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Modeling and Simulation of Unshielded and Shielded Energy Cables in Frequency and Time Domains Y. Weens, N. Idir, R. Bausière, and J. J. Franchaud Laboratoire d’Electrotechnique et d’Electronique de Puissance (L2EP), Université des Sciences et Technologies de Lille, Villeneuve d’Ascq Cedex, Nord 59655, France Switching of semiconductor devices is the main source of electromagnetic interference (EMI) in power static converters. Fast power devices, in particular, generate high levels of emissions in the form of high-frequency currents known as common mode (CM) and differential mode (DM) currents. In power systems, these high-frequency disturbances travel and spread over the power cables. To study the influence of the power cable characteristics on the level of the conducted emissions, it is necessary to construct a precise model of the power cables, taking into account various phenomena that appear when the frequency increases. This paper deals with two-wire unshielded and shielded energy cables. It uses a distributed parameter model that takes into account the skin and proximity effects, and the dielectric losses. The model is simulated and validated (with LT Spice/SwitcherCADIII software) in the frequency and time domains. Index Terms—Dielectric losses, frequency-domain analysis, modeling, power cables, skin effect, time-domain analysis, transmission line circuits.
I. INTRODUCTION
T
HE power converters, the wired connections constitute the spreading paths of the conducted emissions between the source and the load [1]–[4]. In the order to use a circuit analysis tool such as SPICE software to analyze the influence of the power cable on the electromagnetic interference (EMI) level, it is necessary to build a model of the cable with distributed parameters. A preliminary study demonstrated that the use of a transmission line model (low energy signal) proposed by SPICE to model an energy cable did not give satisfactory results because it is necessary to take into account both conductor resistance variation (skin and proximity effects), and also the conductance variation (dielectric losses) between two wires when the frequency increases. In this study, various electrical parameters of the shielded and unshielded power cable models are determined using the finite-element method (FEMM software: finite-element method magnetics). The obtained results are compared to the experimental measurements. The model initially obtained for a 1-m cable length is then used to simulate a 10-m cable length in the frequency and time domains. II. TWO-WIRE UNSHIELDED ENERGY CABLE MODELING The unshielded cable under study is composed of two conductors coated with PVC, and the unit is placed in a rubber sheath. The characteristics of the energy cable under study are indicated in Fig. 1. The various geometrical and electrical parameters of the cable are: r1 r2
0.69 mm; 1.44 mm;
Digital Object Identifier 10.1109/TMAG.2006.874306
Fig. 1. Structure of the two-wire unshielded cable.
r3 S D
4.35 mm; Cross sectional area of the wires mm ; mm; Distance between wires Length of the cable m; Relative permittivity ; Resistivity of conductor material m; Conductivity of the conductor material = 45.94 MS/m; Relative permeability .
A. Determination of the Cable Parameters The electrical parameters per unit length of the cable model in low frequency (here kHz) are determined using electromagnetic solver tools (FEMM). This method makes it possible to determine the inductance and resistance of the wires by taking into account the skin and proximity effects. In order to improve the precision of the model, we used a fine mesh at the interior and around the two conductors. The resistance and inductance values determined with FEMM software are given in Table I. In order to validate the obtained results, we carried out a series of measurements by using an impedance bridge (HP4294A and socket HP16047E). This also allowed us to measure, in low frequency, the parallel element values (between two wires) of the C and G parameters per unit length of the cable. Table I
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TABLE I VALUES OF THE CALCULATED AND MEASURED CABLE PARAMETERS
Fig. 4. Evolution of the cable impedance (1-m length) in short-circuit and open-circuit configurations. Fig. 2. Transmission line model (n: number of basic cells).
