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A Practical Guide for 2025: How Do You Calculate Turn Ratio with 3 Proven Methods?

Sep 2 | INDUSTRY NEWS

Abstract

The transformer turns ratio represents the fundamental relationship between the number of turns in the primary and secondary windings, a parameter that directly governs the transformer’s voltage and current transformation capabilities. Calculating this ratio with precision is foundational to commissioning, routine maintenance, and fault diagnosis of transformers. This document examines the theoretical and practical dimensions of determining the turns ratio. It begins by establishing the physical principles, primarily Faraday’s Law of Induction, that define the transformer’s operation. Three distinct methods for calculation are explored in depth: the direct voltage measurement method, the conceptual physical turn count, and the application of a modern Transformer Turns Ratio (TTR) tester. The analysis extends beyond mere calculation to the interpretation of results, focusing on the industry-accepted tolerance of ±0.5% and how deviations from this standard can indicate specific internal faults such as shorted turns, open circuits, or incorrect tap changer settings. The discourse also addresses the complexities introduced by tap changers and the diagnostic value of related measurements like excitation current and phase angle, providing a comprehensive framework for ensuring a transformer’s operational integrity and reliability.

Key Takeaways

  • The turn ratio is the number of primary turns divided by the number of secondary turns.
  • A deviation greater than ±0.5% from the nameplate ratio often indicates an internal fault.
  • Learn how do you calculate turn ratio using voltage measurements for a basic field check.
  • A TTR tester provides the most accurate and safest method for turn ratio measurement.
  • Testing all tap positions is necessary for transformers equipped with tap changers.
  • High excitation current alongside a bad ratio points towards a winding or core issue.

Table of Contents

Understanding the Transformer Turn Ratio: The Foundational Principle

Before we can embark on the practical steps of calculation, it is profoundly important to first build a mental model of what a transformer is and why this specific ratio holds such a central place in its function. To approach the ‘how’ without first grasping the ‘why’ would be like learning the grammar of a language without any sense of its poetry or purpose.

What is a Transformer and Why Does it Need Windings?

At its heart, a transformer is a device of elegant simplicity. It transfers electrical energy from one alternating-current (AC) circuit to another without a direct electrical connection. Its purpose is to change the voltage and current levels. Imagine you have a powerful, fast-flowing river. A transformer is like a system of gears that can convert that fast, high-force flow into a slower, wider river, or vice-versa, all while conserving the total amount of water (or power).

This transformation is achieved through two or more coils of insulated wire, known as windings, wrapped around a common magnetic core, typically made of laminated iron. The winding that receives energy from the source is called the primary winding. The winding that delivers energy to the load is the secondary winding.

When an AC voltage is applied to the primary winding, it creates a fluctuating magnetic field, or magnetic flux, within the core. This is where the magic, grounded in physics, happens. This changing magnetic flux travels through the core and “cuts across” the turns of the secondary winding. According to Faraday’s Law of Induction, this action induces a voltage in the secondary winding. The key insight is that the voltage induced in each individual turn of the wire is the same for both the primary and secondary windings. Therefore, the total voltage of each winding is directly proportional to the number of turns it has.

Defining the Turn Ratio: More Than Just a Number

The turn ratio, often denoted by ‘n’ or ‘a’, is the ratio of the number of turns in the primary winding (Np) to the number of turns in the secondary winding (Ns).

Turn Ratio (n) = Np / Ns

This is the transformer’s defining characteristic.

  • If Np is greater than Ns (n > 1), it is a step-down transformer. The secondary voltage is lower than the primary voltage, but the secondary current can be higher. This is common in power distribution to bring high-voltage electricity down to a safe level for homes and businesses.
  • If Ns is greater than Np (n < 1), it is a step-up transformer. The secondary voltage is higher than the primary voltage, but the secondary current is lower. This is used at power generation plants to increase the voltage for efficient long-distance transmission.
  • If Np equals Ns (n = 1), it is an isolation transformer. It does not change the voltage level but provides electrical isolation between two circuits for safety or noise reduction.

This ratio is not merely a design specification on a piece of paper; it is the very soul of the transformer’s function, dictating its role within the vast, interconnected electrical grid.

