Fundamental Frequency And Harmonics: What Are They?

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Key learnings:

Fundamental Frequency Definition: The fundamental frequency is defined as the lowest frequency produced by an instrument, also known as the first harmonic.
Harmonics Definition: Harmonics are defined as higher frequency components that are integer multiples of the fundamental frequency, causing waveform distortion.
Sources of Harmonics: Harmonics come from non-linear loads and powerful electronic switching circuits.
Effects of Harmonics: Harmonics can lead to heating, increased temperatures, and misfiring in motors and equipment.
Types of Sequence Harmonics: Harmonics can be positive sequence (same direction as fundamental), negative sequence (opposite direction), or zero sequence (no rotation).

What Are Harmonics?

Harmonics are sinusoidal components whose frequencies are integer multiples of a fundamental frequency. In power-quality work, harmonic components combine with the fundamental to produce a non-sinusoidal voltage or current waveform. They are often unwanted, but harmonic generation and measurement are also used deliberately in signal processing and RF systems.

A harmonic often has lower amplitude than the fundamental in a power system, but this is not part of the definition. Each harmonic has its own magnitude and phase, and a waveform can even have a weak or missing fundamental.

What is Amplitude?

Amplitude describes the size of an alternating quantity. Peak amplitude is the maximum magnitude measured from the reference level; RMS amplitude and peak-to-peak amplitude are different measures and must be labelled explicitly.

Sources of Harmonics

Harmonic current is produced when a load’s current is not proportional to the applied voltage. Examples include a saturating iron-cored inductor, rectifiers, electronic ballasts in fluorescent lights, saturating transformers, discharge lighting and arcing equipment. Inductance alone does not make a load nonlinear.

Power-electronic circuits using a silicon controlled rectifier (SCR), power transistors or other power converters can draw nonlinear current. An electronics drive such as a variable frequency drive (VFD) has a harmonic pattern determined by its rectifier, DC link, modulation and filtering. Some capacitor-input rectifiers draw current near the peaks of the AC supply, but not every converter does. Any non-sinusoidal periodic current can be decomposed into a fundamental and harmonics.

What Are the Effects of Harmonics?

Harmonic frequencies in the power grid cause power quality problems.

Harmonics in power systems increase heating in equipment and conductors and create a pulsating torque in the motors.

Harmonics cause increasing operating temperature and the iron losses (Hysteresis and Eddy current losses) in the AC motors and transformers because hysteresis loss is proportional to the frequency and eddy current loss is proportional to the square of the frequency.

Harmonics can increase losses, torque ripple, vibration and control or protection errors in variable-speed drives. In AC machines, each harmonic order creates a field with its own sequence and synchronous speed. Its effect is therefore not described by substituting the harmonic frequency into Ns=120f/P and calling the result a slip difference.

In an induction motor, winding distribution and slotting create space harmonics. The seventh space harmonic can contribute forward torque near one-seventh synchronous speed and the fifth produces opposing torque, which helps explain crawling. This differs from time harmonics in the supply waveform.

What Are Time Harmonics and Space Harmonics?

Time harmonics are Fourier components of a waveform that is non-sinusoidal as a function of time. They can originate in a source or a nonlinear load and are not always present in the input supply.

Space harmonics are spatial Fourier components caused by winding distribution, slotting, magnetic saturation and machine geometry. They make the air-gap magnetomotive force and flux vary non-sinusoidally with position.

What is Fundamental Frequency?

The fundamental frequency of a periodic waveform is the reciprocal of its period and is also called the first harmonic. In a power system it is normally the nominal supply frequency. In acoustics it often relates to perceived pitch, but the fundamental need not be the strongest spectral component and is not limited to musical instruments.

An alternating quantity’s frequency, f, counts the cycles completed per second and is expressed in hertz (Hz).

$\begin{align*} f= \frac{1}{T} \end{align*}$

How to Find Fundamental Frequency

In a synchronous generator, a two-pole rotor produces one electrical cycle of alternating current or voltage per mechanical revolution. A four-pole rotor produces two electrical cycles per revolution. The generated frequency is therefore:

$\begin{align*} f= \frac{PN}{120} \end{align*}$

Where:

f = Frequency
P = No. of poles of the alternator
N= Speed of the alternator in rpm

The nominal grid frequency is 50Hz in India and 60Hz in the United States. Actual system frequency varies slightly as generation and demand change.

