# Low-cost sensors and circuits for plasma education: characterizing power and
illuminance 111Research supported in part by the Brazilian agency CNPq Grant
305158/2017-1; 305998/2020-0; 421486/2016-3.

Alessandro N. Vargas22footnotemark: 2<EMAIL_ADDRESS>Victor Miller  Ali
Mesbah  Gabriele Neretti  Universidade Tecnológica Federal do Paraná, UTFPR,
Av. Alberto Carazzai 1640, 86300-000 Cornelio Procópio-PR, Brazil.  Department
of Chemical and Biomolecular Engineering,
University of California, Berkeley, CA 94720, USA.  Electric Electronic and
Information Engineering Department,
University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy.

###### Abstract

Industrial applications of plasma have significantly increased beyond
semiconductor manufacturing in recent years. This necessitates training a
skilled workforce in plasma science and technology. However, an essential
challenge to this end stems from the high cost of plasma devices and
diagnostics. The limited access to plasma devices has hindered plasma
education, particularly in the least developed countries. To this end, this
paper demonstrates how low-cost sensors and circuits can be developed to
enable inexpensive plasma experiments in laboratory environments. In
particular, we show how to measure high voltage, current, and power from a
cold-atmospheric plasma discharge. Additionally, we develop a low-cost
illuminance sensor and demonstrate how it can be used to estimate the
corresponding plasma power. The low-cost sensors and electronics presented in
this paper can aid educators in characterizing plasma power versus plasma
illuminance.
Keywords: Cold atmospheric plasma; plasma education; low-cost sensors; low-
cost electronics; plasma illuminance.

##  1 Introduction

Plasma research has received a lot of attention over the last few decades,
owing to its increased applications in science and technology [1, 2, 3, 4].
For example, cold atmospheric plasma has been the critical element in
materials processing [5], toxin reduction in food [6], decontamination and
sterilization [7], degrading pharmaceutical pollutants [8], regulating
combustion engines [9], and increasing agriculture productivity [10].

To cope with the increasing demand for plasma education, educational plasma
laboratories have been established, aiming to foster plasma knowledge [11,
12]. However, creating an educational plasma laboratory is expensive—the cost
can exceed tens of thousands of US dollars [13]. Moreover, creating research
plasma laboratories can go even further—their cost can easily exceed millions
of US dollars [11, 14]. Consequently, many researchers and educators remain
away from plasma experiments, especially those in the least developed
countries. For those individuals, the only option is to resort to simulations
[13].

Even though simulations represent an important educational tool, experienced
educators advocate that students learn much more—and score much higher—when
learning through hands-on experiments [15, 16]. According to them, simulations
do not replace hands-on experiments.

To perform plasma experiments, a high-voltage power supply is needed. Some
researchers and educators have attempted to make it more accessible. For
example, a study describes a low-cost, high-voltage source discharging energy
into a cathode-ray tube [17]. More recently, a study has shown a DC power
supply that generated up to 3kV to create gliding arcs [18]. Another study has
reported how to assemble a low-cost 7kV DC power supply, yet it was not
specifically designed to ignite plasma [19].

A study has documented a low-cost, high-voltage sinusoidal generator with a
peak voltage of up to 20kV [20]. Another study has shown that one can assemble
a nanosecond high-voltage generator with a cost of around U$ 2500 [21];
however, this cost is unaffordable for many individuals. Another device able
to generate high voltage is known as Marx generator [22, 23, 24], yet it is
unclear whether this electronic topology can result in low-cost products.
Together, these studies emphasize the importance of creating affordable high-
voltage power supplies. Note, though, that a power supply is just one of the
items required in plasma experiments.

Another indispensable item in any plasma experiment is sensors for plasma
diagnostics. Plasma diagnostics are used to understand properties of plasma,
such as in assessing the gas composition and temperature through the optical
emission spectroscopy in plasma [25, 26, 27], and in monitoring the voltage
and current in plasma discharges [28, 29]; see also [1, 11, 30, 31] for
further discussion about other types of sensors deployed in plasma
experiments.

Not surprisingly, sensors used in plasma experiments are expensive. For an
illustration, we can mention that a high-voltage probe—key element to monitor
the dynamic behavior of plasma discharge—can cost several thousand US dollars
(see Section 3.2).

