Rev. Sci. Instrum. 89, 105104 (2018); https://doi.org/10.1063/1.5041814 89, 105104

© 2018 Author(s).

A new B-dot probe circuit for magneticdiagnostics of radio frequency dischargesCite as: Rev. Sci. Instrum. 89, 105104 (2018); https://doi.org/10.1063/1.5041814Submitted: 26 May 2018 . Accepted: 18 September 2018 . Published Online: 04 October 2018

Kai Zhao , Yong-Xin Liu , De-Qi Wen, Demetre J. Economou, and You-Nian Wang

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REVIEW OF SCIENTIFIC INSTRUMENTS 89, 105104 (2018)

A new B-dot probe circuit for magnetic diagnosticsof radio frequency discharges

Kai Zhao,1,2 Yong-Xin Liu,1 De-Qi Wen,1 Demetre J. Economou,2 and You-Nian Wang1,a)1Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams (Ministry of Education),School of Physics, Dalian University of Technology, Dalian 116024, China2Plasma Processing Laboratory, Department of Chemical and Biomolecular Engineering, University of Houston,Houston, Texas 77204-4004, USA

(Received 26 May 2018; accepted 18 September 2018; published online 4 October 2018)

Accurate magnetic measurements in radio frequency capacitively coupled plasmas (CCP) are chal-lenging due to the presence of inherently strong electric fields and relatively weak magnetic fields. Inthis work, a new B-dot probe circuit is presented, comprising two variable capacitors in a tunable seriesresonance circuit, with a center-tapped, step-up transformer. The output characteristics of the probeare predicted using two distinct equivalent circuit models, one for the differential mode and the otherfor the common mode. A Helmholtz coil and a Faraday cup are used for experimental validation of thepredicted probe output. By tuning the two variable capacitors in the circuit, the magnetic probe canachieve improved signal-to-noise ratio by amplifying the inductive signal, while suppressing capaci-tive coupling interference. Using the newly designed probe, magnetic measurements in typical CCPare presented. Published by AIP Publishing. https://doi.org/10.1063/1.5041814

I. INTRODUCTION

The B-dot probe is a workhorse for measuring the mag-netic induction in a variety of plasma devices.1–7 Particularlyin inductively coupled plasmas (ICP), magnetic field measure-ments allow visualization of the spatial distribution of radiofrequency (rf) current density, electric field as well as powerdeposition in the reactor, providing insights into importantphysics, such as the anomalous skin effect.8,9 On the otherhand, accurate measurements of the magnetic field in capaci-tively coupled plasmas (CCP) are challenging because of theinherently weak magnetic field, and the so-called capacitivepickup,10 which stems from the capacitive coupling betweenthe oscillating plasma potential and the probe coil, causingserious interference.

In order to improve the signal-to-noise ratio of the B-dotprobe, one has to increase the inductive signal output (i.e.,improve the magnetic sensitivity) and/or suppress the capaci-tive signal output (i.e., enhance the capacitive pickup rejec-tion). Traditionally, one utilizes a step-up transformer toincrease the B-dot output.11 An alternative method is to usean amplifier.12 On the other hand, a common method tosuppress the capacitive signal output is to reduce the poten-tial difference between the plasma and the probe coil, forexample, by using a coaxial cable in series with the probecoil. By comparing several of the most commonly used mag-netic probe designs, Franck et al.13 reported that the center-tapped transformer (CTT) exhibited optimal performance forrejecting capacitive pickup. More recently, Sun et al.14 pro-posed a tunable magnetic probe with a CTT to improve thesignal-to-noise ratio. It was difficult, however, to eliminate theelectrical asymmetry caused by the rotary variable capacitorused.

a)Electronic mail: ynwang@dlut.edu.cn

In this work, we propose a new tunable series reso-nance circuit for a B-dot probe for magnetic diagnostics ofrf discharges. The circuit is designed geometrically sym-metric with respect to the center tap; moreover, we intro-duce two symmetrical variable capacitors to eliminate theelectrical asymmetry. As will be shown below, the electri-cal symmetry can be ensured by tuning these two variablecapacitors, eventually achieving a minimum capacitive inter-ference signal. Meanwhile, the inductive signal can be highlyamplified.

