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Metrol. Meas. Syst., Vol. XXII (2015), No. 4, pp. 503–512.
Metrol. Meas. Syst., Vol. XXII (2015), No. 4, pp. xxx–xxx.
METROLOGY AND MEASUREMENT SYSTEMS
Index 330930, ISSN 0860-8229
www.metrology.pg.gda.pl
NOISE MEASUREMENTS OF RESISTORS WITH THE USE OF DUAL-PHASE
VIRTUAL LOCK-IN TECHNIQUE
Adam Witold Stadler1), Andrzej Kolek1), Zbigniew Zawiślak1), Andrzej Dziedzic2)
1) Rzeszów University of Technology, Department of Electronics Fundamentals, Powstańców Warszawy 12, 35-959 Rzeszów, Poland
( [email protected], +48 17 865 1279, [email protected], [email protected])
2) Wrocław University of Technology, Faculty of Microsystem Electronics and Photonics, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław,
Poland ([email protected])
Abstract
Measurement of low-frequency noise properties of modern electronic components is a very demanding challenge
due to the low magnitude of a noise signal and the limit of a dissipated power. In such a case, an ac technique with
a lock-in amplifier or the use of a low-noise transformer as the first stage in the signal path are common approaches.
A software dual-phase virtual lock-in (VLI) technique has been developed and tested in low-frequency noise
studies of electronic components. VLI means that phase-sensitive detection is processed by a software layer rather
than by an expensive hardware lock-in amplifier. The VLI method has been tested in exploration of noise in
polymer thick-film resistors. Analysis of the obtained noise spectra of voltage fluctuations confirmed that the 1/f
noise caused by resistance fluctuations is the dominant one. The calculated value of the parameter describing the
noise intensity of a resistive material, C = 1·10−21 m3, is consistent with that obtained with the use of a dc method.
On the other hand, it has been observed that the spectra of (excitation independent) resistance noise contain a 1/f
component whose intensity depends on the excitation frequency. The phenomenon has been explained by means
of noise suppression by impedances of the measurement circuit, giving an excellent agreement with the
experimental data.
Keywords: 1/f noise, polymer thick-film resistor, low-frequency noise measurements, virtual lock-in.
© 2015 Polish Academy of Sciences. All rights reserved
1. Introduction
Modern electronic components and materials are subject of continuous improvements by
means of optimizing their internal structure or applying new materials, which is stimulated by
the RoHS directive [1] restricting the use of some elements, e.g. lead and cadmium, as well as
modifying or developing new ways of manufacturing, in order to obtain cheap and reliable
small-size components. Examples of new materials are Pb-free solder alloys [2] for the
interconnection technology in electronic printed circuit boards and modules, and Pb/Cd-free
systems of compatible conductive and resistive pastes for the thick-film technology [3, 4]. In
specialized electronics, the reliability and quality of the components belong to the key
parameters. It is just the low-frequency noise that is commonly considered as the indicator of
the quality and reliability of a device since there is a relationship between the reliability/quality
and noise that has been shown experimentally [5−9]. It is not so surprising, taking into account
that noise, measured as the power spectral density of fluctuations of examined quantity on
terminals of a tested device, relates strongly to inhomogeneity of its internal structure. Hence,
it is very important to carefully select materials of high quality for production of low-noise and
reliable electronic components.
In this work, we focus on experimental studies of the low-frequency noise in modern, smallsize polymer thick-film resistors. The main difficulty concerned with noise measurements is
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Article history: received on May 11, 2015; accepted on Oct. 15, 2015; available online
on Nov. 05, 2015; DOI: 10.1515/mms-2015-0051.
Article history: received on May 11, 2015; accepted on Oct. 15, 2015; available
online
on Dec.
07, 2015;7:31
DOI: AM
10.1515/mms-2015-0051.
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Date
| 5/11/16
A.W.
Stadler,
A. Kolek,
Kolek, et
et al.:
al.: NOISE
NOISE MEASUREMENTS
MEASUREMENTS OF
OF RESISTORS
RESISTORS WITH
WITH THE
THE USE
USE …
…
A.W. Stadler,
Stadler, A.
A.W.
A.
Kolek,
et
al.:
NOISE
MEASUREMENTS
OF
RESISTORS
WITH
THE
USE
…
aa small
small area
area of
of aa resistive
resistive layer
layer which
which limits
limits the
the amount
amount of
of dissipated
dissipated power.
power. In
In order
order to
to improve
improve
the sensitivity
sensitivity of
of aa measurement
measurement setup,
setup, aa transformer
transformer is
is often
often used
used as
as the
the first
first stage
stage of
of the
the signal
signal
the
path [10].
[10]. However,
However, there
there are
are several
several disadvantages
disadvantages of
of the
the method:
method: (i)
(i) the
the need
need for
for rejection
rejection of
of
path
dc offset,
offset, which
which implies
implies measurements
measurements in
in aa balanced
balanced dc
dc bridge,
bridge, (ii)
(ii) the
the −
− depending
depending on
on
aa dc
the
the transformer,
transformer, but
but generally
generally −
− rather
rather narrow
narrow frequency
frequency band
band of
of observation,
observation, (iii)
(iii) the
the transfer
transfer
function of
of the
the signal
signal path
path strongly
strongly depends
depends on
on the
the source
source (i.e.
(i.e. sample)
function
sample) resistance,
resistance, which
which makes
makes
the
the calibration
calibration of
of each
each sample
sample necessary.
necessary. Therefore,
Therefore, in
in this
this work
work we
we consider
consider an
an ac
ac method
method of
of
noise
noise measurements,
measurements, which
which is
is free
free from
from the
the above
above disadvantages,
disadvantages, and
and additionally
additionally makes
makes it
it
possible
possible to
to obtain
obtain spectra
spectra at
at an
an extremely
extremely low
low frequency,
frequency, which
which is
is not
not available
available in
in aa dc
dc method
method
due
due to
to an
an ac
ac coupling
coupling in
in the
the signal
signal path
path and
and contribution
contribution of
of the
the amplifier
amplifier noise.
noise. In
In the
the ac
ac
technique
the
low-frequency
spectrum
of
resistance
fluctuations,
originated
in
the
tested
device,
technique the low-frequency spectrum of resistance fluctuations, originated in the tested device,
excited
excited by
by an
an ac
ac signal,
signal, is
is shifted
shifted to
to the
the side-bands
side-bands around
around the
the carrier
carrier (exciting)
(exciting) frequency
frequency as
as
aa result
of
modulation.
