Low-frequency performance of openable ﬂexible double-loop Rogowski coil

Double-loop Rogowski coils have become increasingly popular in the market because of their high sensitivity and ﬂexible usage compared to their extended single-loop coils. Due to the high number of turns (thousands) and its small turn diameter ( < 0.17 mm), it is very hard to draw 3D models of double-loop coils with graphical software, let alone calculate electrical quantities by simulating them on the computer. To meet market’s needs for designing high-accuracy double-loop coil, a method for calculating its low-frequency coefﬁcient of the double-loop coil is presented. This calculation method is applicable to low-frequency current-carrying conductors of various geometric conﬁgurations. All experimental results verify the methods. Using this method, the errors of ﬁve typical double-loop coils caused by three representative current-carrying conductors are compared. The double-loop coils are also compared with two relative single-loop coils. These methods can quickly and accurately measure their errors of double-loop coil with thousands of turns under different application environments. These methods are very useful for designing highly accurate double-loop coils for a given application.


INTRODUCTION
Rogowski coils, which are also called differential sensors, consist of an air frame and a winding wire [1]. These coils overcome many common shortcomings of a normal sensor, which do not have the abilities to test heavy currents or high-frequency currents due to their frame magnetic saturation [2]. They are popular in the market for measuring various high current or high-frequency current because of their advantages -simple structure, lightweight, wide bandwidth, non-contact and nonmagnetic-saturation of frame [3][4][5][6][7][8][9][10]. The effects of parasitic inductance and capacitance inside coils on output voltage of coils can be neglected as currentcarrying conductor frequencies are low [11], and the coefficient between inductive voltage of two terminals of coils and current-carrying conductor of low-frequency is equal to mutualinductance (the following is abbreviated as M) [12]. The stability of coefficient M is critical to ensure their measurement accuracies [13][14][15].
As shown in Figure 1a, a tubular double-loop Rogowski coil is applied to the railway environment in [15,16], which describes  [11] analyse the performance of the lumped model (LPM) and the distributed model (DPM) by comparing in the time domain and frequency domain. The coefficient M between single-loop coils with circular-frame and long and straight current-carrying conductors is defined as lumped and distributed parameters of LPM and DPM, respectively. Ahmed et al. [18] present a theoretical analysis of the measurement error caused by the structure of the single-loop Rogowski coils. Several non-ideal cases such as non-constant of coil sectional area are discussed in this paper. A magnetic circuit mathematical model is proposed by considering the hollow cylindrical structure of coil shielding. The influence of the changes in air gap and major radius of the nonmagnetic form on the measurement errors is discussed in [19]. Six novel flat, spiral self-integrating Rogowski coils of two layers were designed, calibrated and tested [20]. The coefficient M was studied by the authors. Design, installation, improvement, measurement errors, modelling and compensation methods of single-loop Rogowski coils for high-accuracy measurement currents of current-carrying conductors were presented and analysed in [21][22][23][24][25][26], and the coefficient M of these single-loop coils is an important parameter of these studies.
Drawing the actual structure of double-loop coils with thousands of turns using drawing software is very difficult, let alone dividing them into finite element mesh to be simulated. Some of these reasons include, large contrast of radii of winding wires (about 0.075 mm) to these normal size of current-carrying conductors (can be 120 mm) and large number of turns of these double-loop coils (thousands). This also makes the theoretical research of this paper more meaningful.
When number of turns and frame perimeter are equal, a double-loop coil has larger coefficient M than the coefficient of their single-loop form [15]. The higher the coefficient M, the higher the output voltage of an integrator and coils' sensitivity [8]. This paper presents the anti-interference and the calculation method of the coefficient M of the double-loop Rogowski coil. As many current-carrying conductors of different geometric configurations exist in their application environment, the theoretical calculated methods of coefficient M of double-loop coils with large number of turns and currentcarrying conductors with different geometric configurations are presented. Two flexible double-loop coils of different common terminals connection modes are compared. Two relative double-loop coils and three representative current-carrying conductors are both explored and compared theoretically and experimentally. All experimental results verify the numerical results according to the derivation. The three representative

CALCULATION METHODS
Ignoring influence of coils' inner capacitance and inductance when current frequencies are low, the principle of Rogowski coils is expressed as [27] where, i represents currents in current-carrying conductors, t represents time in seconds, v represents coil's output inductive voltages.
Formula (1) shows that parameter M is the only coefficient between the output voltage and the low-frequency current. When the coefficient M changes, measured v will change, and the measure errors of the current-carrying current will change. M is determined by the structure of coils and geometric configuration of current-carrying conductors [19].
To derive the coefficient M of double-loop coils, two doubleloop coils of different terminal connection modes are set as coil-S ( Figure 2) and coil-T ( Figure 5). It is assumed that the longitudinal magnetic field of the Rogowski coils is zero due to the return loop [28].

