu1} = Z_{pu2}$ , then $S_1 = S_2 = S_{total}/2$ . If $Z_{pu1} eq Z_{pu2}$ , then the load sharing is given by:

$S_1 = S_{total} \frac{Z_{pu2}}{Z_{pu1} + Z_{pu2}}$

$S_2 = S_{total} \frac{Z_{pu1}}{Z_{pu1} + Z_{pu2}}$

A significant difference in per-unit impedance can lead to one transformer being overloaded while the other is underloaded, even if the total load is within the combined capacity.

Three-Phase Transformer Parallel Operation

For three-phase transformers, in addition to the conditions for single-phase transformers, the phase displacement between the primary and secondary voltages must be the same. This is determined by the vector group of the transformer (e.g., Dyn11, Yyn0). Connecting transformers with different vector groups can lead to severe circulating currents and damage.

Why is matching the vector group crucial for parallel operation of three-phase transformers?

To ensure secondary voltages are in phase, preventing circulating currents.

Mitigation of Harmonics

Several methods are employed to mitigate the effects of harmonics:

Harmonic Filters: Passive filters (LC circuits tuned to specific harmonic frequencies) or active filters can be used to absorb or cancel harmonic currents.

Phase Shifting Transformers: Using transformers with specific vector groups (like Dy11) can cancel out certain harmonics (e.g., the 3rd harmonic in Dy connections).

Oversizing Transformers: Designing transformers with a larger capacity than the nominal load can help them handle the increased losses due to harmonics.

Using K-rated Transformers: These transformers are specifically designed to handle the increased heating effects of harmonics.

Learning Resources

Harmonics in Power Systems - An Overview(blog)

Provides a foundational understanding of harmonics, their sources, and effects in power systems.

Transformer Parallel Operation Conditions(blog)

Details the essential conditions required for successful parallel operation of transformers, with explanations.

GATE Electrical Engineering - Power Systems(documentation)

Official syllabus for GATE Electrical Engineering, which includes Power Systems and Machines, providing context for the importance of these topics.

Understanding Harmonics in Electrical Systems(video)

A video tutorial explaining the concept of harmonics, their generation, and their impact on electrical systems.

Parallel Operation of Transformers - GATE Electrical(video)

A video lecture specifically covering the parallel operation of transformers, often with GATE-specific problem-solving approaches.

An authoritative document from IEEE detailing standards and practices for analyzing harmonics in power systems.

Transformer Vector Groups Explained(blog)

Explains the concept of transformer vector groups and their significance, especially for three-phase transformer connections.

Harmonic Distortion - Wikipedia(wikipedia)

A comprehensive overview of harmonic distortion, its mathematical representation, and its effects across various fields.

Power System Harmonics - An Introduction(blog)

An introductory article discussing the causes, effects, and mitigation strategies for harmonics in modern power systems.

Parallel Operation of Three Phase Transformers(blog)

Focuses on the specific requirements and considerations for operating three-phase transformers in parallel.

Harmonics and Parallel Operation