In carrier-based PWM for power converters, if A_r is the peak amplitude of the reference (modulating) signal and A_c is the peak amplitude of the carrier (triangular) signal, the modulation index M is defined as:

Introduction / Context:Carrier-based PWM (pulse-width modulation) compares a low-frequency reference (modulating) signal with a high-frequency triangular carrier to generate switching pulses. The modulation index M quantifies how strongly the reference modulates the carrier, thereby controlling the fundamental output voltage of inverters or converters.Given Data / Assumptions:

• A_r: peak amplitude of the reference signal.
• A_c: peak amplitude of the triangular carrier.
• Linear modulation region is assumed (no overmodulation).

Concept / Approach:In standard sinusoidal PWM, the modulation index is defined as the ratio of the reference amplitude to the carrier amplitude (both peaks). This ratio governs the fundamental component of the output voltage for a given DC bus.Step-by-Step Solution:1) Identify A_r as the modulating signal amplitude.2) Identify A_c as the carrier amplitude.3) Use the definition M = A_r / A_c.Verification / Alternative check:Textbooks and application notes present M = A_r / A_c for sinusoidal PWM; fundamental output is proportional to M in the linear region.Why Other Options Are Wrong:

• A_c / A_r: inverted ratio.
• Product or difference/sum forms: not the standard PWM modulation index definition.
• A_r^2 / A_c: dimensionally inconsistent for the definition.

Common Pitfalls:

• Confusing peak, RMS, or average definitions; M uses peak values.
• Applying the definition outside linear range where distortion occurs.

Final Answer:M = A_r / A_c.