Lock-in amplier 2.3 Phase shifter . . . . . . . . . . . .

8
2.4 Lock-in detector . . . . . . . . . . 8
by Paul Cervenak, Eva-Johanna Hengeler & 2.5 Lock-In Detector with adesigned
David Spataro test signal . . . . . . . . . . . . . . 8
January 14, 2014
3 Analysis 9
The experiment lock-in amplier was performed 3.1 Preamplier . . . . . . . . . . . . . 9
on 2013-12-18 at the advanced practical lab at the 3.1.1 General behaviour . . . . . 9
University of Konstanz. The following pages con- 3.1.2 3 dB frequencies with re-
tain the documentation and results of the experi- spect to the gain factor . . 9
ment. Tutor: Qijun Liang 3.2 Filters . . . . . . . . . . . . . . . . 10
3.2.1 Band-pass lter . . . . . . . 10
3.2.2 Low-pass and high-pass lter 10
Contents 3.2.3 Transmission behaviour of
1 Theoretical basics 1 rectangular signals . . . . . 11
1.1 Basics about electronic systems . . 1 3.2.4 Roll-o for low-pass lter . 12
1.1.1 Complex transfer function 3.3 Phase shifter . . . . . . . . . . . . 12
G(jω) . . . . . . . . . . . . 2 3.4 Lock-in detector . . . . . . . . . . 13
1.1.2 Bode plot of gain/loss over 3.5 Lock-in-Detector with designed test
frequency . . . . . . . . . . 2 signal . . . . . . . . . . . . . . . . 14
1.2 Electronic components . . . . . . . 2 3.5.1 The test signal . . . . . . . 14
1.2.1 Low-pass lter . . . . . . . 2 3.5.2 Pre-amplier and dierent
1.2.2 High-pass lter . . . . . . . 3 lters . . . . . . . . . . . . 14
1.2.3 Band-pass lter and band- 3.5.3 Amplitude detection . . . . 15
stop lter . . . . . . . . . . 3 3.5.4 Lock-In detection . . . . . . 16
1.3 Typical lter congurations . . . . 3 3.5.5 response time . . . . . . . . 16
1.3.1 Maximally at amplitude
within the passband: But- 4 Summary and Conclusions 17
terworth lter . . . . . . 3
1.3.2 Better rectangular fre- References 17
quency response near
cut-o: Chebyshev lter . 4
1.3.3 Maximally at delay within
the passband: Bessel lter 4
1.4 Noise Sources . . . . . . . . . . . . 5
1.4.1 thermal noise . . . . . . . . 5 1 Theoretical basics
1.4.2 shot noise . . . . . . . . . . 5
1.4.3 Flicker noise . . . . . . . . . 5
1.5 Lock-in amplier or PSD (Phase 1.1 Basics about electronic systems
sensitive Detector) . . . . . . . . . 6
1.5.1 How to set up a lock-in am- In the following, we want to talk about periodic
plier . . . . . . . . . . . . 6 signals in electronic systems, especially systems
1.5.2 Mathematical basics . . . . 6 containing lters. It will be important to char-
acterize the input-output behaviour of our system
2 Experimental setups 7 with respect to the amplitude and the phase and
2.1 Preamplier . . . . . . . . . . . . . 7 discuss about the combination of various compo-
2.2 Filters . . . . . . . . . . . . . . . . 7 nents.

1
1.1.1 Complex transfer function G(jω) scaled in dB over the (normalized) frequency in
logarithmic scaling. The second plot is the phase
First of all we want to introduce the complex trans-
plot. It is the plot of the phase shift function θ(ω)
fer function G(jω) dened as linear scaled over the (normalized) frequency in
G(jω) = |G(jω)| · e|jθ(ω) (1) logarithmic scaling. This special visualization al-
| {z } {z }
phase modulation
lows us to see the critical points, like the cut-o
amplitude modulation
frequency, directly. It is also very easy to char-
where θ(ω) is the phase shift function with respect acterize the device by linearizing the response in
to ω . The output signal of the system is the input good approximation. In addition, getting a result-
signal multiplied with the transfer function, which ing plot for any components in series is possible
is modulating amplitude and phase of the signal by adding the plots graphically. For connection in
depending on its frequency. parallel, it is not that easy.

