lab 2 write up

8/2/2019 Lab 2 Write Up

1/17

Lock-in Amplifier

Kevin Oskar Negron 1

1Department of Physics, Cal State University of Fullerton, Fullerton, CA 92831

Abstract. With this experiment we will become acquainted with individual components of the lock-in amplifier:preamplifier, filter, noise generator, phase shifter.

I. INTRODUCTION

A lock-in amplifier is a type of amplifier that canextract a signal with a known carrier wave from anextremely noisy environment. It provides a DC outputproportional to the A C signal under investigation. Thespecial rectifier, which performs this AC to DCconversion, forms the heart of the instrument. It isspecial in that it rectifies only the signal of interestwhile suppressing the effect of noise or interferingcomponents which may accompany that signal. Thetraditional rectifier, which is found in a typical ACvoltmeter, makes no distinction between signal andnoise and produces errors due to rectified noisecomponents. The noise at the input to a lock-inamplifier, however, is not rectified but appears at theoutput as an A C fluctuation. This means that thedesired signal response, now a DC level, can beseparated from the noise accompanying it in the

output by means of a simple low-pass filter. Hence ina lock-in amplifier the final output is not affected bythe presence of noise in the applied signal. This isachieved by supplying it with a reference voltage of the same frequency and with a fixed phaserelationship to that of the signal. This is mostcommonly done by ensuring that they are derivedfrom the same source. The use of such a referencesignal ensures that the instrument will track anychanges in the frequency of the signal of interest,since the reference circuit is locked to it. It is fromthis characteristic that the instrument derives itsname. This inherent tracking ability allows extremelysmall bandwidths to be defined for the purpose of signal-to-noise ratio improvement since there is nofrequency drift , as is the case with analog tunedfilter/rectifier systems. Because of the automatictracking, lock-in amplifiers can give effective Qvalues (a measure of filter selectivity) in excess of 100,000, whereas a normal band pass filter becomesdifficult to use with Q s greater than 50 [1].

II. EXPERIMENT 1.1 COMPONENTS

THE SETUP

For this section we utilized the followingequipment:

y O scilloscope by Tektronixy Lock-in Amplifiery Voltmeter

THE EXPERIMENT

This part of the lab is divided into six parts in whichwe will investigate the different components that thelock-in amplifier has.

Part I: Gain Accuracy In this part we investigate how gain affects voltage

output. We will do this by changing the gain andcomparing it theoretical.

Part II: Pre-Amplifier The preamp module can be used as a non-

inverting, inverting, or differential amplifier with A C-or DC- coupled high-impedance inputs and a gainadjustable from 1 to 1000. Using a 1 kHz input signal,verify the operation of the preamplifier. A s with mostamplifiers above a certain frequency the output of thepreamplifier drops, and the frequency at which thisoccurs depends on the selected gain. A t each gainsetting, measure the frequency at which the outputdrops off by 3 dB relative to the input at 1 kHz.

Part III: Low-Pass Filter In this section we experimented with the effects of the quality factor, Q. To see the effects of Q, choosethe l ow filter , set the cutoff frequency atapproximately 100 Hz, and measure the transferfunction (both magnitude and phase) for bothButterworth and Q = 50. This can be done byconnecting the output of the function generator toboth Channel 1 on the oscilloscope and to the input


2/17

BNC on the filter module. Connect the output of thefilter you chose to Channel 2 on the oscilloscope. Thisconfiguration will allow you to measure easily boththe input and output amplitudes and the relativephase.

Part IV: Phase Shifter

The phase shifter will shift the phase of asinusoidal signal through 360. The coarse knobchanges the phase in 90 increments and the fineknob allows more accurate adjustment of the phase.Examine the operation of the phase shifter, andcarefully record your observations.

Part V: Lock-in\Amplitude Detector We will be using the lock-in detector with an AC-

coupled input signal; adjust the two switches to thesesettings. V erify the operation of this unit byconnecting the output of the function generator toboth the input of the lock-in detector and the input of

the phase shifter. Connect the output of the phaseshifter to the input reference of the lock-in detector.Now examine the output of the lock-in detector onthe oscilloscope. Examine the effect of changing therelative phase of the two signals and adjusting theamplifier gain.

Part VI: Low-Pass Filter-Amplifier Maintaining the setup in the last section, now

connect the output of the lock-in detector to both theoscilloscope and to the input of the low-pass filter.Connect the output of the low-pass filter to a DCvoltmeter to provide a digital readout in addition tothe provided analog scale. Select the 12 dB/octavefilter, turn off the DC offset, and use a 0.3 s timeconstant. Verify the expected DC voltages for each of the four waveforms from the previous section, andthe operation of the DC gain. Measure the DC voltageas a function of the phase shift, and compare totheory.

