Calculating Vrms
Supplement to Lab Oscilloscope
Given a set of data one can find the average simply by adding all the values and dividing the sum
by the total number of values in data set. How does one find the average value, not of a discrete
data set, but of one that is continuous? For example, how can the average of the function
𝐬𝐢𝐧(𝝎𝒕) be determined over one full period? For continuous functions the sums in the average
are replaced by integrals:
1
𝑓)*+ =
∫2 𝑓 (𝑡)𝑑𝑡
1
∫2 𝑑𝑡
1 1
= 5 𝑓(𝑡) 𝑑𝑡
𝑇 2
Recall the definition of the angular frequency 𝜔 =
78
1
, where T is the period of the sine function.
If you carry out the above calculation for function sin(𝜔𝑡) you should get zero. (Sketch one period
of sine curve and see if that make sense.)
The sine function crops up in all areas of physics and daily life. In studying alternating current or
AC circuits, the sine function is often used as the input voltage to the circuit.
For reasons that will become clear as the semester progresses it makes little sense to use the
average of the sine function since it is zero and all results based on it would likewise be zero and
therefore meaningless. To get around this we define a quantity called the root-mean-square. This
can be calculated by taking the square root of the average of the square of the desired function.
The root-mean-square voltage or Vrms is defined as:
1
𝑉=>? = @
∫2 𝑉 7 (𝑡)𝑑𝑡
1
∫2 𝑑𝑡
1 1
= A 5 𝑉 7 (𝑡)𝑑𝑡
𝑇 2
As an example, let’s determine Vrms when the input voltage is 𝑉 (t) = 𝑉2 sin(𝜔𝑡) where
V0 is the amplitude of the voltage signal and w is its angular frequency. Recall 𝜔 = 2𝜋𝑓
where the frequency f is in Hz (1/s) and is what set on signal generator.
The derivation follows:
7
𝑉=>?
=
1 1 7
𝑉27 1
5 𝑉2 sin7 (𝜔𝑡)𝑑𝑡 =
5 (1 − cos(2𝜔𝑡))𝑑𝑡
𝑇 2
2𝑇 2
=
1
𝑉27
1
𝑉27
𝑉27
H𝑡 −
sin(2𝜔𝑡)I =
𝑇=
.
2𝑇
2𝜔
2𝑇
2
2
As a result, 𝑉=>? = K𝑉27 /2 = 𝑉2 / √2 . In other words, the rms voltage is the voltage amplitude
divided by the square root of two. The result follows from the fact that the average value of
sin7 (𝜔𝑡) Sketch a sine squared curve over one period and convince yourself this makes sense.
In this lab you are asked to derive the result for a triangle wave, Fig. A triangle wave can be
described by the function:
4𝑉2
𝑇
𝑡
,
0≤𝑡≤
⎧
4
⎪ 𝑇
−4𝑉2
𝑇
𝑇
3𝑇
𝑉 (𝑡 ) =
H𝑡 − I
,
≤𝑡≤
2
4
4
⎨ 𝑇
3𝑇
⎪ 4𝑉2
(
)
,
≤𝑡≤𝑇
⎩ 𝑇 𝑡−𝑇
4
Use this definition along with the definition of 𝑽𝒓𝒎𝒔 and show that the result for triangle wave
[
will be 𝑽𝒓𝒎𝒔 = \ .
√𝟑