About periodic signals, phase and phase shift

A periodic signal is a signal with the condition
α(t) = α(t+τ ) where τ is a constant. The smallest
τ of a function is called its period. Often used
periodic signals are sine, rectangle and sawtooth
functions. After passing an electronic system it
might occur that like dened before, the output
signal is shifted along the time axis. To dene
some terms about properties of the system let the
test input signal be:
Vin = A1 sin(ωt) (2)
The output signal is: Figure 1: Bode plot of magnication and phase of
a rst order low-pass Butterworth l-
Vout = A1 |G(jω)| sin(ωt + θ(ω)) (3) ter in the typical bode scaling [WMC1]
We dene τP D := − θ(ω)ω and call it phase delay.
We dene τGR := − dω and call it group delay. 1.2 Electronic components
dθ(ω)

If θ(ω) = kω , k constant, we get a linear phase

lter τP D = τGR = k without a phase distortion. 1.2.1 Low-pass lter
Every other phase shift function will have a phase
distortion. A low-pass lter is an electronical AC device to
minimize the transmission for signals above a given
frequency ωc . The low-pass lter of the rst order
1.1.2 Bode plot of gain/loss over frequency
is given by a RC circuit tapping o the output
The Bode plot named after the Amercan en- signal parallel to the capacity. As you intuitively
gineer Hendrik Wade Bode introduced at Bell gets, the capacity blocks for small frqunencies and
Labs in the United States in the 1930s is a tech- the whole potential will be at the resistance. For
nique visualize the transmission of a periodic sig- increasing frequency the potential will devide be-
nal through an electronic system by drawing the tween the capacity and the resistance. So the po-
complex transfer function in a special way. The tential at the resistance become less for high frue-
Bode plot is divided into two dierent plots. The qencies. Later we want to characterize some typ-
rst plot is the magnitude plot. It is the plot of ically used lters for application. An illustration
the absolute value of the complex transfer |G(jω)| on the frequency response is given in Fig. 4.

2
Bandwidth and quality factor Q For character- overlap at the stopband in a parallel circuit the re-
izing the passband of the lter we have to dene a sulting lter will block for frequencies within the
quantity called the bandwidth B . The easiest way common stopband, so we call it band-stop. Fil-
we might say that this bandwidth is dened as the ters with a very small stopband looking like a thin
frequency interval between two zero transmission notch are also called notch lter. Notch lters are
frequency. However, as a skilled experimenter you used to lter signals given by the mains freuquncy
know that these frequencies does not exist in a of 50Hz . Exemplary an illustration of a composed
real experiment. That is why it is better to de- band-pass lter is given at ??
ne the bandwidth as full width at half maximum
(FWHM) measured in the baseband (neglecting
negative transmission frequencies), or in decibel
scale it is the width at 10 · log 12 ≈ −3dB , so it
is also known as 3-dB-halfwidth. Notice that the
3-dB-level is the same than the cut-o frequency.
Hence the bandwidth of a low-pass lter is ωc . For
a band-pass lter, it is the dierence between the
upper and lower cut-o frequency. Figure 2: Cascaded low-pass and high-pass lter
We want to dene a second quantity called the composing a band-pass lter [Will06,
quality factor Q to characterize the damping of a p.17]
system capable of oscillating. The quality factor
is dened as
ωr
Roll-o of multi-stage lters For quantyng the
Q= (4) sharpness of a lters for the frequency response be-
B
tween passband and stopband we dene the roll-o
where ωr is the resonant frequency. L (for loss). It is dened as the steepness of the
absolute value of the transfer function in logarith-
1.2.2 High-pass lter mic scale as we know it from the bode plot. That
A high-pass lter is an electronical AC device to is why it is easy to determine the roll-o for fre-
minimize the transmission for signals above a given quencies well above ωc (for a low-pass) looking at
frequency ωc . The high-pass lter of the rst or- the bode plot. As a consequence of the additivity
der is given by swithing the devices used for the of bode plots of cascaded lters the roll-os of each
low-pass lter. So the electronical properties of lter are also additive.
the devices and the capacitor blocks for small fre-
quencies. For increasing frequencies the capacitor 1.3 Typical lter congurations
gets charged and uncharged and the signal trans-
mission improves. The bode plot is similar to the As we already have seen, perfect lters does not
low-pass lter with the condition the plot is mir- exist. Hence, we want to study a few lter cong-
rored in the cut-o frequency. urations designed for certain applications.

