Measuring relative phase between two waveforms using an oscilloscope

Transcription

1 Measuring relative hase between two waveforms using an oscilloscoe Overview There are a number of ways to measure the hase difference between two voltage waveforms using an oscilloscoe. This document covers four methods and summarizes the advantages and limitations of each. Method Oscilloscoe Requirements Waveform Requirements Advantages Limitations Time-difference channels B a Lissajous v. mode Sinusoidal only D, E a, c Product mode Sinusoidal only A a, b Curvefitting channels Data connectivity A, B, C, D d a: Errors introduced when dc offsets are resent b: Reduced recision near θ = 0, 80 c: Reduced recision near θ = 90, 70 d: Possible incorrect solution A: No manual reading of values from dislay (can be automated B: Works with nonsinusoidal waveforms C: Uses entire waveform information to increase accuracy D: Cool E: No need to measure time scales Exlanations are given to show how each method works. These are given for reference, but understanding them is not necessary to aly the methods.

2 Measuring relative hase between oscilloscoe traces using the timedifference method Oscilloscoe: Two channels (Includes virtually all oscilloscoes. Common frequency and shae Dislay both channels as a function of time. Scale each voltage channel so that each waveform fits in the dislay. Ground or zero each channel searately and adjust the line to the center axis of the dislay. Return to ac couling.. (Otional If you can continuously adjust the voltage er division, scale your waveforms to an even number of divisions. You can then set the zero crossing more accurately by ositioning the waveform between gridlines on the dislay. Pick a feature (e.g., eak or zero crossing for sinusoids, rising or falling transition for square waves to base your time measurements on. The eak of a sinusoid is not affected by dc offsets, but is harder to inoint than the zero crossing. Then follow one of the methods below: Method Method Measure the eriod T between reeats. Digital scoes often measure f = / T automatically. Measure t d, the smallest time difference between occurences of the feature on the two waveforms. The hase difference is then t θ θ = d 360 T Figure. Dual-channel dislay. With either method, the sign of θ is determined by which channel is leading (to the left of the other. In the figure, v leads v. (requires continuous time base scaling Fit one eriod of your waveform to 4, 6, or 9 divisions. Scale the time base by a factor of ten (exand the lot horizontally, so that each division will be 9, 6, or 4, resectively. Count the number of divisions between similar oints on the two waveforms.

3 Measuring relative hase between oscilloscoe traces using the Lissajous (ellise method Oscilloscoe: Able to dislay the voltage of one channel vertically and the other channel horizontally. (Includes virtually all oscilloscoes. Set the oscilloscoe to xy mode. Scale each voltage channel so that the ellise fits in the dislay. (This may be a line if the hase difference is near 0 or 80. Ground or zero each channel searately and adjust the line to the center (vertical or horizontal axis of the dislay. (On analog scoes, you can ground both simultaneously and center the resulting dot. Return to ac couling to dislay the ellise. (Otional If you can continuously adjust the voltage er division, scale your waveforms to an even number of divisions. You can then center the ellise more accurately by ositioning the waveform between gridlines on the dislay. Measure the horizontal width A and zero crossing width C as shown in the figure to the right. The magnitude of the hase difference is then given by ± sin ( C / A to of ellise in QI θ θ = ± [ 80 sin ( C / A ] to of ellise in QII The sign of ( θ θ must be determined by insection of the dual-channel trace. Sinusoidal Common frequency Figure. Lissajous figure. (by Paul Kavan

4 Exlanation: Given two waveforms: v vertical, v horizontal The ellise will cross the horizontal axis at time t 0 when v (t 0 = 0 or At this time the value of v (t 0 will be A trig identity yields* cos ( α + β = cos α cosβ sin α sin β v v ( t = + θ ( t = + θ ωt 0 + θ = n + π t0 = n + π θ ω v ( t0 = cos n + π θ + θ v ( + if is odd t n 0 = ± sin( θ θ if n is even 3 C / A above Now we have two nasty details to take care of. First, we have to be careful taking the inverse sine, since a hase change between 90 and 80 gives rise to the same (C / A ratio as its coangle. To decide which we have, find the times t θ and t when v (t and v (t eak: ωt + θ = 0 t = ω And look at v ( t at that time: v ( t = cos( θ θ Conversely v ( t cos( θ θ = ( θ θ = cos So if v is ositive when v is maximum ositive if θ θ is between 0 and 90. For this case, the to and right side of the ellise will be in Quadrant I. But if v is negative when v maximum ositive, then we need the other angle with the same sine. This means the to of the ellise will be in Quadrant II and the right side in Quadrant I. So if θ θ > 90, then the actual inverse sine is [ 80 sin ( C / A ], where sin - reresents the rincile inverse sine between 0 and 90. Second, what is the sign of ( θ θ? Suose we have a case where two voltages ( t and ( t have the same amlitudes as before but with oosite hase angles: ( t = θ ( t = θ Since cos( α = cos( α, ( t = cos[ ω( t + θ] ( t = cos[ ω( t + θ ] This is the exact same as v ( t and v ( t, but reversed in time. The Lissajous figure will look exactly the same, but the trace will recess in the oosite direction. That means the sign of the hase difference cannot be determined in xy mode. So we need the dual-channel trace to determine the sign of ( θ. θ