Furthermore,
the
contribution
of
the
amplifier
noise
into
result of modulation. Furthermore, the contribution of the amplifier noise into the
the system
system
noise
noise may
may be
be optimized
optimized by
by careful
careful selection
selection of
of the
the value
value of
of the
the carrier
carrier frequency.
frequency. Next,
Next,
the
the demodulation
demodulation is
is usually
usually carried
carried out
out in
in aa lock-in
lock-in amplifier
amplifier by
by the
the phase-sensitive
phase-sensitive detection.
detection.
The
The idea
idea of
of the
the method
method has
has been
been described
described and
and analysed
analysed in
in [11].
[11]. The
The method
method occurred
occurred to
to be
be
helpful
helpful in
in noise
noise exploration
exploration at
at low
low temperature
temperature [12],
[12], down
down to
to 0.3
0.3 K.
K. Moreover,
Moreover, using
using the
the ac
ac
method,
method, either
either the
the background
background and
and total
total noise
noise [11]
[11] or
or the
the excess
excess noise
noise [13]
[13] is
is directly
directly accessible
accessible
at
at the
the output,
output, depending
depending on
on phase
phase angles
angles used
used for
for the
the phase-sensitive
phase-sensitive detection.
detection. Although,
Although, in
in
[12]
a
standard,
i.e.
hardware,
lock-in
amplifier
was
employed
for
the
demodulation,
[12] a standard, i.e. hardware, lock-in amplifier was employed for the demodulation, in
in this
this
work
work we
we show
show aa software
software solution
solution of
of the
the in-phase
in-phase (‘X’)
(‘X’) and
and out-of-phase
out-of-phase (‘Y’,
(‘Y’, the
the quadrature
quadrature
component)
detection,
i.e.
a
virtual
lock-in
(VLI).
Despite
a
lock-in
technique
is
widely
component) detection, i.e. a virtual lock-in (VLI). Despite a lock-in technique is widely used
used in
in
many
many research
research methods,
methods, there
there are
are only
only aa few
few published
published papers
papers that
that describe
describe usefulness
usefulness of
of
the
the virtual
virtual lock-in
lock-in [14−16]
[14−16] and
and none
none of
of them
them considers
considers the
the use
use of
of VLI
VLI in
in noise
noise exploration
exploration or
or
multichannel
multichannel detection
detection at
at different
different sets
sets of
of phase
phase angles.
angles.
2.
2. Virtual
Virtual lock-in
lock-in technique
technique
The
The measurement
measurement setup
setup developed
developed for
for measurements
measurements of
of noise
noise in
in resistive
resistive electronic
electronic
components
is
shown
in
Fig.
1.
It
consists
of
a
typical
ac
Wheatstone
bridge
components is shown in Fig. 1. It consists of a typical ac Wheatstone bridge driven
driven by
by aa remote
remote
controlled
controlled sinusoidal
sinusoidal generator
generator of
of high
high precision
precision and
and stability.
stability. The
The bottom
bottom arms
arms of
of the
the bridge
bridge
with matched
matched resistance,
resistance, while
while the
the upper
upper arms
arms contain
contain
are created
created by
by aa pair
pair of
of the
the samples
samples R
RSS with
are
the noiseless
noiseless wire-wound
wire-wound ballast
ballast resistors
resistors R
RBB (R
(RBB 〉〉
〉〉 R
RSS)) and
and additional
additional components
components (omitted
(omitted in
in
the
the circuit
circuit of
of Fig.
Fig. 1
1 for
for clarity)
clarity) inserted
inserted for
for balancing
balancing the
the bridge.
bridge.
the
Fig.
Fig. 1.
1. The
The setup
setup for
for measurements
measurements of
of noise
noise in
in resistors
resistors with
with the
the use
use of
of aa virtual
virtual lock-in
lock-in technique.
technique.
The
, and its sub-diagonals, V(x−x),, where
where
The voltage
voltage signals
signals from
from the
the bridge
bridge diagonal,
diagonal, V
V(7−7)
(7−7), and its sub-diagonals, V(x−x)
= 2,
2, 3,
3, …6
…6 is
is the
the side
side terminal,
terminal, are
are connected
connected to
to low-noise
low-noise differential
differential preamplifiers
preamplifiers (5186
(5186
xx =
from Signal
Signal Recovery)
followed by
by low-pass
low-pass filters.
filters. Additional
Additional signal
signal paths
paths consisting
consisting of
of
from
Recovery) followed
amplifiers
amplifiers followed
followed by
by low-pass
low-pass (anti-aliasing)
(anti-aliasing) filters
filters are
are needed
needed only
only for
for correlation
correlation
measurements
measurements using
using different
different pairs
pairs of
of voltages
voltages taken
taken from
from the
the bridge
bridge diagonal
diagonal and
and its
its subsub504
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Vol. XXII
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XXII (2015),
(2015), No.
No. 4,
4, pp.
pp. 503–512.
xxx–xxx.
Metrol. Meas. Syst., Vol. XXII (2015), No. 4, pp. xxx–xxx.
diagonals. Next, the signals are coupled to a multichannel data acquisition (daq) device with
diagonals. Next, the signals are coupled to a multichannel data acquisition (daq) device with
simultaneous sampling.
simultaneous sampling.
The bridge was supplied with a sinusoidal signal of frequency around 1 kHz. An additional
The bridge was supplied with a sinusoidal signal of frequency around 1 kHz. An additional
digital lock-in amplifier (model 7265 from Signal Recovery), whose X (in-phase), and Y
digital lock-in amplifier (model 7265 from Signal Recovery), whose X (in-phase), and Y
(quadrature) outputs were also attached to the daq device, was used for verification of the VLI
(quadrature) outputs were also attached to the daq device, was used for verification of the VLI
results. In order to precise control the sampling frequency, an external source was used. Both
results. In order to precise control the sampling frequency, an external source was used. Both
ac signals, one exciting the bridge and the other triggering the daq device, were produced by a
ac signals, one exciting the bridge and the other triggering the daq device, were produced by a
2-channel signal generator with extended stability (model 33512B from Keysight Technology).