The main contribution of this paper is to develop low-cost sensors and
circuits, and demonstrate how these sensors enable measuring (i) high voltage
and current, (ii) plasma power, and (iii) plasma brightness (measured in lux).
All circuits and schematics are provided. All circuits are assembled with low-
cost parts.

This paper does not intend to show technological advances; instead, it focuses
on a detailed description of low-cost sensors for plasma experiments. Even
though the circuits and sensors were invented many decades ago and are
ubiquitous in the electronics industry [32, 33, 34, 35], we bring them
together for the sake of low-cost plasma experiments. Namely, we detail how to
combine low-cost sensors and circuits, how to assemble them, and how to apply
them in plasma experiments. We then demonstrate how the low-cost sensors can
be used to characterize plasma power versus brightness. To the best of the
authors’ knowledge, such an experiment relating plasma power to brightness has
not been done before.

Figure 1: Setup of the plasma experiment. A high-voltage power supply provides
direct current (DC) to the circuit. Plasma is ignited between a needle and a
lower plate when the voltage $u(t)$ exceeds a certain threshold. A light
sensor is conveniently placed to measure plasma brightness. A shunt resistor
is used to help measure the current $i(t)$.

##  2 Experimental setup

This paper focuses on a direct current (DC) gliding arc discharge, a type of
reactor that can generate cold atmospheric plasma [36, 37, 38, 39]. All the
experiments detailed in this paper were performed in the open air and at
atmospheric pressure, a usual configuration for local non-thermal treatments
[38, 39]. Our setup allowed us to ignite plasma between a needle and a plate;
see Fig. 1.

A high-voltage DC power supply generates an electric current $i(t)$ that flows
through the circuit. Plasma is ignited when high voltage $u(t)$ is adjusted to
exceed a certain threshold value. Note that $u(t)$ affects both the needle
voltage $v(t)$ and the current $i(t)$. The power consumed by the plasma
discharge then equals $p(t)=v(t)i(t)$ for all $t>t_{0}$, where $t_{0}$ denotes
the time in which plasma is ignited. Conveniently placed next to the plasma
discharge, a light sensor measures the plasma brightness $\ell(t)$, $t>t_{0}$.

This paper shows how to measure $v(t),i(t),p(t)$, and $\ell(t)$, $t>t_{0}$.

######  Remark 1.

This paper does not cover how to build up a high-voltage direct current (DC)
power supply. As mentioned previously, other studies have reported how to
build high-voltage power supplies using both DC and AC. Implementing such
devices is outside the scope of this paper.

Warning: The experiments described in this paper were carried out with an
industrial high-voltage DC power supply manufactured by Spellman model
SL10PN1200 (Hauppauge, NY, USA), see Section 4. This device complies with the
US regulations for electrical safety and protection. Plasma experiments
involve high voltage, which may even ensue the risk of death. High voltage can
be lethal. For this reason, only trained, qualified individuals should attempt
to deal with high voltage. All the low-cost sensors and electronics described
in this paper manipulate high voltage. Only qualified personnel should attempt
to reproduce the circuits shown in this paper. All the low-cost sensors and
circuits were tested in a laboratory under safe working conditions. No
electric problems were observed during the experiments.

##  3 Low-cost sensors and circuits

This section details how to implement the sensors and circuits necessary to
measure the physical properties shown in Fig. 1.

###  3.1 Low-cost sensor for plasma brightness

Researchers of plasma science often rely on optical emission spectrometers to
measure plasma light properties (e.g., [40], [41]). For example, optical
emission spectrometers can measure plasma intensity as a function of
wavelength so that researchers can point out certain properties of plasma
chemistry [25, 26, 27], [31, Fig. 4.12, p. 62]. Indeed, optical emission
spectrometers are the go-to instruments for plasma emission analysis. But they
are expensive. The starting cost of simple models ranges from thousands of US
dollars. This means that optical emission spectrometers remain inaccessible to
most researchers based in low-income countries and laboratories with meager
financial support.

As an attempt to circumvent this affordability problem, we devised a low-cost
circuitry for a light sensor. This sensor is limited because it does not
measure the full scale of wavelenghts. However, as detailed in the sequel, it
does measure the aggregate of wavelenghts in the form of illuminance.

