Laboratory current transformer based on Rogowski coil

This paper cover the analysis and construction of current to voltage transducer based on Rogowski coil which satisfy the requirements of high-accuracy measurement of AC current (up to 20 A at power supply frequency, with the aiming uncertainty of 100 parts per million). Primary source of ac current uncertainty measured by this type of transducer is nonuniform density of turns which, in case of eccentricity or shift of primary conductor results in deviations of mutual inductance. Self capacitance and self resistance, temperature dependence of coil geometry and electromagnetic interferences affect the accuracy as secondary source of uncertainty. With respect of influencing parametres, the current transducer of aim accuracy can be realized. I. Requests on transformer properties Electromotive force, induced in Rogowski coil with arbitrary, most often flexible toroidal shape which enclose the primary conductor, is proportional to derivation of total magnetic flux in coil, i.e. time derivation of measured current in primary conductor and mutual inductance of that geometrical system: t t i M t t Ψ t e d ) ( d d ) ( d ) ( ⋅ − = − = (1) The current with strictly sinusoidal shape can be measured without integrator but then their frequency should be measured. However, for the purpose of reconstruction of complex current waveform, the appliance of integrator is obligatory. Thus, the integration and amplification of voltage induced in secondary coil is equally important part of complete current transducer (Fig.1). integrator amplifier In view of requests, the laboratory current transformer should be capable for measurement of ac current up to 20 A in frequency range from 20 Hz to 1 kHz, with relative uncertainty in order of 10. Moreover, it must be persistent on influences of electromagnetic interferences and disturbances.. Fig. 1. The principle of ac current measurement by transducer based on Rogowski coil II. Selection of transformer geometry The construction of transformer depends greatly of geometrical relationship between its primary and secondary part. Stability of mutual inductance of this system is achieved by firm mechanical connection between primary conductor and secondary coil, and especially by delicate geometry of secondary turns. Therefore, secondary turns shall be wound on rigid body with accurately known dimensions, which precise construction is the most simplified by adopting standard toroidal geometry of non-magnetic insulator with rectangular cross-section. The version with circular cross-section is not discussed, as the requested accuracy of body dimensions are much harder to achieve. In selection of dimensions and construction of precise coils, the results of the previously analysis [1,2] are taken into consideration. This verdicts can be condense into three elementary claims: 16 IMEKO TC4 Symposium Exploring New Frontiers of Instrumentation and Methods for Electrical and Electronic Measurements Sept. 22-24, 2008, Florence, Italy 1. for obtaining the largest possible mutual inductance, the coil must be positioned as close as possible to primary conductor, and must have greatest cross-section of each turn 2. the coil must be wounded in one layer, and in this case the number of turns depends on diameter of used wire 3. The influence of eventual discontinuity, caused by first and last turn, and varying density of turns along the perimeter of toroid are minimised by centered mutual position and firm mechanical connection between primary conductor and coil III. Conception of the construction relating to influences of strange electromagnetic fields Small achieved mutual inductance of the system consists of coil and conductor indicating its high sensitivity on electromagnetic interferences, as well as possible unwished influence on accuracy in precise current measurements. In principle, the disturbances derived from strange magnetic fields induce additional electromotive force Em in leakage inductance Lm, while the disturbances derived from strange electrical fileds effects as injection of small current IE in measuring circuit through effective parasitic capacitance CE between coil and whole surroundings. (Fig.2). Fig. 2. Equivalent circuit in analysis of influence of strange electromagnetic fields The basic problem is unwished axial contour area of coil, which in electrical manner act as mentioned leakage inductance Lm. Neutralisation of this effect can be performed by using the additional loop with equal crosssection fitted close to (Fig.3a) or into coil. coil additional loop strange magnetic field Fig. 3a. Cancelation of influence of strange ac magnetic field by using an tertiary loop Fig. 3b. Solution with astatic configuration which has two coils wounded in opposite directions Electromotive forces induced by strange magnetic field in this