Fundamental Wave and Harmonics in AC Circuits: Explained with Types, Effects, and Equations

In theoretical analysis, alternating voltages and currents are often assumed to be purely sinusoidal in nature. However, in real-world applications, such ideal waveforms are nearly impossible to achieve. Practical AC waveforms often deviate from the pure sine wave due to various reasons. These non-sinusoidal or distorted waveforms are referred to as complex waveforms.

Complex waveforms arise due to the superposition of multiple sinusoidal waves of different frequencies. This is common in electronics applications such as audio signals, TV transmission, and rectifier outputs.

What Are Harmonics?

A complex waveform can be broken down into:

Fundamental Wave (First Harmonic): The lowest frequency component, denoted by f.
Harmonics: Sinusoidal components with frequencies that are integer multiples of the fundamental frequency (e.g., 2f, 3f, 4f, etc.).

These harmonics can be categorized as:

Even Harmonics: Frequencies like 2f, 4f, 6f, etc.
Odd Harmonics: Frequencies like 3f, 5f, 7f, etc.

In terms of angular frequency (ω), harmonics appear as:

Even: 2ω, 4ω, 6ω, …
Odd: 3ω, 5ω, 7ω, …

Formation of Complex Waveforms

When fundamental and harmonic waves combine, the resulting waveform is a complex wave. The shape of this waveform depends on:

Whether the harmonics are in phase or out of phase with the fundamental.
The amplitude of each harmonic.

Case Example:

If the second harmonic has the same amplitude as the fundamental and is in phase, the resulting waveform will show a clear 100% second harmonic distortion.

General Equation of a Complex Wave

A complex wave consisting of the fundamental and multiple harmonics can be mathematically expressed as:

Voltage Equation:

e(t) = E1m·sin(ωt + Ψ1) + E2m·sin(2ωt + Ψ2) + ... + Enm·sin(nωt + Ψn)

Current Equation:

i(t) = I1m·sin(ωt + φ1) + I2m·sin(2ωt + φ2) + ... + Inm·sin(nωt + φn)

Where:

E1m, E2m, ..., Enm are the peak values of harmonics.
Ψ1, Ψ2, ..., Ψn are phase angles for each voltage harmonic.
φ1, φ2, ..., φn are phase angles for each current harmonic.

Harmonics in Single-Phase AC Circuits

(a) Pure Resistance (R)

Each harmonic component of voltage produces its own current independently.
Resulting current waveform mirrors the voltage waveform.
Harmonic content remains unchanged in the current.

(b) Pure Inductance (L)

Reactance increases with frequency: X₁ = ωL, X₂ = 2ωL, ..., Xₙ = nωL
Current lags voltage by 90° for all harmonics.
Higher harmonic currents are suppressed (i.e., nth harmonic current = 1/n of the voltage harmonic).
The current waveform is less distorted than the voltage waveform.

(c) Pure Capacitance (C)

Reactance decreases with frequency: X₁ = 1/ωC, X₂ = 1/2ωC, ..., Xₙ = 1/nωC
Current leads voltage by 90° for all harmonics.
Harmonic current is amplified (i.e., nth harmonic current = n times the voltage harmonic).
The current waveform is more distorted than the voltage waveform.

🔄 Comparison: Inductors reduce distortion in current; capacitors increase it.

Selective Resonance Due to Harmonics

When a circuit with both inductance and capacitance is exposed to a complex voltage, it may resonate at a specific harmonic frequency—this is known as Selective Resonance.

Types:

Series Resonance: Can cause large harmonic currents even with small harmonic voltages.
Parallel Resonance: Minimizes current drawn from the supply at resonance.

This effect is critical in harmonic analysis and suppression. It is also used in practical harmonic detection using:

Variable inductors and capacitors
Non-inductive resistors
Oscillographs

These tools help identify and analyze the presence of specific harmonics in a waveform.

Harmonics in Three-Phase AC Systems

In three-phase systems, harmonics behave similarly to single-phase systems. However:

Even harmonics are typically absent.
Odd harmonics, especially triplen harmonics (multiples of 3rd like 3rd, 9th, 15th), require special attention as they can cause:

Neutral conductor heating
Transformer core saturation
Distortion in power quality

Therefore, harmonic suppression techniques are crucial in three-phase power systems to maintain system stability and performance.

