RF Cascade Workbook

Copyright

1996 -
2016

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Kirt
Blattenberger,

BSEE - KB3UON

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When plotted as voltage (V) as a function of phase (θ), a triangle wave looks similar to the figure to the right.
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:

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.

**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.

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

Also see Sinewave Voltages and Square Wave Voltages pages.

As the name implies, V

V

Since finding a full derivation of the formulas for root-mean-square (V

**So,**
≈ 0.577 V_{pk}

= 0.57735026918962576450914878050196

= 1.7320508075688772935274463415059

As the name implies, V

As with the V

**So,**
≈ 0.5 V_{pk}