Question

The frequency modulation of a 3-\(\phi\) sinusoidal pulse width modulation is an odd multiple of 3. Then the output line voltage contains

• A. \( 5^{th}, 7^{th}, 9^{th}, 11^{th}, \dots \) harmonics
• B. \( 3^{rd}, 5^{th}, 7^{th}, 9^{th}, \dots \) harmonics
• C. \( 5^{th}, 7^{th}, 11^{th}, 13^{th}, \dots \) harmonics
• D. \( 7^{th}, 11^{th}, 15^{th}, \dots \) harmonics

Hint:

In 3-\(\phi\) SPWM, triplen harmonics (3rd, 9th...) cancel out in line voltage; dominant harmonics are \( 6k \pm 1 \).

Answer 1

The Correct Option is D

In a three-phase (3-\(\phi\)) sinusoidal pulse width modulation (SPWM) system, the frequency modulation index \(m_f\) is defined as:

\[m_f = \frac{f_c}{f_m}\]

Where:

• \(f_c\) = carrier frequency
• \(f_m\) = modulating frequency

When \(m_f\) is an odd multiple of 3, such as \(3k\), where \(k\) is an odd integer, the system achieves a specific harmonic cancellation.

The harmonics in the line voltage of a 3-phase system can be described in terms of the harmonic number \(n\). Typically, harmonics are generated at frequencies of \(n \cdot f_m\), where \(n\) is an integer. For a 3-\(\phi\) SPWM with \(m_f\) as an odd multiple of 3, certain lower order harmonics cancel out due to the specific frequency modulations. The characteristic harmonics that remain in the line voltage are:

• The harmonics of the order \((6k \pm 1)\). These represent the non-triplen harmonics, which do not get canceled out.

For \(m_f\) being an odd multiple of 3, the order of harmonics is:

• \(7^{th}, 11^{th}, 13^{th}, 17^{th}, \dots\)

However, when considering only those harmonics that appear in all line-to-line voltages (due to cancellation phenomena in 3-\(\phi\) systems), the specific leading harmonics are \(7^{th}, 11^{th}, 15^{th}, \ldots\)

The correct choice reflects this knowledge:

\(7^{th}, 11^{th}, 15^{th}, \dots\) harmonics.