RMS value of rectangular wave of period T, having a value of +V for a duration T1(<T) and -V for the duration T-T₁=T₂ equals 

Explanation:

RMS Value of Rectangular Wave:

The root mean square (RMS) value of a waveform is a measure of the effective value or the equivalent DC value of the waveform. It is particularly useful in electrical engineering as it helps in determining the amount of power delivered by a periodic waveform. For a rectangular wave with alternating positive and negative values, the RMS value can be derived as follows:

Given:

• A rectangular wave with period T.
• Amplitude of the wave is +V for a duration T₁ and -V for the remaining duration T₂ = T - T₁.

Derivation of RMS Value:

The RMS value of a periodic waveform is defined as:

RMS Value = √(1/T ∫₀^T [f(t)]² dt)

For the given rectangular wave:

• During the interval 0 ≤ t ≤ T₁, the value of the waveform is +V.
• During the interval T₁ < t ≤ T, the value of the waveform is -V.

Thus, the RMS value can be calculated as:

RMS Value = √(1/T [∫₀^T₁ (+V)² dt + ∫_T1^T (-V)² dt])

Since (+V)² = (-V)² = V², the expression simplifies to:

RMS Value = √(1/T [∫₀^T₁ V² dt + ∫_T1^T V² dt])

Breaking it into two parts:

RMS Value = √(1/T [V²∫₀^T₁ dt + V²∫_T1^T dt])

The integrals represent the time intervals T₁ and T₂ (where T₂ = T - T₁), so:

RMS Value = √(1/T [V²T₁ + V²T₂])

Substituting T₂ = T - T₁:

RMS Value = √(1/T [V²T₁ + V²(T - T₁)])

Simplifying further:

RMS Value = √(1/T [V²T])

RMS Value = √(V²)

RMS Value = V

Hence, the RMS value of the rectangular wave is V.

Correct Option:

Option 1: V

This option correctly represents the RMS value of the given rectangular wave.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: T₁ - T₂ / T

This option incorrectly assumes that the RMS value depends on the difference between T₁ and T₂. However, the RMS value is derived from the square of the waveform values over the entire period and is independent of the relative durations of T₁ and T₂, as long as the waveform alternates symmetrically.

Option 3: V / √2

This option is incorrect because V / √2 is the RMS value for a sinusoidal waveform, not a rectangular wave. The rectangular wave has a constant amplitude +V and -V, leading to an RMS value equal to the amplitude V.

Option 4: T₁ / T₂

This option is unrelated to the calculation of RMS value. The ratio of T₁ to T₂ does not influence the RMS value of the waveform.

Conclusion:

The RMS value of a rectangular wave with amplitude ±V is equal to the amplitude V, as derived. Understanding the RMS calculation for different waveforms is essential for analyzing their power delivery capabilities in electrical and electronic systems.