How to Calculate Harmonics

For a periodic voltage or current, Fourier analysis represents the waveform as a fundamental component plus sinusoidal components at integer multiples of the fundamental frequency. Each component has a magnitude and phase.

The nth harmonic has frequency nf, so the harmonic frequencies are 2f, 3f, 4f, 5f and so on.

For a 50Hz fundamental, the second harmonic is 100Hz (2 × 50), the third harmonic is 150Hz (3 × 50), the fifth harmonic is 250Hz and the seventh harmonic is 350Hz.

Harmonics Calculation Formula

The first harmonic is the fundamental-frequency component. These protected equations show zero-phase sine terms; a general Fourier series also allows phase angles and may include a DC component.

$\begin{align*} V_1= V_m_a_x sin(\omega t) &= V_m_a_x sin(2\pi f t) \end{align*}$

2nd Harmonic: It is two times the fundamental frequency.

$\begin{align*} V_2 = V_2_m_a_x sin(2\omega t) = V_2_m_a_x sin(2*2\pi f t) = V_2_m_a_x sin(4\pi f t) \end{align*}$

3rd Harmonic: It is three times the fundamental frequency.

$\begin{align*} V_3 = V_3_m_a_x sin(3\omega t) = V_3_m_a_x sin(3*2\pi f t) = V_3_m_a_x sin(6\pi f t) \end{align*}$

4th Harmonic: It is four times the fundamental frequency.

$\begin{align*} V_4 = V_4_m_a_x sin(4\omega t) = V_4_m_a_x sin(4*2\pi f t) = V_4_m_a_x sin(8\pi f t) \end{align*}$

And for

The nth harmonic is n times the fundamental frequency. In the protected equation below, the final sine argument should be 2nπft; mπft is a legacy typesetting error.

$\begin{align*} V_n = V_n_m_a_x sin(n\omega t) = V_n_m_a_x sin(n*2\pi f t) = V_n_m_a_x sin(m\pi f t) \end{align*}$

A periodic waveform can be reconstructed by adding its fundamental and harmonic components. The protected expression below is a simplified zero-phase form.

$\begin{align*} V_T_o_t_a_l = V_1 + V_2 +V_3 + V_4+.............+ V_n \end{align*}$

$\begin{align*} V_T_o_t_a_l = V_1_m_a_x sin(2\pi f t) + V_2_m_a_x sin(4\pi f t) + V_3_m_a_x sin(6\pi f t) + .............. + \\ V_n_m_a_x sin(m\pi f t) \end{align*}$

Types of Harmonics

Harmonics can be classified by order and, in a balanced three-phase system, by phase sequence. For a 50Hz fundamental, the second harmonic is 100Hz. Sequence describes how a harmonic’s three phase quantities rotate relative to the positive-sequence fundamental.

There are three sequence classes for balanced harmonic components.

Positive sequence harmonic: These components rotate in the same direction as the positive-sequence fundamental. Their orders follow 3k + 1, such as the 1st, 4th, 7th and 10th.

Negative sequence harmonic: These components rotate opposite to the positive-sequence fundamental. Their orders follow 3k + 2, such as the 2nd, 5th, 8th and 11th.

Zero sequence harmonics: The three phase quantities are in phase, so they have no rotating sequence. Their orders are multiples of three, such as the 3rd, 6th, 9th and 12th, and are called triplen harmonics. Half-wave symmetry normally suppresses the even orders.

Effects of Positive, Negative and Zero Sequence Harmonics

Positive-sequence harmonic currents add RMS current and frequency-dependent losses in conductors, power lines and transformers. Their direction of rotation does not itself explain the heating.

Negative-sequence harmonics create an oppositely rotating rotating magnetic field. In induction motors, this field produces opposing torque, extra rotor loss, heating and torque pulsation.

Zero-sequence harmonic currents from single-phase loads add arithmetically in a shared neutral instead of cancelling. For one balanced triplen component, the neutral component is three times that harmonic’s phase component; total neutral current is not necessarily three times total phase current.

Current Harmonics

In an ideal AC power system, voltage and current are sinusoidal at the fundamental frequency, normally 50Hz or 60Hz. Real waveforms can also contain harmonic and interharmonic components.

