Calculating Harmonics in AC Circuits

Lets imagine a smooth alternating wave, harmonics are like ripples on top of that wave, making it look messy and uneven. These ripples are higher pitched sounds that mess with the original sound.

1. Perfect AC waveforms: The ideal shape for voltage or current in an AC circuit is a smooth, continuous wave called a sine wave.
2. Non-linear loads and distortion: In reality, some devices in circuits can distort this perfect wave. These devices are called non-linear loads, and their presence creates additional higher-frequency components on top of the fundamental frequency. These extra frequencies are the harmonics.
3. Resistors and perfect waves: Resistors, unlike some other components behave the same way in AC and DC circuits. They allow the current to follow the voltage changes smoothly, resulting in a pure sine wave with no distortion.

Forget perfect sine waves….. In real circuits, many devices don’t play by the ideal rules. These are called non-linear devices.

Because they’ are not proportional (current doesn’t exactly follow voltage changes), the smooth and perfect sine wave we expect (voltage or current) gets distorted. This distorted wave, with its bumps and wiggles, is called a non-sinusoidal waveform or a complex waveform.

Think of common culprits like transformers with iron cores, light dimmers (electronic ballasts), or even generators.

They all create these complex waveforms, where the current might be all messed up even if the voltage starts out as a perfect sine wave.

Even more electronic components can mess with perfect sine waves. Devices like power converters, rectifiers, and transistors use switching to control power. Imagine a switch that rapidly cuts on and off, chopping the smooth sine wave of the AC supply into pieces.

Since these switches only turn on for short bursts, often at the peak of the voltage wave, the current they draw gets distorted. This chopped-up current waveform is no longer a sine wave, and it introduces harmonics.

Here’s the key: These complex, distorted waveforms can be mathematically broken down into simpler pieces. One piece is the original, undistorted frequency (the fundamental), and the other pieces are a series of higher-frequency waves (the harmonics).

These harmonics are what cause the distortion in the original wave.

What is a Fundamental Waveform

The fundamental waveform (also known as the first harmonic) is a special type of wave called a sinusoidal waveform. This wave vibrates at a specific rate, which is called the supply frequency.

This special wave is the fundamental frequency (the lowest or base frequency, denoted by ƒ). It acts like the foundation for the entire complex waveform.

Because of this, the time it takes for the complex waveform to complete one cycle (its periodic time denoted by Τ) is the same as the time it takes for the fundamental frequency to complete one cycle.

The image shows an example of a basic fundamental (or first harmonic) AC waveform.

Understanding the Parts of a Sinusoidal Waveform:

• Vmax, stands for “peak value in volts.” It represents the highest (positive) or lowest (negative) voltage that the wave reaches.
• ƒ (frequency in Hertz Hz) represents the waveforms frequency. Frequency refers to how often the wave completes one cycle (up and down) each second. It’s measured in Hertz (Hz), which means cycles per second.

Sine Waves and Frequency:

A sinusoidal waveform is a type of wave where the voltage (or current) keeps going up and down smoothly. This up-and-down pattern follows a specific mathematical function called a sine function.

The frequency (ƒ) of the wave tells you how many of these complete cycles happen every second. Basically it tells you how fast the wave oscillates (vibrates) back and forth.

For example, the fundamental frequency in the United Kingdom is set at 50Hz. This means, the voltage or current completes 50 full cycles (up and down) each second. In the United States, the fundamental frequency is 60Hz so the wave cycles 60 times per second.

In power systems harmonics refer to voltage or current waveforms with frequencies that are integer multiples of the fundamental frequency.

These additional frequencies arise, due to the presence of non-linear loads which don’t draw current proportionally to the applied voltage.

Consider the fundamental frequency denoted by f. The harmonics then appear at multiples of f, such as the second harmonic (2f), third harmonic (3f), and so on.

For instance in a 50 Hz system, the second harmonic would be at 100 Hz (2 x 50 Hz), the third harmonic at 150 Hz (3 x 50 Hz), and so forth.

Likewise, in a 60 Hz system, the second, third, fourth, and fifth harmonics would be at 120 Hz, 180 Hz, 240 Hz, and 300 Hz respectively.

Basically harmonics represent a spectrum of additional frequencies riding on top of the fundamental one.

These additional components can significantly impact power system performance.

Complex Waveforms Resulting from Harmonics

Look at the red waveforms. These are the shapes a device (the load) would actually see, due to the extra frequencies (harmonics) being added to the main frequency (fundamental).

The main frequency is also called the 1st harmonic. This means a 2nd harmonic has a frequency twice as high, a 3rd harmonic has a frequency three times as high, and so on, as shown on the left.

The red shapes on the right show the final, combined waveform. This is what happens when you add the main frequency (1st harmonic) to the other harmonic frequencies. The important thing to remember is that the final shape depends on two things:

1. How many harmonics are there? (How many extra frequencies are present?)
2. How strong are they? (What are the amplitudes of the harmonics?)

