gar
Senior Member
- Location
- Ann Arbor, Michigan
- Occupation
- EE
161106-1135 EST
This was brought up in the RS485 thread, but needs its own thread.
RMS measurements apply to AC, DC, and stationary random signals, and are meaningful in the same way with all of these signals.
For any type of signal the statistics must be stationary.
For meaningfulness there is no qualification needed for DC, pick any arbitrary time period for averaging. For a periodic signal an integral number of cycles are required for the averaging period. For a sine wave this can be half cycles. For a random signal the averaging time needs to be long enough that the reading error from one averaging period to another is less than some desired level. This is bandwidth dependent.
For a resistive load an RMS measurement is a way of having the same numeric reading of voltage or current for the same heating effect in the resistor.
An electrodynamometer meter can be calibrated on DC and when connected to an AC circuit within the bandwidth capability of the meter be calibrated correctly for the AC signal.
For those that have not studied calculus you have to accept the following as being accurate.
RMS refers to a math operation that squares an instantaneous value, and averages this over some period. This results in a single value of which the square root is taken to provide a single output value.
Mathematically integration is a means of summing an infinite number of infinitesimal values to obtain a finite value. For a sine wave the integral of sine^2 wt dt (call it sin squared) is
t/2 - (sin 2wt)/4w. Where t is a time variable, and w is a constant (in Greek symbols omega) equal to 2*Pi*f where Pi = 3.1416 and f is the frequency in Hz.
This equation is evaluated at a beginning time t=T1 and an end time t=T2 and the difference between T2 and T1 is taken. For simplicity let T1=0. Note that 2wt is really a double frequency value. The result is
T2/2 - sin 2wT2/4w
the average value (mean) is
T2/(T2*2) - sin 2wT2/(4w*T2) or
1/2 - 1/2 (sin 2wT2/(2w*T2) ) = (1/2)*(1 - sin 2wT2/(2w*T2) )
Note that sine can never exceed +/-1. Thus, the above equation is always positive.
Unless you do the integration over an integral number of half cycles there a a fluctuation in the value. If averaging is done over many cycles, then the effect of the fluctuation can be small with respect to the desired reading.
When the square root of the average (1/2) is taken, then the result is 0.707 . Does that look familiar?.
Perform the same calculation on a DC signal and the result is 1.
I have to leave. See if there are mistakes. Others can try to clarify my comments.
.
This was brought up in the RS485 thread, but needs its own thread.
RMS measurements apply to AC, DC, and stationary random signals, and are meaningful in the same way with all of these signals.
For any type of signal the statistics must be stationary.
For meaningfulness there is no qualification needed for DC, pick any arbitrary time period for averaging. For a periodic signal an integral number of cycles are required for the averaging period. For a sine wave this can be half cycles. For a random signal the averaging time needs to be long enough that the reading error from one averaging period to another is less than some desired level. This is bandwidth dependent.
For a resistive load an RMS measurement is a way of having the same numeric reading of voltage or current for the same heating effect in the resistor.
An electrodynamometer meter can be calibrated on DC and when connected to an AC circuit within the bandwidth capability of the meter be calibrated correctly for the AC signal.
For those that have not studied calculus you have to accept the following as being accurate.
RMS refers to a math operation that squares an instantaneous value, and averages this over some period. This results in a single value of which the square root is taken to provide a single output value.
Mathematically integration is a means of summing an infinite number of infinitesimal values to obtain a finite value. For a sine wave the integral of sine^2 wt dt (call it sin squared) is
t/2 - (sin 2wt)/4w. Where t is a time variable, and w is a constant (in Greek symbols omega) equal to 2*Pi*f where Pi = 3.1416 and f is the frequency in Hz.
This equation is evaluated at a beginning time t=T1 and an end time t=T2 and the difference between T2 and T1 is taken. For simplicity let T1=0. Note that 2wt is really a double frequency value. The result is
T2/2 - sin 2wT2/4w
the average value (mean) is
T2/(T2*2) - sin 2wT2/(4w*T2) or
1/2 - 1/2 (sin 2wT2/(2w*T2) ) = (1/2)*(1 - sin 2wT2/(2w*T2) )
Note that sine can never exceed +/-1. Thus, the above equation is always positive.
Unless you do the integration over an integral number of half cycles there a a fluctuation in the value. If averaging is done over many cycles, then the effect of the fluctuation can be small with respect to the desired reading.
When the square root of the average (1/2) is taken, then the result is 0.707 . Does that look familiar?.
Perform the same calculation on a DC signal and the result is 1.
I have to leave. See if there are mistakes. Others can try to clarify my comments.
.