Modeling and Analyzing the Mutual Inductance of Rogowski Coils of Arbitrary Skeleton

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Article Modeling and Analyzing the Mutual Inductance of Rogowski Coils of Arbitrary Skeleton

Xiaoyu Liu , Hui Huang * and Chaoqun Jiao School of Electrical Engineering, Beijing Jiaotong University, Beijing 100044, China * Correspondence: [email protected]; Tel.: +86-186-0196-8811

Received: 1 July 2019; Accepted: 31 July 2019; Published: 2 August 2019

Abstract: There are Rogowski coils of various shapes in the on-site measurement, and it is difficult to calculate the electrical quantities of Rogowski coils of curved skeleton and circular cross-section by simulation software. This paper proposes a theoretical derivation to calculate the mutual inductance between the conductors of any shape and Rogowski coils with skeletons of any shape. Based on the derivation, the influence of four skeleton shapes of Rogowski coils and four shapes of the primary conductors on the mutual inductance of Rogowski coils are studied by the comparison between the ideal cases and some non-ideal ones. The gap and gap compensation of the openable Rogowski coils are also considered. Experiments verify the numerical results according to the derivation. It is shown that to reduce the errors of the measurement the circular skeleton deformation should be avoided, the coil’s skeleton should be with curved angle, the primary conductor should be as straight as possible and should go through the center of the skeletons vertically. Furthermore, for the Rogowski coils of the rectangular skeleton, we propose a new skeleton structure to reduce the deviation influence of the primary conductors.

Keywords: Rogowski coil; shape of the skeleton; current measurement; shape of the primary conductors

1. Introduction Rogowski coils, which are also called current measurement coils or differential current sensors, are designed to measure different types of alternating and transient current (from tens to several thousands of amperes) [1]. Traditional electromagnetic current sensors cannot accurately measure high Ampere currents because of the saturation problem of the magnetic core [2], while the Rogowski coil is different: Its skeleton is made of non-ferromagnetic materials, and its turns evenly wound on the skeleton [3]. The Rogowski coil has been widely used in multiple industries [4,5] due to its characteristics—simplicity, low cost, light weight, linearity, dynamic bandwidth, non-invasive and immunity for core saturation [6–10]. The operating principle of the Rogowski coil is based on Ampere and Faraday laws [11]. When a Rogowski coil is around an alternating current-carrying conductor, the coil generates an inductive voltage [1]. Mutual inductance is a crucial parameter between the inductive voltage and the primary conductors [12–15]. Therefore, to study the influence quantities on mutual inductance is important to ensure the accuracy of Rogowski coil [16,17]. Antonio Cataliotti et al. present the results of an experimental study dealing with the characterization and error compensation of a commercial Rogowski coil. The influence of the position of the primary conductor on the errors of experimental results is analyzed [18]. Mirko Marracci et al. present the results of an analytical and experimental study of the critical parameters affecting the mutual inductance between the straight primary conductors and the Rogowski coil with the rectangular cross-section of the circular skeleton, while the study focuses on the case when the primary conductors

Sensors 2019, 19, 3397; doi:10.3390/s19153397 www.mdpi.com/journal/sensors Sensors 2019, 19, 3397 2 of 16 are straight and located in the hole of the coil, the shape of skeleton is circular and the cross-section of the circular skeleton is rectangular [19]. The effects of the geometrical parameters on high frequency performance of Rogowski coil are studied in [20], and the mutual inductance is also given of the Rogowski coils with the rectangular, circular and oval cross-sections and the primary conductors of the infinite length in the paper. In [21], the mutual inductance between the discrete Rogowski coils and the conductors with the circular and rectangular cross-sections is calculated by the magnetic vector potential method. The influence of the section parameters, the tilt angles, and the eccentricity of the conductors on the errors of the mutual inductance is analyzed, when their conductors are straight and of the infinite length and their shapes of the cross-sections are rectangular. The influence of the adjacent external conductors on the mutual inductance of the Rogowski coil is studied in [2], and when the primary conductors are straight and of the infinite length. Mengyuan Xu et al. analyze the influence of the position of the primary conductors on the measurement errors of a circular array of the magnetic sensors for current measurement, and the straight and infinitely long primary conductors are located inside of the circular array of the magnetic sensors [22]. A special experiment is carried out to prove that the mutual inductance of PCB Rogowski coil is immune to the variation of the center of the plasma current in [4], and the formula of mutual inductance applies to the cases with the infinite-length primary conductors and the uniform magnetic field on the skeleton section. Luka Ferkovic´ et al. present the influence of the primary conductor from the central position of the Rogowski coil, and their studies focus on the case with the infinite-length straight conductors and the Rogowski coil with the rectangular cross-sections [23,24]. Mario Chiampi et al. investigate the effects of some influence quantities on flexible Rogowski coil, and the numerical model is based on the magnetic vector potential for cases with the straight primary conductor and the circular skeleton of the Rogowski coil [25]. Tao et al. present the calculation and measurement of mutual inductance of the PCB Rogowski coil under several conditions, including the eccentricity from axis, external parallel and perpendicular of the primary conductors [26]. Unfortunately, it is difficult to get the mutual inductance of the Rogowski coil by 3D electromagnetic simulation software because of the large number of its turns and the huge difference in the sizes between its enameled wire (usually the diameter of the cross-section is less than 0.17 mm) and the primary conductors (usually the side of the cross-section is larger than 120 mm). Therefore, the theoretical derivation and experimental study for the mutual inductance between the conductor and the Rogowski coil are effective approaches to figure out the influence quantities. Since there are various shapes of the conductor and Rogowski coil in the on-site measurement [27] (Figure1 shows some cases), it is crucial to figure out the mutual inductance between the Rogowski coil with the skeleton of any shape and the primary conductor of different shapes in order to reduce the measurement errors. Here, we present a mathematical model to calculate the mutual inductance between the Rogowski coil with the skeleton of any shape and the primary conductor of any shape. Moreover, we compare the results of the experimental and numerical calculation, which are based on our theoretical derivation for 3 kinds of Rogowski coils and the conductors with several common shapes. Moreover, we present a new structure of rectangular skeletons with arc angle corner to reduce the errors. The numerical calculation results show that this structure can greatly decrease the errors (about 12.6%) of mutual inductance caused by the position and cross-section of the primary conductors. Sensors 2019, 19, x FOR PEER REVIEW 3 of 16 Sensors 2019, 19, 3397 3 of 16

(a) (b)

FigureFigure 1.1. ConductorsConductors andand RogowskiRogowski coils. coils. ( a()a) Conductors Conductors with with some some shapes; shapes; (b )(b ﬁve) five Rogowski Rogowski coils coils of diofﬀ differenterent structures; structures; (c) non-circular(c) non-circular Rogowski Rogowski coil; coil; (d) ( rectangulard) rectangular Rogowski Rogowski coil. coil.

