RMS versus Average, A Second Look - John Dunn, Consultant, Ambertec, P.E., P.C.

We saw before that for a zero to +E square pulse train, that the RMS value equals the average value divided by the square root of the duty cycle. The narrower the duty cycle gets, the higher the ratio of the RMS to the average becomes.

Here, we look instead at the ratios of RMS to average for sine tips of different conduction angles.

In each case below, the waveform goes from zero to +100 and back again to zero along sinusoidal trajectories. The RMS and average values for several conduction angles are calculated and their ratios are compared.

 
As with the narrower duty cycle square pulse trains, the smaller (narrower) conduction angles here also yield higher ratios of RMS to average values.

Comments

[John Dunn]

This is the GWBASIC code that gave the above results.

10 CLS:SCREEN 9:COLOR 15,1:PI=3.14159265#:XSTART=50:YSTART=120
20 PRINT "save "+CHR$(34)+"rmstoavg.bas"+CHR$(34):PRINT:PK=100:ST=-40
30 PRINT "save "+CHR$(34)+"a:\rmstoavg.bas"+CHR$(34):PRINT:PRINT
40 C$=" ###°":D$="Avg = ##.## RMS = ##.## RMS/Avg = ##.###":GOTO 70
50 C=XSTART+THETA:D=(320-Y*.25-YSTART):IF KK<>0 THEN LINE (A,B)-(C,D)
60 A=C:B=D:KK=1:RETURN
70 FOR PHI=180 TO 20 STEP ST:CT=0:AV=0:RMS=0:THMAX=180*3:KK=0
80 FOR THETA=0 TO THMAX STEP 1
90 Y=PK*(ABS(SIN(THETA*PI/180))-SIN((180-PHI)*PI/360)):IF Y<0 THEN Y=0
100 Y=Y/(SIN(90*PI/180)-SIN((180-PHI)*PI/360)):REM Peak normalization
110 AV=AV+Y:RMS=RMS+Y^2:CT=CT+1:GOSUB 50:NEXT THETA:REM Accumulations
120 CT=CT-1:AV=AV/CT:RMS=SQR(RMS/CT):RA=RMS/AV
130 LOCATE (10+PHI/ST),8:PRINT USING C$;PHI:YSTART=YSTART-30
140 LOCATE (10+PHI/ST),16:PRINT USING D$;AV,RMS,RA:NEXT PHI