1.2.3 Band-pass lter and band-stop lter 1.3.1 Maximally at amplitude within the
passband: Butterworth lter
By connection a low-pass and a high-pass lter in
series we will get a multiplication of the transfer The Butterworth lter named after the British
function what is the same as a summation of both physicist Stephen Butterworth is a lter de-
bode plots. So it is easy to see that the result- signed for getting a nearly constant transmission
ing lter has a passband at the common passband behaviour at the passband. The application of this
of both lters. That is why we call it band-pass. lter is good for small frequencies at the expense
For connecting a low-pass and a high-pass with an of the frequencies near the cut-o frequency ωc .

3
The transfer function is given by: by introducing a ripple in the transmission prole.
The transfer function is given by:
1
|Gn (jω)| =   2n 1/2 (5) 1
1 + ωωc |Gn (jω)| =   2n 1/2 (6)
ω
1 + 2 Cn2 ωc
For frequencies well above ωc the roll-o of an
decade is given by ∆L = −20ndB . where Cn is the Chebyshev polynomial of the
rst kind of degree n, and  is a constant inter-
preted as the passband ribble.
The ripple value RdB , often expressed in dB, is
given with the condition:
RdB = 10 · log(2 + 1) (7)
There also exist an Chebyshev type II which one
is not interesting for out experiment, so we will not
spend any explenation for it. A typical frequency
response for a low-pass Chebyshev lter is given
at Fig. 4.
A short summary of the properties of the Cheby-
shev lter:

• a wavy frequency response within the pass-

band
Figure 3: Transfer functions for Butterworth
lters of increasing orders becoming • a steep slope at the cut-o frequency
steeper at the cut-o [Horo94, p.269]
• the bigger the ripple the steeper the slope of
the transfer function at the cut-o frequency
A short summary of the properties of the But-
increasing with higher order
terworth lter:
• for a decreasing ripple the lter approaches a
• monotonous amplitude response over all fre-
Butterworth lter
quencies
• there is no constant group delay within the
• fast bending behaviour at the cut-o fre-
passband
quency, getting better for increasing order
• group delay depending on frequency 1.3.3 Maximally at delay within the
passband: Bessel lter
• signicant overshooting at the step response
for increasing order The Bessel lter named after the German math-
ematician Friedrich Bessel is a lter designed for
• the phase response has a small nonlinearity a good rectangle transmission behaviour, it is poor
for applications depending on a drop behaviour at
1.3.2 Better rectangular frequency response the cut-o frequency. Generally, these lters are
near cut-o: Chebyshev lter used for analysis transient properties are present.
A low-pass Bessel lter is characterized by:
The Chebyshev lter named after the Russian
mathematician Pafnuti Lwowitsch Chebyshev is θn (0)
a lter designed for getting a sharp bend at cut-o |Gn (jω)| = (8)
θn (jω/ω0 )

4
where θn is the reverse Bessel polynomial. 1.4.1 thermal noise
Thermal noise (even called Johnsonnoise) is most
frequently noticed by using a resistor. That kind
of electronic noise is caused by the Brownian mo-
tion. The noise voltage is proportional to the ther-
mal Energy E = kB T and the bandwidth ∆f of
the measurement. This is described by the rela-
tion:
(9)
p
Vnoise = 4kB T ∆f R

Where:
kB = 's constant
T = absolute temperature
R = resistance
∆f = eective bandwidth