The Significance of an Accurate Turn Ratio in Power Systems

Why does a technician in 2025, standing in a substation, need to be so concerned with verifying this number? An incorrect turn ratio is a symptom of a deep-seated problem. A transformer is designed with a precise ratio to deliver a specific voltage. If the measured ratio deviates from the nameplate value, it signifies that the physical integrity of the windings has been compromised.

This could be due to a short circuit between turns, caused by insulation failure from overheating, electrical surges, or mechanical stress. Even a single shorted turn can cause a significant circulating current, leading to extreme localized heating. If left undetected, this condition can escalate catastrophically, resulting in the complete failure of the transformer, costly outages, and significant safety hazards.

Therefore, performing a turn ratio test is a fundamental health check. It allows us to peer inside the transformer’s sealed tank and confirm that its internal geometry—the precise count of its windings—is exactly as it should be. It is a predictive measure, a way to catch incipient faults before they lead to disaster.

Method 1: The Fundamental Voltage Ratio Calculation

The most direct and intuitive way to approach the question of how do you calculate turn ratio is by working with the very quantities the transformer is designed to change: voltage. This method stems directly from the ideal transformer equation.

The Underlying Physics: Faraday’s Law of Induction in Action

As we touched upon, the voltage induced in a winding is proportional to the number of turns and the rate of change of the magnetic flux (dΦ/dt).

For the primary winding: Vp = Np * (dΦ/dt) For the secondary winding: Vs = Ns * (dΦ/dt)

In an ideal transformer, the magnetic flux linking both windings is the same. If we divide the first equation by the second, the (dΦ/dt) term cancels out, leaving us with a beautifully simple relationship:

Vp / Vs = Np / Ns

This tells us that the ratio of the voltages is equal to the ratio of the turns. This is the theoretical basis for our first method. By measuring the primary and secondary voltages, we can infer the turn ratio without ever needing to see or count the physical windings.

Step-by-Step Guide to Calculating Turn Ratio from Voltage

Let us walk through this process as if we were in the field. The transformer must be de-energized and isolated from the power system before any testing can begin.

  1. Safety First: Confirm the transformer is completely disconnected from all external power sources, both high and low voltage sides. Follow all local lockout/tagout (LOTO) procedures. The workspace should be secure and clear of hazards.
  2. Identify Windings: Clearly identify the primary (High Voltage or HV) and secondary (Low Voltage or LV) terminals of the transformer. These are typically marked as H1, H2… for the primary and X1, X2… for the secondary.
  3. Apply Excitation Voltage: Connect a stable, known AC voltage source to the primary (HV) terminals. It is vital to use a voltage that is low enough to be safe but high enough to produce a measurable reading on the secondary side. A voltage between 120 VAC and 240 VAC from a portable generator or isolated supply is often sufficient. Do not use the transformer’s rated high voltage.
  4. Measure Primary Voltage: Using a calibrated and appropriately rated voltmeter, measure the exact voltage being applied across the primary terminals (Vp). Record this value. Even if your source is nominally 120V, it might actually be 118.7V, and this precision matters.
  5. Measure Secondary Voltage: Simultaneously or immediately after, use the same voltmeter (or another calibrated one) to measure the voltage induced across the secondary terminals (Vs). Record this value.
  6. Perform the Calculation: Divide the measured primary voltage by the measured secondary voltage. The result is your calculated turn ratio.

Example: Suppose you apply 240.5 Volts to the primary winding of a distribution transformer. You then measure 12.1 Volts on the secondary winding.

Turn Ratio (n) = Vp / Vs = 240.5 V / 12.1 V ≈ 19.88

You would then compare this calculated value of 19.88 to the ratio specified on the transformer’s nameplate.

Necessary Equipment and Safety Precautions for Voltage Measurement

This method, while simple in principle, requires care and the right tools.

Equipment / Precaution Description Purpose
Voltmeter A calibrated, high-quality digital multimeter (DMM) with a true RMS rating. To ensure accurate measurement of AC voltages, which may not be perfect sine waves.
AC Voltage Source A portable, isolated power supply or generator. Must be stable. To provide a safe and controlled excitation voltage for the test.
Insulated Leads High-quality test leads with secure connectors (alligator clips or probes). To safely connect the source and voltmeter to the transformer terminals without risk of shock or short circuits.
Personal Protective Equipment (PPE) Safety glasses, voltage-rated gloves, and flame-resistant clothing. To protect the technician from potential electrical arcs or flashes, even at lower test voltages.
Lockout/Tagout (LOTO) Kit Locks, tags, and hasps. To physically prevent the re-energization of the transformer while work is being performed, ensuring personnel safety.