2.2. Theoretical Derivation of Mutual InductanceInductance TheThe basicbasic RogowskiRogowski coilcoil relationrelation isis [[28]28] d𝑖di 𝑣v = = −𝑀M d𝑡 (1)(1) − dt where v and i are the instantaneous values of the induce voltage of the Rogowski coil and the primary v i wherecurrent respectively,and are the and instantaneous M is the mutual values inductance of the induce between voltage the Rogowski of the Rogowski coil and coilthe andprimary the primaryconductor. current respectively, and M is the mutual inductance between the Rogowski coil and the primary conductor. 2.1. Rogowski Coils with Arbitrary Skeleton and Straight Conductors 2.1. Rogowski Coils with Arbitrary Skeleton and Straight Conductors To study the mutual inductance between the arbitrary primary conductor and the Rogowski coil To study the mutual inductance between the arbitrary primary conductor and the Rogowski coil of the arbitrary shape skeleton, we begin with what goes between the straight conductor and the of the arbitrary shape skeleton, we begin with what goes between the straight conductor and the Rogowski coil with the arbitrary shape skeleton. Rogowski coil with the arbitrary shape skeleton. As shown in Figure 2, the cross-section of the skeleton is a circle with radius r. We pick the center As shown in Figure2, the cross-section of the skeleton is a circle with radius r. We pick the center of the cross-section of a turn as O’. O” is the projection of point O’ onto the xOy plane. The angle of the cross-section of a turn as O’.O” is the projection of point O’ onto the xOy plane. The angle between x-axis and O’ O” is 𝛽, as shown in Figure 2. Then we pick a point P on the cross-section between x-axis and O’ O” is β, as shown in Figure2. Then we pick a point P on the cross-section arbitrarily. A(𝑎, 𝑏, 𝑐) and B(𝑑, 𝑒, 𝑓) are the ends of the straight conductor. 𝑙 is the vertical distance arbitrarily. A(a, b, c) and B(d, e, f ) are the ends of the straight conductor. lP is the vertical distance from from point P to the straight conductor. The angle between the conductor AB and line AP is 𝜃1, while point P to the straight conductor. The angle between the conductor AB and line AP is θ1, while the the angle between the conductor BA and line BP is 𝜃2. N is the number of the turns of a Rogowski angle between the conductor BA and line BP is θ . N is the number of the turns of a Rogowski coil. coil. 2

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θ 2

y β

θ 1

FigureFigure 2. StructureStructure diagram diagram of of a Rogowski a Rogowski coil ofcoil arbitrary of arbitrary shape andshape a ﬁnite-length and a finite-length straight conductor. straight conductor. Here, we assume that the parameter equation of the center line of the distorted coil skeleton is: Here, we assume that the parameter equation of the center line of the distorted coil skeleton is:  ( )  x = χ β  𝑥 = 𝜒(𝛽)  y = $(β) (0 < β 2π) (2)  ≤ 𝑦z = = ψ𝜛((𝛽β)) (0 < 𝛽 ≤ 2𝜋) (2) 𝑧 = 𝜓(𝛽)

TheThe normal normal vector vector of of the the coil coil cross-section cross-section where where point P𝑃 isis located located is is→e 𝑒β⃗(χ(𝜒0(β(𝛽),)$,𝜛0(β()𝛽,)ψ,𝜓′(𝛽))(β)). . 0 TheThe coordinate coordinate equation equation of of point point P P on on the the cross-section cross-section of of the the turn turn which which corresponds corresponds to to 𝑂′(𝜒O (χ(𝛽(β)),𝜛, $((𝛽β)),, 𝜓(𝛽))ψ(β)) isis 0 𝜛(𝛽)cos𝑡 𝜒(𝛽)𝜓(𝛽)𝑠𝑖𝑛𝑡  $ (β) cos t χ (β)ψ (β)sint  ⎧𝑥 = 𝜒(𝛽) +𝑟( 0 + 0 0 )  xP = χ(β) + r( q + q )  ⎪ 2 2 2  ⎪ (χ𝜒(β())𝛽)+ (+$(𝜛β))(𝛽) (χ𝜒 ((β𝛽))ψ𝜓 ((β𝛽)))2++($𝜛(β(𝛽)ψ)𝜓(β(𝛽)))2++((χ𝜒(β())𝛽)2+(+$𝜛(β))(𝛽2))  ⎪ 0 0 0 0 0 0 0 0  ( ) ( ) ( ) !  ⎪ −𝜒( ) 𝛽 cos𝑡 𝜔( 𝛽) 𝜓( 𝛽) 𝑠𝑖𝑛𝑡 t  𝑦 = 𝜛(𝛽) +𝑟( χ0 β cos t + ω0 β ψ0 β sint ) 0<𝑡≤2𝜋,0 < 2π,  y = $(β) + r( q − + q ) ≤ P 2 (3)(3)  ⎨ 𝜒 (𝛽2) + 𝜛 (𝛽2) 𝜒 (𝛽)𝜓 (𝛽)2 + 𝜛 (𝛽)𝜓 (𝛽)2 + 𝜒 (𝛽)2 + 𝜛 (𝛽2) 00 << 𝑟r ≤ 𝑟 r  (χ0(β)) +($0(β)) (χ0(β)ψ0(β)) +($0(β)ψ0(β)) +((χ0(β)) +($0(β)) ) 0  ⎪ ≤  2 2  ⎪ r((χ0(β)) +($0(β𝑟())𝜒)sint(𝛽) + 𝜛 (𝛽) )𝑠𝑖𝑛𝑡  zP =⎪ψ(β) q 𝑧 = 𝜓(𝛽) −  ⎪ 2 2 2 2 2  − (χ (β)ψ (β)) +($ (β)ψ (β)) +((χ (β)) +($ (β)) ) ⎩ 0 0 𝜒0 (𝛽)𝜓0 (𝛽) + 𝜛0 (𝛽)𝜓 (𝛽)0 + 𝜒 (𝛽) + 𝜛 (𝛽) TheThe magnetic magnetic density density magnitude magnitude of of point point P P generated generated by by the the current current 𝐼I flowingﬂowing in in the the straight straight conductorconductor AB AB is is given given by by [29]: [29]:

𝐵 = 𝜇𝐼(cos𝜃 − cos𝜃)4𝜋𝑙⁄ (4) BP = µ I(cos θ cos θ )/4πlP (4) 0 1 − 2 −7 where 𝜇 is the permeability of air, which equals 4 × 10 H/m. where µ is the permeability⃗ of air, which equals 4 10 7 H/m. The0 normal vector of 𝐵 is × − The normal vector of →B is P 𝑒⃗ = (𝑒⃗ ×𝑃𝐴⃗)/𝑒⃗ ×𝑃𝐴⃗ (5) → → →e B = (→el PA)/ →el PA (5) According to Equations (4) and (5), we can× get the| projection× | of the magnetic density onto the According to Equations (4) and (5), we can get the projection of the magnetic density onto the normal vector 𝑒⃗ normal vector e→ β 𝐵 = 𝐵 ∙ ((𝑒⃗ ∙𝑒⃗)(𝑒 ⃗ |𝑒⃗ | (6) BP = BP ((→e →e B)/( →e →e B )) (6) β · β· β In order to get the magnetic flux of the cross-section of the turn, which corresponds to angle 𝛽, In order to get the magnetic ﬂux of the cross-section of the turn, which corresponds to angle we divide 𝑟 and 2π into equal parts 𝑛 and 𝑛, respectively. As shown in Figure 3, we get that β, we divide r and 2π into equal parts n and n , respectively. As shown in Figure3, we get that ∆𝑟 = ,∆𝑡 0= . 𝐵 (𝑖 = 11,2,… , 𝑛 2,𝑗 = 1,2,…,𝑛 ) means the magnetic flux of point r0 2π (.)(.) ∆r = n , ∆t = n . BPβ(i 0.5)(j 0.5) (i = 1, 2, ... , n1, j = 1, 2, ... , n2) means the magnetic ﬂux of 1 2 − − Ppoint located P located in the insection the section (𝑟 = ( (𝑖−0.5r = (i)∆𝑟,0.5 𝑡)∆r , = t (𝑗= (−j 0.5)∆𝑡0.5). )∆t). − −

