How do you determine an RMS value of a complex wave?

Answer 1

To explain the basics behind rms of complex waves and harmonics. Let V = Volt, I= Current, P = Average power, and rms or RMS as Route Mean Square (both are acceptable) and let e.m.f. be Electromotive Force. Then rms values for V or I can be obtained in the following manner. An electrical quantity such as V or I (be careful for P) in a pure sine wave will have a rms value of, lets use V as argument:

Vrms = Vpeak/sqr(2)

This mean that the rms is the average of the absolute value of the sine wave. Not of the sine wave it self. The average value of a sine wave is 0. It's important to know it is the absolute value.

If we say for the sake of simplicity:

i = I x sin(a+b) then i2 = I2.sin2(a+b)

Since the average value is the sine amplitude divided by sqr(2), it makes sense to say that:

average of sine2 = 1/2 [the square root falls away (sqr(2)2 = 2)]

Thus, sin2(a+b) will have an average value of a 1/2, it will always be 1/2 since a sine wave will always have a peak value of 1 and 12 = 1

so let us say: An e.m.f produce a fundamental and another two harmonics, an e.m.f will normally produce odd harmonics such as 1st, 3rd and 5th

i = I1.sin(a+b1) + I3.sin(a+b3) + I5.sin(a+b5) +..... In.sin(a+bn)

i2rms = (1/2*I12)+(1/2*I32)+(1/2*I52)+...+(1/2*In2)

remove the power from i

irms = sqr[(1/2*I12)+(1/2*I32)+(1/2*I52)+...+(1/2*In2)]

remove 1/2 as common and the result will be:

irms = sqr[{(I12)+(I32)+(I52)+...+(In2)}/2]

Or what appears to be more conventional, remove 1/2 and apply Sqr(2) below the line

then it looks more like rms calculation :)

irms = sqr[(I12)+(I32)+(I52)+...+(In2)] /sqr(2)

Just another tip: I and V are done exactly the same but power is differenent since sqr(2) x sqr(2) = 2 and not sqr(2) so just to keep it simple do it as Pavr=Irms x Vrms

so fist find Irms and Vrms before power, there might be short cut once you know this well, but for now it's best to do it like this and not to confuse you.

To determine reactances and impedances, it's best to calculate each harmonic separately like

XL1 = 2 x (pi) x f1 x L

XL3 = 2 x (pi) x f3 x L

XL5 = 2 x (pi) x f5 x L

Z1=sqr(R2+XL12)

Z3=sqr(R2+XL32)

Z5=sqr(R2+XL52)

Power due to harmonic is Ph = Eh(rms) x Ih(rms) x cos(a) where a is the V-I phase shift

Ptotal = sum(Ph1+Ph3+Ph5+...)

Then power factor PF= Ptotal / [Eh(rms) x Ih(rms)]

Resonance is still at Xc=XL but only with individual harmonics, such as Xc1=XL1;Xc3=XL3;...