101410114 lissajous method for phase and frequency measurements

7/27/2019 101410114 Lissajous Method for Phase and Frequency Measurements

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Phase Measurements on the OscilloscopeIn many physical phenomena resulting in oscillating moti011, currents, voltages, etc. it is necessaryto determine a phase shift present in one of the parameters. In this experiment we will investigate thephase shift of a sinusoidal current produced in a simple circuit powered by a sinusoidal voltage. The

phase shift of the current will be measured relative to the phase ofthe voltage.THEORY:In this experiment we will measure the phase shift two different ways. First we will produce aLissajous pattem by using our dual beam oscilloscope as an X-Y display device, then we look at thethe phase difference directly by putting the signals into the oscilloscope channels 1 and 2 andmeasuring the phase shift in terms of the time base.Method 1 The Lissajous Figure

The two sinusoids we wish to compare may be represented by:

where is the phase shift to be determined. Notice that V1 may or may not be equal to V2. Thesinusoids are written out as Vx and Vy because we will put the two voltages into the vertical (y) and thehorizontal (x) axes of the oscilloscope. Begin by deriving a relation between the phase shift and theobservable voltages so that we can anticipate what will be seen on the scope.Expand Vy = V2( sinrotcos + cosrotsin) and substitute using sinrot = VxN 1 to getv2Vy= ------ VxCOS +V2cosrotsin

V1This equation can now be solved for cosrot so that we havev2cosrot =

Vy- --- VxCOS V1

Now substitute in sin2rot + cos2rot(Vy- {V2Nt}Vxcos)

2

and

1 to obtain

1 = ---------------------------- + v 2N 12V/sin2

which reduces to

grouping like terms gives us

since sinrot = VxN

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V/V.Zsin2 = V12V/ + V/V / - 2V1V2VxVyeosor finally v2y Vx2 VxVy

sin2 = + ------ - 2------- cosV / v.Z V1V2

This is the equation ofan ellipse! The derivation shows that two sinusoidal functions oftime, in this casevoltages, when plotted against each other will appear as an ellipse. In particular, ifwe consider therelation when Vx or Vy =0:IfVx =O then sin = V/V2 and ifVy =O then sin = VxN1

Note that we are at the corresponding intercepts for Vy or VxThe diagram shows how the ellipse is produced on the oscilloscope. Note that we only ge t llbecause the + indicates the direction the beam is moving, and this is not observable. The pattem is anexample of a Lissajous figure. Lissajous figures result when we plot , or physically drive, twosinusoids perpendicular to each other. We are investigating an example where rox = roy, but in generalLissajous figures are produced even if ox = roy. In this method we will measure both Vy, V2 and Vx, V1and use sin = V/V , and VxN1 corresponding to V x and Vy = O (intercept) to determine ll. F o rimproved accuracy it is better to measure a larger range such as 2Vy and 2Vx.

I /o r-

- o

"- ,\ V z ~ M t + 3 / ;' \ I~ //1 :IT = 21T

6 )

-' , / I

vt( f l /fv ./- - - - - -- 0 r ------ y /- ,.' X .//I ,.,. .. A-

,..,."'"' I

I

t =o -} o17


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Method 1: Dual Channel Scope lnputsIn this case we put the driving voltage sinusoid and the resulting voltage Vc into channels 1 and 2 ofthe oscilloscope respectively. The time bases can then be observed to find the phase shift in terms of atime difference that can be related to a phase shift angle as a fraction of the period T. It is importantto watch your grounding in this set up because the function generator and scope have a commonground through the power connection.PROCEDURE:Set up your circuit as shown below:

c= 0.02J.LFR=22k

The sine wave, or driving voltage is provided by the "function generator. Set the function generator toproduce a 400HZ sine wave with -2 V peak-to-peak voltage.

The basic circuit will be the same will be the same for both methods so we will do the measurementsin sequence. First we will use the Lissajous figure then we will use the time base phase shift. Make ameasurement for tive frequencies, 200, 400, 600, 800, and 1000 Hz.

Set the oscilloscope on internai trigger, with the function generator output connected to channel 1ofthe scope. Now set the output of the function generator by adjusting the amplitude on the front ofthe function generator until the sine wave on the scope is 2 V peak-to-peak.PART 1: Connect the output o f the function generator to the circuit as shown above as E. Now connect

the channel 2 probe across R making sure the clip is connected to a common ground.Switch the scope to X-Y mode making sure the probes are 1X inputs. Adjust the Y-axisgain so the ellipse is easily measured. Record the Vy, V2, Vx and V1 data for each frequencyin data table 1.

PART 2: Now switch out ofthe X-Y mode operation, and measure the phase shift for each frequencyby comparing the two sinusoids on the time base. Record this data as .::1t. Be careful to setup your scope trigger so that the signal from channel 1 (function generator) is the trigger.Note since you are monitoring VR this experiment actually makes a measurement of thecurrent i which we expect to be out o f phase with the driving voltage.

PART 3: Adding sinusoids is much like adding vectors where the direction of the vector is just asimportant as the magnitude. The important quantities in the adding of sinusoids are theamplitude (magnitude) and the phase (direction). In this experiment it is useful to investigatehow the amplitudes of Vc, VR and E are related. In order to make this observation anddetermine if it is frequency dependent, measure the amplitudes o vc, vRand E at 200, 600and 1000 Hz. In order to do this C and R must be switched in the circuit beca use of theground connection. Recorda table ofVc and VR for the three frequencies.

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PRESENTATION AND CONCLUSIONS:The presentation should begin with the calculations needed to find the phase shifts from therecorded data. For the Lissajous figure method use the average value ofthe X-axis and Y-axis data,

and for the time shift use Atff and convert to radians. Organize your results in a table similar to thatshown below.For each frequency compare the result for each method including an estimated error. Is thephase shift frequency dependent? For each case write out VR both as a phasor andas a time domainquantity. Also write E as a phasor and time domain quantity. Evaluate how the voltages add: Vc+ VR oras Vc2 + VR2? Which sum is more appropriate?

DATA TABLES:Lissajous Pattem Data

frequency Y. Vt VJVt Vy Yz v;vl ave V._/Vu

200Hz

400600800

1000

Direct Time "Shift" Measurementfrequency T (sec) At = 21tAt/T

200Hz

400600800

1000

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Adding Sinusoidal Voltages

frequency Vc VR ...JVc2 + VR2

200Hz

600

1000

20

101410114 lissajous method for phase and frequency measurements

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