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ri r i−1ri Δ Δr ri−1t Δr Δt t =0 O t =0 O

Figure 3. The magnetic field of one point on a circular cross-section of one turn. Figure 3. TheThe magnetic magnetic field field of one point on a circular cross-section of one turn. According to Equations (3) and (6), the magnetic flux of the cross-section of the kth turn correspondingAccording to to angle Equations 𝛽 is (3) and (6), the magnetic flux flux of the cross-section of the kth turn corresponding to angle β𝛽is is Xn1 Xn2 𝛷 = 𝐵 𝑟∆𝑟∆𝑡 (7) Φβk = B(.)(.)Pβ(i 0.5)(j 0.5)r∆r∆t (7) − − 𝛷 = i = 1 j = 1 𝐵(.)(.) 𝑟∆𝑟∆𝑡 (7) According to Equation (7), the magnetic fluxflux of all the turns of the Rogowski coilcoil isis According to Equation (7), the magnetic flux of all the turns of the Rogowski coil is N 𝛷 =X 𝛷 (8) Φ = Φβk (8) 𝛷 = 𝛷 (8) k = 1 where 𝛽k is the angel 𝛽 corresponding to the kth turn of the Rogowski coil. whereAccording β𝛽kk isis thethe angelangelto Equation β𝛽corresponding corresponding (8), mutualto to inductancethe thek kthth turn turn M of ofof the thethe Rogowski RogowskiRogowski coil. coil.coil with the straight primary conductorAccording is [21] to Equation (8), mutual inductance M of the Rogowski coil with the straight primary conductor is [21] conductor is [21] Φ 𝑀 = Φ𝐼 (9) M = Φ (9) 𝑀 = I𝐼 (9) 2.2. Rogowski Rogowski Coils Coils with with Arbitrary Arbitrary Skeleton and Curved Conductors 2.2. Rogowski Coils with Arbitrary Skeleton and Curved Conductors To study the influenceinfluence of the curvilinear conductor in the on-site measurement on mutual inductanceTo study of of the the Rogowski influence coilof the of curvilineararbitrary shap shape conductore skeleton, in thewe deduceon-site measurementan expression onof mutual inductance betweenbetweenof the Rogowski thethe Rogowski Rogowski coil coilof coil arbitrary of of arbitrary arbitrary shap shape eshape skeleton, skeleton skeleton we and deduceand the the curvilinear ancurvilinear expression conductor conductor of mutual in this in thisinductancesection, section, as shown between as shown in Figurethe in RogowskiFigure4. 4. coil of arbitrary shape skeleton and the curvilinear conductor in this section, as shown in Figure 4.

y β y β

Figure 4. Structure diagram of a Rogowski coil and a finite-length curvilinear conductor. Figure 4. Structure diagram of a Rogowski coil and a finite-length curvilinear conductor. DivideFigure the conductor 4. Structure into diagram infinitesimal of a Rogowski elements. coil and The a magneticfinite-length field curvilinear generated conductor. by the s th current Divide the conductor into infinitesimal elements. The magnetic field generated by the sth current element of the curve at point P is given by [30,31] elementDivide of the the curve conductor at po intinto P infinitesimalis given by [30,31] elements. The magnetic field generated by the sth current element of the curve at point P is given by [30,31] →⃗ →⃗ 3 𝐵Bpm = (𝜇 (µ0𝐼𝑑𝑙Idls ×𝑙l0))(4𝜋/(4π |l𝑙0 | )) (10)(10) × s s 𝐵 = (𝜇 𝐼𝑑𝑙⃗ ×𝑙⃗ )(4𝜋 |𝑙 |) (10) The magnetic flux on the turn corresponding to angle 𝛽 generated by the sth current element is

Φ, Theand magneticwe can get flux Φ on the from turn Equations corresponding (8) and to (10). angle The 𝛽 magngeneratedetic flux by theof the sth wholecurrent turn element of the is Φ, and we can get Φ from Equations (8) and (10). The magnetic flux of the whole turn of the

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The magnetic flux on the turn corresponding to angle β generated by the sth current element is Φβs, Sensors 2019, 19, x FOR PEER REVIEW 6 of 16 and we can get Φβs from Equations (8) and (10). The magnetic flux of the whole turn of the Rogowski coil generated by the sth current element is Φs, and we can get Φs from Equation (8). The magnetic Rogowski coil generated by the sth current element is Φ, and we can get Φ from Equation (8). The flux of the whole coil generated by the whole conductor is given by magnetic flux of the whole coil generated by the whole conductor is given by X Φ = Φ (11) Φ = Φs (11) s Mutual inductanceinductanceM Mof of the the Rogowski Rogowski coil withcoil with the curvilinear the curvilinear primary primary conductor conductor can be obtained can be accordingobtained according to Equation to Equation (9). (9).

3. Experimental Platform To verify the derivation which we deduced above, we build the experimental platform, as shown in FiguresFigures5 and5 and6, to6, study to study the mutual the mutual inductance inductance between between the Rogowski the Rogowski coils of non-circularcoils of non-circular skeleton andskeleton the primaryand the primary conductors conductors of any shape, of any size shape, and size position. and position. We wind We three wind Rogowski three Rogowski coils by coils the electricby the electric winding winding machine machine to simulate to simulate the condition the condition in which in which a flexible a flexible Rogowski Rogowski coil deforms coil deforms under externalunder external forces. forces. Coil A Coil is of A circularis of circular skeleton skeleton and and gap gap compensation, compensation, while while both both Coil Coil Band B and Coil Coil C areC are of ovalof oval skeleton skeleton and and gap gap compensation compensation [25], [25], as shown as shown in Figure in Figure5. The 5. threeThe three Rogowski Rogowski coils havecoils thehave same the same circumstance, circumstance, cross-sections cross-sections and turn and numbers.turn numbers. The three The three Rogowski Rogowski coils arecoils fixed are onfixed three on acrylicthree acrylic plates plates with with the test the holestest holes to hold to hold the the coils coils in shapein shape and and place. place. The The parameters parameters ofof thethe three Rogowski coils are shown in Table1 1:: N is the number of the turns of the Rogowski coil, r is the radius of the cross-section cross-section of of the the coil coil skeleton, skeleton, LgL isg isthe the length length of ofthe the gap gap of the of theRogowski Rogowski coil, coil, Ncg isN cgtheis turn the turnnumber number of the of gap the compensation, gap compensation, and 𝛽 and is βtheg is angle the angle between between the line the from line the from middle the middle point of point the ofgap the to gapthe tocenter the centerof the skeleton of the skeleton O and Ox-axis. and xTable-axis. 2 Table shows2 shows the inform the informationation of the oftest the holes test which holes whichare located are located in the primary in the primary conductors conductors of the fixed of the plates, fixed while plates, holes while label holes I1, label I2, I3, I1, …, I2, I11 I3, and... , O1, I11 andO2, O3, O1, …, O2, O6 O3, represent... , O6 representthe inner theholes inner and holes the outsid and thee holes outside respectively holes respectively on the fixed on theplate. fixed In order plate. Into orderreduce to the reduce interference the interference of the coils, of the the coils, nearest the nearest charged charged body bodyis 2.5 ism 2.5 away m away from from the coils the coils in the in theexperiments. experiments.

(a) (b)

Figure 5. Fixed plates with test holes. ( (aa)) The The fixed fixed plate plate of of Coil Coil A; ( b) the fixed fixed plate of Coil B; (c) the fixedfixed plate of Coil C; (d)) thethe picturepicture ofof thethe testtest plate.plate.