Limitations and Potential Inaccuracies of the Voltage Method

While beautifully straightforward, this method is not without its shortcomings. The ideal equation Vp/Vs = Np/Ns assumes a perfect transformer. Real-world transformers have losses and imperfections.

  • Winding Resistance: The copper windings have some resistance, which causes a small voltage drop, especially under load (though we are testing under no-load).
  • Flux Leakage: Not all of the magnetic flux created by the primary links with the secondary. This “leakage flux” means the induced secondary voltage is slightly less than the ideal value, which can skew the calculation.
  • Voltmeter Inaccuracy: The precision of your result is entirely dependent on the accuracy of your voltmeter. An uncalibrated or low-quality meter can introduce significant errors.
  • Source Fluctuation: If the AC voltage source is not stable, the Vp and Vs measurements taken at different moments can lead to an incorrect ratio.

For a quick confidence check, this method is valuable. For official commissioning or in-depth fault diagnostics, however, its potential for small errors makes it less reliable than more specialized approaches.

Method 2: The Winding Turns Count—A Theoretical Ideal

To truly appreciate the elegance of modern testing equipment, it is useful to consider the most literal, albeit impractical, method: physically counting the turns. This is a thought experiment for the field technician but a reality for the transformer designer and manufacturer.

The Theoretical Purity of Counting Physical Turns

In a perfect world, if you wanted to know the turn ratio, you would simply unwind the primary coil, counting each loop of wire, and then do the same for the secondary coil. The resulting ratio of those two numbers would be the absolute, undeniable turn ratio. There would be no need for voltage measurements, no concerns about flux leakage or winding resistance. The result would be an integer ratio, pure and simple.

This method represents the “ground truth.” Every other method is, in essence, an attempt to deduce this physical reality through indirect electrical measurements. Companies that specialize in manufacturing high-quality electrical testing equipment are founded on a deep understanding of these first principles, ensuring their instruments provide results that closely mirror this physical truth. You can learn more about the philosophy of such experienced manufacturers who bridge the gap between theoretical ideals and practical application.

Why This Method is a Thought Experiment for Most Transformers

Now, let’s return to the reality of a technician standing before a sealed, oil-filled power transformer weighing several tons. The idea of disassembling it, carefully unwinding potentially thousands of turns of fine copper wire without damaging the delicate insulation, and then reassembling it perfectly is not just impractical; it is impossible and destructive.

The windings are tightly packed, insulated with paper, and submerged in insulating oil within a welded steel tank. Accessing them would require completely dismantling the transformer, a process reserved for major factory overhauls or forensic failure analysis. For routine testing, this method is purely a conceptual exercise.

Application in Manufacturing and Design Stages

Where this method finds its real-world application is on the factory floor. During the manufacturing process, the windings are created on a winding machine that precisely counts each turn as it is wrapped around the former. The design engineer specifies “Np = 2000 turns” and “Ns = 100 turns,” and the machinery executes this with high precision.

Quality control checks during this phase may involve verifying the count before the windings are assembled onto the core and placed in the tank. This ensures that the transformer is built to the correct specification from the very beginning. For the end-user, however, the physical turns are a given, sealed away and inaccessible. We must rely on electrical methods to verify that this built-in ratio has remained intact over the transformer’s service life.

Method 3: Utilizing a Transformer Turns Ratio (TTR) Tester for Ultimate Precision

Given the potential inaccuracies of the basic voltage method and the impossibility of the physical count, the industry has developed a specialized instrument for this exact purpose: the Transformer Turns Ratio (TTR) tester. This is the professional standard for answering the question of how do you calculate turn ratio.

What is a TTR Tester and How Does it Work?

A TTR tester is a portable electronic instrument designed to measure the turn ratio with very high accuracy. It improves upon the simple voltage method in several key ways.

Instead of relying on a potentially unstable external source, a TTR tester generates its own, very stable, low-voltage AC signal (typically 8V, 40V, or 100V). It applies this known voltage (Vp) to the high-voltage winding of the transformer. Then, using highly sensitive and precise circuitry, it measures the resulting induced voltage (Vs) on the low-voltage winding.