2-channel signal generator with extended stability (model 33512B from Keysight Technology).
The software layer running on a PC takes care of (i) continuous acquisition of consecutive
The software layer running on a PC takes care of (i) continuous acquisition of consecutive
time records from the daq module, (ii) real-time digital signal processing (calculation of
time records from the daq module, (ii) real-time digital signal processing (calculation of
waveforms, spectra and/or cross-spectra using the FFT algorithm, phase-sensitive detection),
waveforms, spectra and/or cross-spectra using the FFT algorithm, phase-sensitive detection),
(iii) presentation of the results in a graphical form on the PC screen during the experiment, (iv)
(iii) presentation of the results in a graphical form on the PC screen during the experiment, (iv)
recording data in files for further analysis, and (v) controlling the auxiliary equipment (digital
recording data in files for further analysis, and (v) controlling the auxiliary equipment (digital
voltmeters, a temperature controller/monitor, and a signal generator) via the GPIB interface
voltmeters, a temperature controller/monitor, and a signal generator) via the GPIB interface
[17]. In a dc method, the power spectral density SV of voltage fluctuations for both biased and
[17]. In a dc method, the power spectral density SV of voltage fluctuations for both biased and
unbiased (V = 0) samples has to be obtained in separate experiments in order to get the excess
unbiased (V = 0) samples has to be obtained in separate experiments in order to get the excess
noise:
noise:
SVex = SV – SV=0,
(1)
SVex = SV – SV=0,
(1)
which is the subject of interest. The background noise, SV=0, consists of (i) the thermal noise of
which is the subject of interest. The background noise, SV=0, consists of (i) the thermal noise of
the bridge and (ii) the amplifier noise. However, in the ac technique it is possible to obtain both
the bridge and (ii) the amplifier noise. However, in the ac technique it is possible to obtain both
of signals detected at the
SV and SV = 0 simultaneously, calculating the spectra SV00 and SV90
SV and SV = 0 simultaneously, calculating the spectra SV and SV90 of signals detected at the
phase angles 0° and 90°, respectively [11], e.g. using a dual-phase lock-in amplifier.
phase angles 0° and 90°, respectively [11],45e.g. using a dual-phase lock-in amplifier.
Furthermore, calculating the cross-spectrum, SVV
45 , of signals detected at the phase angles +45°
Furthermore, calculating the cross-spectrum, SVV
, of signals detected at the phase angles +45°
and −45° referenced to the signal driving the bridge, one may obtain directly [13]:
and −45° referenced to the signal driving the bridge, one may obtain directly [13]:
45
SVex = 2Re( SVV
(2)
45 ).
SVex = 2Re( SVV
).
(2)
Both methods have been implemented and tested in the developed VLI.
Both methods have been implemented and tested in the developed VLI.
It is worth to note, that in our configuration of the bridge (Fig. 1), the voltage from the bridge
It is worth to note, that in our configuration of the bridge (Fig. 1), the voltage from the bridge
diagonal includes fluctuations originated in both samples. Hence, SV observed on the bridge
diagonal includes fluctuations originated in both samples. Hence, SV observed on the bridge
diagonal is the doubled SV of one sample, assuming that (i) both samples have been carefully
diagonal is the doubled SV of one sample, assuming that (i) both samples have been carefully
selected with matched resistance and have the same statistical properties, (ii) the bridge is
selected with matched resistance and have the same statistical properties, (ii) the bridge is
balanced and therefore the ac current I (rms value) flowing through the samples is the same.
balanced and therefore the ac current I (rms value) flowing through the samples is the same.
Thus, SV(7−7) = 2SV(7−1), where SV(7−1) is the power spectral density of voltage fluctuations in one
Thus, SV(7−7) = 2SV(7−1), where SV(7−1) is the power spectral density of voltage fluctuations in one
sample.
sample.
3. Experiment
3. Experiment
As the subject of studies a pair of polymer thick-film resistors (PTFRs), with matched
As the subject of studies a pair of polymer thick-film resistors (PTFRs), with matched
resistance, prepared on the same substrate, has been taken. Resistive layers of PTFRs have been
resistance, prepared on the same substrate, has been taken. Resistive layers of PTFRs have been
made of ED7100_200Ω − carbon-based composition from Electra Polymers Ltd on FR-4
made of ED7100_200Ω − carbon-based composition from Electra Polymers Ltd on FR-4
laminate. ED7100 is a commercial ink – ElectraΩd’or™, based on epoxy resin (200 Ω/□, fillers:
laminate. ED7100 is a commercial ink – ElectraΩd’or™, based on epoxy resin (200 Ω/□, fillers:
carbon black and graphite powders). FR4 Laminate Sheet is a glass fabric reinforcement in an
carbon black and graphite powders). FR4 Laminate Sheet is a glass fabric reinforcement in an
epoxy resin binder. It is the most versatile laminate with an extremely high mechanical strength,
epoxy resin binder. It is the most versatile laminate with an extremely high mechanical strength,
good dielectric loss properties, and good electric strength properties under both dry and humid
good dielectric loss properties, and good electric strength properties under both dry and humid
conditions − and therefore very often used on an industrial scale for the production of printed
conditions − and therefore very often used on an industrial scale for the production of printed
circuit boards. Polymer thick resistive films have been screen-printed on bare Cu contacts. The
circuit boards. Polymer thick resistive films have been screen-printed on bare Cu contacts. The
samples have been prepared as multi-terminal PTFRs in which the rectangular resistive layer
samples have been prepared as multi-terminal PTFRs in which the rectangular resistive layer
(length L = 15 mm, width w = 0.5 mm and thickness d = 20 µm) terminated on the opposite
(length L = 15 mm, width w = 0.5 mm and thickness d = 20 µm) terminated on the opposite
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505
A.W. Stadler,
Stadler, A.
A.W.