###  3.2 High-voltage probe and current probe for measuring plasma power

Measuring voltage and current in plasma discharge is not a novel research
topic. Researchers have been doing so for many decades; see, for instance, the
voltage-current curves in [1, Fig. 14.2, p. 538], [42, Fig. 1], and [43, Fig.
2]. Voltage and current combined determine the plasma power, a key information
for researchers. For example, researchers monitor power because it changes the
plasma properties when in contact with a target material skin [44, 45] and in
material processing [46, 47].

Researchers have almost exclusively documented the use of industrial probes to
characterize the voltage-current relationship in plasma. For instance,
experiments were made with the probe model P6015A in [28, 29, 48], model P5210
in [49], model TCP312A in [50], and model TCP0020 in [51], all of the probes
manufactured by Tektronix (Beaverton, OR, USA).

Industrial probes, like the aforementioned ones, are precise and extremely
useful. However, they are expensive. This paper demonstrates that it is
possible to assemble reliable, low-cost probes, as described next.

Figure 2: Schematics of the high-voltage probe. When high-voltage input is
applied in $+V_{i}$, a low voltage appears in the output terminal $+V_{o}$. A
series of RC filters treats the corresponding signal and defines the
attenuation $V_{o}/V_{i}$. The value of the components $C_{0},\ldots,C_{n}$
and $R_{0},\ldots,R_{n}$ varies according to the project specification.

####  3.2.1 Low-cost high-voltage probe

Studies have documented that a simple, low-cost resistive voltage divider can
effectively measure high voltage [32, 33]. As suggested by the authors of
[32], the scheme of Fig. 2 suffices for measuring high voltage. Measurements
from the circuit of Fig. 2 can generate low-distortion signals for frequencies
of up to 1 MHz (see Fig. 9 in [32]). This finding is confirmed in another
study [33]. We followed these studies and implemented the circuit of Fig. 2.

Implementation: The circuit of Fig. 2 was assembled in the laboratory with
$C_{1}=\cdots=C_{n}$ and $R_{1}=\cdots=R_{n}$. This condition allows us to
write (see Appendix)

$\frac{V_{o}}{V_{i}}=\frac{R_{0}}{nR_{1}+R_{0}},\quad\text{and}\qquad
C_{0}=\frac{C_{1}R_{1}}{R_{0}},$

where $V_{i}$ represents the high-voltage input and $V_{o}$ denotes the low-
voltage output. Our low-cost probe has attenuation close to 1000:1. To obtain
this attenuation, we set $n=5$, and picked up off-the-shelf components with
values $C_{i}=15$pF, $R_{i}=10$M$\Omega$, $i=1\ldots,5$, $C_{0}=3$nF, and
$R_{0}=52.8$K$\Omega$. All resistors and capacitors were rated for high
voltage for safety reasons (see further details in Appendix). The assembled
probe is depicted in Fig. 3. Epoxy resin covers the components as these
chemicals increase the insulation between the components’ terminals, creating
extra protection against arcs or accidental short circuits.

Figure 3: Low-cost high-voltage probe.  Figure 4: Configuration of the shunt
resistor. This scheme allows us to measure the current that flows through the
plasma discharge.

####  3.2.2 Low-cost current probe

When a resistor is connected in series with a circuit, we immediately observe
a voltage drop in the resistor’s terminals. This is the working principle of
the shunt resistor (e.g., [52, p. 31], [53]), a classical current sensor [52,
p. 31]. A shunt resistor can measure both alternating current (AC) and direct
current (DC).

The shunt resistor has been a popular solution to measure current in plasma
experiments (e.g., [54, Fig. 1], [29, Fig. 1], [55, Fig. 1]). As usual in the
plasma literature, the shunt resistor connects the metal plate and the ground
(Fig. 4). The shunt resistor’s value should be small enough to consume as
little plasma power as possible.