But that’s not all! The order in which these extra frequencies hit the main frequency (their phase relationship), also plays a role in shaping the final waveform.

Sounds can be deceiving! A smooth, complex wave you hear is actually a combination of simpler waves. The first and most important wave is called the fundamental frequency.

Think of it as the base note. Added on top are other waves, called harmonics, which are multiples of the fundamental frequency (like whole number multiples of a musical note).

Each harmonic has its own strength (peak value) and timing (phase angle).

For instance imagine the fundamental frequency is like a drumbeat (E = Vmax(2πƒt)).

The second harmonic would be like another instrument playing the same beat twice as fast (E2 = V2max(2ωt)),

The third harmonic three times as fast (E3 = V3max(3ωt)), and so on.

The strength (V_max) of each harmonic can vary, influencing the overall sound.

This principle is why different instruments sound uniquely, even when playing the same note. The specific mix of harmonics they produce, creates their characteristic sound, or timbre.

Putting all these waves together, the equation for a complex waveform becomes:

• E_T = E₁ + E₂ + E₃ + —–E_(n) so on.
• E_T = V_1maxsin(2πfτ) + V_2maxsin(4πfτ) + V_3maxsin(6πfτ)…so on

Harmonics are extra frequencies layered on top of the main electrical wave (fundamental frequency). We describe them in two ways:

1. By name and frequency: Like musical notes, harmonics have an order (2nd, 3rd, etc.) and a frequency thats a multiple of the fundamental (e.g., a 2nd harmonic of a 100 Hz fundamental is 200 Hz).
2. By sequence: This applies specifically to 3-phase systems with four wires. Here, harmonics are categorized based on how they “rotate” compared to the main wave.

Imagine the fundamental wave as a spinning reference point. Now picture the harmonics as additional waves spinning around it. There are two types:

• Positive sequence harmonics (4th, 7th, 10th, etc.): These harmonics spin in the same direction (forward) as the main wave.
• Negative sequence harmonics (2nd, 5th, 8th, etc.): These harmonics spin in the opposite direction (backward) to the main wave.

Knowing the sequence of a harmonic is important for understanding its impact on the overall health of a 3-phase power system.

Not all harmonics are created equal! Here’s why:

• Positive sequence harmonics (4th, 7th, 10th…): These add their energy to the main wave causing equipment like wires, transformers, and conductors to overheat. It’s like adding more traffic to an already crowded road leading to congestion and breakdowns.
• Negative sequence harmonics (2nd, 5th, 8th…): These are the troublemakers. They spin in the opposite direction compared to the main wave. In a motor, this creates a tug of war between the main magnetic field and the negative sequence one, weakening the overall field. This weakens the motor’s ability to turn (torque) and can lead to problems.
• Triplen harmonics (3rd, 6th, 9th…): These have a special twist, they don’t rotate at all. They flow between the phases and the neutral or ground connection. Although they can still cause issues, they differ from the other two types.

Positive and negative sequence harmonics partially cancel each other out in a balanced system. But, triplen harmonics (3rd, 6th, 9th…) are a different story.

Unlike the others, they dont cancel and instead add up in the neutral wire. Since this wire carries current from all three phases, the triplen harmonic current can be as much as three times the current of a single phase at the fundamental frequency! This overload makes the neutral wire less efficient and prone to overheating.

Heres a quick recap of the sequence effects for a common 50 Hz fundamental frequency:

Sequencing of Harmonics

The concepts of positive, negative, and triplen harmonics hold true even for systems with a fundamental frequency of 60 Hz.

Conclusions:

Harmonics: The Uninvited Guests in Your Power System

Imagine your electrical circuit humming along at its normal frequency (50 Hz or 60 Hz). Suddenly we have these uninvited guests, these are harmonics, higher frequency waves that distort the smooth flow of electricity.

These unwelcome visitors werent a big issue in the past. But, with the rise of electronic devices like motor drives, switching power supplies, and energy-saving fluorescent lights, they’ve become more common. Why? Because these devices don’t draw power as smoothly as traditional equipment.

Here’s the problem, harmonics combine with the fundamental frequency, creating a distorted and complex waveform. This distorted wave, like a tangled mess of wires, can wreak havoc on your power system.

The severity of the distortion depends on the “guests”:

• Their number: More harmonics means more chaos.
• Their type: Low-order harmonics (2nd to 19th) are the most disruptive, especially the “triplen harmonics” (multiples of 3).
• Their shape: The specific shape of each harmonic wave influences the overall distortion.

In short, harmonics are a modern day challenge for electrical systems. By understanding their presence and impact, we can take steps, to mitigate them and keep our power flowing smoothly.

References: Understanding Power System Harmonics – Baylor University

Harmonics