Because both the source voltage and the measurement circuitry are contained within the same calibrated instrument, the potential for error is dramatically reduced. The device performs the calculation Vp / Vs internally and displays the ratio directly on its screen, often to four or five decimal places.

Furthermore, advanced TTR testers do more than just calculate the voltage ratio. They also measure:

  • Excitation Current: The small current required to magnetize the core. An abnormally high excitation current can indicate a problem with the windings or the magnetic core itself, such as shorted turns or a core fault.
  • Phase Angle Deviation: In an ideal transformer, the primary and secondary voltages are perfectly in phase (or 180 degrees out of phase). A TTR tester measures the small deviation from this ideal, which can provide clues about the transformer’s magnetic properties and winding configuration.

A Detailed Walkthrough of Using a Modern TTR Tester

Using a modern TTR tester is a systematic process designed for safety and accuracy. Let’s outline the typical steps:

  1. De-energize and Isolate: As with any transformer testing, this is the first and most critical step. The transformer must be completely disconnected from the power grid, and all LOTO procedures must be strictly followed.
  2. Connect the Leads: TTR testers come with specialized lead sets, clearly marked for the HV (H) and LV (X) windings. These leads often use robust clamps for a secure connection. Connect the H leads to the primary terminals and the X leads to the secondary terminals of the transformer. For three-phase transformers, you will connect to the corresponding phases (e.g., H1/H2 and X1/X2 for the A-phase).
  3. Configure the Tester: Power on the TTR tester. You will need to enter some basic information from the transformer’s nameplate. This includes the rated primary and secondary voltages. This allows the instrument to calculate the expected nameplate ratio and display the percentage deviation of the measured value, which is the most important result. For three-phase transformers, you will also need to select the correct vector group (e.g., Dyn11, YNyn0), which tells the tester the expected phase relationship between the windings.
  4. Execute the Test: Once configured, you simply press the “Test” button. The instrument automatically applies the test voltage, measures the secondary voltage and excitation current, performs the calculations, and displays the results in a matter of seconds.
  5. Test All Taps: If the transformer has a tap changer, the turn ratio test must be performed on every single tap position. The tap changer alters the number of turns in one of the windings to make small adjustments to the output voltage. Testing each tap ensures that the tap changer mechanism and the tapped sections of the winding are all intact.
  6. Record and Analyze: Record the measured ratio, percentage deviation, and excitation current for each phase and each tap position. A modern tester will often store these results internally for later download to a computer for analysis and report generation.

The use of such specialized transformer turns ratio testers has become the industry benchmark for its safety, speed, and unparalleled accuracy.

Comparison of Turn Ratio Calculation Methods

Feature Method 1: Voltage Ratio Method 2: Physical Count Method 3: TTR Tester
Principle Vp / Vs ≈ Np / Ns Direct counting of Np and Ns Precise measurement of Vp / Vs
Accuracy Low to Medium Absolute (Theoretical) Very High (typically < 0.1%)
Practicality Feasible for field checks Impossible for in-service units Standard industry practice
Safety Moderate risk (uses higher voltage) Destructive High safety (uses low voltage)
Diagnostics Basic ratio only N/A Ratio, % Deviation, Excitation Current, Phase Angle
Equipment Voltmeter, AC Source N/A (requires factory disassembly) Dedicated TTR instrument

As the table illustrates, the TTR tester method provides a superior combination of accuracy, safety, and diagnostic depth, making it the preferred choice for professional applications.

Analyzing the Results: What Do Your Turn Ratio Calculations Mean?

Obtaining a number is only half the battle. The true skill lies in interpreting that number in the context of the transformer’s health. What separates a good result from a bad one?

The Industry Standard: Understanding the ±0.5% Tolerance

The accepted industry standard, as outlined in documents like IEEE C57.12.00, states that the measured turn ratio should not deviate from the calculated nameplate ratio by more than one-half of one percent (±0.5%).

Let’s break this down. First, you calculate the nameplate ratio by dividing the rated primary voltage by the rated secondary voltage. For example, a transformer rated 13,800V / 480V has a nameplate ratio of 13800 / 480 = 28.75.

Your TTR tester then measures the actual ratio. Let’s say it measures 28.71.

To find the percentage deviation, you use the formula: % Deviation = [(Measured Ratio – Nameplate Ratio) / Nameplate Ratio] * 100 % Deviation = [(28.71 – 28.75) / 28.75] * 100 = -0.14%

Since -0.14% is well within the ±0.5% window, this result is considered a “pass.” It indicates that the windings of this transformer are in excellent condition.