Fig. 3. Elementary cell of the cable model (32 cells/meter). Fig. 5. Modeling of the skin and proximity effect by an R-L ladder network (elementary cell).
presents a comparison of calculated and measured cable parameter values. We notice a slight difference between these values. In the following section, the experimental parameters are used to simulate the proposed cable model. B. Modeling of the Two-Wire Unshielded Energy Cable Generally, there are three elementary cells, which may be used to model the wired connections as shown in Fig. 2. In this study, the L circuit [Fig. 2(b)] has been chosen to model the two-wire unshielded and shielded energy cables [5], [6]. Initially, we simulated the cable model by using the values of the parameters R, L, C, and G previously determined in low frequency. To simulate the 1-m cable length, a different number of elementary cells are used (4, 8, 16, 32, and 64). For 32 cells and over (see Fig. 3), the relative error on the two resonant frequencies is lower than 1.5% (8.5% for 16 cells). The evolution of the cable impedance according to the frequency of a 1-m cable length in short-circuit (sc) and open circuit (oc) is represented in Fig. 4. A comparison of the simulation and experimental results shows that the resonance frequencies are close to each other; however, one notes a great difference between the impedance values at these frequencies. To improve this model, it is necessary to take into account the evolution of the various parameters of the cable model when frequency increases by taking into account the skin and proximity effects, and the dielectric losses in the cable.
1) Modeling of the Skin and Proximity Effects: There are various methods to model the skin effect in a conductor [7]–[9]. To model the evolution of conductor resistance when the frequency increases (skin and proximity effects), an R-L ladder circuit shown in Fig. 5 is used. The various values of the elements of this circuit were obtained by using a mathematical solver tool. Fig. 6 represents the simulation results of the R-L ladder network. A comparison of these results with those obtained with FEMM software in the frequency band varying from 100 kHz to 110 MHz shows good agreement. The conductor inductance value of the ladder network has a negligible influence on the global wire inductance as shown in Fig. 6. 2) Modeling of the Dielectric Losses: Generally, to model the transmission lines (low energy signal) the conductance is neglected. Given the dimensions of the energy cable (dimensions of PVC and rubber) and the levels of the rated currents and voltages, it is necessary to take into account the conductance and its evolution when frequency varies. As before, an R-C ladder circuit (Fig. 7), is used to model the evolution of the conductance (dielectric losses) when the frequency increases. The various values of the elements of this circuit were obtained by using a mathematical solver tool. The results of the simulation of the R-C ladder network are shown in Fig. 8. A comparison of the simulation results with measured data (using the impedance bridge) shows good concordance. The capacitance value of the ladder network has a
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Fig. 6. Evolution of the resistance and inductance of one wire when the frequency increases (ladder circuit and FEMM). Fig. 9. Diagram of the circuit of an elementary cell taking into account the skin and proximity effects, and the dielectric losses.
Fig. 7. Modeling of the dielectric losses by an R-C ladder network (elementary cell).
Fig. 10. Evolution of the cable impedance (1-m length) in short-circuit and open-circuit configurations.
Fig. 8. Evolution of the conductance and capacitance between two wires when the frequency varies (ladder circuit and measurement).
negligible influence on the global capacitance of the cable as shown in Fig. 8. C. Simulation of 1-m and 10-m Cable Length 1) 1-m Cable Length: A model of one cell (32cells/meter) of the two-wire unshielded cable, taking into account the skin and proximity effects, and the dielectric losses is represented in Fig. 9. Fig. 10 represents a comparison between the simulation results of the cable impedance in the frequency domain and experimental measurements with impedance bridge HP4294A, socket HP16047E and high-frequency balun (Fig. 11). These results show good agreement between simulation and measurement in open circuit configuration, and a slight instantaneous frequency deviation for the test in short-circuit configuration.
Fig. 11. Cable connection to the impedance analyzer.
To validate the obtained energy cable model in the time domain, the cable input is fed by a voltage pulse generator (HP 8114A) to model the power switch transitions, and the output is loaded by a 750 planar resistor. The experimental setup is shown in Fig. 12. The current in the cable is measured with a high-frequency probe current (Tektronix TCP202). The cable model simulation
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Fig. 14. Evolution of the cable impedance (10-m length) in short-circuit and open-circuit configurations.
Fig. 12. (a) Experimental setup. (b) Pulse voltage waveform.
Fig. 15. Current waveforms in the 10-m cable length.