This paper is organized as follows. The details of the B-dotprobe design are described in Sec. II. In Sec. III, by employingtwo distinct equivalent circuit models and well-defined exper-iments, we present the output characteristics of the magneticprobe. Magnetic induction measurements in a typical CCPreactor are shown in Sec. IV before concluding.

II. PROBE DESIGN

Miniaturization, high magnetic sensitivity, and highcapacitive pickup rejection are the three most desirable charac-teristics of a magnetic probe. To achieve these characteristics,one must pay particular attention to the following: (i) probecoil design, (ii) probe shaft and enclosure, (iii) electrostaticshielding, and (iv) signal processing circuit.

Figure 1(a) illustrates a schematic of the magnetic probe.The probe head is a two-turn circular coil with a diameter ofd0 = 3 mm, made of 0.51 mm-diameter enameled copper wire.The two ends of the coil are twisted in pairs and welded to theinner conductors of two 60 cm-length semirigid coaxial cables(SFT50-1). To reduce the influence of plasma potential oscil-lations on the probe shaft, the coaxial cables are wrapped witha grounded aluminum foil and then inserted into a supportingquartz tube. The other ends of the two cables are terminatedby the subminiature version A (SMA) connectors, which can

0034-6748/2018/89(10)/105104/6/$30.00 89, 105104-1 Published by AIP Publishing.

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FIG. 1. Schematics of the magnetic probe (a) and the signal processing circuit(b). Variable capacitors C1 and C2 are tuned to obtain electrical symmetry.

be connected to the input ports of the signal processing circuit[Fig. 1(b)].

The signal processing circuit is primarily composed of atunable series resonance circuit coupled with a center-tapped,step-up transformer. A sketch of the center-tapped, step-uptransformer is illustrated in Fig. 2. It consists of a 2-turn pri-mary winding and a 3-turn secondary winding. Both windingsare made of enamelled copper wire 0.8 mm in diameter, woundon a circular spool frame with an outer diameter of 70 mm.A center tap connects the geometric center of the primarywinding to ground. Electrical measurements were carriedout using an RLC meter (Tonghui TH2822C), with standard“short” and “open” corrections, at room temperature. Theinductances of the primary and secondary windings were 1.78and 2.75 µH, respectively. The primary winding features twovariable capacitors, forming a series resonant circuit. Thesecapacitors are placed in geometrically symmetric positionswith respect to the center tap, in order to promote circuitsymmetry. The secondary winding is connected to the outputport of the circuit. A Faraday shield, fabricated by a single-sided printed circuit board (PCB) with a comb-like pattern,is placed between the primary and secondary windings tosuppress capacitive coupling.

FIG. 2. Sketch of the center-tapped, step-up transformer. The turns ratio of theprimary-to-secondary windings of the transformer is 1:1:3. A Faraday shieldis placed between the primary and secondary windings to suppress capacitivecoupling. No ferrite core is employed.

III. PROBE OUTPUT CHARACTERISTICS

To gain insight into the output characteristics of theB-dot probe, two distinct equivalent circuit models are used.The output signal of the probe can be regarded as a superpo-sition of two components: (i) the inductive component causedby the time derivative of the magnetic flux [the differentialmode (DM) signal] and (ii) the capacitive component result-ing from capacitive coupling between the plasma and the probecoil [the common mode (CM) signal]. In this section, we willbriefly describe these two modes followed by modeling andexperimental results.