If the measured ratio was 28.50, the calculation would be: % Deviation = [(28.50 – 28.75) / 28.75] * 100 = -0.87%

A deviation of -0.87% is outside the ±0.5% limit. This is a “fail” result and immediately signals a problem that requires further investigation.

Diagnosing Faults: Shorted Turns, Open Circuits, and Incorrect Taps

A failing turn ratio test is a powerful diagnostic clue. The nature of the deviation can point to specific types of faults.

  • Shorted Turns: This is the most common and dangerous fault detected by a turn ratio test. When the insulation between adjacent turns in a winding fails, a short circuit is created. This effectively removes one or more turns from the winding, which directly alters the Np/Ns ratio. A short in the primary (Np) will decrease Np, causing the ratio to go down. A short in the secondary (Ns) will decrease Ns, causing the ratio (Np/Ns) to go up. A key accompanying symptom is a dramatically increased excitation current, as the shorted turn acts like a secondary winding with a huge load, drawing a lot of current.
  • Open Circuits: If there is a break in a winding or a faulty connection at a terminal, the TTR tester will be unable to measure a secondary voltage or will read an infinite impedance. The test will fail, and the device will typically display an “open circuit” or “no reading” error. This is a definitive indication of a connection issue or a broken conductor.
  • Incorrect Tap Settings or Connections: Sometimes, a test fails simply because the internal connections for a tap position are incorrect, or the tap changer mechanism is not making proper contact. This is why testing all tap positions is so important. If 10 out of 11 taps pass but one fails, it points directly to a problem with that specific tap. For three-phase transformers, incorrect wiring of the vector group during manufacturing or repair can also lead to bizarre ratio and phase angle readings.

The Role of Polarity and Phase Angle in Three-Phase Transformers

For single-phase transformers, the relationship is straightforward. For three-phase transformers, things become more complex due to the different ways the windings can be connected (Wye or Delta) and the resulting phase shifts between the primary and secondary voltages.

This is where the vector group (e.g., Dyn11) becomes vital. It tells the TTR tester the expected phase shift. For example, a “Dyn11” transformer has a Delta-connected primary, a Wye-connected secondary, and the secondary voltage lags the primary voltage by 330 degrees (or leads by 30 degrees).

A modern TTR tester will measure this phase displacement. If the measured displacement matches the expected value for the vector group, it confirms the transformer is wired correctly. If the ratio is correct but the phase angle is wrong, it could indicate a mis-wired winding or that the transformer was incorrectly identified.

Advanced Considerations in Turn Ratio Testing

For those seeking a deeper understanding, the process involves more than just connecting leads and pressing a button. Several factors can influence the results and require careful consideration.

Testing Transformers with Tap Changers

Most large power transformers are equipped with tap changers to adjust the turn ratio and regulate the output voltage. There are two main types:

  • De-Energized Tap Changer (DETC): These are simple switches that can only be operated when the transformer is completely offline. The testing procedure is straightforward: you perform a TTR test at each physical tap position, moving the switch manually between each test.
  • Load Tap Changer (LTC): These are complex mechanical devices designed to change taps while the transformer is energized and under load. Testing an LTC is more involved. You must test each tap position, and it’s also good practice to test the “bridging” positions where the LTC is momentarily transitioning between taps. The excitation current readings are particularly valuable here, as a spike in current during a transition can indicate worn contacts within the LTC mechanism.

For both types, the goal is to verify that the percentage deviation is within ±0.5% for every single tap position.

The Impact of Temperature and Excitation Current

While the turn ratio itself is theoretically independent of temperature, the physical properties of the transformer are not. The resistance of the copper windings increases with temperature. While a good TTR tester is designed to minimize the effect of resistance, very significant temperature differences between tests could cause minor variations. For consistency, it is best to record the oil temperature at the time of the test.

The excitation current, however, is much more revealing. As mentioned, a high excitation current is a strong indicator of a winding or core problem. Comparing the excitation current across the three phases of a three-phase transformer is a powerful diagnostic technique. In a healthy transformer, the two outer phases will have a similar, slightly higher excitation current, while the center phase will have a lower current due to a more efficient magnetic path. If one phase shows a dramatically higher current than the others (e.g., 50% or more), it strongly suggests a fault on that specific phase, even if the turn ratio is still within limits.

