tiempo promedio

7/30/2019 Tiempo Promedio

1/10

I

1

^ 1

i

" "" " '1

BRUEL& KJAR

NftRUM - DENMARK


2/10

By C G . Wahrman, A/S Briie! & Kjser, Naerum, Denmark.

Averaging Time of Measurements

SignalRunning Integration

Stepwise Integration

Time averaging together with some sort of weighting are the two most important processes in thedata reduction which is necessary in almost any measurement of a signal. For signals involvingonly high frequencies the time averaging normally causes no problems, but for low frequencysignals or single pulses it is more important to know the influence of the averaging process andof a fin ite averaging time. T he main purpose of this paper is to compare two simple types ofaveraging and find out how to choose the parameters to get most equal results.

Mathematically the most simple averaging is a true integration of thesignal over a fini te time T , the resultbeing divided by T . Ref. fig.1. If onewants to follow the signal later on,the integration interval could bemoved continuously giving always anaveraging of the last T seconds ofthe signal. This is called running integration in the following. However,this requires a complete memory ofthe last T seconds of the signal to beable to throw away the values justrunning out of the integration period.In most cases it is therefore preferredto do the integration stepwise infixed time intervals, thereby obtaining sample values of the curve thatthe running integration would give.This is called stepwise integration inthe following.Integration t ime

Time171338Fig, 1. True Integra tion of a Signa l


3/10

whichRM A / V

V |o| u

V = R C $ +UU ( t) =/ " ( t - t ) P ( t ) d - r

P ( i} = e*p (-i/RC)

Fig. 2. Some formulas for RC -ave raging.171339

RC Averogln

True integration

F ig . 3 . We ight ing curves (TimewiseJ Integra tion t i me

ISRC-1ow-Another way of averaging,simpler electronically is thepass filter. For this system a simplefirst order differential equation canbe written, ref. fig.2. The impulse isa decaying differential equation, ref,fig.2. The impulse response is a decaying exponential. The response toan arbitrary signal is the convolutionof the signal and the impulseThis can be considered themulti-integralresponse.same as integrating the signalplied with a weighting function,which is the impulse response foldedaround to negative times, ref. fig.3.In this figure is also shown the timeweighting function of true integration, which is 1/T inside the integration interval and zero outside. Thescales of the two curves are arbitrary,but they are predisposed to the results of the following by having Tequivalent to 2 RC instead of 1 RC.Actually the differences between thetwo curves are better distributed inthis way as the two areas outside the

0.135 and 0.153outside therectangle arethe onlysingle areawith T - RC would bewhereasrectangle0.368.In fig.4 is shown the result of RC-averaging of the same signal whichwas used in fiq.1.

171340

Exponential Weighted Signal

TimeFjg i RC A veraging of a Signal2 171341


4/10

0d B

U5

10

RC- Aver aq inq

V l + { 2 n f - R C ) 2B=l / iRC =j t f0 / 2

15--

20- .

25 -

True Integrationsin(f-T)f- T

0,1 02 1 *

Effe ctive Energy Ba ndwidth B

0.5 1

Fig,5, Weighting curves (F reque ncy-w ise)

3 L 5 6Relative frequencyf-T or f-2R C

in 342

Instead of comparing the two typesbasis of their timebe corn-frequencyof averaging onweighting curves they canpared on basis of theirweighting curves, ref. fig.5. Again thescales of the two curves are arbitrary,but if T is chosen equal to 2 RC thetwo curves fit nicely with the sametwo limiting lines in this double logarithmic plot. However, it is moreimportant that on this basis the twoeffective power bandwidths are thesame. Actually this is the main reason for choosing 2 RC as the equivalent averaging time. If for instance aband of randomarrowsquared orresult in otherthe isband

noise isways rectifieda DC plus a low frequence Dana of random noise, ref.fig.6. If this is filtered by two averaging processes with the same loweffective power bandwidth, the standard deviation of the output fluctuations wi ll be the same. (This is further described in lit. ref 1.)

BOriginal narrow band random noise spectrum

f

Now, the information wanted from the signal is normally not a simple average. Pure AC signals have zeroaverage independent of the size of the signal. Thereforesome amplitude weighting of the instantaneous values isintroduced. E lectronically the simplest method will be toconsider only the values with one sign; or those withnegative sign could be turned around thus consideringonly the absolute or numerical values, ref. fig.7.

DC-component

fResulting Spectrum after squaring

Effective Energy Ba ndw idth^Averaging filter response

fEnlarged vie w of the low frequency part of above s pectrum.