Figure 4: Dierent 1st order low-pass lter in com- The noise voltage as a function of frequency is
parision [Horo94, p.270] nearly constant. That at spectrum is also called
white noise. To minimize this expression we can
either reduce the temperature (not usually practi-
The properties of the Bessel lter are: cal) or limit the bandwidth of the measurement.
• an optimized rectangle transmission be-
1.4.2 shot noise
haviour within the passband
The current ow consists of moving charge carri-
• which is equal to a nearly constant group de- ers. However current has a intrinsic noise in its
lay within the passband nature. The uctuation due to the stochastically
process of generation/recombination of holes and
• a nearly linear phase response within the pass- electrons. That phenomenon is called shot noise
band and is irreducible. But at large currents is the
percentage uctuation quite smaller than at small
• poor bending behaviour at the cut-o fre- currents. We use the expression:
quency
(10)
p
Inoise = 2eI∆f

1.4 Noise Sources Where:

e = elementary charge
At the laboratory we are confronted with two dif- I = rms current
ferent types of noise: intrinsic and external noise. ∆f = eective bandwidth
The external noise is due to the environment such
as galactic noise, muons or ground loops. The in-
1.4.3 Flicker noise
uence of these parameters could be reduced by
careful attention to grounding and shielding. Flicker noise is inversely proportional to the fre-
The intrinsic noise, is caused by physical processes, quency. Therefore it is often called f1 noise. It
so we cannot get rid of it. But if we understand occurs in almost all electronic devices but the ori-
about it, we are able to minimize it. Because of gin of that kind of noise is still under study.
that I will list the most important sorts of intrinsic The total random noise results of the individual
noise: contributions.

5
1.5 Lock-in amplier or PSD (Phase 1.5.2 Mathematical basics
sensitive Detector) These lock-in measurements require a frequency
Lock-in ampliers are used to measure and de- reference. The reference signal is usually a phase
tect small AC signals. Small means in this case shiftable square wave signal R(t) with period ωr ,
a few nano volts. You need it e.g for very accurate shifted on the time axis by a phase shift Φ. We can
measuremeants of signals being disturbed by noise assume our incoming signal has the shape of a sine
sources which may be much larger than the signal wave, because Fouriers theorem is teaching, that
you want to measure. The mechanics of lock-in every signal can be written as a sum of dierent
ampliers are called PSD or phase-sensitive detec- sine und cosine functions and no other component
tion. It is used to single out the component of the matters for operating on the input signal because
signal at a specic phase and frequency. Simply the scalar product is zero.
spoken, a lock-in-amplier is mixing a measured For our calculation the following denitions apply:
signal with a reference signal calculates the dier- ∞
sin((2n − 1)ωr t + Φ)
ential quotient. In an ideal case, this quotient will
X
R(t) = Vr ·
be zero. In [SRS1] there is a beautiful example 2n − 1
n=1
why it can be useful to utilize a PSD. S(t) = Vs · sin(ωr t)

1.5.1 How to set up a lock-in amplier In this case the square wave signal R(t) can be
approximated by R(t) ≈ Vr · sin(ωr t)) to simplify
In the picture below from [NI1] we can see a simple the problem. The result of the multiplication of
experimental design of a lock-in amplier: both signals is:

R(t) · S(t) = Vr sin(ωr t + Φ) · AS sin(ωr t)

Vr Vs
= (cos(ω− t − Φ) − cos(ω− t + Φ))
2

where ω± = ωr ± ωs .

Figure 5: Experimental design of a lock-in ampli- The PSD presents us two AC signals with the fre-
er quency of ω± . The reference signal is chosen in
such a way that ω− ≈ 0 and ω+  ω− . That
is the reason why ω+ is ltered by the low-pass
With a preamplier the signal (with the noise while ω+ still passes. To reduce the usual statis-
source) is amplied rst; the detectors will no- tically distributed noise, the low-pass oers only
tice the signals from the input and the reference a small bandwidth that can pass. The low-pass
oscillator. In the refernce signal there is also a supplies the following DC signal:
phase shift before the signals are multiplied and
Vs Vr
then tey are integrated by a low-pass lter. In- V (t) = cos(Φ)
tegrated means the time-related averaging in this 2
case, what implies a constant time constant peri- We know the reference amplitude and could de-
odic signal and gets very problematic for increas- termine the phase shift. The output DC signal
ing integration times. Also, sometimes an attenu- is proportional to the incoming signal amplitude.
ator is used to weaken a signal instead to amplify Figure 6 shows the signals before and after multi-
it at the signal input. plying and integrating.