Double-loop coil-S and current-carrying conductor of straight
First, establish a mathematical model to deduce the coefficient M between coil-S and the current-carrying conductor of  Figure 2, longitudinal centreline of a double-loop frame coincides with z-axis. Blue circular-line is a geometric centreline projection of the double-loop frame on xOy plane. R 1 represents the distance between coordinate origin and projection line. Their circular section radii of the spiral frame are r 0 . Current on current-carrying conductor is I. Current-carrying conductors' terminal coordinates are A (a, b, c) and B (d, e, f), respectively. As shown in Figure 3, the distance from any dot Q on the turn section to the conductor is l p . Q' is the projection of the dot Q on the xOy plane, and l is the distance from the dot Q' to the z-axis. β is the angle between line OQ and the positive x-axis. ⃗ e l is the unit vector of the straight conductor. θ 1 and θ 2 are the angles between ⃗ e l and the line AQ, QB, respectively. Magnetic field B Q in dot Q of measured currents is presented in [29]. The unit vector of the magnetic field B Q is ⃖⃖⃗ e BQ : The spatial parameter equation of the spiral geometric centreline of the frame is where h is the equivalent pitch of the coil's spiral frame. As shown in Figure 2a Unit vector ⃗ e of turn corresponding to β: Projection B Q of ⃖⃗ B Q on unit vector ⃗ e : Expression of dot Q on β turn: where n represents loop sum of coil's frame. is the angle between the line QO' and the xOy plane of the coordinate system. ζ = 0 means the case QO' is parallel to the xOy plane.

Double-loop coil-S and bent current-carrying conductor
First, establish the bent conductor's space parameter equation. The bent conductor can be located in any position in the coordinate system. Then dividing the bent conductor into small current units. As shown in Figure 4, the sth unit's magnetic field B Qs on point Q is Based on Equation (8), double-loop coils' magnetic flux ∅ s produced by sth unit can be calculated. Based on spatial parameter equations of bent current-carrying conductors, double-loop coils' ∅ produced by bent current-carrying conductors:

Double-loop coil-T and current-carrying conductor
As shown in Figure 5, the centreline of the frame of coil-T consists of five arc partsĈD,DE,ÊF ,FG ,ĜH . The magnetic flux of the five arc parts generated by the current-carrying con- respectively. Based on the above derivation method, the magnetic flux of the five arc parts of coil-T can be calculated. R 1 is the radii of the semicirclê DE andFG . R 2 is the distance between the origin of coordinates and the projection of the midpoint of the gap to the xOy plane. Magnetic flux ∅ of conductors through coil-T is M between coil-T and current-carrying conductors of different geometric configurations can be calculated based on Equations (8) and (12).

EXPERIMENTAL EQUIPMENT
Relative experiments were conducted to verify the calculation method and to explore the effects of current-carrying conductors on double-loop coils. An adjustable voltage source, a resistor (1 Ω) and a current-carrying insulated conductor together constitute a sinusoidal alternating current loop. The adjustable voltage source is used to output the sinusoid voltage (< 10 V) with the frequency range <5 kHz. The frequency of the current in the loop is 1.1 kHz, and the amplitude is less than 10 A. The screen of the adjustable voltage source shows the frequency and amplitude of the current. Output inductive voltage between the terminals of coil was collected by DAQ NI 6002. DAQ transmits the signal to PC with LabVIEW software with filter function. Coil-S and coil-T are fixed on the plastic plate with test holes.
The labels of the holes are shown in Figure 6b. The parameters of coil-S, coil-T and the locations of the test holes are shown in Tables 1 and 2, respectively. d represents the distance from holes' centres to origin coordinates. Angle among line from origin coordinates to hole's centres and x-axis's positive direction is Г. The radius of the turns wire is equal to 0.085 mm. As shown in Figure 5a, L g is the arc gap length of the centreline of the coil's frame. Considering normal thickness of openable coils' two plastic connections, distance L g among double-loop coil-T's two terminals is 2 mm. The midpoint of the gap is located in the x-axis. The radius R 1 and the distance R 2 are shown in Figure 5c. Coil-SU is the single-loop coil which is formed by the expansion of the double-loop coil-S.