A. Kolek,
Kolek, et
et al.:
al.: NOISE
NOISE MEASUREMENTS
MEASUREMENTS OF
OF RESISTORS
RESISTORS WITH
WITH THE
THE USE
USE …
…
sides with the current terminals (see Fig. 1). Additional, equally spaced along the layer, voltage
probes have been formed for detailed examinations. Such a shape of samples has been already
used in the previous experiments [18]. The resistance of the selected samples was RS ≈ 11.1 kΩ.
Noise measurements have been carried out at the room temperature, T = 300 K. The carrier
frequency, fcarrier, of a sinusoidal waveform (exiting the bridge) ranged from 325 Hz to 50025
Hz, while the bandwidth of the preamplifiers was 1 MHz. The sampling frequency, fs, varied
from 32 kHz to 512 kHz, the record duration, T, ranged from 4 s to 128 s and the number of
samples per record, Ns = Tfs (Ns ranged from 2 M to 8 M samples), were selected so as to hold
fs/fcarrier > 10. However, there are several important technical limitations, that had to be taken
into account, like the maximal sampling frequency accepted by the DAQ device (fsmax = 2 MHz)
and a long time of FFT calculations for a large number Ns. The latter limits the minimal record
duration, especially in the case of large Ns.
a)
b)
10-12
0
VLI: SV
Power spectral density, V2/Hz
VLI: S
10-13
VLI: 2Re(S45
)
VV
0
7265: SV
7265: S90
V
-14
10
SV5186
10-15
sample voltage
4.8 mV
9.7 mV
16.6 mV
38.6 mV
0.12 V
0.33 V
0.66 V
~1/f
10-10
90
V
Excess noise, SVex, V2/Hz
~1/f
10-11
10-12
10-13
10-14
10-15
10-16
4kT(2RS) = 3.5⋅10-16 V2/Hz
100m
1
thermal noise
10-16
10
Frequency f, Hz
100
100m
1
10
100
Frequency f, Hz
Fig. 2. a) Comparison of the noise spectra obtained in different ways at the sample voltage 16.6 mV.
The dotted (pure 1/f noise) and dashed (thermal noise) lines have been added for reference.
45 ), for different sample voltages.
b) The selected excess noise spectra from VLI, i.e. 2Re( SVV
The dash-dot-dot line is the plot of the amplifier noise.
The average spectra (Nav = 20) of voltage fluctuations, taken from the bridge diagonal,
detected by the VLI and the traditional lock-in amplifier have been compared in the plot of Fig.
2a (solid lines). In fact, the respective spectra obtained with the use of the hardware lock-in
amplifier (labelled “7265” in the plot’s legend) and those obtained with the VLI are almost
identical, making lines overlap each other. However, for the frequency above 25 Hz the
rejection by the 7265 internal filter is apparent. It is worth to note, that all spectra shown in Fig.
2a were calculated from the same time records. The obvious advantage of the VLI is that the
detection for different sets of phase angles is carried out on the same time records, which is
impossible in a single standard lock-in amplifier. Additionally, the power spectral density of
thermal noise 4kT(2RS) = 3.5·10−16 V2/Hz, where k is the Boltzmann constant and T − the
ambient temperature, calculated for the whole bridge resistance (2RS), has been plotted for
reference (dashed line). Another, the dash-dot-dot line, labelled SV5186, which is the plot of the
amplifier noise (model 5186 from Signal recovery) referred to the input, has been added to give
an imagination about the noise properties of the signal path.
506
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Metrol. Meas.
Meas. Syst.,
Syst., Vol.
Vol. XXII
Metrol.
XXII (2015),
(2015), No.
No. 4,
4, pp.
pp. 503–512.
xxx–xxx.
4. Noise of polymer thick-film resistors
Looking at the collection of spectra shown in Fig. 2b it is apparent that the 1/f noise is the
dominant noise component. Hence, the product of frequency and excess noise averaged over
the frequency band ∆f, < f SVex > ∆f, is a convenient measure of noise intensity. Its square
dependence on the sample voltage, shown in Fig. 3a (open symbols), is the evidence that the
observed noise is caused by resistance fluctuations. The solid line is the approximation of the
noise intensity for the frequency band 1–10 Hz with the squared voltage. The data from
measurements with a dc method [17] (solid symbols) have been added for comparison. Both ac
and dc methods give consistent results.
The quadratic dependence of noise intensity on the sample voltage is the basis for evaluation
of the material noise intensity, C − the parameter describing noise properties of the material
itself − which is independent of the frequency, sample volume Ω, and voltage V:
(3)
C = <f SVex>∆f V−2Ω.
It is worth to note, that in our experiment SVex is obtained for the voltage signal taken from
the bridge diagonal, which gathers fluctuations from both samples, and therefore the voltage V
and the volume Ω in the general definition (3) should be substituted by the voltage across both
samples connected in serial, V = 2V(7−1), and the doubled volume of the single sample,
respectively. The value C = 1·10−21 m3 obtained for the studied samples has been found to be
in a good agreement with 8.2·10−22 m3 obtained with a dc method [17], confirming that both
methods give consistent results (see Fig. 3a).
The parameter C describes the noise properties of a resistive material itself, and therefore
may be used for comparison of noise properties of bulk and layered materials. Another
advantage of the parameter C [m3] is that it enables calculation of measurable quantities, like
SV or SR= SV/I2, provided that the geometrical sizes of the samples and the bias (voltage or
current) are known. However, C strongly depends on the free carrier concentration in the
sample. In view of the experiments on photoconductive and volume scaling, it turns out that the
factor 1/N is crucial [19, 20], where N is the total number of carriers. Therefore, another
parameter, defined as the ratio of 1/f noise normalized for bias, frequency and unit volume to
resistivity (ρ), K = C/ρ is used, even if N is unknown. It was originally introduced in [21] for
layered materials as K = Cus/Rsh, where Cus= C/d is the noise intensity normalized per unit area
and Rsh = ρ/d is the sheet resistance. The parameter K is widely used as a figure of merit for
both homogeneous and inhomogeneous materials, even if they are made in different
technologies. The obtained C-value and ρ = 7.4·10−3 Ωm for PTFRs studied in this work, lead
to K = 1.4·10−7 µm2/Ω, which is a typical value for percolation in thick-film resistors and
conductive polymers [18, 20, 22]. It is a well-known fact that the inhomogeneous electric fields
and current crowding on a microscopic scale result in high K-values and apparent αH (Hooge
parameter) [22, 23]. It is in line with the conclusions given in [24], where exhaustive studies of
electrical properties of polymer resistors with carbon black and/or graphite fillers, have been
carried out and conduction as well as noise generation mechanisms have been explained within
the framework of percolation theory. Furthermore, analysing the plot shown in Fig. 3a we may
conclude that using the ac method the noise exploration is possible at a significantly smaller
excitation magnitude comparing with that of a dc method. It is of great importance in lowtemperature studies, in which preventing the sample from self-heating is the key issue [12].