A. Differential mode (DM) signal output

A schematic of the equivalent circuit for the DM signal isshown in Fig. 3(a). The induced voltage in the probe coil canbe considered an ideal voltage source U id(t) = −jωnaB0e jωt ,where ω and B0 represent the angular frequency and theamplitude of the time-varying magnetic field, respectively.Furthermore, n and a denote the number of turns and thecross-sectional area of the probe coil, respectively. The pri-mary circuit consists of two coaxial cables (Z1 and Z2), twovariable capacitors (C1 and C2), and the inductance (Lp) andresistance (Rp) of the primary winding. The secondary cir-cuit includes a coaxial cable (Z3), and the inductance (Ls)and resistance (Rs) of the secondary winding. The load is theinput impedance Z in of the oscilloscope. The coaxial cablesare assumed to be lossless so that the propagation constant ofthe transmission line is purely imaginary (γ = jβ, where β isthe phase-shift coefficient). Furthermore, in the presence of theFaraday shield, the capacitive coupling between the primaryand secondary windings of the transformer is neglected in thiscircuit model.

FIG. 3. (a) Schematic diagram of the equivalent circuit for the differentialmode signal, with all outer conductors of the coaxial cables assumed to begrounded; model predictions (lines) and experimental data (points) of the DMsignal output (Uod and φod ) as a function of C2, for different values of C1:(b) C1 = 110 pF, (c) C1 = 170 pF, and (d) C1 = 230 pF.

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The DM signal output is investigated experimentally usinga Helmholtz coil, for producing a well-defined magnetic field.The Helmholtz coil is constructed with a pair of single-turnloops with a radius R0 = 35 mm, separated by 35 mm. A13.56 MHz sinusoidal voltage (0–20 V), produced by afunction generator (Tektronix AFG3252C), is applied to theHelmholtz coil. The probe coil is positioned at the center ofthe Helmholtz coil to detect the magnetic induction.

The output signal of the B-dot probe is recorded by usingan oscilloscope (LeCroy, Waverunner 104Xi, 1 GHz analogbandwidth, 10 Gs/s). Simultaneously, the current Icoil in theHelmholtz coil is measured by using a current monitor (Pear-son 6585, 250 MHz bandwidth), through which the magneticfield amplitude B0 at the center of the Helmholtz coil can becalculated by B0 = (4/5)3/2µ0NIcoil/R0. Here, µ0 = 4π × 10−7

T m/A is the permeability in free space, and N represents thenumber of turns of the Helmholtz coil. The coil current is fixedat Icoil = 128 mA for all conditions. Based on the above, B0

and the induced voltage amplitude in the probe coil, U id , areestimated to be 3.29 µT and 3.96 mV, respectively. The mag-netic sensitivity, determined by U id /B0, is about 1.2 mV/µT,quite a small value.

Figures 3(b)–3(d) show model predictions (lines) andexperimental data (points) of the variation of the DM out-put voltage amplitude Uod and the relative phase φod [withrespect to the current Icoil(t) in the Helmholtz coil] as afunction of C2, for different values of C1. Reasonably goodagreement between modeling and experiment, for both Uod

and φod , is observed. For every value of C1 tested, Uod

goes through a maximum (Uod ,max) as a function of C2,at which φod exhibits an abrupt change. In quantitativeterms, Uod ,max can reach 42 mV by tuning C2 for givenC1, which is more than one order of magnitude greater thanU id (∼3.96 mV); correspondingly, the magnetic sensitivity isimproved from 1.2 to 12.8 mV/µT. These results are indica-tive of a typical series resonance characteristic. By increas-ing C2, the circuit transitions from capacitive reactance toinductive reactance. Resonance occurs at the so-called “res-onance point” where the inductive reactance is equal to thecapacitive reactance, resulting in minimum impedance and,thus, maximum current. Apart from the series resonance cir-cuit, the step-up transformer also plays an important role inamplifying Uod .

B. Common mode (CM) signal output

A schematic diagram of the equivalent circuit for the CMsignal is illustrated in Fig. 4. Compared to the DM case, onlythe primary circuit is changed. In this scenario, the fluctuatingpotential difference between plasma and probe coil can berepresented by an ideal voltage source U ic(t). The capacitivecurrent flows through two branches, i.e., a-c and b-d, and thengoes via the center tap to ground. In the a-c branch, the coaxialcable (Z1), variable capacitor (C1), and half of the inductance(Lp/2) and half of the resistance (Rp/2) in the primary windingare in series. A symmetric, series connection involves Z2, C2,Lp/2, and Rp/2 in the b-d branch.