III. EXPERIMENT 1.2 REALEXPERIMENT

THE SETUP

For this section we utilized the followingequipment:

y O scilloscope by Tektronixy Lock-in Amplifiery Function Generatory Voltmetery Wire

y Breadboard with grounding knoby Grounding devicey 1 resistory 10 resistory 100 resistory 1 resistor

THE EXPERIMENT

This section is split into four parts.Part I: Measurement One

Set the function generator to provide anapproximate 10 Hz, 1 V pp sine wave, and wire theconnections to measure VA . Adjust the phase shifterto provide the maximum positive DC output voltage.Using the relationship between the DC output voltageand the actual peak-to-peak input voltage you foundearlier, adjust the input voltage and/or attenuators so

that the actual input is 1V

pp . With these settings,record the DC output voltage and gain settings used.Now measure VB using the same settings on thephase shifter. As V B is much smaller than VA , you willneed to increase the gain on the amplifier. Rememberto monitor the output of the lock-in detector for signsof clipping. Now record the DC output voltage and thegain settings used. Use these measurements tocalculate the resistance of the wire. Repeat thisprocedure at frequencies of 30 Hz, 100 Hz, 330 Hz, 1kHz, 3 kHz, 10 kHz, and 30 kHz.

Part II: Measurement TwoUsing a fixed frequency, 330 Hz, follow the same

procedure as before; measure the resistance of thewire at input peak-to-peak voltages V A of 1 V , 300 m V ,100 m V , 30 m V , 10 m V , 6 m V , and 2 m V .

Part III: Measurement ThreeNow use a fixed frequency of 300 Hz and follow

the same procedure as before; measure theresistance of the wire at input peak-to-peak voltagesV A of 1 V , 300 m V , 100 m V , 30 m V , 10 m V , 6 m V , and2 m V .

Part IV: Measurement Four Insert a 1 k isolation resistor between the

connection at V B and the input to the preamplifier.

Using a 6 m V input peak-to-peak voltage at VA at afixed frequency of 330 Hz and a 3 s time constant onthe low-pass filter, measure the resistance of the wireand confirm that your measurement agrees roughlywith your previous measurements. Decrease the timeconstant to 0.3 s and note that the output voltage isfluctuating. Using a 0.3 s time constant, record 10measurements of the voltage, each separated by afew seconds (which is many time constants),


3/17

introducing as little bias into the measurements aspossible. Calculate the resistance of the wire for eachof these 10 measurements, and calculate thestandard deviation of these 10 measurements. Repeat

these measurements using isolation resistances of 10k , 100 k , and 1M .

IV. RESULTS AND DISCUSSION

EXPERIMENT 1.1

RESULTS

Part I: Gain Accuracy

Vin (Volt) V out(Volt)

GainExpected

Gain Measured(Vout /V in)

Error (%) :=(Expected-Measured)/Expected

0.103 0.108 1 1.05 50.103 0.208 2 2.02 10.103 0.502 5 5.05 10.103 1.02 10 9.90 10.103 2.12 20 20.6 30.103 5.20 50 50.5 10.103 10.2 100 99.0 10.103 20.6 200 200 0

0.0428 19.8 500 463 7.40.0212 18.2 1000 858 14.2

Part II : Pre-Amplifier

The table is of the gain setting and the -3dB crossover frequency.

Gain -3dB Frequency (kHz)1 13902 9705 660

10 56320 34050 150

100 40200 9.4

500 3.61000 2.0


4/17

Part III: Low-Pass Filter

Below is a table for the Quality factor Q=0.707 and a cutoff frequency of 100Hz.

Frequency (Hz) V in (Volts) V out (Volts) t (ms) (Degrees)

20 1.06 1.06 23.2 360 of t=167 o 30 1.06 1.06 14.8 16050 1.06 1.02 7.4 13375 1.06 0.880 4.0 10880 1.06 0.860 3.4 97.9 90 1.06 0.780 2.8 90.7

100 1.06 0.648 2.2 79.2110 1.06 0.576 2.0 79.2125 1.06 0.488 1.4 63.0150 1.06 0.360 1.1 59.4200 1.06 0.224 .52 37.4

The table below is for the Quality factor Q=50 and a cutoff frequency of 100Hz.