As shown in Figure 6, we use ITECH IT7626 AC programmable power supply to generate the sinusoidal voltage signal with frequency of 100Hz and amplitude of 0~10 V. The resistance value of the resistance is 1 Ω. The power supply, the resistance and the primary conductor with the insulation form the AC current loop. The amplifier OP07 is to amplify the induction voltage signal at both ends of the Rogowski coils and to transmit the amplified signal to the acquisition card NI 6002 DAQ. The

Sensors 2019, 19, x FOR PEER REVIEW 7 of 16 amplificationSensors 2019, 19, 3397coefficient of the amplifier is 100. DAQ transmits signal to PC with LabVIEW software7 of 16 with filter function.

FigureFigure 6. 6. ExperimentalExperimental platform. platform. (a) (Schematica) Schematic diagram; diagram; (b) (powerb) power supply supply and andthe theresistor; resistor; (c) experimental(c) experimental equipment equipment in th ine thelaboratory laboratory environment. environment.

Table 1. Parameters of the three test coils. Table 1. Parameters of the three test coils. Coil Label N r (mm) a (mm) b (mm) L (mm) N β Coil Label N r (mm) a (mm) b (mm) Lg (mm) Ncgcg 𝜷𝒈 g CoilCoil A A 1604 1604 4.3 4.3 63.8 63.8 63.8 63.8 2 10 10 0° 0◦ CoilCoil B B 1604 1604 4.3 4.3 57.3 57.3 70 70 2 10 10 0° 0◦ Coil C 1604 4.3 45.2 80 2 10 0 Coil C 1604 4.3 45.2 80 2 10 0° ◦

Table 2. Location information of the holes of the ﬁxed plates. Table 2. Location information of the holes of the fixed plates.

CoilCoil Label Label Coil Coil A Coil Coil B B Coil Coil C C PositionPosition x (mm)(mm) yy (mm)(mm) x (mm)x (mm) y (mm)y (mm) x (mm)x (mm) y (mm)y (mm) I1I1 00 0 0 0 00 00 00 0 I2I2 2020 20 20 20 20 20 20 20 2020 20 I3 0 30 0 30 0 30 I3 0 −−30 0 −30− 0 −30− I4 51 0 44 0 32 0 I5I4 5151 0 0 4444 0 0 32 320 0 − − − I6I5 − 3651 0 36 −44 35 0 35−32 30 0 30 I7I6 3636 3636 3535 35 35 30 30 30 30 − − − − − − I8I7 − 036 −36 51 −35 0 −35 51 −30 0−30 60 I9 20 25 20 25 20 25 I8 − 0 51 0− 51 0− 60 I10 32 0 0 30 0 30 − I11I9 − 3220 25 0 −20 - 25 -−20 -25 - O1I10 − 7732 0 0 0 70 30 0 0 58 30 0 O2I11 5432 0 54 - 54 - 54 - 49 - 48 O3O1 77 0 0 77 70 0 0 8358 00 93 O4 77 0 70 0 58 0 O2 −54 54 54− 54 49− 48 O5 54 54 54 54 49 49 − − − − − − O6O3 950 77 0 0 8883 00 7693 0 O4 −77 0 −70 0 −58 0 O5 −54 −54 −54 −54 −49 −49 As shown inO6 Figure 6, we95 use ITECH0 IT762688 AC programmable0 power76 supply0 to generate the sinusoidal voltage signal with frequency of 100Hz and amplitude of 0~10 V. The resistance value of 4.the Results resistance of Numerical is 1 Ω. The Calculation power supply, and the Experiments resistance and the primary conductor with the insulation form the AC current loop. The amplifier OP07 is to amplify the induction voltage signal at both endsTo of theverify Rogowski the mutual coils andinductance to transmit formula the amplified we deduced signal toabove, the acquisition we calculated card NI the 6002 mutual DAQ. inductanceThe amplification between coe theffi Rogowskicient of the coils amplifier of the circular, is 100. oval DAQ and transmits rectangular signal skeleton to PC and with the LabVIEW primary conductorssoftware with of several filter function. common shape by MATLAB code according to the derivation in Section 2. The

SensorsSensors 20192019,, 1919,, x 3397 FOR PEER REVIEW 8 8of of 16 16 results of the calculation of the Rogowski coils of the oval and circular skeletons were verified by the 4. Results of Numerical Calculation and Experiments experiments. TheTo verify circumference the mutual of inductance the center formulaline of the we Rogo deducedwski above, coils of we the calculated rectangular the mutualand oval inductance skeleton webetween chose thewas Rogowski the same coilsas that of theof the circular, Rogowski oval andcoil rectangularof the circular skeleton skeleton and (Coil the primaryA). The conductorsnumber of turnsof several and the common radius shape of the by cross-section MATLAB code of the according Rogowski to the coil derivation of the rectangular in Section skeleton2. The results were ofequal the tocalculation those of Coil of the A, Rogowskirespectively. coils Moreover, of the oval we and assu circularmed that skeletons the turns were were verified uniformly by the distributed experiments. on the skeleton.The circumference We defined of Coil the D center as the line Rogowski of the Rogowski coil of the coils rectangular of the rectangular skeleton, and ovalthe length skeleton of thewe quadrilateral chose was the center same line as that was of 50.1 the mm. Rogowski coil of the circular skeleton (Coil A). The number of turnsM and0, which the radius has been of the studied cross-section in [32],of presents the Rogowski the mutual coil of inductance the rectangular of the skeleton Rogowski were coil equal of the to circularthose of skeleton Coil A, respectively. in the ideal Moreover, case (the weprimary assumed conductor that the of turns infinite-length were uniformly goes distributed perpendicularly on the throughskeleton. the We center defined of Coilthe DRogowski as the Rogowski coil of the coil ci ofrcular therectangular cross-section skeleton, and circular and the skeleton) length of as the a standardquadrilateral to normalize center line the was resu 50.1lts of mm. the nonideal cases. AllM0 ,calculation which has beenresults studied by MATLAB in [32], presentscodes are the prefixed mutual inductancewith “Cal.” ofand the the Rogowski experimental coil of measurementthe circular skeleton results inare the prefixed ideal case with (the “Mea”. primary All conductor center lines of infinite-length of the Rogowski goes coil perpendicularly skeleton in Figuresthrough 7, the 10, center 12, 15 of and the 18 Rogowski are on the coil xOy of thecoordinate circular cross-sectionplane. and circular skeleton) as a standard to normalize the results of the nonideal cases. 4.1. RogowskiAll calculation Coils of Noncircular results by Skelet MATLABon and codesLong Straight are prefixed Primary with Conductors “Cal.” and the experimental measurement results are prefixed with “Mea”. All center lines of the Rogowski coil skeleton in While the primary conductor is apt to tilt or deviate from the center line (z-axis as shown in Figures 7, 10, 12, 15 and 18 are on the xOy coordinate plane. Figure 7) of the Rogowski coil, the circular skeleton of the flexile Rogowski coils are easy to deform under4.1. Rogowski the external Coils offorce Noncircular when the Skeleton Rogowski and Longcoils Straightare used. Primary We studied Conductors the influence of the straight primary conductor tilted or deviated from the center of the skeleton on the mutual inductance of the RogowskiWhile coils the primaryof the oval conductor or rectangular is apt toskeleton tilt ors. deviate All thefrom Rogowski the center coils linehave ( zboth-axis gap as shown and gap in compensationFigure7) of the in Rogowski this section. coil, the circular skeleton of the flexile Rogowski coils are easy to deform underAs the shown external in Figure force when7, the theintersection Rogowski of coils the areprimary used. conductor We studied AB the and influence the xOy of plane the straight is 𝑂. Parameterprimary conductor 𝜆 is the tiltedratio of or A deviatedO’ and AB; from 𝜆 the =center 0.5 means of the that skeleton the middle on the of mutual the primary inductance conductor of the isRogowski on the xOy coils plane; of the L is oval the orlength rectangular of the primary skeletons. conductor, All the Rogowskiand L = 5000 coils mm, have and both α is gapthe angle and gap of thecompensation primary conductor in this section. tilted from the z-axis.

