Hello again,
Just to note, the correct answer is Irms=I*sqrt(2/3) as noted by some of
the other posts. This has been verified using four different techniques,
two of which will be shown here.
Here are two methods to calculate the rms value of that wave.
The first method uses the definition of an rms value, and the
second method (far below) uses a shortcut knowing something
about the properties of rms values for different wave shapes.
Just to note, the wave being worked on here is as follows:
From t=0 to T/2 the wave goes from -I to +I (slanted), and
from t=T/2 to T the wave is a constant equal to +I.
METHOD #1
For the slanted part of the wave, we can use the slope intercept
form of a line to get the required equation of that line from
0 to T/2:
i=mx+b
The slope m is:
m=(i2-i1)/(t2-t1)=(I-(-I))/(T/2-0)=2*2*I/T=4*I/T
(note: the way we use this is similar to the "Two Point Form" of a line)
The intercept constant b is simply:
b=-I
and x is of course equal to t:
x=t
Substituting all these into the slope intercept form of a line above:
i=mx+b
and calling the amplitude I, we get:
i=(4*I/T)*t+(-I)=4*t*I/T-I
So now we have the equation for the slanted line segment:
is=4*t*I/T-I
The equation for the horizontal section is simply:
ih=I
Now squaring both we get:
is^2=I^2*(T^2-8*t*T+16*t^2)/T^2
and
ih^2=I^2
Now integrating is^2 from 0 to T/2 we get:
iis=(I^2*T)/6
and integrating ih from T/2 to T we get:
iih=(I^2*T)/2
The rms value is now:
Irms=sqrt(1/T*(iis+iih))=sqrt(1/T*I^2*T*(1/6+1/2))=sqrt(I^2*(2/3))=I*sqrt(2/3)
METHOD #2:
The way they approached this problem in that linked equation picture was
different however. They used the rms property of triangle waves and of
simple horizontal (pulse) waves, and the fact that rms values add as
the square root of the sum of squares of the individual rms components
times their individual durations.
The component of the triangle wave is:
c1=I/sqrt(3)
(note that you have to know this property of triangular shaped waves
beforehand in order to start the solution the way they did in that pic)
and the component of the pulse wave is:
c2=I
and after squaring we get
c1=I^2/3
and
c2=I^2
and now taking into account that each component only exists for 1/2 the
total time period we multiply by 1/2:
c1=(1/2)*I^2/3
and
c2=(1/2)*I^2
and now we add them:
sum=(1/2)*(I^2/3+I^2)=(1/2)*I^2*(4/3)=I^2*(2/3)
and now taking the square root we get:
Irms=sqrt(I^2*(2/3))=I*sqrt(2/3)
In the original document, another way of taking into account that the waves
are only present for 1/2 the total time period is to multiply by duration T/2 and
then we have to also multiply by 1/T.