Experimentally, the CM signal output was investigatedby using a Faraday cup, which is employed to produce a

FIG. 4. (a) Schematic diagram of the equivalent circuit for the common modesignal, with all outer conductors of the coaxial cables assumed to be grounded;model predictions (lines) and experimental data (points) of the CM signaloutput (Uoc and φoc) as a function of C2, for different values of C1: (b) C1 =110 pF, (c) C1 = 170 pF, and (d) C1 = 230 pF.

homogeneous electric field. The Faraday cup is 160 mm inlength and 40 mm in inner diameter, and it is mechanicallyfastened to a grounded box. A 13.56 MHz rf voltage gener-ated by using a power amplifier is applied to the Faraday cup.The probe coil is inserted into the center of the Faraday cupto detect the capacitive signal. The voltage amplitude Ucup onthe Faraday cup is measured by using a high-voltage probe(Tektronix P6015A). Here, Ucup is kept at 100 V for all condi-tions. Accordingly, assuming that the probe coil has a 1.5 pFcoupling to the Faraday cup/plasma, the amplitude of U ic(t) isestimated to be about 102 mV.

Figures 4(b)–4(d) display the dependence of the CM out-put [voltage amplitude Uoc and phase φoc with respect to thevoltage Ucup(t)] on C2, for different values of C1. Again areasonably good agreement between model predictions andexperimental data is observed. The variation of Uoc with C2 ismore complicated compared to Uod [see Figs. 3(b)–3(d)]. AtC1 = 110 pF, Uoc is observed to first slowly decrease with C2

and then to rise sharply, going through a maximum. However,at larger C1 (i.e., 170 pF and 230 pF), Uoc first increases withC2, reaches a maximum, and then goes through a minimum, abehavior not seen at C1 = 110 pF.

From Fig. 4, one can draw the following conclusions.(i) For given C1, two extreme values (a maximum Uoc ,max

and a minimum Uoc ,min) occur in the curve of Uoc vs. C2.Around each extreme value, an abrupt change in φoc is clearlyobserved. (ii) In the vicinity of Uoc ,max, the abrupt change inφoc is supposed to be caused by electrical resonance; i.e., byincreasing C2, an impedance transition occurs, leading to anabrupt change in the current phase, which is very similar to thecases in Fig. 3. (iii) In the vicinity of Uoc ,min, the abrupt changein φoc is related to the change in the net current direction inthe primary winding.

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Results presented thus far demonstrate that for given C1,Uoc ,min can reach a minimum value (∼0) by tuning C2. Underthe condition Uoc ,min = 0, the branch currents Ip1 (in a-cbranch) and Ip2 (in b-d branch) are identical both in ampli-tude and in phase so that they cancel out, leading to sup-pression of capacitive pickup. In short, the main role of thetwo variable capacitors is to balance the capacitive currentsflowing through the two branches to the ground, when oper-ated in the CM mode. In addition, as important elementsin the series resonant circuit, these capacitors also play arole in amplifying the inductive signal, when operated in theDM mode.

In order to achieve the highest signal-to-noise ratio,Uod ,max in the DM mode and Uoc ,min in the CM case must occurat the same value of C2. This can be achieved, according to theresults shown in Figs. 3 and 4. In Figs. 3(b)–3(d), with increas-ing C1, the corresponding C2 for obtaining Uod ,max graduallydecreases. By contrast, in Figs. 4(b)–4(d), with increasingC1, the corresponding C2 for obtaining Uoc ,min graduallyincreases. Thus, by increasing C1, there is a certain value ofC2 at which Uod ,max and Uoc ,min occur simultaneously; this isthe optimal operating point. Figure 5 illustrates modeling andexperimental data for obtaining the optimal operating point,which occurs at C1 = C2 = 155 pF.