Current harmonics arise when a nonlinear load draws current that is not proportional to the applied voltage. Sources include electronic transformers, discharge lighting, saturated magnetic devices, electronic ballasts in fluorescent lights, computer power supplies, laser printers, half-wave rectifiers and SMPS units. A PLC or refrigerator may be nonlinear because of its input electronics, not simply because of its function. Power converters using IGBTs or MOSFETs can also produce harmonic current.

These loads distort the current waveform, which can then be described as a fundamental component plus harmonic components.

What is Linear Load?

A linear electrical load has a proportional voltage-current relationship. With sinusoidal voltage, its steady-state current is sinusoidal at the same frequency, although the current may lead or lag the voltage.

What is Non-Linear Load?

A nonlinear load does not have a proportional voltage-current relationship. Its periodic current normally contains the supply fundamental plus harmonic components, rather than lacking the supply frequency.

Effects of Current Harmonics

Harmonic currents increase I²R losses and frequency-dependent stray losses in the transformer winding. They increase total RMS current and can reduce the transformer’s capacity for fundamental-frequency load current. A harmonic can exchange active power when its voltage and current components have an in-phase component, so harmonic current is not inherently non-power-producing.

Triplen harmonic currents from balanced phase-to-neutral loads are in phase and add in the shared neutral. This can make the neutral current large even when the fundamental phase currents cancel; the third-harmonic component has three times the fundamental frequency.

Nonlinear loads are the main source of harmonic current. A linear load supplied by distorted voltage can also draw harmonic current according to its impedance at each harmonic frequency.

Harmonic current flowing through system impedance produces harmonic voltage drops and can distort the voltage at the point of connection.

Voltage Harmonics

Voltage harmonics may come from the source or from harmonic current flowing through system impedance. The resulting harmonic voltage drops distort the voltage waveform.

Nonlinear loads often draw harmonic-rich current. The amount of voltage distortion depends on the harmonic currents and the network impedance at those frequencies.

Effects of Voltage Harmonics

Harmonic voltage can increase eddy current losses in motors and transformers, which raises operating temperature. Harmonic voltages applied to a stator also induce higher-frequency rotor currents and extra losses.

A fifth-harmonic voltage has negative sequence and creates a counter-rotating field in an induction motor. It produces opposing torque, heating and torque pulsation, but it does not normally reverse a running motor by itself.

Harmonic voltages can increase torque pulsation and vibration in single-phase and three-phase motors. Sustained vibration can increase bearing wear.

Odd and Even Order Harmonics

Odd-order harmonics occur at f, 3f, 5f, 7f and so on; the fundamental is the first harmonic. Even-order harmonics occur at 2f, 4f, 6f, 8f and so on.

Half-wave symmetry, where the negative half-cycle is the inverse of the positive half-cycle, suppresses even-order harmonics. It does not make individual positive and negative cycles cancel. Many power system waveforms are approximately half-wave symmetric, so odd orders often dominate. Only zero-sequence triplen components add directly in a shared neutral.

Nonlinear loads often produce odd-order harmonics. Measurable even-order content indicates that the waveform lacks half-wave symmetry, which can result from unequal positive and negative half-cycles or a DC offset.

Time-varying equipment such as electric arc furnaces and arc welders can generate interharmonics, whose frequencies are not integer multiples of the fundamental. Symmetric Transformer magnetizing currents normally contain mainly odd harmonics; asymmetry can introduce even harmonics.

Difference Between Harmonics and Overtones

In acoustics, an overtone is any frequency component above the fundamental. The first overtone equals the second harmonic only when that component is an integer multiple of the fundamental. Harmonics are frequency components, not necessarily stationary waves or resonant frequencies. The protected legacy table below should be read with those corrections.

Harmonics	Overtones
A harmonic is an integral multiplication of the fundamental frequency.	An overtone is defined as any frequency which is greater than the fundamental frequency.
Harmonics starts counting from the fundamental frequency.	Overtones start counting after the fundamental frequency and starts counting from the harmonics.
All harmonics are stationary waves.	Overtones may be stationary not stationary waves. Those overtones which match the frequencies of the harmonic acts as a stationary wave.
Harmonics are a resonant frequency.	Overtones are also a resonant frequency.
Example. The first harmonic is a Fundamental frequency (f). Second harmonic is two times the fundamental frequency (2f).	Example. Second harmonic (2f) is the first overtone. Third harmonic (3f) is the second overtone.