Why Is the RMS Coefficient of a Triangle Wave \(\frac{1}{2\sqrt{3}}\)?

[gulsen]

I've seen some coefficent such as $$\frac{1}{2\sqrt 3}$$ for triangle waves. Who ordered that? How can I derieve it? It doesn't seem to be $$\sqrt{ \frac{1}{T} \int_0^T {V^2 dt} }$$ because I found something like that for triangle wave: $$V_m 2\sqrt 2 f$$ (am I wrong?? or I'm calculating the wrong thing??)

 

[gulsen]

Whoops! Miscalculation! It's $$\frac{V_m}{\sqrt 3}$$. Gotta sleep, I guess...

 

[quicknote]

The coefficient of \frac{1}{2\sqrt 3} for triangle waves is a result of the mathematical derivation of the root mean square (RMS) value of a triangle wave. The RMS value is a measure of the average power of a signal and is calculated by taking the square root of the mean of the squared values of the signal.

To derive this coefficient, we first need to understand the properties of a triangle wave. A triangle wave is a periodic signal with a triangular shape, where the amplitude increases linearly from 0 to a maximum value and then decreases back to 0. This shape can be described by the equation V(t) = \frac{2V_m}{T}t, where V_m is the maximum amplitude and T is the period of the wave.

To calculate the RMS value of a triangle wave, we need to square the signal and take the mean over one period. This can be represented mathematically as:

RMS = \sqrt{\frac{1}{T} \int_0^T {V^2 dt}}

Substituting the equation for V(t) into this formula, we get:

RMS = \sqrt{\frac{1}{T} \int_0^T {(\frac{2V_m}{T}t)^2 dt}}

Solving this integral, we get:

RMS = \sqrt{\frac{4V_m^2}{3T^3} \int_0^T {t^2 dt}}

= \sqrt{\frac{4V_m^2}{3T^3} [\frac{t^3}{3}]_0^T}

= \frac{V_m}{T\sqrt{3}}

= \frac{1}{2\sqrt{3}}V_m

Therefore, the coefficient of \frac{1}{2\sqrt 3} is derived from the mathematical calculation of the RMS value of a triangle wave. It is not the same as the calculation for a sinusoidal wave, which results in a coefficient of \frac{1}{\sqrt 2}. It is important to note that this coefficient is not arbitrary and is a result of the properties of a triangle wave.