6
 were told to be curious what would happen if we
increase the voltage of the output signal.

2.2 Filters

Figure 6: Illustration of multitplication and inte-

gration of the signals
Figure 8: Setup with the lters [Teach09, p.35]

2 Experimental setups Maybe the most important components are the l-
First we had to analyze all of the components the ters we had to examine. So we tried to do it care-
lock-in amplier is made of. So we had for each fully to get good data.
component measurements with dierent setups to We started with the band-pass lter Fig. 8 and
test them. searched for the resonance frequency. Therefore,
we measured the phase shift of the band-pass over
2.1 Preamplier the input frequency and the gain over the fre-
quency. We did this for dierent types of the lter.
The same procedure goes for the low-pass lter.

As told in [Teach09] we had to choose a square

wave as the input signal. Our task then was to ex-
amine the inuence of the dierent types or Q fac-
tors of the lters.These measurements were done
with the low-pass lter.

Figure 7: Setup of the preamplier [Teach09, p.34] Last we had to test the dierence between the
6dB/oct and 12dB/oct roll-o. We again mea-
sured the gain and phase shift over the input fre-
In Fig. 7 you can see how we built the setup ac-
quency. The experimental setup is shown in Fig.
cording to [Teach09].
9:
With this setup we can measure both: the phase
shift and the signal gain of the preamplier. We
had to plot the results of our measurement in a
bode diagram. There the phase shift and gain is
ploted against the input frequency. We also had
to look for the 3dB roll-o at a high frequency.
This 3dB frequency is when the maximum voltage
is half of the input voltage and therefore half of
the amplitude of the input signal. These measure-
ments are done for dierent gain levels. We also Figure 9: Setup with the lters [Teach09, p.37]

7
2.3 Phase shifter

To examine how the phase shift depends on the

frequency we built the setup shown in Fig. 10 and
measured the phase shift for dierent frequencies.

Figure 12: Lock-In detector with a designed test

signal [Teach09, p.38]

Instead of 50Hz we used a 60Hz input signal be-

cause the frequency of the electric current is 50
Hz in Germany. We created this test signal with
Figure 10: Phase shifter [Teach09, p.38] a signal-to-noice ratio of 1/1 dened by the rms
voltage. The we changed the parameters as high-
Pass or low-Pass lter and band-pass lter with
dierent lter types and compared the results.

2.4 Lock-in detector Switching back to band-pass lter we measure the

signal-to-noise ratio for dierent parameters. We
compared measurements with signal and noise to
To realize a lock-in detector we made the setup
only the noise. Also we had a switch to attenuate
like shown in Fig. ?? and measured the dierence
the signal by 100. The we had to test dierent
of the output signal for various phase shifts. The
quality factors also. Therefore one should have
output signals are measured with phase shifts of
used the amplitude detection. This measurement
0◦ , 45◦ , 90◦ and 180◦ .
had to be repeated for the lock-in detection mode.
Then we had to search for the smallest S/N ratio
by identifying an oset after the signal is switched
o.
Last we measured the response times for the
12dB/oct and 6db/oct roll-o and varying the
time constants at the output of the lter.

Figure 11: Lock-in detector [Teach09, p.37]

2.5 Lock-In Detector with adesigned test

signal

Last part of our experimental setups is the lock-in

detector with a designed test signal:

8
3 Analysis

During our measurements we got data which we

want to discuss below. First of all we have to un-
derstand the used components. We will start with
the preamplier.

3.1 Preamplier

3.1.1 General behaviour

Figure 14: Bodeplot of phase over frequency of
We visualized our results in a Bodeplot like you the used preamplier
see in Fig. 13 and 14. For frequencies below 10
kHz the gain behaviour like as you can see in Fig.
14 is very at. So the intensity of the signal is 3.1.2 3 dB frequencies with respect to the
nearly the same than the input. At a freuquency gain factor
of 550 kHz the maximum gain is 1.81 so we have The next quantity we measured were the 3dB fre-
the resonance frequnecy at that point. For increas- quencies of the preamplier. For a preamplier
ing freqencies the amplier is going to block the it is not very important to know this qunatity of
signal, so it becomes totally useless. For frequnen- half intensity, because that is not the eect why
cies below 200 kHz as you can see the phase rela- we used an amplier, but for getting more prac-
tion between the input and output signal is nearly tice we characterized this frequency array above
constant. So for frequencies below 10 kHz we get the resonance frequency.
all the properties we expect from a preamplier.
So we know that the measurements below 10 kHz
will not be inuence stongly by the preamplier
and will be free from measurement errors caused
by the amplier.