CALCULATED AND MEASUREMENT RESULTS
To explore low-frequency performance of a double-loop coil's sensitivity and accuracy, it is important to figure out the coefficient M of double-loop coil-S and coil-T through measurement and calculated results by MATLAB codes, which are written based on the calculated methods. All calculated size parameters of double-loop coil and parameters of current-carrying conductor are equal to corresponding size parameters of their real test double-loop coil and current-carrying conductor used in measurement respectively. The current-carrying conductor includes the straight conductors of finite-length (as shown in Figure 2a), the straight conductors with right angle (as shown in Figure 9) and the bent conductors (as shown in Figure 12). All calculation results are prefixed with 'Cal'. and the experimental measurement results are prefixed with 'Meas'. To better explore the effects of current-carrying conductor on a double-loop coil, four typical single-loop and double-loop coils are calculated and compared with double-loop coil-T and coil-S. The four typical coils include two lower turn density double-loop coils (coil-S1 and coil-T1) which correspond to coil-S and coil-T, respectively; a single-loop coil (coil-U) which has the same frame radius (R 1 ) and turn density with coil-S has gap compensation [30]; and a deformed coil (coil-T2) corresponding to coil-T. The parameters of the four coils are shown in Table 3.

Case of double-loop coils and straight-long current-carrying conductors
Because the current-carrying conductor of straight-long is prone to lean and shift as coils are applied, influence of lean and displaced current-carrying conductor on double-loop coil's coefficient M is explored quantificationally. As shown in Figures 2 and 5, the length of the straight current-carrying conductor is 5000 mm, and the midpoint of the conductor AB is placed on plane z = 0. The angle between the unit vector ⃖⃖⃗ e AB and z-axis is always equal to zero, and the angle between the unit vector ⃖⃖⃗ e AB and the positive z-axis is α. As shown in Figure 7, the current-carrying conductor moves within the coils along x and y directions, and the step sizes are Δx = 1 mm and Δy = 1 mm, respectively. Because the structure of the two coils is symmetric around the yOz plane, it is reasonable to only consider the region (x > 0). M 0 is the reference value of the coefficient M of each coils when the conductor AB is located in the centre of the coils vertically. As shown in Figure 8, the experimental results agree with the numerical results. The numerical results of double-loop coil-S and single-loop coil-U in Figure 8 show that the influence of the current-carrying conductors on coil-S is less than coil-U. The low turn density double-loop coils have larger influence than the high-density double-loop coils. When the straight conductor located in the inner holes of the coils as shown in Figure 8a, the coefficient M of coil-S and T remain basically unchanged

Case of double-loop coil and current-carrying conductors of right angle
In this section, we explore the effect of right angle currentcarrying conductors on coefficient M of coil-S and coil-T. As shown in Figure 9, the length of the conductors AB, BC, CD and DA is 118 mm and the length of the conductors DE and CG is 59 mm. Conductor AB is parallel to the z-axis and the midpoint of the conductor AB is on plane xOy. The plane of ABCD conductor is perpendicular to the z-axis. The conductors EF and GH are bound together tightly. The amplitude of current on the conductor EF and GH is equal but the phase is opposite. So the sum of the magnetic field of the conductors EF and GH at any point is 0. Their reference values of M 0 of coil-S and coil-T are these values as conductor of right angle go through centre I1of coil-S and coil-T, respectively.
As shown in Figure 10, the conductor of right angle moves within coils along with x and y directions, and step sizes are   Figure 11, the experimental results agree with the numerical results. The reference value M 0 is the coefficient M value of corresponding coils as infinite-long conductor of straight which vertically went through coils' centre (I1). As currentcarrying conductor of square form shifts within these coils, M/M 0 (0.9921) of coil-S and coil-T experiences a greater change than that of the straight conductor (0.9932). Antiinterference ability of coil-S is better than coil-T in this case. The current-carrying conductors of square form have less influence on single-loop coils than the double-loop coils. Currentcarrying conductors of square form also have greater effect on low turn density coils than high turn density coils, whether square-form conductors are outside coils or inside coils.