Frequency (Hz) V in (Volts) V out (Volts) t (ms) (Degrees)20 1.06 1.16 25 180

30 1.06 1.34 16 173

50 1.06 2.88 10 180

55 1.06 4.32 9.4 186

5 10 5 0 100 500 1000Gain

5

10

50

100

5 00

1000

Frequency kHzPreamp Gain Setting versus the 3 dB Frequency


5/17

60 1.06 10.6 8.8 190

62 1.06 24.0 4.2 93.7

63 1.06 23.6 2.8 63.5

65 1.06 15.0 0.80 18.7

70 1.06 4.60 0.60 15.1

75 1.06 2.80 0.30 8.1

80 1.06 1.86 0 0

90 1.06 1.10 0.40 13.0100 1.06 .760 0.40 14.4110 1.06 .580 0.40 15.8125 1.06 .400 0.40 18.0150 1.06 .280 0.16 8.64200 1.06 .180 0.16 11.5

Part IV : Phase Shifter

Below are values for the read phase shift , V out , t, and the measured phase shift ex for the input sine

signal of 1 V at 1kHz.

(Degrees) V out (mV) t ( s) ex (Degrees)0 888 20 7.2

15 696 50 1830 560 80 28.845 504 150 5460 560 210 75.6

10050 20030 15070Frequency Hz

0.51.0

2.0

5.0

10.0

20.0

Output Voltage VoltsOutput Voltage Volts versus Frequency Hz

Q .707

Q 50


6/17


7/17


8/17

Part V : Lock-in\Amplitude Detector

Below is a table of phase shift, , of the input reference of the lock-in detector, and V out for a 1kHz44m V pp signal with a gain of 2.

(Degrees) V out (mV)0 50

45 7890 90

135 74180 54225 78270 92315 76

Below is a table of read Gain, and V out , and the measured Gain, for a given phase of theinput reference of the lock-in detector of 90 o and a 44m V pp sine signal.

Gain V out (mV) Measured Gain2 90 2.055 208 4.73

10 416 9.4520 900 20.550 2120 48.2

100 4240 96.4


9/17

200 8600 195500 20600 468

Part VI : Low-Pass Filter-Amplifier

Below is a table of phase shift, , DC output andV

pp , for a 1.04V

pp signal.

(Degrees) DC Volts V pp (Volts)0 0 2.08

45 -0.55 1.1290 -0.62 2.08

135 -0.29 1.04180 0 1.56225 0.56 1.96270 0.62 1.52315 0.28 1.04

Below are values for V in, DC output and the ratio of the two for the Low-Pass Filter- Amplifier.

Vin (AC Volts) V out (DC Volts) Ratio (V in/V out )200 0.12 0.60500 0.31 0.62700 0.45 0.643

1000 0.62 0.621300 0.81 0.6231500 0.94 0.627

D ISCUSSION AN D REMARKS

Part I: Gain Accuracy The gain, for the most part, was close to what the theoretical value should be. This is very crucial since for the

rest of the laboratory we will be using i t.Part II: Pre-Amplifier

Here is the code I used to produce the plot in Mathematica:

.

We see that as the gain increases the crossover frequency decreases. We also can see that there is a pointwhere it rapidly decreases, around a gain of 50, and then it stabilizes a little as the gain becomes much larger.

Part III: Low-Pass Filter Here is the code I used to produce the plot in Mathematica:


10/17

lowppassqpt707

It seems that the higher quality factor, Q, had a greater range in filtering out signals. For the Q equal to 50, ithad the highest V out at 62Hz whereas for the Q=.707 the highest recorded was 1.06 like the input voltage.

Part IV: Phase Shifter The phase shifter modified the period of the sinusoidal wav, as seen in the snap shot. Notice also that the

absolute amplitude remained the same through each of the shifts. The absolute phase shift is changes with thechange in frequency.

Part V: Lock-in\Amplitude Detector As we modified the phase shift the output voltage cycled, from the data is seems that the output voltage

increased from the phase shift of zero to ninety and decreased from ninety to 180 and then increased when

shifting to 270. Modifying the gain also yielded the results we had hoped, expected; it multiplied our input signalby the gain factor.

Part VI: Low-Pass Filter-Amplifier The maximum measured DC output was .62 DC Volts this was accompanied with an input VPP of 1.04 V olts.

EXPERIMENT 1.2

RESULTS

Part I: Measurement One

The dimensions of the wire are 0.49mm in diameter and 19.9cm in length.

The table is of the frequency of the input the value of V A the output DC voltage, the gain, and thecalculated value of V B, Rwire and the resistivity.