(a) (b)

FigureFigure 7. 7. ConductorConductor nonvertically throughthrough RogowskiRogowski coils coils ( xOy(xOyplane) plane) (λ (𝜆= 0.5 = 0.5).). (a ()a Rogowski) Rogowski coil coil of ofoval oval skeleton; skeleton; (b )(b Rogowski) Rogowski coil coil of of rectangular rectangular skeleton. skeleton.

WeAs shownshow both in Figure the calculation7, the intersection and experimental of the primary results conductor of several ABcases and in theFigurexOy 8. planeAs we is canO0 . see,Parameter they areλ is matched. the ratio ofNormalized AO’ and AB; mutualλ = 0.5inductanmeansce that of thethe middleRogowski of the coil primary of the conductor oval skeleton is on increasesthe xOy plane; (up toL 1.01)is the as length the tilted of the angle primary α increases conductor, when and theL primary= 5000 mm, conductor and α isgoes the through angle of the the innerprimary holes conductor I1 and I4. tilted While from in the z-inneraxis. holes I2 and I8, it decreases as the angle α increases. Under the sameWe showcondition, both thethe calculationmutual inductance and experimental changes (up results to 7%) of severalwhen the cases skeleton in Figure of the8. AsRogowski we can coilsee, is they deformed. are matched. Figure Normalized8e,f shows the mutual interference inductance of the of external the Rogowski conductor coil in of the the coils oval when skeleton the skeletonsincreases are (up deformed. to 1.01) as Results the tilted show angle thatα increasesserious de whenformation the primary of coil skeleton conductor should goes be through avoided, the andinner the holes primary I1 and conductor I4. While should in the avoid inner holesbeing I2close and to I8, the it decreasesgap of the asRogowski the angle coil.α increases. Under

Sensors 2019, 19, 3397 9 of 16 the same condition, the mutual inductance changes (up to 7%) when the skeleton of the Rogowski coil is deformed. Figure8e,f shows the interference of the external conductor in the coils when the skeletons are deformed. Results show that serious deformation of coil skeleton should be avoided, Sensorsand the 2019 primary, 19, x FOR conductor PEER REVIEW should avoid being close to the gap of the Rogowski coil. 9 of 16

(a) (b)

(e) (f)

FigureFigure 8. Normalized mutual mutual inductance inductance vs. vs. tilted tilted angl anglee when when the the straight primary conductor does notnot go go vertically vertically through through different diﬀerent posi positionstions of of Coil Coil A, A, Coil Coil B, B, Coil Coil C. C. ( (aa)) At At inner inner hole hole I1; I1; ( (bb)) at at inner inner holehole I2; I2; ( cc)) at at inner inner hole hole I4; I4; ( (d)) at at inner inner hole I8; ( e) at outer hole O1; ( f) at outer hole O3.

ForFor Coil Coil D D whose whose skeleton skeleton is is rectangular, rectangular, we we show show the the calculation calculation results results in in Figure Figure 99.. Moreover, Moreover, we can see the mutual inductance of the Rogowski coil of the rectangular skeleton is is larger (about 3.075%)3.075%) thanthan thatthat of of the the coil coil of theof the circular circular skeleton skeleton (with (with the same the skeletonsame skeleton circumference circumference and number and numberof turns). of The turns). closer The the closer primary the conductorprimary conducto to the cornerr to the of thecorner rectangular of the rectangular skeleton, the skeleton, greater the greatermutual inductancethe mutual changes. inductance The mutualchanges. inductance The mutual of the inductance Rogowski coilof the of theRogowski rectangular coil skeleton of the rectangularﬂuctuates more skeleton than fluctuates that of the more oval orthan circular that of skeleton. the oval or circular skeleton.

SensorsSensors 20192019,, 1919,, x 3397 FOR PEER REVIEW 1010 of of 16 16 Sensors 2019, 19, x FOR PEER REVIEW 10 of 16 Sensors 2019, 19, x FOR PEER REVIEW 10 of 16

(a) (b) (a) (b) (a) (b) Figure 9. Calculated normalized mutual inductance vs. position when the primary conductor goes Figure 9. Calculated normalized mutual inductance vs. position when the primary conductor goes FigurenonverticallyFigure 9. 9. CalculatedCalculated through normalized normalizedthe Coil D. mutual ( mutuala) α = 0°;inductance inductance (b) α = 45°. vs vs. . position position when when the the primary primary conductor conductor goes goes nonvertically through the Coil D. (a) α = 0°; (b) α = 45°. nonverticallynonvertically through through the the Coil Coil D. D. (a (a) )αα == 0°;0 ◦(;(b)b α) α= =45°.45 ◦. 4.2. Rogowski Coils of Noncircular Skeleton and External Conductors Parallel to the Coil Plane(xOy) 4.2. Rogowski Coils of Noncircular Skeleton and External Conductors Parallel to the Coil Plane(xOy) 4.2.4.2. Rogowski Rogowski Coils Coils of of Noncircular Noncircular Skeleton Skeleton and and Ex Externalternal Conductors Conductors Parallel Parallel to to the the Coil Coil Plane(xOy) Plane(xOy) In this section, we study the sensitivity of the Rogowski coils of different skeleton shape to the In this section, we study the sensitivity of the Rogowski coils of different skeleton shape to the externalInIn this thisconductor section, section, weparallel we study study to the the sensitivity sensitivitycoils. As shownof of the the Rogowskiin Rogowski Figure 10,coils coils the of ofexternal different diﬀerent conductor skeleton skeleton shapeis shape parallel to to thethe to external conductor parallel to the coils. As shown in Figure 10, the external conductor is parallel to externaltheexternal y-axis conductor conductor and the distance parallel parallel hto of the the coils. conductor As shownshown to the inin coils FigureFigure is 8 10 mm.10,, the the The external external length conductor ofconductor the external is is parallel parallel conductor to to the the y-axis and the distance h of the conductor to the coils is 8 mm. The length of the external conductor theisy-axis 5000 y-axis andmm. and the the distance distanceh of h of the the conductor conductor to to the the coils coils is is 8 8 mm. mm. The The length length of of the the external external conductor conductor is is 5000 mm. is5000 5000 mm. mm.

FigureFigure 10. 10. TheThe external external conductor conductor parallel parallel to to the the plane plane of of the the Rogowski Rogowski coils coils of of the the oval oval skeleton. skeleton. Figure 10. The external conductor parallel to the plane of the Rogowski coils of the oval skeleton. Figure 10. The external conductor parallel to the plane of the Rogowski coils of the oval skeleton. AccordingAccording to to the the results results shown shown in in Figure Figure 11,11, the the pa parallelrallel conductor conductor has has little little effect eﬀect on on the the mutual mutual According to the results shown in Figure 11, the parallel conductor has little effect on the mutual inductanceinductanceAccording eveneven to for forthe the resultsthe deformed deformed shown skeleton inskeleton Figure case. 11,case. Moreover,the Moreover, parallel the conductor parallelthe parallel conductor has littleconductor effect should onshould not the be mutual not placed be inductance even for the deformed skeleton case. Moreover, the parallel conductor should not be inductanceplacedright above right even theabove planefor the the ofplane deformed the Rogowskiof the skeletonRogowski coils. case. coils. Moreover, the parallel conductor should not be placed right above the plane of the Rogowski coils. placed right above the plane of the Rogowski coils.

