True RMS Vs Area Under The Curve

Question

I'm not sure if this is the correct place to ask the question, but here goes...

Using my DS1104z scope...

I measure a 60 Hz sine wave AC voltage and read 120 Vrms for a single waveform

but when I measure a single waveform using Math/ABS Function ( Displaying negative portion of waveform as positive )

I get and Area ( under the curve ) Of 1.81 Vs

Using the calculation

1.81 Vs X 60 Hz = 108.6 Volts

Shouldn't the Vrms And Calculated Area Voltage Match? or am I missing something?

Thanks,

Bill

Answer 1

What you're missing is the definition of RMS: $$V_{RMS} = \sqrt{\frac{1}{τ}\int_t^{t+τ}V(t)^2\ dt}$$

What you're calling the calculated area value is $$\frac{1}{τ}\int_t^{t+τ}{|V(t)|\ dt},$$

which is not the RMS of the signal.

(in math terms, I believe you would say the square root and integral operators don't commute, so you can't just pull the square root into the integral and cancel it out with the square.)

Answer 2

\$\sqrt{\int_0^T s^2(t)\, \mathrm dt} \ne \int_0^T \lvert s(t)\rvert\,\mathrm dt\$, simple as that.