Another interesting conclusion concerning noise sources in PTFRs is that the noise
generation mechanism is linear, i.e. the dependence of noise intensities on both ac and dc
excitation is the same, making points overlap each other in the plot of Fig. 3a. It is contrary to
the nonlinear mechanisms that make the noise intensity for an ac excitation much smaller (even
2–3 decades) than that for a dc excitation [25, 26].
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507
A.W. Stadler,
Stadler, A.
A.W.
A. Kolek,
Kolek, et
et al.:
al.: NOISE
NOISE MEASUREMENTS
MEASUREMENTS OF
OF RESISTORS
RESISTORS WITH
WITH THE
THE USE
USE …
…
A.W. Stadler, A. Kolek, et al.: NOISE MEASUREMENTS OF RESISTORS WITH THE USE …
b)
b)
10-11
10-11
10-12
10-12
10-11
10-11
frequency band ∆f
frequency band ∆f
0.1 -1 Hz
0.1 -1 Hz
1 -10 Hz
1 -10 Hz
10 -100 Hz
10 -100 Hz
1 - 10 Hz (DC)
1 - 10 Hz (DC)
10-13
10-13
10-14
10-14
10
10
10-13
10-13
V(4-4)
90
S
V(4-4)
S90
V(4-4)
90
S
V(4-4)V(7-7)
S90
~1/f
~1/f
V(4-4)V(7-7)
SSVex(1-7)
Vex(1-7)
10-14
10-14
SV5186
SV5186
-15
-15
10
10
V2
1.4⋅10-11
1.4⋅10-11V2
-15
-15
SSVex(1-4)
Vex(1-4)
SS00V(4-4)
10-12
10-12
2
SSV(4-4)V(7-7)
, ,VV2/Hz
/Hz
V(4-4)V(7-7)
22
Noise
Noiseintensity,
intensity,<f⋅S
<f⋅S
>>, V
,V
Vex
Vex ∆f∆f
a)
a)
10-16
10-16
10-16
10-16
0.01
0.01
0.1
0.1
1
1
-17
-17
10
10
10m
4kT(2R4-1) = 1.8⋅10-16
V2/Hz
4kT(2R4-1) = 1.8⋅10-16 V2/Hz
100m
1
10
100
10m
100m
1
10
100
Sample voltage V(7-1), V
Frequency
Sample voltage V(7-1), V
Frequency f,
f, Hz
Hz
Fig. 3. a) The noise intensity as a function of the sample voltage in different frequency bands (open symbols)
Fig. 3. a) The noise intensity as a function of the sample voltage in different frequency bands (open symbols)
and the approximation for the frequency band 1–10 Hz with voltage square (line). The solid symbols, added
and the approximation for the frequency band 1–10 Hz with voltage square (line). The solid symbols, added
for comparison, denote the data obtained in a dc method. b) The spectra obtained in different ways
for comparison, denote the data obtained in a dc method. b) The spectra obtained in different ways
for the sectors 1−4 of the resistive layer at the sample voltage V(7−1) = 48 mV, i.e. V(4−1) = 24 mV.
for the sectors 1−4 of the resistive layer at the sample voltage V(7−1) = 48 mV, i.e. V(4−1) = 24 mV.
The dashed line indicating the thermal noise of the sectors (1−4) and the dotted line showing
The dashed line indicating the thermal noise of the sectors (1−4) and the dotted line showing
the pure 1/f noise as well as the amplifier noise SV5186 (dash-dotted line) have been added for reference.
the pure 1/f noise as well as the amplifier noise SV5186 (dash-dotted line) have been added for reference.
5.
5. Correlation
Correlation method
method
An
An important
important advantage
advantage of
of the
the ac
ac method
method is
is the
the possibility
possibility of
of exploration
exploration of
of noise
noise spectra
spectra at
at
the
limit
of
extremely
low
frequency,
limited
only
by
the
period
of
observation.
Furthermore,
the limit of extremely low frequency, limited only by the period of observation. Furthermore,
the
the contribution
contribution of
of the
the amplifier
amplifier noise
noise in
in the
the ac
ac method
method depends
depends on
on the
the carrier
carrier frequency,
frequency, aa
90 . It is worth to note, that for
careful
selection
of
which
may
minimize
the
contribution
to
S
90
careful selection of which may minimize the contribution to SVV . It is worth to note, that for
the
amplifier,
the 5186
5186
amplifier, used
used in
in the
the studies,
studies, the
the power
power spectral
spectral density
density of
of its
its noise
noise at
at 325
325 Hz
Hz is
is
−17 V2/Hz (see Fig. 2) and sharply rises for frequencies below 10 Hz. However, a further
1.7·10
−17
2
1.7·10 V /Hz (see Fig. 2) and sharply rises for frequencies below 10 Hz. However, a further
reduction
reduction of
of the
the amplifier
amplifier noise
noise contribution
contribution is
is still
still possible,
possible, e.g.
e.g. by
by the
the use
use of
of aa correlation
correlation
90
90
technique
for
measurements,
i.e.
calculating
the
cross-spectrum
of
signals
detected
at
S
technique for SVV
measurements,
i.e.
calculating
the
cross-spectrum
of
signals
detected
at 90°
90°
VV
in
two
independent
signal
paths.