F requency(Hz)

VA(Volts)

DCVoltage

Gain V B (Volts)DC/(.62*Gain)

R wire ( ) Resistivity ( *m)=A*R/L

10 1 0.81 10000 130 V 0.130645161 1.23738E-07 30 1 1.49 2 5 000 96.1 V 0.096129032 9.10465E-08

100 1 1. 5 2 2 5 000 98.1 V 0.098064516 9.28797E-08 330 1 1.82 2 5 000 117 V 0.117419355 1.11211E-07

1000 1 1.69 2 5 000 109 V 0.109032258 1.03268E-07 3000 1 1.03 2 5 000 66. 5 V 0.066451613 6.29382E-08

10000 1 0.42 2 5 000 27.1 V 0.027096774 2.56641E-08 30000 1 0.66 5 0000 21.3 V 0.021290323 2.01647E-08


11/17

Element Resistivity at 20 C( .m)

Aluminum 2.82 x 10-8

Constantan 4.9 x 10-7Copper 1.7 x 10-8

Germanium 4.6 x 10-1

Gold 2.44 x 10-8Iron 1.0 x 10-7

Glass 1010 to 1014Lead 2.2 x 10-7

Manganin 4.82 x 10-7

Mercury 9.8 x 10-7

Silicon 6.40 x 102Platinum 1.1 x 10-7

Silver 1.59 x 10-8

Tungsten 5.6 x 10-8

Part II: Measurement Two

The table is of the values for an input signal of 330Hz.

V A (mVolts) DC Voltage Gain V B ( Volts) R wire ( )1 000 0.723 1 0000 11 7 0. 11 7300 0. 198 1 0000 3 1 .9 0.1 061 00 0.0 9 2 1 0000 14 .8 0.148

10 5 0 100 5 00 1000 5 0001 104Frequency kHz

0.02

0.04

0.06

0.08

0.10

0.12

RW ire

R esistance of the W ire as a Function of Frequency


12/17

30 0. 155 5 0000 5 .00 0. 1 671 0 0.0 55 5 0000 1 .77 0. 1 776 0.0 11 25 000 0.7 1 0 0. 118 2 0.02 5 1 00000 0. 4 03 0.202

Part III: Measurement Three

The table is of values for an input signal of 300Hz. Note for values at and below 10m V input the errorwas above 100%.

VA (mVolt) DC Voltage Gain V B ( Volt) R wire ( )1000 0.81 10000 131 0.131300 0.28 10000 45.2 0.151100 0.23 25000 14.8 0.14830 0.065 25000 4.19 0.14010 0.025 50000 0.806 0.0806

0.00 5 0.010 0.0 50 0.100 0. 500 1.000V A Volts

0.0 5

0.10

0.1 5

0.20

R WireR esistance of the W ire as a Function of V A


13/17

Part IV: Measurement

With a 1k isolation resistor and a 6m V pp 330Hz signal with a low pass filter time constant of = 3, wemeasured the output DC voltage of 0.011 V at a gain of 50000. These values give us a resistance of thewire to be 0.1183 .

Below are ten, eleven for Riso=1M , values of the DC output (measured in m V ) for a 6m V pp 330Hz signalwith a low pass filter time constant of = 0.3.

Riso=1kGain=50K

10.4m V

12.0m V

13.6m V

7.0m V

8.9m V

15.8m V

8.6m V

11.9m V

11.9m V

11.1m V

N/ A

Riso=10kGain=100K

31.7m V

5.6m V

-0.7m V

10.7m V

16.7m V

2.7m V

-10.8m V

3.4m V

7.6m V

2.7m V

N/ A

Riso=100kGain=100K

-10.7m V

9.9m V

-1.9m V

1.5m V

35.8m V

11.4m V

-16.6m V

-12.2m V

12.8m V

-10.8m V

N/ A

Riso=1MGain=100K

19.0m V

12.3m V

8.5m V

-4.5m V

-22.8m V

-17.5m V

1.9m V

-10.5m V

6.8m V

6.1m V

4.5m V

0.02 .0 5 0.10 0.20 0. 5 0 1.00V A Volts

0.02

0.04

0.06

0.08

0.10

0.12

0.14

RWire

R esistance of the W ire as a Function of V A


14/17

D ISCUSSION AN D REMARKS

Part I: Measurement OneThe code used to produce the plot in Mathematica:

.62

d

.42

.81

.66

.