Figure 15: Characteristic 3 dB gain response of

dierent gains over frequency

We measured the curve shown in Fig. 15. In com-

parison to the reference meseasement given by the
Figure 13: Bodeplot of gain over frequency of the TeachSpin manual we got the main tendency very
used preamplier well. For a gain of 1 the 3 dB frequency is well

9
above the former measured resonance frequency, known 308 Hz. As the Q factor is dened for in-
so the roll-o is very poor. For increasing gains creasing Q, the bandwidth besomes smaller. For
the frequencies lowers and the range of good in- the CHEB, BESS and BUTT lter, the Q factors
tensity transmission becomes smaller. It might be are smaller than 1.
better to reduce the frequency range for measure-
ment to 5 kHz.

For higher input voltages the oscilloscope shows

a signal with clippings because the amplication
has a maximum output voltage, so the measured
signal of the lock-in amplier will be distorbed for
these signals. So for the experiment it is important
to keep that in mind and to use moderate high
voltages.

3.2 Filters
In the next step we want to learn more about the
used lters in our experiment.

3.2.1 Band-pass lter

Figure 17: Gain response of various band-pass l-
In Fig. 16 it is given the phase response of a band- ter types
pass lter. As you can see the phase shift moves
from the negative to the positive by becoming zero
in the resonance frequency which is around 308 Hz.
The phase response is not linear, so there is a phase
distorsion by the lter.

3.2.2 Low-pass and high-pass lter

In Fig. 18 and 19 you can see that the used lters

are dierent in their frequency dependencies. Es-
pecially the Chebyshev lter shows a signicant
overshoot at the cut-o which is caused by the rip-
ple. As we described in the theoretical basics, this
Figure 16: Phase response of a band-pass lter ripple improves the bending behaviour of the curve
and the roll-o becomes steeper. The Bessel l-
Visualizing the gain response for this lter, we will ter shows a poor roll-o because it is optimized for
get the bahoviour shown in Fig. 17. There are a good rectangular transmission. The low-pass l-
various types of bandpass lters. All of them peak ter response is the mirrored result of the high-pass
in the same resonance frequency at the already lter.

10
 Figure 20: Reactangular signal before and after
Figure 18: Gain response of BUTT, BESS and passing a Butterworth low-pass
CHEB low-pass lter

Figure 21: Reactangular signal before and after

Figure 19: Gain response of BUTT, BESS and passing a Bessel low-pass
CHEB high-pass lter

3.2.3 Transmission behaviour of rectangular

signals

Figure 22: Reactangular signal before and after

passing a Chebychev low-pass

11
 As you can see in the gures above the But-
terworth lter shows a moderate result for the
transmission behaviour. The Bessel lter does
a better job because it is optimize for this task
by a maximally at delay within the passband.
For increasing Q factors the gain response is not
constant in the cutt-o array, so the rectangular
signal described as a Fourier series is modulated
unsteady, so the rectangular signal becomes more
and more a single frequency signal by amplifying
the single frequency.

Figure 23: Reactangular signal before and after 3.2.4 Roll-o for low-pass lter
passing a low-pass with Q = 2 As a special feature the roll-o of the low-pass
is abjustable with two values: 6 dB/oct and 12
dB/oct. The gain response given in Fig. 26 shows
the signifacantly dierent curves of the low-pass
with both roll-o setting. As we set before the
steepness of the roll-o array for the 12 dB curve
is apparent twice as steep as the 6 dB curve.

Figure 24: Reactangular signal before and after

passing a low-pass with Q = 5

Figure 26: Adjustable roll-o with 6dB/oct and

12dB/oct for a low-pass lter

The measured data correlate very good with the

[Teach09] reference.