Next,
expanding
the
concept
by
the
use
of
a
multi-terminal
in two independent signal paths. Next, expanding the concept by the use of a multi-terminal
configuration
configuration of
of the
the sample
sample in
in conjunction
conjunction with
with the
the correlation
correlation method
method makes
makes it
it possible
possible to
to
localize
noise
components
originated
in
different
parts
of
the
multi-terminal
device.
localize noise components originated in different parts of the multi-terminal device. For
For
0
90
example,
and
example, calculating
calculating the
the cross-spectra
cross-spectra SV0 ( x − x )V ( 7 − 7 ) and
and SV90( x − x )V ( 7 − 7 ) ,, where
where V
V(7−7)
(7−7) and
SV ( x − x )V ( 7 − 7 )
SV ( x − x )V ( 7 − 7 )
V
are voltages taken from the bridge diagonal (terminals 7−7) and its sub-diagonal x−x, the
V(x−x)
(x−x) are voltages taken from the bridge diagonal (terminals 7−7) and its sub-diagonal x−x, the
of the sectors of resistive layers, located between contacts 1 and x, may be
excess
excess noise
noise S
SVex(1−x)
Vex(1−x) of the sectors of resistive layers, located between contacts 1 and x, may be
obtained
as:
obtained as:
90
0
(4)
S
(1− x ) = SV0 ( x − x )V (7−7) − SV90( x − x )V (7−7) .
(4)
SVex
Vex (1− x ) = SV ( x − x )V (7−7) − SV ( x − x )V (7−7) .
It
It is
is worth
worth to
to note,
note, that
that the
the method
method described
described by
by (2)
(2) could
could not
not be
be used
used in
in this
this case.
case. The
The
advantage
of
the
above
concept
has
been
demonstrated
in
Fig.
3b
where
the
spectra
obtained
advantage of the above concept has been demonstrated in Fig. 3b where the spectra obtained in
in
90
different
was measured
measured
different ways
ways have
have been
been gathered
gathered for
for comparison.
comparison. Since
Since the
the spectrum
spectrum SV90( 4 − 4 ) was
SV ( 4 − 4 )
on
on the
the side-contacts,
side-contacts, it
it includes
includes −
− apart
apart from
from the
the noise
noise originated
originated in
in the
the resistive
resistive layer
layer located
located
90
between
side-contacts
1
and
4
−
also
the
noise
of
voltage
contacts.
Therefore,
S
gives
between side-contacts 1 and 4 − also the noise of voltage contacts. Therefore, V90( 4 − 4 ) gives
SV ( 4 − 4 )
508
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Metrol. Meas.
Meas. Syst.,
Syst., Vol.
Metrol.
Vol. XXII
XXII (2015),
(2015), No.
No. 4,
4, pp.
pp. 503–512.
xxx–xxx.
Metrol. Meas. Syst., Vol. XXII (2015), No. 4, pp. xxx–xxx.
the overestimated value of noise magnitude. On the contrary, the cross-spectrum SV90
the overestimated value of noise magnitude. On the contrary, the cross-spectrum SV90(( 44 −− 44 ))VV (( 77 −− 77 ))
gives the correct value of thermal noise of sectors 1−4, evidencing usefulness of the correlation
gives the correct value of thermal noise of sectors 1−4, evidencing usefulness of the correlation
technique. The resistance R(4−1) = 5.5 kΩ, involved in calculation of the thermal noise
technique. The resistance2 R(4−1) = 5.5 kΩ, involved in calculation of the thermal noise
V /Hz, has been measured using four-wire method, R(4−1) = V(4−1)/I(7−1),
4kT(2R(4−1)) = 1.8·10−16
4kT(2R(4−1)) = 1.8·10−16 V2/Hz, has been measured using four-wire method, R(4−1) = V(4−1)/I(7−1),
where V(4−1) is the voltage drop on terminals 1 and 4 measured under the current excitation I(7−1)
the current excitation
where V(4−1) is the voltage drop on terminals 1 and 4 measured under
0 I(7−1)
flowing from contact 7 to contact 1. The excess noise 2Re( SV45
( 4 − 4 )V ( 4 − 4 ) ) and SV0 ( 4 − 4 )
45
flowing from contact 7 to contact 1. The excess noise 2Re( SV ( 4 − 4 )V ( 4 − 4 ) ) and SV ( 4 − 4 )
measured directly on the sub-diagonal (4−4) also lead to overestimate the noise magnitude of
measured directly on the sub-diagonal (4−4) also lead to overestimate the noise magnitude of
sectors (1−4).
sectors (1−4).
6. Noise suppression
6. Noise suppression
During the noise exploration of PTFRs, it has been observed that the spectra of background
During
noisethose
exploration
PTFRs,
it has been
that the 1/f
spectra
of background
noise,
apartthefrom
of whiteofnoise,
contained
alsoobserved
an unexpected
component,
whose
noise,
apart
from
those
of
white
noise,
contained
also
an
unexpected
1/f
component,
whose
intensity has been found to be strongly dependent on the carrier frequency. It has been shown
intensity has been found to be90strongly dependent on the carrier frequency. It has been shown
in Fig. 4, where the spectra SV90 , obtained at the fixed sample voltage of V(7−1) = 0.31 V and at
in Fig. 4, where the spectra SV , obtained at the fixed sample voltage of V(7−1) = 0.31 V and at
different carrier frequencies, have been gathered. The excess noise component,
different
frequencies, have been gathered. The excess noise component,
90
90carrier
kT (2RS ) , has been calculated and plotted in the inset. The explanation of the
SVex
90 = SV90 − 4
SVex
= SV − 4kT (2RS ) , has been calculated and plotted in the inset. The explanation of the
is involved in calculation of SVex (see (4)).
phenomenon is of great importance since SV90
phenomenon is of great importance since SV90 is involved in calculation of SVex (see (4)).