FigureFigure 11. 11. NormalizedNormalized mutual mutual inductance inductance vs. vs. position position pr primaryimary conductor conductor of of the the when when the the conductors conductors Figure 11. Normalized mutual inductance vs. position primary conductor of the when the conductors Figureareare parallel parallel 11. Normalized to to the the Rogowski Rogowski mutual coils coils inductance plane. plane. vs. position primary conductor of the when the conductors are parallel to the Rogowski coils plane. are parallel to the Rogowski coils plane. 4.3. Rogowski Coils of Noncircular Skeleton and D-Shaped Conductors 4.3. Rogowski Coils of Noncircular Skeleton and D-Shaped Conductors 4.3. Rogowski Coils of Noncircular Skeleton and D-Shaped Conductors

Sensors 2019, 19, 3397 11 of 16

Sensors 2019, 19, x FOR PEER REVIEW 11 of 16 Sensors4.3. Rogowski 2019, 19, x CoilsFOR PEER of Noncircular REVIEW Skeleton and D-Shaped Conductors 11 of 16 In this section, we explore the influence of the primary conductor with the right angles on the InIn this this section, section, we we explore explore the the influence inﬂuence of of th thee primary primary conductor conductor with with the the right right angles angles on on the the mutual inductance of the Rogowski coils. As shown in Figure 12, the lengths of the conductor section mutualmutual inductance inductance of of the the Rogowski Rogowski coils. coils. As As shown shown in in Figure Figure 12, 12 ,the the lengths lengths of of the the conductor conductor section section AB, AA’, A’C and BC are all equal to 117.8 mm. The lengths of BB’, CC’ and AO’ are equal to 58.9 AB,AB, AA’, AA’, A’C andand BCBC areare all all equal equal to to 117.8 117.8 mm. mm. The The lengths lengths of of BB’, BB’, CC’ CC’ and and AO’ AO’ are are equal equal to 58.9 to 58.9 mm. mm. The conductors DB’ and D’C’ are tightly tied together, and the magnitude of the currents in DB’ mm.The conductorsThe conductors DB’ DB’ and and D’C’ D’C’ are tightlyare tightly tied tied together, together, and and the magnitudethe magnitude of the of currentsthe currents in DB’ in DB’ and and C’D’ are equal while their directions are opposite. Therefore, the sum of the magnetic flux andC’D’ C’D’ are equalare equal while while their directionstheir directions are opposite. are oppo Therefore,site. Therefore, the sum the of thesum magnetic of the magnetic ﬂux produced flux produced by DB’ and C’D’ is zero. producedby DB’ and by C’D’ DB’ isand zero. C’D’ is zero.

(a) (b) (a) (b) FigureFigure 12. 12. TheThe D-shaped D-shaped primary primary conductor vertically through Rogowski coils (λ𝜆 = = 0.50.5).). ( (aa)) The The Figure 12. The D-shaped primary conductor vertically through Rogowski coils (𝜆 = 0.5). (a) The RogowskiRogowski coil coil of of the the oval oval skeleton; skeleton; ( (bb)) the the Rogowski Rogowski coil coil with with rectangular rectangular skeleton. skeleton. Rogowski coil of the oval skeleton; (b) the Rogowski coil with rectangular skeleton. WithWith the D-shaped primaryprimary conductor,conductor, we we show show both both the the calculation calculation and and experimental experimental results results for With the D-shaped primary conductor, we show both the calculation and experimental results forthe the Rogowski Rogowski coils coils of of the the circular circular or or oval oval skeleton skeleton in in Figure Figure 13 13.. AsAs shownshown inin Figure 13a,13a, the the mutual mutual for the Rogowski coils of the circular or oval skeleton in Figure 13. As shown in Figure 13a, the mutual inductanceinductance between between the the D-shaped D-shaped primary primary conductor conductor and and the the coils coils changes changes (up (up to to 0.3%) 0.3%) when when the the inductance between the D-shaped primary conductor and the coils changes (up to 0.3%) when the skeletonskeleton of of Rogowski Rogowski coil coil is is deformed. deformed. The The mutual mutual inductance inductance of of the the D-shaped D-shaped conductor conductor is is greater greater skeleton of Rogowski coil is deformed. The mutual inductance of the D-shaped conductor is greater thanthan that that of of the the long long straight straight primary primary conductor. conductor. When When the the primary primary conductor conductor goes goes through through the the inner inner than that of the long straight primary conductor. When the primary conductor goes through the inner holes,holes, the D-shapedD-shaped conductorconductor has has greater greater eﬀ ecteffect on theon mutualthe mutual inductance inductance than thatthan of that the longof the straight long holes, the D-shaped conductor has greater effect on the mutual inductance than that of the long straightconductor, conductor, while the while outside the D-shapedoutside D-shaped conductor conductor has less has eﬀect. less effect. straight conductor, while the outside D-shaped conductor has less effect.

(a) (b) (a) (b) Figure 13. Normalized mutual inductance between the Rogowski coils and the D-shaped primary FigureFigure 13. 13. NormalizedNormalized mutual mutual inductance inductance between between the the Rogowski Rogowski coils coils and and the the D-shaped D-shaped primary primary conductors. (a) The D-shaped conductor through the inner holds; (b) the D-shaped conductor through conductors.conductors. (a (a) )The The D-shaped D-shaped conductor conductor th throughrough the the inner inner holds; holds; (b (b) )the the D-shaped D-shaped conductor conductor through through the outer side holds. thethe outer outer side side holds. holds. Also with a D-shaped primary conductor, we show the calculation results for the Rogowski coil AlsoAlso with with a a D-shaped D-shaped primary primary conductor, conductor, we we show show the the calculation calculation results results for for the the Rogowski Rogowski coil coil of the rectangular skeleton in Figure 14. The mutual inductance between the Rogowski coil of the ofof the the rectangular rectangular skeleton skeleton in in Figure Figure 14.14. The The mutu mutualal inductance inductance between between the the Rogowski Rogowski coil coil of of the the rectangular skeleton and the D-shaped primary conductor is larger than that of the long straight rectangularrectangular skeleton skeleton and and the the D-shaped D-shaped primary primary conduc conductortor is is larger larger than than that that of of the the long long straight straight conductor and Coil A. The influence of the D-shaped conductor on the mutual inductance of Coil D conductorconductor and and Coil Coil A. A. The influence inﬂuence of thethe D-shapedD-shaped conductorconductor onon thethe mutual mutual inductance inductance of of Coil Coil D D is is larger than that of the long straight primary conductor when the position of the primary conductor islarger larger than than that that of of the the long long straight straight primaryprimary conductorconductor whenwhen the position of of the the primary primary conductor conductor changes. The closer the D-shaped conductor to the corner of the rectangular skeleton, the greater the changes. The closer the D-shaped conductor to the corner of the rectangular skeleton, the greater the influence on the mutual inductance (up to 25%). influence on the mutual inductance (up to 25%).

Sensors 2019, 19, x FOR PEER REVIEW 12 of 16

Sensors 2019, 19, 3397 12 of 16

changes. The closer the D-shaped conductor to the corner of the rectangular skeleton, the greater the Sensorsinﬂuence 2019, on19, x the FOR mutual PEER REVIEW inductance (up to 25%). 12 of 16

Figure 14. Calculation normalized mutual inductance between the Rogowski coil of rectangular skeleton and the D-shaped primary conductor.