two loops are canceled by its properly serial connection. The same performance, but with double transducer sensitivity (i.e. double mutual inductance) can be obtained by astatic configuration (Fig.3b). In this case, the addition of electromotive forces induced by measured current in primary conductor is given by two identical coils which are wounded in opposite directions. Here are the coil terminals with same polarity of electromotive force induced by measured current pointed by dots, while the terminals with the same polarity of electromotive force induced by interferences pointed by asterisk. On the 16 IMEKO TC4 Symposium Exploring New Frontiers of Instrumentation and Methods for Electrical and Electronic Measurements Sept. 22-24, 2008, Florence, Italy other side, owing to large area of secondary coils (approximately 500 cm) it can be expect relatively high sensitivity on disturbances of capacitive type induced by strange electric field. Thus, the conception with two parallel and identical measuring channels is adopted. Their opposite output voltages are lead to voltage amplifier of instrumentation type (Fig.4), whose common-mode rejection ratio (CMRR) is very high . Fig. 4. Conception of current measurement by two parallel measuring chanells If those chanells are well balanced (which can be achived by careful designing), the transformer would be immunity on influences and disturbances caused by strange electromagnetic fields. In this manner, the usage of electromagnetic shields and screens are avoided. A. Primary circuit of the transformer The main parameter in designing of geometry and estimating of primary conductor is density of measured current and waste heat, so the conductor should be treated as dissipative element of the system. The thermal stabilisation of entire transducer is more simplest task if the heat dissipation are minimised, which is reduce the temperature gradient in the transformer volume. The highest expected current are IPM = 20 A and the crosssection of the primary conductor must be relatively large, so the diameter of 18 mm was chosen. As the primary conductor also serves like a frame of the coil centrators, it must be precise processed, so the best choise of material was duralumine alloy (AlCu5Mg1). According to room required for secondary coils, the lenght of the conductor is 350 mm. The influence of temperature variations on transformer properties is controled by dissipation limit, which is 50 mW at highest expected current. B. Secondary circuit of the transformer Design of the secondary part of transformer was based on desired electrical properties and geometrical parameters. Like starting point, the transimpedance (i.e. sensitivity) ZT of 1,25 mV/A at frequency f = 50 Hz was chosen. Thus, the mutual inductance is: μH 4 π 2 ≈ ⋅ = f Z M T T (2) As the rejection of interferences is solved by double astatic construction (i.e. secondary is composed of four coils), the system of conductor and single coil should have mutual inductance of 1 μH. In real case, the effective cross-section of each turn is somewhat larger than rectangular cross-section of toroidal body, so the mutual inductance could be expressed as: ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + ⋅ + ⋅ ⋅ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ = d r d r d h N r r h N M U V 0 U V 0 2 2 ln 2π ' ' ln 2π ' μ μ , (3) where N is number of turns, d is wire diameter, h is body thickness, while rU and rV are inner and outer body radius, respectively. Furhermore, it is easily to verify that the inductance of toroidal coil is L = N·M. With known desired mutual inductance, the estimation should be start with consideration of influences which might became obstacle in the process of production. In the first place, this is wire diameter, which should be chosed in respect to precision of winding, as well as inner diameter of coil body because of winding technology. As the coil must 16 IMEKO TC4 Symposium Exploring New Frontiers of Instrumentation and Methods for Electrical and Electronic Measurements Sept. 22-24, 2008, Florence, Italy be single-layered, the wire should be of satisfactory strenght, but also sufficiently thin so the aimed number of turns (i.e. mutual inductance) can be realized. The optimal diameter d of copper wire was settled at roughly 0,3 mm so, owing to thickness of insulation lacquer, the external diameter of wire is d = 0,322 mm). The coil is performed on the body of toroidal shape, made of polymethyl methacrylate (plexiglas), which was proven as good choice during the construction of test probes, which was used in verification of analitycal methods for estimation of mutual inductance of real Rogowski transducers. Due to standardized dimension of plexiglas sheets, the height of toroidal body is h = 19,24 mm, while the inner and outer radius is rU = 15 mm and rV = 37 mm, respectively (Fig.5).