10-12
10-12
Carrier frequency
Hz
Carrier2525
frequency
5025 Hz
Hz
2525
10025Hz
Hz
5025
20025 Hz
Hz
10025
40025 Hz
Hz
20025
~1/f
-13
S90
S, 90
V2,/Hz
V2/Hz
Vex Vex
10
2
SV90S,V90V,2V
/Hz
/Hz
10-13
10-13
~1/f
-13
10
-14
10
-14
10
40025 Hz
-15
10
-15
10
-16
10-14
10-14
10
-16
100m
1
10
100
1k
100m
1
10 f, Hz
Frequency
Frequency f, Hz
100
1k
10
10-15
10-15
10-16
10-16100m
100m
1
1
10
10
100
100
1k
1k
10k
10k
Frequency f, Hz
Frequency f, Hz
Fig. 4. The selected spectra of SV90(7 − 7) for the sample voltage fixed at V(7−1) = 0.31 V and for different carrier
Fig. 4. The selected spectra of SV90(7 − 7) for the sample voltage fixed at V(7−1) = 0.31 V and for different carrier
frequencies. Inset: the excess noise component. The dashed line indicating the thermal noise of the samples
frequencies. Inset:
component.
The 1/f
dashed
indicating
the for
thermal
noise of the samples
andthe
the excess
dotted noise
line showing
the pure
noiseline
have
been added
reference.
and the dotted line showing the pure 1/f noise have been added for reference.
0
90
90
The analysis of the noise intensities, s 00 =< fSVex
0 > ∆f , s 90 =< fSVex
90 > ∆f , plotted in Fig. 5a,
The analysis of the noise intensities, s =< fSVex
> ∆f , s =< fSVex
> ∆f , plotted in Fig. 5a,
0
90
reveals that both SV0 and SV90 components depend on the carrier frequency, although the
reveals that both SV and SV components depend on the carrier frequency, although the
is more pronounced. The noise model for the bridge circuit of Fig. 1 is
dependence of SV90
dependence of SV90 is more pronounced. The noise model for the bridge circuit of Fig. 1 is
proposed in order to explain the observed dependence. The equivalent circuit consisting of (i)
proposed
order
to explain
the observed
dependence.
The equivalent
circuit consisting
of (i)
the capacitance
the
ballastin
and
sample
resistances,
(ii) the equivalent
capacitance,
Ceq, including
the
ballast
and
sample
resistances,
(ii)
the
equivalent
capacitance,
C
eq, including the capacitance
of the circuit, wire connecting the output voltage to the amplifier, the preamplifier input, and
of
wire connecting
the in
output
voltage
to the amplifier,
the preamplifier
input,
and
(iii)the
thecircuit,
noise source
Un, is shown
Fig. 5b.
The circuit
acts as an impedance
voltage
divider,
(iii)
the
noise
source
U
n, is shown in Fig. 5b. The circuit acts as an impedance voltage divider,
yielding:
yielding:
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509
A.W.
al.: NOISE
NOISE MEASUREMENTS OF
WITH THE
THE USE …
…
A.W. Stadler,
Stadler, A.
A. Kolek,
Kolek, et
et al.:
OF RESISTORS
RESISTORS WITH
A.W.
Stadler,
A.
Kolek,
et
al.: NOISE MEASUREMENTS
MEASUREMENTS OF
RESISTORS
WITH THE USE
USE …
(R |||| RRS ))C
U
1 − j 2πf
U amp
amp = 1 − j 2πf carrier
carrier (RB
B
S Ceq
eq ,
(5)
=
,
(5)
2
U
2
(
)
11 +
2
π
f
R
||
R
C
U nn
carrier
B
S
eq
+ 2πf carrier (RB || RS )Ceq
where:
|| RS )C eq is the time constant of the circuit. Switching to power spectral densities,
where: ((R
RB
B || RS )C eq is the time constant of the circuit. Switching to power spectral densities,
we
we arrive
arrive at:
at:
[[
]]
0
0
S
−
SV
−2
2.
(R |||| RRS ))C
V =
= 2
2π
πff carrier
.
90
carrier (RB
B
S Ceq
eq
90
S
SV
[[
]]
(6)
(6)
V
a)
a)
b)
b)
10-12
10-12
s00
s
s90
Frequency band ∆f
s90 Frequency band ∆f
0.125-1.25 Hz
0.125-1.25 Hz
1-10 Hz
1-10 Hz
10-100 Hz
10-100 Hz
100-1000 Hz
100-1000 Hz
10-14
10-14
1044
10
Frequency band ∆f
Frequency
band ∆fHz
0.125-1.25
0.125-1.25
1-10 Hz Hz
1-10 HzHz
10-100
10-100 HzHz
100-1000
100-1000 Hz
1033
10
10-15
10-15
1022
10
10-16
10-16
RS
RS
1000
10 1k
1k
1k
1k
Uamp
Uamp
Ceq
Ceq
1011
10
10-17
10-17
10-18
10-18
RB
RB
Noise
Noise
intensity
intensity
ratio
ratio
s0/s
s090/s90
2 2
Noise
Noiseintensity
intensitys0s, 0s, 90s,90V
,V
10-13
10-13
(b)
(b)
10k
10k f
Carrier frequency
, Hz
Carrier frequency fcarrier
, Hz
carrier
10k
10k
100k
100k
Un
Un
100k
100k
*
Carrier frequency, Hz
Carrier frequency, Hz
Fig. 5. a) The intensities of excess noise, calculated for SV00 (solid symbols) and SV90
(open symbols) in different
Fig. 5. a) The intensities of excess noise, calculated for SV (solid symbols) and SV90 (open symbols) in different
frequency bands, as a function of the carrier frequency. Inset: the noise intensity ratio (symbols) as a function
frequency bands, as a function of the carrier frequency. Inset: the noise intensity ratio (symbols) as a function
of the carrier frequency. The line is the plot of r.h.s. of (6) with values of parameters given in the text.
of the carrier frequency. The line is the plot of r.h.s. of (6) with values of parameters given in the text.
b) The noise model for the circuit of Fig. 1.
b) The noise model for the circuit of Fig. 1.
The
with
The comparison
comparison of
of the
the model
model prediction
prediction
with the
the experimental
experimental data
data is
is shown
shown in
in the
the inset
inset of
of
0/s90, calculated for different frequency bands is plotted
Fig.