It is worth mentioning that, in addition to the two variablecapacitors (C1 and C2), the output characteristics of the B-dotprobe can be significantly affected by the length of the coaxialcables (Z1, Z2, and Z3) and the turns of the transformer wind-ings, which determine the inductance of the circuit and thus theresulting resonant behavior. In addition, changing the coaxialcables and the transformer windings can shift the optimal oper-ating point. Therefore, the performance to suppress capacitivepickup can also be affected. On the other hand, the amplifi-cation of the B-dot probe depends not only on the couplingcoefficient of the transformer but also on the quality factorof the LC resonance circuit. Hence, proper sizes of the coax-ial cables and transformer windings are crucial for achieving

FIG. 5. Model predictions (lines) and experimental data (points) of C2 (atdifferent values of C1) for obtaining Uod ,max in the DM case and Uoc ,min inthe CM case. The intersection between the solid and dashed lines defines theoptimal operating point.

a satisfactory/stable performance of the device. Furthermore,it should be noted that the optimal operating point stronglydepends on the working frequency and is also sensitive tochanges in temperature.

IV. MAGNETIC MEASUREMENTS IN A CCP REACTOR

The B-dot probe was tested in a typical CCP reactor15,16

to further evaluate its performance. An argon plasma was sus-tained at 3 Pa, by a 13.56 MHz power supply with a fixed outputpower of 100 W. The diameter of the parallel plate electrodeswas 21 cm. The measurement point was located at r = 10 cmclose to the electrode edge. In order to clarify the crucial rolesof the probe circuit, a comparative experiment was performedunder three different circuit conditions: (i) without any tun-ing capacitor, (ii) with a single tuning capacitor, and (iii) withboth tuning capacitors. For each situation, we measured theraw signals from the B-dot probe for three different probe ori-entations; i.e., the plane of the probe coil is perpendicular tothe electrodes (φ = ±π/2 to measure the azimuthal componentof the magnetic induction Bϕ) and parallel to the electrodes(φ = 0 to measure the axial component of the magnetic induc-tion Bz). Measurement results are presented in Fig. 6 anddiscussed below.

Case I: without any tuning capacitor. In this case, thecapacitors C1 and C2 illustrated in Figs. 3(a) and 4(a) wereremoved and replaced by a connecting wire. Without the

FIG. 6. Comparisons of the raw signals from the B-dot probe output underthree different circuit conditions: (a) without any tuning capacitor, (b)with a single tuning capacitor, and (c) with both tuning capacitors. Mea-surements were performed in a CCP reactor for three probe orientations(φ = 0, ±π/2).

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tuning capacitors, the output signals [Fig. 6(a)] are observedto be quite weak as compared to those in the other two cases[Figs. 6(b) and 6(c)]. This could be mainly attributed to a rela-tively large inductive reactance ( jωLp ∼ 152Ω) of the primarywinding in the transformer, which leads to a quite small cur-rent/voltage in the primary circuit and, hence, a small inducedvoltage in the secondary circuit. On the basis of the equiva-lent circuit of the DM signal, taking an approximate magneticfield intensity (Bϕ ∼ 4.5 µT) into account, we estimate thatthe amplitude of the DM signal is less than 1 mV, quite asmall value. This estimation is in accordance with our mea-surement results. In Fig. 6(a), the signals for all orientationsexhibit similar values both in amplitudes and in phases, indi-cating that the signals detected are mainly caused by the CMinterference, whereas the contribution from the DM compo-nent is fairly small. These CM signals could be due to the slightasymmetries in the center-tapped, step-up transformer or eventhe B-dot probe itself.