The circuit of Fig. 5 allows us to measure the plasma current. The circuit
contains a low-cost operational amplifier TL072 and a low-cost voltage
reference TL431, both manufactured by Texas Instruments (Dallas, TX, USA). The
upper part of Fig. 5 shows a buffer stage that ensures high impedance for the
signal $v_{S}$; the lower part generates a DC offset fixed at $+1.25$V. The
reason for generating this DC offset is related to a limitation on the
microcontroller analog input, as detailed next.

Most microcontrollers have analog-to-digital converters that can handle only
positive voltages. In this case, a microcontroller cannot read a negative
voltage on the shunt voltage $v_{S}$. However, we can add the negative signal
$v_{S}$ to the DC offset of $1.25$V and send the corresponding signal to the
microcontroller if $v_{S}+1.25$ is positive. This strategy allows us to
consider negative voltages on $v_{S}$ up to $-1.25$V.

(Implementation). The circuits of Figs. 4–6 were implemented in the
laboratory. A shunt resistor of $23$ $\Omega$ was used (it was the only
precise resistor available during the experiments). From Fig. 5, we have
v1$=v_{S}$ and v2=$1.25$V. Both v1 and v2 were connected in the corresponding
ports of the circuit shown in Fig. 6, which in turn guarantees that
v3=v1+v2=$v_{S}+1.25$.

Figure 5: Auxiliary circuit for the current sensor. Upper circuit: buffer for
the signal $v_{S}$ from a shunt resistor. Lower circuit: DC offset reference
fixed at $1.25$V.  Figure 6: Non-inverting summing amplifier. The signal at
the point v3 equals v1+v2.

####  3.2.3 Light-dependent resistor

A light-dependent resistor (LDR) is a device that changes its resistance
according to the light intensity (e.g., [56, Sec. 14.4], [35, Sec. 3.5],[57,
Sec. 1.6.2]). LDR is a semiconductor device manufactured so that its two-
terminal resistance decreases when the light intensity hitting its surface
increases. The relationship between resistance and light intensity is
nonlinear, as manufacturers indicate (e.g., [58], [59, p. 2]).

This paper shows that LDR sensors are appropriate for plasma experiments.
Indeed, the reasons for using LDR in plasma experiments are as follows. First,
LDR is small—it has round faces under a few millimeters in size, taking up a
small space in plasma reactors. Second, LDR is cheap—its price is less than
three US dollars each. Third, LDR is easy to use and behaves like a classical
resistor. These features have led to the widespread of LDR in industry, such
as in the circuits that control the street lights [60, Sec. 2.1.1] and that
measure radiotherapy irradiation [61].

As with any semiconductor-based component, LDR has no calibration. Namely,
during the manufacturing process, each LDR sensor is made unique. As a result,
each LDR inherits a particular light-sensing characteristic that can differ
widely from other LDRs manufactured in the same industrial line. Two off-the-
shelf LDR sensors—manufactured by the same company and sold under the same
part number—might behave completely differently.

Manufacturers openly disclose the unmatched properties of their LDRs in
manuals; see, for example, how the resistance of an LDR can vary drastically
from component to component, as indicated in the datasheet [62, Fig. 1].
Therefore, a fair use of LDR mandates a calibration. The next section presents
a method to calibrate LDRs, borrowing ideas from [58].

Figure 7: LDR amplifier circuit.

###  3.3 Calibration of a light-dependent resistor

The calibration experiment was performed with the circuit of Fig. 7 (this
circuit is borrowed from [34, Fig. 2.3, p.63]). The LDR was manufactured by
Advanced Photonics, photoresistor model PDV-P9008 (Palo Alto, CA, USA).

Figure 8: Setup for the calibration of a light-dependent resistor (LDR). The
sensor on the right represents an industrial ready-to-use illuminance meter.

To calibrate this LDR, we followed the procedure suggested in [58]. The author
of [58] suggests using a boom stand to hold the light source; see Fig. 8. Our
boom stand had a holding arm that could move up and down. A clamp was adjusted
to set the arm’s height when necessary. The arm was attached to a small
sliding table that could move freely in the horizontal direction. Glued at the
lower part of the sliding table was a small automotive light bulb manufactured
by Sylvania model 1157A (Wilmington, MA, USA).