Re lative standard devia tion on RMS '.* - 1 - 12\/BT 2VB-2RC

Fig. 6. Averaging of squared narrow band random nois e. 3171343


5/10

Absolute Average Mean Square

Fig.7, Rectifying Characteristics (Fixed) 777344

Mathematically this process is not sosimple as it looks, because integrationhas to be split in positive andositivemuch negative parts. It is much simpler toconsider a weighting proportional tothe value, that is to use a MeanSquare averaging. While the MS valuemathematically is more useful, it ismore difficult to obtain it electronically. O ften a thermal converter isused, as the heating of a current in ato the MS ofesistor is proportionathe current. Other makeurrent. Other circu its maKe apolygonal approximation to the parabolic characteristic of fig.7 byof a network consistina ofonsistingand still otherlog. and log. in. conversion to obtain the squaring.

means OT adiodes and resistors,circuits utilize lin.

Direct Method :RMS = VMS

U =VV2V'-U R C , ^ 1

Feedback Method:RMS =

U =MSTRMSI

Y _ 2U= MTRMSI

or

Tav - 2 RCi =Fig.S.FbrmuLas for two methods

V2 M _ R r dUV2_ U2_.RC Idiy 2l2 dt

RC2of obtaining RMS values.

171345

l

In most cases the R oot Mean Squarethe value preferred to the MS. Its me vaiue preterred tocan be obtained by simple squareroot extraction of the MS value, butit can also be obtained in anotherIf the MSway.dividedisby the by some meanstheutput signalsame result is obtained, ref. fig.8.

Such a division is for instance donein a polygonalby letting the approximating circuitproportionalref. to

cornerthe

points move

notoutput voltage,fig.9. Although the MS value canbe found anywhere in thewill anywherebehave circuit, It will behave exactly as thefirs t type of roo t mean squaring, astheong asloaded. parabola is not over-

RM S auasi RMS Peak

F ig.9. R ectifying Cha racteristics (Moveable)1/134E

Very often sufficient accurate RMS values can be made by a simple polygonal approximationhaving only two corners as shown in fig.9. Such a circuit may be called a quasi RMS circuit (ref.lit. 2 and 3). If the ratio between the charging and the discharging resistors is approximately 1 : 3,then the error w il l be less than 1 dB for signals with crest factors up to 3 and signals assinewaves, squarewaves, triangular waves, and gaussia random noise will show only a few per centerror.If the ratio between the charging and discharging resistor is decreased to a very low value asshown in fig.9, the circuit turns into a (quasi) peak circuit, the better the smaller the ratio is.

4


6/10

Int. max.

- 1 0

E ffectivn E nergy Bandjwidth B

0.2 0.5 1 2 3 4 5 62 f - T a vFig.10. Low frequency rippeL for M S an d RM S a verag ing of s inewave s

171347

dB

0

- 2

- 4

- 6 -

- 80.5 1 3 4 5 G"f-Tav

Fig.11. Low frequency rippel for MS and RMS averaging of pulsewaves

As the MS and RMS is the mostinteresting values we will start considering these. In fig5 was shown thefrequency responses of the two typesof averaaina. This will also be theaveraging.frequency versus modula-esponsestion frequency for a high frequencycarrier weakly amplitude modulatedby a low frequency sinewave. However, another type of frequencyresponse can be made, namely theripple obtained when measuring thelow frequencies alone. In fig. 10 isshown the result for sinewaves whilefig.11 shows the result for a wave ofnarrow pulses. It is seen that in mostparts of the figures the ripple fortrue integration is with in the ripplefor RC-averaging, This is an argumentfor choosing a longer timeconstantand especially fig-11 shows thatT a v = RC is a better choice from thispoint of view.

171348

5
http://fig.11/http://fig.11/http://fig.11/


7/10

MS

R M S

0dB

Log .( R ) M S - 5

-10

^ ^ ^ Jf j"JJ"Jj"J""J^ 1^S/////////////ff//////////////// A/ /

IF//9J- Jyi f / /if////if/////W///

V///////////v/v/ ////M" JJJJJy"J JJ J^ *w/w////]w A

viiifiiV^lf\ ^ ^ %\-^\yyy/\ S W\ |

L JAveraging Time RC 2 Time

nput Running Int. Stepwise Int. R C- Ave -F ig , 12- Step responses of MS and R MS averaging

171349

Another aspect of interest is the behaviour of the two types of averagingof rectangular pulses or stepshown firstfig. 12 areof thesinglefunctions.the responses OT tne mean square.For true integration they are straightsloping lines, whereas for RC-aver-aging both the rising and the fallingresponse are equal exponential curveswith time constant RC^