Case II: with a single tuning capacitor. In this case, onlycapacitor C1 was removed and replaced by a connecting wire.Before our experiments, the B-dot probe was tested and cali-brated using the Helmholtz coil. By tuning C2, we operated theprobe circuit at the resonance frequency of 13.56 MHz, i.e., asame maximum DM signal as case III was achieved. Even so,one can clearly identify from Fig. 6(b) that the amplitude ofthe output signal in the orientation φ = 0 is sufficiently largecompared to the signals in the orientations φ = ±π/2, suggest-ing that the CM interference is still considerable. Due to thecontamination of the DM signal by the CM interference, thesignal waveforms in the orientations φ = ±π/2 become highlyasymmetric. In practice, when operated at the resonance point,both DM and CM signals can be greatly amplified. Due to theabsence of C1, the circuit asymmetry of the two branches in a-cand b-d, as shown in Fig. 4(a), generates two extremely unbal-anced capacitive currents flowing through the two branches tothe ground and, thus, a large net current occurs in the primarywinding of the transformer, significantly contributing to theoutput signal.

Case III: with both tuning capacitors. In order to makethe probe circuit symmetric, both capacitors were employedin this case. As mentioned before, by tuning capacitors C1

and C2, we operated the B-dot probe at the optimal operat-ing point; namely, using the Helmholtz coil and the Faradaycup, Uod ,max in the DM mode and Uoc ,min in the CM casecan be achieved simultaneously. Observed from Fig. 6(c),one can find that the signals in the orientations φ = π/2 andφ = −π/2 are almost equal in amplitude but out of phase witheach other. This coincides with the fact that reversing the exter-nal magnetic field will result in reversal of the polarity of theinduced voltage in the probe coil, but the amplitude will remainunchanged. In addition, the amplitude of the output signal inthe orientation φ = 0 is observed to be negligible comparedto the signals in the orientations φ = ±π/2 in accordance withthe fact that the axial magnetic field component Bz should bezero in a CCP reactor. The results presented thus demonstratedthat with the two-tuning-capacitor circuit the B-dot probecan achieve an improved signal-to-noise ratio by amplifyingthe inductive signal, while suppressing capacitive couplinginterference.

V. CONCLUSIONS

In summary, a new B-dot probe circuit was developedfor magnetic diagnostics of radio frequency (rf) discharges.To improve the magnetic sensitivity and enhance the capac-itive pickup rejection, attention was paid to the probe coildesign, probe shaft and enclosure, electrostatic shielding,as well as the signal processing circuit. The latter wasprincipally composed of a tunable series resonance circuitfeaturing two variable capacitors, combined with a center-tapped, step-up transformer. In addition, a Faraday shieldwas placed between the primary and secondary windings ofthe transformer to further suppress electrostatic capacitivecoupling.

We developed two distinct equivalent circuit models tostudy the output signal of the B-dot probe, including the dif-ferential mode (DM) component caused by the time derivativeof the magnetic flux and the common mode (CM) compo-nent resulting from capacitive coupling between the oscillatingplasma potential and the probe coil. For experimental vali-dation, we used a Helmholtz coil to produce a well-definedmagnetic field and a Faraday cup to produce a homoge-neous electric field. The output characteristics of the B-dotprobe were predicted by the circuit models and validatedby the experiments. By tuning the two variable capacitorsin the circuit, the DM output voltage Uod could be highlyamplified (as compared to the original output voltage ofthe probe coil), while the CM output voltage Uoc could besubstantially reduced, yielding an improved signal-to-noiseratio. Furthermore, the optimal operating condition for achiev-ing the highest signal-to-noise ratio was determined throughmodeling.

The B-dot probe was tested in a typical CCP reactor tofurther evaluate its performance. By performing a compara-tive experiment under three different circuit conditions, i.e.,without any tuning capacitor, with a single tuning capacitor,and with both tuning capacitors, the crucial characteristics ofthe probe circuit were demonstrated.

ACKNOWLEDGMENTS

This work was supported by the National Natural Sci-ence Foundation of China (NSFC) (Grant Nos. 11335004and 11722541). D.J.E. acknowledges support from theNational Science Foundation (No. PHY-1500518) and theDepartment of Energy, Office of Fusion Energy Science(No. DE-SC0014132). K.Z. acknowledges financial supportfrom the China Scholarship Council. The authors grate-fully acknowledge Peng-Cheng Du and Xiang-Yu Wang forexperimental assistance.

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