Notice also that the formula has a Gain, this is due because of a miscount; we did this to amend the error wehad done. A lso in Mathematica the table did look like a table, I could put it in input form here but I choose not tosince it is aesthetically unpleasant. The material seems to be very close to A luminum or Iron, my guess it isaluminum. The table is from [2]. From the plot, the resistance of the wire seems to be affected by range

frequencies, resonance?, and after the frequency gets really high the resistivity is low.

Part II: Measurement TwoHere is the code I used to produce the plot in Mathematica:

.We see that initially as we increase V A, the resistance of the wire increases and then steadily start to decrease

until it picks up around .500 V olts.

10 8 10 5 0.01 10 10 4 RI solated

10 5

0.001

0.1

10

1000

Standard DeviationStandard Deviation of the Measurements of the R esistance of the W ire as a Function of Isolation R esistance


15/17

Part III: Measurement Three

Here is the code I used to produce the plot in Mathematica:

;

It seem that the change in frequency has increases DC voltage, for the most part, and also affected R Wire , for

the most part it is larger than before. We still get the rapid increase with a decreasing rate of change.

Part IV: Measurement Four For the standard deviation, Mathematica has a built-in function. There was one problem with that function,

when trying to plot it along with resistance, initially it gave the standard deviation as a string that is why in thecode I did not use data to plot the log-log plot instead I put the values manually. The dense curve is the squareroot of x. We had a problem with displaying the proper number of point and curve; the problem was the range, itdid not cover sufficient lower point for the theoretical value, square root of x.

The code used in Mathematica:

.62 .5

.006

.0104.012

.0136.007

.0089

.0086

.0119

.0158

.0111

.0119

.0056.0007

.0107

.0167

.0027.0108

.0034

.0076

.0027

.0107.0099.0019

.0015

.0358

.0114

.0166

.0122.0128.0108

.019850.0123.0045.0228.0175

.0019

.0105.0068.0061.0045


16/17

IX. CONCLUSION

In this lab became aquatinted with most of thecomponents of the lock-in amplifier. Lock-inamplifiers use a frequency mixer to convert thesignal's phase and amplitude to a DC voltage signal.The device is can used to measure phase shift, evenwhen the signals are large and of high signal-to-noiseratio, and do not need further improvement.It can also recover signals at low signal-to-noise ratios.As far as the experiment goes, not much could havebeen done to improve our results, with one exceptionin measurement four of experiment 1.2.

EXPERIMENT 1.1

There is not much to say here. The data wasconsistent with what we had expected. The goal of

this section was to be aquatinted with the materials,this was a success. Most of the error was systematic;the rounding gave us mistake in this section of thelaboratory. If we had more precise equipment, wecould have had lower our error. A lternatively wecould have had simulated the experiment withMultisim 11.0 by National Instruments. We couldhave simulated the lock-in amplifier component bycomponent through its schematic. The program has abuilt in noise receiver and generator. With this wecould have had verified the data, and the simulationdata is very accurate.

EXPERIMENT 1.2

The results for the first three measurements werefair. The sample size of the data was very small; toimprove the results and data it would be best if wetook more data points, which due to time constraintswe didn t. Measurement four results were not reallygreat, in fact they were horrible. We were off the

theoretical value. There were many factors that leadto this huge error, first the wire was very susceptibleto the surrounding; if touched, accidentally or not, themeasured DC output would change dramatically. Wealso had to ground it, but this in itself proved to be ahassle. Since resistance of the wire was so small,coming in contact with any other material made theDC voltage change even if the source was grounded.To go around this problem the most logical stepwould have had been to solder the wire. Anotherproblem was the data collection; as we increased theisolated resistance, just for the last two, the DCoutput was varying greatly. The data collection wastaken by covering the voltmeter and counting fiveseconds and looking and whatever number we hadwe wrote down. To improve this experiment, weshould have had soldered the wire to a groundedsource, for less movement.

ACKNOWLEDGMENTS

The author would like to thank all the people whomade this lab possible, Dr. Smith for guiding us inthrough part of the lab, Danny O rton and TimothyPlett for being extraordinary and reliable lab partners.The calculations were made using Mathematica 8.0.1by Wolfram.

RE F ERENCES

1. P. Horowitz and W. Hill, The Art of Electronics 2 nd edition, New York: Cambridge Press, 1989

2. Giancoli, Douglas C., Physics, 4th Ed, Prentice Hall,(1995).

3. P. Horowitz and W. Hill, Students Manual for The Art of Electronics, New York: Cambridge Press,1989


17/17

lab 2 write up

Documents