Case of double-loop coil and bent current-carrying conductors of circular form
There are bent current-carrying conductors exist in coils' application environment. The errors of double-loop coil-S and coil-T caused by the bent conductor are explored. As shown in Figure 12, the diameter of the circular-form conductor is 2R ′ (2R ′ = 150 mm). The conductor plane is perpendicular to the z-axis. The conductor of circular form is symmetric about the xOy plane. The conductors EF and GH are bound together tightly. The amplitude of current on the conductors EF and GH are equal but the phase is opposite. So the sum of the magnetic field of the conductors EF and GH at any point is 0. Reference values M 0 of coil-S and coil-T are those values as bent conduc-    Figure 14, the experimental results agree with the numerical results. Reference value M 0 is corresponding coils' coefficient M as infinite-long current-carrying conductor of straight vertically went through coils' centre (I1). As bent current-carrying conductors of circular form move in coils, M/M 0 (0.991) of coil-S and coil-T changes are larger than that of the straight conductor (0.9932). M/M 0 of the coil-U changes are larger than that of the double-loop coils. Errors of coil-T and coil-T1 caused by outside conductors of circular form are greater than errors of coil-S1, coil-S and coil-U.

Comparison of double-loop coils and double-loop coil
Sensitivity of double-loop coil and relative single-loop coil is compared. The single-loop Rogowski coil has the same length and number of turns as coil-S and coil-T, respectively. As shown in Figure 15, M 0 represents cases when coil is single-loop, and infinite-long conductors of straight go through coils' centre (I1). M 0 of the line of Meas. Coil-S is the coefficient M of the test coil coil-SU. L, D and O in the x-axis and current-carrying conductor is straight (as shown in Figures 2 and 5, α = 0 • , L = 5000 mm), the square-form current-carrying conductor (as shown in Figure 9), the circular-form current-carrying conductor (as shown in Figure 12), respectively.

FIGURE 15
M/M 0 between double-loop coil and three current-carrying conductors with different geometry As shown in Figure 15, M of the double-loop coils is about twice that of the single-loop coils. The coefficient M of coil-S changes slightly with the shape change of the currentcarrying conductor (2.0028-2.0031), but M of coil-T changes as change of current-carrying conductors' geometry (from 1.9982 to 2.004).

Influence of the deformation of double-loop coil-T
Since the double-loop coil is easily deformed and the inner loop and external loop are not tied closely together during its Because the segment DC of the square-form conductor of the case IV. B passes coil-T2's outer-loop when square-form current-carrying conductors were located in holes I2' and O4', numerical data of these cases are not shown in Figure 16. M 0 is the value of corresponding coils when the infinite-long straight conductor which vertically located in the inner hole I1 of coils. Figure 16 shows that when the size of the inner-loop and the outer-loop of coil-T changes and the conductors near the gap between the coil's two terminals, the error is less than 0.2% when the conductor is long and straight, the error is less than 0.65% when the conductor is circular form, the error is less than 0.75% when the conductor is square form. However, larger errors will be observed when the conductor is near the gap of coil-T2.

CONCLUSION
This paper provides calculation methods of low-frequency coefficient M among double-loop coil and current-carrying conductor of various geometry. All experimental results verify the calculation results according to the derivation. Effects on several typical double-loop and single-loop coils caused by current-carrying conductors of straight and long, square form and circular form are calculated and compared. These coils include coils of higher turn density, coils of lower turn density, deformed coil-T, single-loop and double-loop coils. Numerical and experimental low-frequency data show that (1) Double-loop coils' sensitivity is about twice that their extended single-loop frame; (2) when the long-straight conductors are vertically go through hole of coil-S, the position of the conductors have no effect on coil-S; (3) almost no error of coil-S caused by inclination of current-carrying conductor of long-straight and goes through the centre of coils; (4) the long-straight conductors have almost no effect on coil-T when the conductors are not close to the gap; (5) great errors of double-loop coil are caused by current-carrying conductor of square-form (the maximum errors almost up to 0.8%); (6) the circular-form current-carrying conductors have large influence on the double-loop coil-S (the maximum errors almost up to 3.5%); (7) almost no error of coil-S is generated by outer-loop current-carrying conductor, but large errors of coil-T are caused as the narrow distance between the current-carrying conductor and gap of coil-T; (8) the square-form and circularform current-carrying conductors cause greater effects on high turn density double-loop coils than low turn density coils. The current-carrying conductors of long-straight cause less errors of high turn density double-loop coils than errors of low turn density coils; (9) bent current-carrying conductors cause larger errors of single-loop coils than errors of double-loop one; (10) outer-loop conductors cause larger errors of double-loop coil of lower turn density than errors of double-loop coils of higher turn density; (11) coil-T2 is more susceptible than coil-T when the conductor is close to their gap.