Our boom stand allowed us to position the light bulb at any point. In
particular, we aligned the light bulb with points $A$ and $B$, corresponding
to the perpendicular position of two sensors fixed at the ground: (A) LDR
sensor (model PDV-P9008) and (B) industrial light meter sensor, manufactured
by Urceri model MT-912 (China—sold at Amazon.com), which allowed us to measure
illuminance (lux).

Figure 9: Experimental data from the LDR calibration in logarithm scale. The
dots represent real data measured in the laboratory. The curve represents a
third-order polynomial that fits the data.

All the calibration experiments were performed in a dark room. We applied
values from 10V to 18V to the light bulb and moved it from point A to point B.
The arm’s height was decreased in steps until the light bulb reached a few
millimeters away from the two sensors. The corresponding experimental data is
summarized in Fig. 9. As the data plotted in Fig. 9 suggest, measurements from
the LDR can be approximated by a third-order polynomial, which reads as

$\Delta(s)=a_{3}s^{3}+a_{2}s^{2}+a_{1}s+a_{0},$ (1)

where the values for $a_{0},\ldots,a_{3}$ are available in Table 1 (see
Appendix).

Because (1) follows a logarithm scale, we now introduce its linear
counterpart. Let $\ell$ denote the illuminance measurement from the LDR, and
consider $v$=v4 as in Fig. 7. Setting $\ln(\ell)=\Delta(s)$ and $\ln(v)=s$, we
have from (1) that

$\ln(\ell)=a_{3}\ln(v)^{3}+a_{2}\ln(v)^{2}+a_{1}\ln(v)+a_{0},$

which is equivalent to

$\ell=\exp\left(a_{3}\ln(v)^{3}+a_{2}\ln(v)^{2}+a_{1}\ln(v)+a_{0}\right).$ (2)

As detailed in the next section, the formula in (1) was programmed in a
microcontroller to convert voltage to illuminance. For illustration, suppose
that the microcontroller reads +1V at the point v4. It then sets $v=1$ in the
expression (1) to obtain $\ell=12.5952$ (lux); see further details in Remark
2.

###  3.4 Measuring plasma power

A microcontroller Arduino, model Due (Monza, Lombardia, Italy), was used to
process experimental data. Arduino is a popular brand that shares free, open-
hardware platforms, allowing users to modify its hardware and software as much
as needed. Arduino Due has twelve analog-to-digital converter (DAC) ports and
two digital-to-analog converters (ADC). The DAC ports can handle voltage from
0V to 3.3V; the ADC ports can generate voltage from 0.55V to 2.75V. Arduino
Due belongs to the family of low-cost microcontrollers (see www.arduino.cc).

In the laboratory, the low-cost high-voltage probe measured the voltage $v(t)$
at the needle (see Section 3.2.1); and the low-cost current probe measured the
current $i(t)$ flowing through the plasma (see Section 3.2.2). Information
from both probes feeds the two DAC ports of the Arduino Due, which then
computes the plasma power $p(t)=v(t)i(t)$, for all $t>t_{0}$.

Arduino Due sends real-time data to a computer screen, allowing users to
monitor the plasma power. For example, Fig. 10 shows a snapshot of
measurements taken from both Arduino Due and an oscilloscope Tektronix model
TDS2022B (high-voltage probe Tektronix P6015A on Channel 1 and oscilloscope
probe reading the voltage on shunt resistor on Channel 2). As indicated in
Fig. 10, the consumed power was $p(t)=0.498\times 10^{3}\times 0.0366=18.227$
(watts) according to Arduino Due; and it was $p(t)=0.479\times
10^{3}\times(0.877/23)=18.264$ (watts) according to the industrial probes.
These numbers suggest that the power computed by our low-cost approach differs
less than 2% from the industrial instrument.

Figure 10: Low-cost sensors and electronics for plasma experiments. Sensors
measured real-time data from a plasma discharge and sent the data to Arduino
Due. A laptop received data from Arduino Due and projected on-screen
information about the voltage $v(t)$, current $i(t)$, power consumed by the
plasma $p(t)$, and plasma brightness $\ell(t)$. Figure 11: Plasma experiment.
A light-dependent sensor (LDR) was positioned close to the plasma discharge to
measure plasma brightness while the power consumed by the plasma was
monitored.