3.3 Phase shifter

We took our datas and ploted them with a semilog
frequency axes to show how the phase shift corre-
sponds to the frequency. We also took the pic-
Figure 25: Reactangular signal before and after ture for the phase shift over frequency plot from
passing a low-pass with Q = 10 [Teach09] to campare our results with.

12
 Figure 29: 0◦
Figure 27: Phase shift over frequency plot

Figure 30: 45◦

Figure 28: Phase shift over frequency plot refer-
ence from [Teach09]

We saw, that the phase shift is dependant on the

frequqncy, we have to keep this in mind for the
Lock-In Amplier later. We also notice that this
dependancy is of a logarithmical form. We can
assume that the equation
φ = k · log(f ) + s (11)
will give us the results for the phase shift φ with
certain parameters k and s.

3.4 Lock-in detector

We did the measurements for 0◦ , 45◦ , 90◦ and 180◦
and got these results: Figure 31: 90◦

13
 Figure 32: 180◦
Figure 33: [SN]
We notice that the rst and the forth plot are
nearly inverted. We also notice that the signal
is inverted after half of a period and then is in- After that, we put the test signal into the pre-
verted for the same amount of time. So we only amplier. Therefore we grounded the minus input.
have a shifted absolute value or negative absolute While observing the digital output of the oscillo-
value of a sin funtion. At other phase shifts the scope, we use dierent gains and switched between
inversion of the function is at other values of the ac and dc coupling. The dc coupled signal shows
sine function. At 90◦ phase shift the sine/cosine clipping starting from the gain of 100 while the ac
function is inverted at its maximum/minimum to coupled one does not show clipping or saturation
a minimum/maximum. At other phase shifts like for any gain.
in the gure with 45◦ phase shift you cannot easily
notice what happens there, but a small hint gives 3.5.2 Pre-amplier and dierent lters
the inversion at a point where the sine/cosine fun- In this section we take a look at the output signal
tion is between minimum and zero. So you can of the band-, low- and high-pass lter. At rst we
guess that the phase shift is of the value the pic- looked at the high-pass lter and could not spot
ture shows. This behaviour is also explained in the a sine wave for every Q-factor. That is not re-
theoretical part. ally surprising, because the most of the generating
noise is a high frequently one. So we cannot get
3.5 Lock-in-Detector with designed test the signal without the noise with help of the high-
signal pass.
The low-pass in contrast zero the higher frequently
3.5.1 The test signal
noise (above 60 Hz) if we use the right modulation.
Now we generate a test signal. Therefore we use a With a low quality factor we could generate a quite
signal frequency of 60 Hz instead of 50 Hz to avoid similar signal to the original. As well we had a sat-
interference with the supply voltage. isfying level of signicance for the bandpass, too,
First of all, we adjust the oscillator amplitude until while we use the frequency range that contains 60
the rms voltage equals the noise voltage. So we Hz. But we got in addition a oscillation of the am-
get a signal to noise ratio S/N = 1/1. The original plitude, which implies that there is a strong static
signal is still visible. But if we reduce the signal noise signal near 60Hz. The reason could be the
by a factor of ten, we can't spot the signal out of 50Hz power supply, but then we should have the
the noise any more. In the following picture you oscillation with the low-pass, too.
can see the primer sine-wave and a combined with Unfortunately we have not saved the pictures, so
a signal to noise ratio 1/1. we cannot get sure, if we oversight a amplitude

14
modulation or the system has a second, constant Back to the old time constant (t=0.1), we vary
noise just above 60 Hz. the Q-factor. First picture shows a lower, 37 a
Q-factor of 50.
3.5.3 Amplitude detection
To examine the noisy signal, we use the ampli-
tude detection in this section. Therefore we cre-
ate a 1/30 test signal, use the bandpass lter and
observe the output signal of the low-pass ampli-
er. On a digital oscilloscope we can see the sig-
nal sweeping across the screen. When the signal
passed the half way, we have to switch o the sig-
nal (by dividing the signal amplitude by 100). The
following discusses the dierent setting possibili-
ties of Q-factor, time constant and gain.