Below the RMS responses. For trueintegration the rise and fall curvesare equal parabolas, but for RC-aver-aging the two curves are no longerthe same. The curve is notisingexponential, while the falling curve isbuttill exponentialconstant RC2 =another reasonhaving a

2 R C T . Thisfortime

' av 2 RC! =giveschoosing the

theC2 as the decaytime constant can easily be measuredwhereas inthe movableincircuitMSdoes notconfusiontheexist.with

parabolatime constant RC-i(It is due to thistwo time constantsthat the designation "averaging timehas been used in this paper)./ /

F inally is shown the responses with logarithmic ordinates. Here the RC-averaging decay is astraight line, wh ile the rise curve looks even steeper than in the middle figure . In all three figuresthe choise T a v = 2 RC1 seems to give reasonable equa lity between true integration andRC-averaging. As mentioned above the moving parabola RMS circuit will behave exactly as thefixed parabola circuit, as long as the parabola is not overloaded. This must, however, always bethe case at the beginning of the rise curve, where the parabola is very smallresponse is therefore shown in double logarithmic coordinates.response is reached approximately, when the response is a factor of C below the final level, where.C is the crest factor capability*. That is rectangular pulses of longer duration than about T a v /C 2are integrated as the ideal RMS circuit should do. It is also observable that measuring short singleRC-i = 1/2 RC2 would be a better choice when comparing true integration wi th

In fig. 13 the riseIt is seen that the ideal RMS

pulses T a v =RC-averaging. F or such pulses it is, howeveand shape than just the energy of the pulse. rma

Another aspect which is importantwhen comparing averaging processes,is their ability to follow slow changesin signal levels. This is for instanceimportant when measuring frequencewithesponses a sweeping smewave.

Fig.13.R ising step response of RM S circ uits with Limited crest factor capability C. 171350* In an earlier paper (4) the crest factor capability is defined in a different way corresponding to twice the values used here.6


8/10

LevelPeak Resonance

F i g . 14. Response to slow leve l changes

Ant i resonanceTime

171351

from fi there wi ls it appearsbe a certain delay which depends onthe slope of the curve. The mostimportant parts of a curve like thatare normally the peaks and notches,and here the averaging also produceserrors in the levels. Normally thesepeaks and notches are not sharp cor-shown on the first part ofsersthe curvewhich but round resonances,as parabolasnear their tops respectively bottoms.can be treated

n fig.15 the resulting delay is shownIts function of the slope.that the delayfor low

appearsis approximately1/2 T a v for low slopes for both run-integration andis another strong argument forchoosing T a v = 2 RCi = RC2. It alsoat the RC averaqinq is

ningThisRC-averaging.

appears tnat tne H U averagingfaster in the rising and slower in thethan the

1 8,7 10Fig.15. Delay for slow changes. MS and RMS .

fallingand thatfollow athan 8.7 dB/T

runningit will nevedownwardntegration,be able toslope of more

av

20 Retativ Slope: 50dB/Tw

171352

1 2 8.7 10 20 50Relative slope: dB/T avFig.16. E rrors on P eaks ond Valleys. MS and R MS

Fig.16 shows the errors on the sharpand vallevs as a functioneaksthe valleysof the as a runc tion ofsides. The curveslopesare based on the assumption that theleading slope has had sufficient duration to obtain the delays shown inthethat previous figure. AgainRC-averaging cannot we seefollowdown into a valley with slopes abovedB/T a v. However, even for slopesup to 5 - 6 dB/Tavbetter than RC-averaging isstepwiseandhe worst

with highthan running integration.case,slopes

integrationfor thein

it is evenpeaksbetter

171353 1


9/10

2 5Relative equiv averaging bandw. : J fllf'5Bw.

F ig.17 shows the errors on resonances and antiresonances as functionof the relative equivalent averagingbandwidth: T 3V S /Bw where S isthe sweep speed and Bw is the 3dBbandwidth. The shape of the anti-resonances is parabolic and not exponential as the valleys used in fig.16and therefore there is no verticalasymptote on the curve for RC-aver-aging. Again it is seen that the RC-averaging fits within the shaded areaof stepwise integration and is evenbetter than running integration forresonances at the right end of thecurve.

Fig. 17. Errors on Resonances and Antiresonances . MS and RM S.171354

Delay1.0

0.75

0.5

0.25

0

Stepwise Integration

7

tiempo promedio

Documents