Electrical circuits lecture notes b. Tech

Triangular wave: 
When plotted as voltage (V) as a function of phase (θ), a triangle wave looks similar to the 
figure to the above. The waveform repeats every 2π radians (360°), and is symmetrical about 
the voltage axis (when no DC offset is present). Voltage and current exhibiting cyclic 
behavior is referred to as alternating; i.e., alternating current (AC). One full cycle is shown 
here. The basic equation for a triangle wave is as follows: 
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. 
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. 
≈ 0.5 V
pk

---

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.