Conclusion

Understanding harmonics and complex waveforms is vital for analyzing and designing efficient AC systems. Engineers must consider the behavior of harmonics in various circuit elements and configurations to ensure minimal distortion and optimized performance.

FAQs on Fundamental Wave and Harmonics

Q1. What is a fundamental wave in an AC circuit?

The fundamental wave is the basic sine wave component of a complex waveform. It has the lowest frequency (denoted as 'f') and serves as the reference for identifying all other harmonics in an alternating current (AC) system.

Q2. What are harmonics in electrical systems?

Harmonics are sinusoidal components of a complex waveform whose frequencies are integral multiples of the fundamental frequency, such as 2f, 3f, 4f, etc. They cause waveform distortion and can impact the performance of electrical machines and power systems.

Q3. What is the difference between even and odd harmonics?

Even harmonics have frequencies that are even multiples of the fundamental (e.g., 2f, 4f, 6f).
Odd harmonics have frequencies that are odd multiples of the fundamental (e.g., 3f, 5f, 7f).
Odd harmonics are more common in practical electrical systems, while even harmonics are typically minimized.

Q4. How are complex waveforms formed?

Complex waveforms are formed by the superposition of multiple sinusoidal waves, including the fundamental wave and its harmonics. These waveforms are found in AC circuits, rectifier outputs, audio signals, and electronic devices.

Q5. What is the general equation of a complex waveform?

A complex waveform can be expressed as the sum of sinusoidal functions of different frequencies:

$e (t) = E_{1} \sin (ω t + Ψ_{1}) + E_{2} \sin (2 ω t + Ψ_{2}) + \dots + E_{n} \sin (n ω t + Ψ_{n})$

Here, $E_n$ are the amplitudes, $n\omega$ are the harmonic frequencies, and $\Psi_n$ are the phase angles.

Q6. What is the impact of harmonics in single-phase AC circuits?

In single-phase linear AC circuits, harmonics in the voltage waveform generate corresponding harmonics in the current waveform. The overall current is the algebraic sum of individual harmonic currents based on the principle of superposition.

Q7. How do harmonics affect pure resistance circuits?

In a pure resistive circuit, the current waveform mirrors the voltage waveform. Thus, the harmonic distortion in current is identical to that in the voltage waveform.

Q8. What is the effect of harmonics in inductive circuits?

In a pure inductance circuit, the harmonic current lags the voltage by 90°. The higher the harmonic order, the lower its current amplitude. This makes the current waveform less distorted compared to the voltage waveform.

Q9. How do capacitive circuits react to harmonics?

In a pure capacitive circuit, harmonic currents lead the voltage by 90°. The amplitude of harmonic current is directly proportional to the harmonic order, making the current waveform more distorted than the voltage waveform.

Q10. What is selective resonance due to harmonics?

Selective resonance occurs in circuits containing both inductance and capacitance, where the circuit resonates at a particular harmonic frequency. This can lead to high current in series resonance or minimum supply current in parallel resonance, even if harmonic voltages are small.

Q11. How are harmonics analyzed using resonance circuits?

Harmonic analysis is performed using a resonant circuit consisting of variable inductance, capacitance, and resistance connected in series. By tuning to different harmonic frequencies, harmonic content in a waveform is identified using instruments like oscillographs.

Q12. What are triple-n harmonics in three-phase systems?

Triple-n harmonics are harmonics whose order is a multiple of 3 (like 3rd, 6th, 9th, etc.). In three-phase electrical systems, they are particularly important because they can cause neutral current buildup and require special attention during harmonic analysis and filtering.

Q13. Are even harmonics present in three-phase systems?

Even harmonics are usually absent or minimized in balanced three-phase systems due to symmetrical waveform generation. Most of the distortion comes from odd and triple-n harmonics.

Q14. Why is harmonic analysis important in electrical engineering?

Harmonic analysis helps in identifying power quality issues, improving system efficiency, protecting electrical equipment, and designing filters and resonant circuits in power electronics and industrial electrical systems.