4.4. Rogowski Coils of Noncircular Skeleton and O-Shaped Conductors In this section, we study the mutual inductance between the curved primary conductor and the Rogowski coil with deformed skeleton. As shown in Figure 15, 2𝑅 = 150 mm is the diameter of the O-shaped conductor. The conductor plane is parallel to the xOz plane and is symmetric about the xOy. 𝑂′′ is the intersection of the conductor and the xOy plane. FigureFigure 14. 14. CalculationCalculation normalized normalized mutual mutual inductance inductance between between the the Rogowski Rogowski coil coil of of rectangular rectangular Both the calculation and experimental results are shown in Figure 16. From Figure 16a, we can skeletonskeleton and and the the D-shaped D-shaped primary primary conductor. conductor. see that the mutual inductance of the O-shaped conductor is greater (0.2%) than that of the long 4.4.straight4.4. Rogowski Rogowski primary Coils Coils conductor.of of Noncircular Noncircular The Skeleton Skeleton error andof and the O-Shaped O-Shaped mutual Conductors Conductors inductance between the O-shaped primary conductor and the Rogowski coil is greater (up to 0.3%) when the skeleton of the Rogowski coil is InIn this this section, section, we we study study the the mutual mutual inductan inductancece between between the the curved curved primary primary conductor conductor and and the the deformed. The primary conductor should not be close to the gap of= the Rogowski coil. The outside RogowskiRogowski coil coil with with deformed skeleton.skeleton. AsAs shownshown inin FigureFigure 15 15,, 2 R2𝑅0 =150 150mm mmis theis the diameter diameter of theof O-shaped conductor.conductor Thehas less conductor effect on plane the is mutual parallel inductance to the xOz ofplane the andRogowski is symmetric coil than about that the ofxOy the. theoutside O-shaped long conductor.straight conductor. The conductor The skeleton plane is parallelof the flexible to the xOz Rogowski plane and coil is shouldsymmetric be keptabout from the O00 is𝑂′′ the intersection of the conductor and the xOy plane. xOydeforming.. is the intersection of the conductor and the xOy plane. Both the calculation and experimental results are shown in Figure 16. From Figure 16a, we can see that the mutual inductance of the O-shaped conductor is greater (0.2%) than that of the long straight primary conductor. The error of the mutual inductance between the O-shaped primary conductor and the Rogowski coil is greater (up to 0.3%) when the skeleton of the Rogowski coil is deformed. The primary conductor should not be close to the gap of the Rogowski coil. The outside O-shaped conductor has less effect on the mutual inductance of the Rogowski coil than that of the outside long straight conductor. The skeleton of the flexible Rogowski coil should be kept from deforming.

(a) (b)

Figure 15. Curved conductor going vertically through the coils. ( (aa)) The The Rogowski Rogowski coil coil of of the oval skeleton; ( b) the Rogowski coil of the rectangular skeleton.skeleton.

Both the calculation and experimental results are shown in Figure 16. From Figure 16a, we can see that the mutual inductance of the O-shaped conductor is greater (0.2%) than that of the long straight primary conductor. The error of the mutual inductance between the O-shaped primary conductor

and the Rogowski coil is greater (up to 0.3%) when the skeleton of the Rogowski coil is deformed. (a) (b) The primary conductor should not be close to the gap of the Rogowski coil. The outside O-shaped conductorFigure has15. Curved less eff ectconductor on the mutualgoing vertically inductance through of the the Rogowski coils. (a) The coil Rogowski than that coil of theof the outside oval long straightskeleton; conductor. (b) the Rogowski The skeleton coil of of the the rectangular flexible Rogowski skeleton. coil should be kept from deforming. Figure 17 shows that the mutual inductance between the Rogowski coil of the rectangular skeleton and O-shaped primary conductor is larger than that of the long straight conductor. The influence of the O-shaped conductor on the mutual inductance is larger than that of the long straight primary conductor when the position of the primary conductor changes. The closer the O-shaped conductor to (a) (b) the corner of the rectangular skeleton, the greater the influence on the mutual inductance (up to 30%). The curved primary conductor should be avoided in this case.

(a) (b)

Sensors 2019, 19, x FOR PEER REVIEW 12 of 16

Figure 14. Calculation normalized mutual inductance between the Rogowski coil of rectangular skeleton and the D-shaped primary conductor.

(a) (b) Sensors 2019, 19, x FOR PEER REVIEW 13 of 16 Figure 15. Curved conductor going vertically through the coils. (a) The Rogowski coil of the oval Sensors 2019, 19, 3397 13 of 16 Figureskeleton; 16. ( bNormalized) the Rogowski mutual coil ofinductance the rectangular between skeleton. the Rogowski coils and the O-shaped primary conductors. (a) The O-shaped conductor going through the inner holes; (b) the O-shaped conductor going through the outer holes.

Figure 17 shows that the mutual inductance between the Rogowski coil of the rectangular Sensorsskeleton 2019, and19, x FORO-shaped PEER REVIEW primary conductor is larger than that of the long straight conductor.13 of The16 influence of the O-shaped conductor on the mutual inductance is larger than that of the long straight Figure 16. Normalized mutual inductance between the Rogowski coils and the O-shaped primary primary conductor when the position of the primary conductor changes. The closer the O-shaped conductors. (a) The O-shaped conductor going through the inner holes; (b) the O-shaped conductor conductor to the corner of the rectangular skeleton, the greater the influence on the mutual going through the outer holes. inductance (up to 30%). The curved primary conductor should be avoided in this case. Figure 17 shows that the mutual inductance between the Rogowski coil of the rectangular skeleton and O-shaped primary conductor is larger than that of the long straight conductor. The influence of the O-shaped(a) conductor on the mutual inductance is larger than(b) that of the long straight primary conductor when the position of the primary conductor changes. The closer the O-shaped conductorFigure to 16. theNormalized corner of mutual the rectangular inductance betweenskeleton, the the Rogowski greater coils the and influence the O-shaped on the primary mutual inductanceconductors. (up to ( a30%).) The O-shapedThe curved conductor primary going conductor through should the inner be holes;avoided (b) thein this O-shaped case. conductor going through the outer holes.

Figure 17. Calculation of the normalized mutual inductance between the Rogowski coil of the rectangular skeleton and the O-shaped primary conductors.