##  4 Experiments: characterizing plasma power versus illuminance

The plasma experiments were carried out under the scheme shown in Fig. 1. A
high-voltage direct current (DC) power supply, manufactured by Spellman model
SL10PN1200 (Hauppauge, NY, USA), was used to generate up to +10kV (100mA) in
one terminal of a ballast resistor of $R=120$K$\Omega$. This resistor limits
the maximum discharge current to below 120mA, the maximum value the DC power
supply can provide [38, p. 219]. The other resistor’s terminal was connected
to a tungsten electrode needle (it contained 2% lanthanide) with a diameter of
1.016 millimeters, manufactured by Diamong Ground Products Inc. (Newbury Park,
CA, USA).

A stainless steel plate was placed below the electrode needle, with a distance
of about five millimeters. A light-dependent resistor (LDR), already described
in Section 3.3, was placed about eight millimeters away from the plasma
discharge.

The needle-to-plate configuration of Fig. 1 formed a plasma discharge (see
Fig. 11). While plasma was under production, the needle-to-plate discharge was
kept inside a chemical fume hood for safety. All the experiments were
conducted in the open air, with the temperature at about $24^{\circ}$C and at
atmospheric pressure.

Arduino Due processed the voltage corresponding to the plasma brightness
(i.e., illuminance) from the LDR (see Section 3.3), following the expression
(1). Arduino Due also processed the voltage $v(t)$ and the current $i(t)$ (see
Section 3.4) to compute the power plasma $p(t)=v(t)i(t)$, for all $t>t_{0}$.

Figure 12: Experimental data from a plasma discharge. A third-order polynomial
(blue) fits the experimental data (dot points).

The DC high-voltage power supply was manipulated to vary voltage between 0V
and 10kV. Plasma was ignited when the voltage crossed a threshold of about
4.5kV. Once the plasma was ignited, it remained in production even under
voltages moderately below the threshold. All the experimental data were
recorded and are freely available for download; see Remark 2 in connection.

Fig. 12 summarizes the corresponding experimental data. It shows the relation
between the power consumed by the plasma discharge and the plasma brightness.
The curve in Fig. 12 is a third-order polynomial, approximated through
experimental data (see Section 3.3)

$\ell=\exp\left(a_{3}\ln(p)^{3}+a_{2}\ln(p)^{2}+a_{1}\ln(p)+a_{0}\right),$ (3)

where $p$ represents the plasma power (watts), and $\ell$ denotes the plasma
brightness (lux); the constants $a_{0},\ldots,a_{3}$ are available in Table 1
(see Appendix).

Experimental data suggest that the polynomial in (3) can represent plasma
power versus plasma illuminance (Fig. 12). However, the polynomial does not
capture the transient behavior of plasma dynamics. For example, Fig. 12 shows
a minor glitch in the experimental data, corresponding to data recorded once
the plasma was ignited. In other words, Arduino Due started recording the
illuminance and power once the plasma ignited. Thus, in Fig. 12, the transient
points correspond to the black dots located away from the blue curve.

In summary, the expression in (3) does characterize the relation between the
plasma power and plasma brightness, yet it comes from measurements made with
low-cost sensors and electronics. This evidence illustrates the potential of
low-cost electronics for plasma experiments. Besides, researchers and
educators can estimate the plasma power by measuring the plasma luminescence
and vice versa, thus dispensing expensive industrial solutions.

Our initiative helps spread the knowledge of experimental plasma—in particular
plasma power and plasma brightness—among laboratories with a limited budget,
facilitating education on plasma.

######  Remark 2.

All the files used in this manuscript—all files containing data, source code,
schematics, and circuit layouts—are freely available on GitHub at
https://github.com/labcontrol-data/plasma-lowcost Table 2 (Appendix) shows a
final bill of materials used in this project.

##  5 Concluding remarks

This paper showed how to use low-cost sensors and electronics can be used to
characterize plasma experiments. With a budget of less than U$ 120, suitable
circuits were assembled to measure high voltage, current, plasma power, and
plasma illuminance. Such low-cost solutions can represent a paradigm shift in
plasma education and research, particularly in under-resourced communities.