Troubleshooting Common Issues During Turn Ratio Testing

Even with the best equipment, field testing can present challenges.

  • Unstable Readings: If the ratio or excitation current readings are fluctuating, it often points to a poor connection. Check that the clamps are making a firm, clean connection to the transformer terminals, free of paint or corrosion. It could also indicate residual magnetism in the core.
  • Residual Magnetism: If a transformer was recently subjected to DC testing (like a winding resistance test) or a fault, the core can become magnetized. This residual magnetism can interfere with the TTR test, causing unstable or inaccurate readings, particularly for excitation current. The solution is to demagnetize the transformer core using specialized equipment or a specific procedure with a DC source.
  • Electrostatic Interference: In a live substation, high voltages on nearby equipment can induce “ghost” voltages in the test leads, interfering with the TTR tester’s sensitive measurements. Ensure proper grounding of the tester and the transformer tank. Using shielded test leads can also help mitigate this issue.

Frequently Asked Questions (FAQ)

1. What is an acceptable turn ratio test result? A result is considered acceptable or “passing” if the measured turn ratio is within ±0.5% of the ratio calculated from the transformer’s nameplate voltages.

2. Can I calculate the turn ratio on a live transformer? Absolutely not. A turn ratio test must only be performed after the transformer has been completely de-energized, disconnected from all power sources, and proper lockout/tagout procedures have been implemented. Applying test equipment to a live transformer is extremely dangerous and will destroy the instrument.

3. Why do I need to test all the taps on a tap changer? Each tap position represents a different physical connection and a different number of turns. A fault, such as a broken conductor or a bad contact, may only exist on a single tap. Testing every position is the only way to ensure the entire winding and the tap changer mechanism are functioning correctly.

4. What does a very high excitation current mean during a TTR test? A significantly high excitation current, especially when compared to the other phases or to previous test results, is a strong red flag. It most often points to shorted turns in the winding or a problem with the magnetic core, such as a shorted lamination.

5. How is the turn ratio of a three-phase transformer calculated? The principle is the same, but it’s done on a per-phase basis. The TTR tester is connected to one phase of the HV winding and the corresponding phase of the LV winding. The test is then repeated for the other two phases. The results for all three phases should be consistent and within the ±0.5% tolerance.

6. My voltage ratio calculation doesn’t match the nameplate. What should I do? If your simple voltage ratio calculation deviates significantly, it warrants a more accurate test. The first step would be to re-verify your measurements. If the discrepancy remains, it is highly recommended to use a proper TTR tester to confirm the finding. A deviation beyond the 0.5% tolerance indicates a likely internal fault.

7. Does the turn ratio change over time? For a healthy transformer, the turn ratio should never change. It is fixed by the physical number of wire turns. A change in the measured turn ratio indicates that a physical change has occurred inside the transformer, which is almost always a sign of a fault.

Conclusion

The inquiry, “how do you calculate turn ratio,” leads us down a path from fundamental electrical principles to the realities of professional field diagnostics. We have seen that while a simple voltage measurement can provide a rough estimate, it is fraught with potential inaccuracies. The true, physical ratio of turns is sealed within the transformer, inaccessible to us. This leaves us with the definitive modern method: the use of a dedicated Transformer Turns Ratio tester.

This instrument provides the accuracy, safety, and diagnostic depth required to assess a transformer’s health with confidence. By comparing the measured ratio to the nameplate value and adhering to the strict ±0.5% tolerance, technicians can verify the integrity of the windings. When combined with an analysis of excitation current and phase angle, the TTR test becomes a powerful tool for detecting incipient faults like shorted turns, open circuits, and tap changer problems long before they can lead to catastrophic failure. Understanding and correctly applying these test methods are not just technical skills; they are essential practices for ensuring the safety, reliability, and longevity of our critical power infrastructure. For those responsible for maintaining these vital assets, exploring a full range of power testing equipment is a step toward greater diagnostic capability and system security.

References

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IEEE. (2010). IEEE standard test code for liquid-immersed distribution, power, and regulating transformers (IEEE Std C57.12.90-2010). Institute of Electrical and Electronics Engineers.

Kulkarni, S. V., & Khaparde, S. A. (2004). Transformer engineering: Design, technology, and diagnostics. CRC Press.

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