Figure 36: t=0.1, Q=5, Gain=50

With the lower Q-factor the signal in- and de-

creases faster then the other.

Figure 34: t=0.1, Q=20, Gain=50

Now we vary only the time constant. You can

see, that the signal needs a quite longer time to
decrease, but you can hardly nd oscillations in
the signal.

Figure 37: t=0.1, Q=50, Gain=50

Here will be noted, that the decrease oscillate in

37, like it did in section 3.2.2. But until now the
amplitude does not change. At the next gure
we changed the gain value. Now the output
amplitude increases (you have to take a look at
the scale, which changed from 500mV to 1V).
Figure 35: t=1, Q=20, Gain=50

15
 Figure 38: t=0.1, Q=20, Gain=200 Figure 40: t=0.1, Q=20, Gain=50, S/N = 1/70

3.5.4 Lock-In detection

To make lock-in detection we have to adjust the 3.5.5 response time
phase correctly. Now we turn back on the noise
and set the low-pass amplier output on the fol- In our last part of analysis, we focus on the
lowing setting: time constant = 0.1, roll o = response time. This is the time, the signal need
6db/oct. Now we make half way sweeping like we to reach the end voltage by half way sweeping.
did by using the amplitude detection. Using the digital oscilloscope we can determine
this time, you see it in the following picture.

Figure 39: t=0.1, Q=20, Gain=50, S/N = 1/30

We tried dierent settings for lock-in detection, Figure 41: measurement of the response time
too. The result equals the study of amplitude
detection. With a higher gain increases the end
voltage, a larger response time only raises the time To get a functional correlation we measure the
to reach the end voltage and a higher Q-factor response time for three dierent time constants,
brought more vibrations. t ∈ {1, 3, 10}. Besides we repeat the measure-
But contrary to the amplitude detection, we can ment for a roll o of 12dB/oct instead of 6dB/oct.
detect a smaller ratio than 1/30. The minimum The result is diagrammed in the following gure .
ratio, where we could recognize a discontinuity There we can identify a linear dependency of the
was by a ratio of 1/70. You see it in the picture time constant and the response time. By repeating
below. the experiment with a dierent roll o we obtain
a similar plot where only the slope is dierent. 37

16
 [Will06] Williams, Athur B.; Taylor, Fred
J.: Electronic lter design handbook.
McGraw-Hill, 4th edition, 2006
[SRS1] Stanford Research Systems: About
Lock-In Ampliers. PDF-File
from 2013-12-09 from https:
//fp.physik.uni-konstanz.de/docs/
LOKIN/AboutLIAs.pdf
[NI1] JPG-File from 2013-12-09 from
http://www.ni.com/cms/images/
devzone/tut/2006-12-08_112403.jpg
[WMC1] Wikimedia Commons: Butterworth
lter bode plot. PNG-File from 2013-
12-10 from http://upload.wikimedia.
4 Summary and Conclusions org/wikipedia/commons/thumb/6/66/
Butterworth_filter_bode_plot.png/
• the gain of the preamp is nearly constant till 640px-Butterworth_filter_bode_
200 kHz, phase phase relation is constant till plot.png
10 kHz
[Teach09] TeachSpin Inc: SPLIA1-A Sig-
• for high gain factors the 3 dB frequency is nal Processor / Lock-In Amplier.
around 10 kHz, so the frequency range for PDF-File from 2014-01-10 from
measurement should be limited at 5 kHz https://fp.physik.uni-konstanz.
de/docs/LOKIN/Manual_LockIn2.pdf
• the used lter fulll the expected theoretical
dened properties [SN] PNG-File from 2014-01-13 from
http://www.businessesgrow.com/
• the phase shift of the phase shifter is propor- wp-content/uploads/2010/03/
tional to log(f) SignalNoise.png

• the lock-in detectors works like theoretically

discussed
• the signal to noise ratio is okay up to
S/N=1/70

References
[Horo94] Horowitz, Paul; Hill Wineld: The
Art of Electronics. Cambridge University
Press, 2nd edition, 1994
[Alex13] Alexander, Charles K.; Sadiku
Matthew N. o.: Fundamentals of Elec-
tric Circuits, McGraw-Hill, 5th edition,
2013

17