5 Measuring relative hase between oscilloscoe traces using the roduct method Oscilloscoe: Automatic amlitude measurement (referably rms value for each channel. Dislay the roduct of the two channels and calculate its dc offset automatically. This method was develoed using the Tektronix 0B oscilloscoe. Sinusoidal Common frequency Dislay the two traces in voltage v. time mode, along with the roduct trace in voltage v. time mode. Use ac couling. Fit the traces in the screen vertically. Dislay the MEAN value (dc offset v math,dc of the roduct trace. Show about 0 of its eriods to ensure that the calculation is not affected by artial waveforms at the beginning and end of the trace. You may also want to average the samling to reduce error. Dislay the rms amlitude of each of the channels ( rms and rms. Using rms (rather than eak-to-eak means that the scoe has averaged over the waveform. The hase difference is then* vmath, dc vmath, dc 8vmath, dc cos ( θ θ = = = The sign of ( Exlanation: θ rms rms θ still has to be determined by insection of the dual-channel trace. Given two waveforms: v ( t = + θ v ( t = + θ Their roduct is v ( t = + θ ( ωt + θ math cos Alying the trig identity cos( α + β + cos( α β yields The hase difference determines the dc cos α cosβ = v math ( t = [ cos( θ θ + + θ + θ ] offset of the math trace: v math, dc = cos( θ θ *This method is least recise when the hase difference is nearly 0 or 80, cases where cosθ is not sensitive to θ.

6 Figure 3. Oscilloscoe dislay for the roduct method showing the dual-channel dislay and the roduct trace, along with the values for rms, rms, and v math,dc as calculated by the oscilloscoe. Here θ θ = cos [ 0.07 /( ] = Since the eak in channel is to the left of the eak in channel, v leads v. Figure 4. Sreadsheet used for curvefitting method. Regions are color coded according to function: instructions, reference, data, results, error figure (between guess and exeriment, arameter guesses (calculated from data, fit arameters. Aroximate values of the fit arameters must sulied before using the Solve function.

7 Measuring relative hase between oscilloscoe traces using the curvefitting method Oscilloscoe: Dual channel caability Caability to transfer data oints to a comuter This method was develoed using the Tektronix 0B oscilloscoe. It has been imlemented using OenChoice Deskto software and an Excel sreadsheet called OscilloscoePhaseFinder--0--OenChoice.xls Common frequency and shae Dislay the two traces in voltage v. time mode. Scale the dislay to show at least one full eriod and maximize the waveform sizes without cliing. Oen the OenChoice Deskto software. Acquire the dislay to the comuter and coy it to the cliboard. Oen the sreadsheet mentioned above (or create your own and aste the data into the aroriate lace. In the curvefitting sreadsheet or other software, set initial guesses for the amlitudes and, dc voltage offsets v dc and v dc, and eak times t and t for each channel, as well as the frequency. Initiate the fit routine. The sreadsheet or other software must then calculate voltage v. time for each channel using the above arameters and comare those values with the measured voltages. It then needs to seek values for the above arameters that roduce the closest match between the calculated and measured values. Once the eak times are known for each channel, the software can comute the hase difference according to* θ θ = ω( t t, where t, t, and ω are results taken from the sreadsheet. This can be remaed into a desired range by adding or subtracting multiles of 360, deending on the alication. For instance, a second-order lowass filter would usually have hase ranging from 0 (at low frequency to 360 (the asymtotic value at high frequency. Exlanation: Given two waveforms: v ( t = vdc + + θ v ( t = v + + θ dc The waveforms eak at times t and t : ωt + θ = 0 ωt + θ = 0 Substracting the lower equation from the uer, ( t = ω t *Note that dc offsets in the signal don t cause inaccurate hase measurements they are accounted for in the fit rocedure. θ θ