5a,
where
the
noise
intensity
ratio,
s
0
90
Fig. 5a, where the noise intensity ratio, s /s , calculated for different frequency bands is plotted
as
as aa function
function of
of the
the carrier
carrier frequency
frequency (open
(open symbols).
symbols). The
The solid
solid line
line is
is the
the plot
plot of
of r.h.s.
r.h.s. of
of (6)
(6)
for
C
eq =180 pF, RS = 11.12 kΩ, RB = 100 kΩ, that were directly measured by the precision
for Ceq =180 pF, RS = 11.12 kΩ, RB = 100 kΩ, that were directly measured by the precision
RLC
RLC meter
meter (E4890A
(E4890A from
from Agilent),
Agilent), giving
giving an
an excellent
excellent agreement
agreement with
with the
the experimental
experimental data
data
in
a
wide
range
of
carrier
frequencies.
Hence,
we
conclude
that
the
observed
in a wide range of carrier frequencies. Hence, we conclude that the observed 1/f
1/f component
component of
of
90 relates to a phase change of both in-phase and quadrature signal components, caused by
SSV90
relates
to
a
phase
change
of
both
in-phase
and
quadrature
signal
components,
caused
by
V
impedances
impedances in
in the
the measurement
measurement circuit.
circuit. The
The most
most significant
significant contribution
contribution to
to the
the capacitance
capacitance
C
eq is the voltage transfer line (approx. 100 pF), indicating possible optimization of the circuit
Ceq is the voltage transfer line (approx. 100 pF), indicating possible optimization of the circuit
in
in order
order to
to minimize
minimize the
the analyzed
analyzed effect.
effect. Another
Another possibility
possibility of
of improvements
improvements in
in the
the properties
properties
of
the
ac
bridge
is
replacing
the
variable
resistor
and
the
capacitor,
used
for
balancing
of the ac bridge is replacing the variable resistor and the capacitor, used for balancing the
the bridge,
bridge,
with aa simple
simple low-noise
low-noise resistor,
resistor, and
and controlling
controlling the
the balance
balance by
by the
the use
use of
of aa two-channel
two-channel sine
sine
with
waveform
generator
with
precise
programming
of
amplitude
and
phase
coupling.
waveform generator with precise programming of amplitude and phase coupling.
7.
7. Conclusions
Conclusions
A
A method
method of
of noise
noise measurements
measurements with
with the
the use
use of
of an
an ac
ac excitation
excitation of
of the
the tested
tested device
device has
has
been
developed.
The
progress
in
the
technique
is
in
using
a
multichannel
daq
module
been developed. The progress in the technique is in using a multichannel daq module and
and
software
software for
for multichannel
multichannel phase
phase sensitive
sensitive detection
detection instead
instead of
of an
an expensive
expensive lock-in
lock-in amplifier.
amplifier.
In
this
way,
it
is
possible
to
carry
out
the
phase-sensitive
detection
for
many
signals
In this way, it is possible to carry out the phase-sensitive detection for many signals at
at different
different
sets
sets of
of phase
phase angles
angles simultaneously,
simultaneously, which
which is
is not
not possible
possible when
when using
using aa standard
standard lock-in
lock-in
amplifier.
amplifier. The
The developed
developed multichannel
multichannel virtual
virtual lock-in
lock-in technique
technique has
has been
been shown
shown to
to be
be helpful
helpful
510
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Metrol. Meas.
Meas. Syst.,
Syst., Vol.
Vol. XXII
Metrol.
XXII (2015),
(2015), No.
No. 4,
4, pp.
pp. 503–512.
xxx–xxx.
for noise exploration in polymer thick-film resistors. The method has been verified with the use
of (i) a traditional (stand-alone) lock-in amplifier for phase-sensitive detection and (ii) a typical
dc method of noise measurement. All methods give consistent results. It has been confirmed
that the 1/f noise caused by resistance fluctuations is the dominant noise component in PTFRs
made of ED7100 resistive composition with Cu contacts. The value of noise parameter
describing material noise intensity, C = 1·10−21 m3, found in this work is in a good agreement
with 8.2·10−22 m3 obtained with a dc method [17]. Furthermore, the parameter K = 1.4·10−7
µm2/Ω is typical for percolation in thick-film resistors and conductive polymers. A very
interesting observation was, that the background noise component obtained with the ac method
includes an unexpected 1/f component. The phenomenon has been analysed and explained as
the noise suppression by impedance of the measurement circuit, giving an excellent agreement
with the experimental data. The multichannel VLI technique may replace hardware instruments,
making the functionality more flexible and powerful. It is a very promising technique in noise
characterization of low-ohmic samples and/or using an extremely low frequency which is
outside the limit of a dc method. Moreover, using a multichannel daq-board, the noise signals
from several pairs of sample terminals may be detected, which makes it possible to use the
correlation method not only for reduction of system noise, but also for localization of noise
sources in different parts of the resistor [27] and evaluation of the contact noise, giving
information about the quality and interactions at the resistive/conductive layers’ interface [18].
Experiments in which an ac method has been used in conjunction with a correlation technique
are extremely rare due to the need of employing several expensive hardware lock-in amplifiers.
We hope that the approach, described in the work, will break the barrier.
The developed VLI is easy to expand in order to increase its functionality, like e.g. evaluation
of the 3rd (or any other) harmonic index [28, 29] which is also used for testing the reliability
and quality of electronic components, instead of rather difficult noise measurements, as well as
impedance analysis at the mHz-frequency range which is not available in the standard
impedance analysers and meters. It has been also possible to use the ac method in exploration
of the noise properties of nonlinear and other passive electronic components like capacitors and
inductors. Furthermore, comparing noise intensity dependencies on dc and ac excitations we
may recognize nonlinearity of conduction and/or noise generation mechanisms [25, 26].
Acknowledgements
The work has been supported from Grant DEC-2011/01/B/ST7/06564 funded by National
Science Centre (Poland) and from statutory activity from Department of Electronics
Fundamentals of Rzeszow University of Technology and Wroclaw University of Technology.
Studies have been performed with the use of an equipment purchased in the project
No POPW.01.03.00-18-012/09 from the Structural Funds, The Development of Eastern Poland
Operational Programme co-financed by the European Union, the European Regional
Development Fund.
References
[1]
Directive 2002/95/EC of the European Parliament and of the Council of 27 Jan. 2003 on the restriction of
the use of certain hazardous substances in electrical and electronic equipment.
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