These are common terms when we speak in terms of voltage and current waveforms, but what that means?

To better understand that let’s consider as example one resistance of 10R supplied by a continuous voltage of 10V. Its power can be calculated as: P = V * I, where V is the voltage at it terminals and I is it current across. According with the Ohm’s law the current can be calculated as I = V / R. So, its power will be P = V^{2} / R, or 10W.

Let’s suppose this resistor will be driven by the same voltage, but with a duty cycle D of 50%. That mean an average voltage of 5V which, according with the previous equation will lead to a dissipated power of 2.5W.

Other approach can be used: the power during the on time will be 10W and during the off time is 0W, giving an average power of 5W.

So, what’s the right way and where’s that difference?

By analyzing in more detail, the first approach would be right if the current was independent of the voltage (like a constant current source) and it power variation linear. That happens for example in an ideal diode, where it forward voltage is independent of its current. How in a resistor the current change proportional to it voltage, it power variation is quadratic.

With that appears the term RMS, which means the equivalent DC voltage that leads to the same power dissipation in a resistor.

**How we can calculate the RMS value?**

In that example it could be easily calculated by considering an average dissipated power of 5W in a 10R resistor, which means a value of 7.1V.

By definition, the RMS (Root Mean Square) is the square root of the mean square and can be calculated as:

#### RMS value of a periodic waveform

Where T is the period of the waveform

#### RMS value of a discrete number of points with same interval

Where n is the number of points sampled at the same interval

#### RMS value of a discrete number of points with different intervals

Where Tn is each time duration and T is it repetition period

For the example above, how the on and off time have the same duration can be used the second equation where n=2, x_{1}=10 and x_{2}=0. For different on and off times can be used the third equation, where x_{1} and x_{2} have the same values and T_{1}/T = 0.5 and T_{2}/T is also 0.5. For both cases the result is the same: 7.1V RMS.

### Common waveforms in power converters

Here are the equations deducted for the definition above for the waveforms most commonly found in power converters:

#### Sine Waveform

#### Square Waveform

#### Triangle Waveform

#### Trapezoidal Waveform

For more complex shapes can be decomposed in partial shapes, calculated it RMS value and combined again with the equation for the discrete number of points either with the same or different time interval.

This is a simple approach to understand the meaning of the RMS value and some useful equations to solve most of the common waveforms in power converters.

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