4.5. Improved Rogowski Coils of Rectangular Skeleton It is shown in Figure 18 that the mutual inductance of the Rogowski coil of the rectangular skeleton of the right angle turns large when the primary conductor deviate s from the center of the skeleton.FigureFigure Errors17. 17. CalculationCalculation of the mutual of ofthe theinductance normalized normalized of mutual the mutual coils indu inductance withctance the betweenright between-angle the the skeletonRogowski Rogowski are coil coilalso of of producedthe the fromrectangularrectangular the changes skeleton skeleton of the and andcross the the -O-shapedsection O-shaped of primary primarythe primary conductors. conductors. conductors [21]. Here, we propose a new structure of the Rogowski coil of the rectangular skeleton which can 4.5. Improved Rogowski Coils of Rectangular Skeleton 4.5.greatly Improved reduce Rogowski the deviation Coils of Rectangular influence. AsSkeleton shown in Figure 18a, we changed the right angle of the It is shown in Figure 18 that the mutual inductance of the Rogowski coil of the rectangular skeleton rectangularIt is shown skeleton in Figure into an18 arcthat angle, the mutual and the inductance circumstance of theand Rogowski turns number coil of the centerrectangular line of of the right angle turns large when the primary conductor deviates from the center of the skeleton. skeletonthe skeleton of the with right an angle arc angle turns continues large when to bethe the pr imarysame asconductor that of those deviates with from a right the angle. center 푟 푐of is the the Errors of the mutual inductance of the coils with the right-angle skeleton are also produced from the skeleton.radius of Errors the arc of angles.the mutual To study inductance the influence of the co ofils curvature with the radiusright-angle 푟푐 on skeleton mutual are inductance, also produced we set changes of the cross-section of the primary conductors [21]. fromdifferent the changes radius of푟푐 theas showncross-section in Figure of the19. primary conductors [21]. Here, we propose a new structure of the Rogowski coil of the rectangular skeleton which can greatly reduce the deviation influence. As shown in Figure 18a, we changed the right angle of the rectangular skeleton intoCenter lanine oarcf angle,Arc angle and the circumstanceCen andter line turnsof R numberight angle of the center line of skeleton skeleton the skeleton with an arc angle continues to be the same as that of those with a right angle. 𝑟 is the radius of the arc angles. To study the influence of curvature radius 𝑟 on mutual inductance, we set O O y y different radius 𝑟 as shown in Figure 19. B(-50.1mm,0mm) B(-53mm,0mm)

A(44mm,44mm) A(44mm,44mm)

x x (a) (b)

FigureFigure 18. SkeletonsSkeletons with with circular circular cross cross-section-section of of the the Rogowski Rogowski coil. coil. (a) (Skeletona) Skeleton with with arc arcangle; angle; (b) (sbkeleton) skeleton with with right right angle. angle.

Here, we propose a new structure of the Rogowski coil of the rectangular skeleton which can greatly reduce the deviation inﬂuence. As shown in Figure 18a, we changed the right angle of the (a) (b)

Figure 18. Skeletons with circular cross-section of the Rogowski coil. (a) Skeleton with arc angle; (b) skeleton with right angle.

Sensors 2019, 19, 3397 14 of 16 rectangular skeleton into an arc angle, and the circumstance and turns number of the center line of the skeleton with an arc angle continues to be the same as that of those with a right angle. rc is the radius of the arc angles. To study the inﬂuence of curvature radius rc on mutual inductance, we set diﬀerent radiusSensors 2019rc as, 19 shown, x FOR inPEER Figure REVIEW 19. 14 of 16

(a) (b)

FigureFigure 19.19. Calculation results results of of the the normalized normalized mutual mutual inductance inductance between between the the Rogowski Rogowski coil coil of the of therectangular rectangular skeleton skeleton and and the thelong long straight straight primary primary conductor. conductor. (a) Skeleton (a) Skeleton with with an arc an angle arc angle and α = 0°; 𝑟 = 13.369 mm andprimary primary conductor conductor ( (α = 0◦; rc = 13.369);mm (b)); skeleton (b) skeleton with withan arc an angle arc angle and andtilt angle tilt angle of the of 𝑟 = 13.369 mm theprimary primary conductor conductor (α (=α 45°;= 45 ◦; rc = 13.369 mm); ();c) ( cskeleton) skeleton with with an an arc arc angle angle and thethe primaryprimary conductorconductor ( α(α= = 00°;◦; r𝑟c==15.9155 mm);); ( (d) skeleton with an arc angleangle andand thethe primaryprimary conductorconductor ( α(α= = 00°;◦; r𝑟c = =22.2817 22.2817 mm). mm).

WeWe showshow thethe calculationcalculation resultsresults ofof thethe newnew structurestructure withwith didifferentﬀerent tilttilt anglesangles ofof thethe primaryprimary conductorconductor inin FigureFigure 1919.. Compared with thethe correspondingcorresponding resultsresults ofof thethe right-angleright-angle structurestructure inin FigureFigure9 9,, the the new new structure structure is is stronger stronger to to keep keep the the mutual mutual inductance inductance even even when when the the position position of of the the primaryprimary conductorconductor changes.changes. ForFor example,example, atat thethe cornercorner positionposition (point(point A:A: 4444 mm,mm, 4444 mm),mm), thethe mutualmutual inductanceinductance ofof thethe right-angleright-angle structurestructure changeschanges 13%13% (shown(shown inin FigureFigure9 9)) while while that that of of the the arc arc angle angle changeschanges 1.4%.1.4%. Furthermore, Furthermore, the the new new structure structure is not is sensitive not sensitive to the tiltto anglethe tilt of theangle primary of the conductor primary sinceconductor Panels since a and Panels b of Figure a and 19 b are of alike.Figure Figure 19 are 19 alike.a,c,d showsFigure that 19a,c,d the largershows the that curvature the larger radius the is,curvature the less theradius displacement is, the less ofthe the displacement conductors aofﬀ ectthe theconductors mutual inductance affect the mutual of the Rogowskiinductance coils of the of rectangularRogowski coils skeleton. of rectangular skeleton.

5.5. ConclusionsConclusions InIn thisthis paper,paper, wewe establishedestablished aa mathematicmathematic modelmodel toto calculatecalculate thethe mutualmutual inductanceinductance betweenbetween thethe RogowskiRogowski coilscoils ofof skeletonsskeletons ofof arbitraryarbitrary shapeshape andand thethe primaryprimary conductorsconductors ofof arbitraryarbitrary shapeshape atat arbitraryarbitrary positionposition to verify verify the the correctness correctness of of the the de derivationrivation and and the the influence inﬂuence of ofthe the skeletons skeletons on the on themutual mutual inductance inductance of the of theRogowski Rogowski coils. coils. The The mu mutualtual inductance inductance between between the the Rogowski Rogowski coils coils of different skeleton shapes and conductors of common shapes are calculated by MATLAB codes which we have written based on the mathematic models. Some calculated results were verified by the experiments. For the Rogowski coil of the rectangular skeleton, we propose a new structure to reduce the deviation influence. According to the calculation and experimental results, there are some ways to reduce the error of mutual inductance: Deformation should be avoided in the skeletons of flexible Rogowski coils, the primary conductors should be located close to the center of the skeletons and the primary conductors

Sensors 2019, 19, 3397 15 of 16 of different skeleton shapes and conductors of common shapes are calculated by MATLAB codes which we have written based on the mathematic models. Some calculated results were verified by the experiments. For the Rogowski coil of the rectangular skeleton, we propose a new structure to reduce the deviation influence. According to the calculation and experimental results, there are some ways to reduce the error of mutual inductance: Deformation should be avoided in the skeletons of flexible Rogowski coils, the primary conductors should be located close to the center of the skeletons and the primary conductors should be as straight as possible. The rectangular skeletons with the arc angle of the Rogowski coils can also greatly reduce the change of the mutual inductance when the position and shape of the primary conductor change. Although we have presented the test results of the common cases in this paper, the model we have established above is suitable for any shape and position. This allows engineers to design optimal Rogowski coils according to the shapes and sizes of the primary conductors without doing many experiments.

Author Contributions: Conceptualization, X.L. and H.H.; data curation, X.L.; formal analysis, X.L.; funding acquisition, C.J.; investigation, X.L.; methodology, X.L.; project administration, C.J.; resources, X.L.; software, X.L.; supervision, H.H.; visualization, C.J.; writing—original draft, X.L.; writing—review and editing, H.H. Funding: The authors acknowledge the support of the Science and Technology Project of State Grid Corporation of China under Grant NYB17201800314, and in part by Accuenergy (China) Ltd. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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