## Appendix

Frequency evaluation of the high-voltage probe shown in Fig. 2. It is well-
known that a resistor and a capacitor—when in parallel—create impedance equal
to (e.g., [63, p. 1526])

$Z_{i}(s)=\frac{R_{i}}{1+R_{i}C_{i}s},\quad i=0,\ldots,n,$

where ‘$s$’ denotes the complex variable in the Laplace domain (e.g., [64, Ch.
3]). The input-output relation of the schematics of Fig. 2 equals to

$G(s)=\frac{V_{o}(s)}{V_{i}(s)}=\frac{Z_{0}(s)}{\sum_{i=0}^{n}Z_{i}(s)}.$ (4)

Substituting $C_{1}=\ldots=C_{n}$ and $R_{1}=\ldots=R_{n}$ in (4) yields

$G(s)=\frac{R_{0}(1+R_{1}C_{1}s)}{nR_{1}+R_{0}+(nR_{0}R_{1}C_{0}+R_{0}R_{1}C_{1})s}.$
(5)

In (5), we substitute

$C_{0}=\frac{R_{1}C_{1}}{R_{0}},$

to obtain

$G(s)=\frac{R_{0}(1+R_{1}C_{1}s)}{nR_{1}+R_{0}+(nR_{1}^{2}C_{1}+R_{0}R_{1}C_{1})s}.$

Because $G(s)$ equals to

$\frac{(1+R_{1}C_{1}s)}{\frac{nR_{1}+R_{0}}{R_{0}}+\frac{(nR_{1}^{2}C_{1}+R_{0}R_{1}C_{1})}{R_{0}}s}=\frac{(1+R_{1}C_{1}s)}{\frac{nR_{1}+R_{0}}{R_{0}}(1+R_{1}C_{1})s},$

we have that

$G(s)=\frac{R_{0}}{nR_{1}+R_{0}}.$ (6)

Note in (6) that $G(s)$ does not depend on the variable $s$. As a result, the
input-output gain of the probe remains constant for all possible frequencies,
and the phase shift equals zero for all possible frequencies [64, Ch. 6.1].

Table 1: Parameters representing two equations: (i) LDR sensor function as in (1), and (ii) plasma power versus plasma brightness as in (3).  Parameter | Equation (1) | Equation (3)
---|---|---
$a_{0}$ | $2.533317$ | $-11.413655$
$a_{1}$ | $1.960146$ | $12.323756$
$a_{2}$ | $2.118486$ | $-3.966212$
$a_{3}$ | $2.101649$ | $0.454388$
Table 2: Bill of materials. Cost of the parts to assemble the low-cost sensors and electronics to measure plasma power and plasma brightness. Item | Description | Unit cost (US$)
---|---|---
R0 |  R0-a and R0-b in parallel to obtain 52.8K$\Omega$ | –
R0-a | Resistor 56K$\Omega$ RL07S563GRE5 (1/4 watt) | 0.37
R0-b |  Resistor 931K$\Omega$ RN55D9313FB14 (1/8 watt) | 0.52
R1–R5 | Resistor 10M$\Omega$ VR68000001005JAC00 | 0.54
R6 |  Resistor 2.2K$\Omega$ 1/4W 2% | 0.30
R7–R12 | Resistor 10K$\Omega$ 1/4W 2% | 0.30
R13 |  Resistor 1K$\Omega$ 1/4W 2% | 0.30
R14 | Resistor 3.3K$\Omega$ 1/4W 2% | 0.30
C0 | Capacitor 1500pF 2kV 594-F152K53Y5RP63K7R | 0.36
C1–C5 | Capacitor 15pF 3kV 564R30GAQ15 | 0.80
C6 |  Electrolytic Capacitor 3$\mu$F 50V | 0.11
C7 | Ceramic Capacitor 100pF | 0.34
LDR | PDV-P9008 | 2.46
Shunt | Resistor 23$\Omega$ CCF6023R7FKE36 | 0.70
TL072 | Operational amplifier | 0.69
TL431 | Precision Programmable Reference | 0.63
Battery | 9V Battery (two-set package) | 12.00
Sensor | Light meter Urceri MT-912 | 30.00
Needle | Tungsten Electrode 2% Lanthanated | 10.00
PCB | Solderable breadboard | 5.00
Arduino | Arduino model Due | 40.00

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