Triangle Wave Voltages - V_pk, V_pk-pk, V_avg, V_rms

for 0 ≤ θ < π/2

There are a number of ways in which the amplitude of a triangle wave is referenced, usually as peak voltage (V_pk or V_p), peak-to-peak voltage (V_pp or V_p-p or V_pkpk or V_pk-pk), average voltage (V_av or V_avg), and root-mean-square voltage (V_rms). Peak voltage and peak-to-peak voltage are apparent by looking at the above plot. Root-mean-square and average voltage are not so apparent.

Also see Sinewave Voltages and Square Wave Voltages pages.

Root-Mean-Square Voltage (V_rms)

As the name implies, V_rms is calculated by taking the square root of the mean average of the square of the voltage in an appropriately chosen interval. In the case of symmetrical waveforms like the triangle wave, a quarter cycle faithfully represents all four quarter cycles of the waveform. Therefore, it is acceptable to choose the first quarter cycle, which goes from 0 radians (0°) through π/2 radians (90°).

V_rms is the value indicated by the vast majority of AC voltmeters. It is the value that, when applied across a resistance, produces that same amount of heat that a direct current (DC) voltage of the same magnitude would produce. For example, 1 V applied across a 1 Ω resistor produces 1 W of heat. A 1 V_rms triangle wave applied across a 1 Ω resistor also produces 1 W of heat. That 1 V_rms triangle wave has a peak voltage of √3 V (≈1.732 V), and a peak-to-peak voltage of 2√3 V  (≈3.464 V).

Since finding a full derivation of the formulas for root-mean-square (V_rms) voltage is difficult, it is done here for you.

So, ≈ 0.577 V_pk

= 0.57735026918962576450914878050196

= 1.7320508075688772935274463415059

Average Voltage (V_avg)

As the name implies, V_avg is calculated by taking the average of the voltage in an appropriately chosen interval. In the case of symmetrical waveforms like the triangle wave, a quarter cycle faithfully represents all four quarter cycles of the waveform. Therefore, it is acceptable to choose the first quarter cycle, which goes from 0 radians (0°) through π/2 radians (90°).

As with the V_rms formula, a full derivation for the V_avg formula is given here as well.

So, ≈ 0.5 V_pk