Fig. 13. Current waveforms in the 1-m cable length.
results in the time domain are represented in Fig. 13. These results show good concordance between the simulation and the measured data, with a slight delay of 5 ns. 2) 10-m Cable Length: In order to validate the model obtained over a longer cable length, we simulated a 10-m cable length by increasing the number of cells of the overall model. The simulation results of the cable model (10-m) in short-circuit and open-circuit configurations are shown in Fig. 14. These results show good agreement in frequency with a slight difference in the amplitudes between simulation and measurement. In the same way as before, we carried out a test in the time domain. The simulation and experimental results of the current waveform in the cable are represented in Fig. 15. These results show good concordance between the simulation and the measured data, with a slight delay of 3 ns. III. TWO-WIRE SHIELDED ENERGY CABLE MODELING In this section, the proposed method is applied to model a 10-m shielded power cable. The two-wire shielded energy cable under study is composed of two conductors coated with PVC and a shield made from the same material as the conductors is placed around the wires. The entire unit is placed in a rubber
Fig. 16. Structure of the two-wire shielded cable.
sheath. The geometrical characteristics of the energy cable under study are indicated in Fig. 16. A model of one elementary cell of the shield cable is shown in Fig. 17. The same model of the transmission line circuit (L circuit) is used. Three types of impedances are used to model the shield energy cable. — Impedance Zl, which represents the conductor inductance and resistance. We noted that the shield cable required an inductance coupling coefficient “K” between conductors. — Impedance Zi, which is used to model capacitance and conductance (dielectric losses) between the two conductors.
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TABLE II VALUES OF THE CALCULATED AND MEASURED CABLE PARAMETERS
Fig. 17. Model of the two-wire shielded cable.
Fig. 18. Circuit configurations used for the determination of the coupling coefficient “K” between conductors. (a) Common mode. (b) Differential mode.
— Impedance Zb, which is used to model capacitance and conductance (dielectric losses) between each conductor and the shield. Values of L and K are calculated from the input inductance expressions obtained from two configuration tests (Fig. 18), in common mode and differential mode, respectively:
In the following section, we describe the method used to determine the shield cable parameters. A. Determination of the Cable Parameters The electrical parameters per unit length of the cable model in low frequency are also determined using the finite-element method and the obtained results are compared to the experimental measurements. As for the unshielded cable, FEMM software is used to determine the conductor inductance and resistance by taking into account the skin and the proximity effects. The resistance and inductance values determined with FEMM are given in Table II. In order to validate the obtained results, we carried out a series of measurements by using the experimental setup described previously. Determining the cable model parameters required two test configurations: one in common mode and the other in differential mode for the cable in both short-circuit and open-circuit configurations. The various calculations and measurements of the cable parameters (per unit length) are presented in Table II. A simulation of the shielded cable using a transmission line model (low energy signal) proposed by SPICE did not give satisfactory results and it is necessary to take into account various phenomena that appear when the frequency increases.
Fig. 19. Evolution of the resistance and inductance of one wire according to the frequency.
B. Modeling of the Two-Wire Shielded Energy Cable As for the unshielded cable, we recorded the evolution of the cable parameters and we noted that all of them vary according to the frequency. The same network circuits are used to model the variations of the cable parameters when the frequency is increased. To model the evolution of the resistance when the frequency varies (skin and proximity effects) an R-L ladder circuit is used. The various values of the elements of this circuit were obtained using a mathematical solver tool. A comparison between the R-L network, FEMM software, and experimental results are presented in Fig. 19. We observe good agreement between experimental and simulation results. We notice a slight difference between the measured values and those obtained with FEMM software. The precision of the results given by FEMM software depends on the accuracy of the measurement of the geometrical parameters of the cable, which remain difficult to measure, and on the electric parameters (permittivity ) which the manufacturer does not always provide. As before, an R-C ladder circuit is used to model the evolution of the conductance between the two wires and between each conductor and shield. The simulation of the R-C ladder network and the experimental results of the capacitance and conductance between two wires (Fig. 20), and between each wire and shield (Fig. 21) show good concordance. However, we could not use software FEMM to obtain the cable capacities and conductances because it is necessary to know the permittivity and resistivity and their evolution when the frequency increases.
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Fig. 20. Evolution of the conductance and capacitance between two wires when the frequency varies (1-m length).
Fig. 23. Evolution of the cable impedance in common mode.
Fig. 21. Evolution of the conductance and capacitance between wires and the shield when the frequency varies (1-m length).
Fig. 24. Evolution of the cable impedance in differential mode.
Fig. 22. Diagram of the circuit of an elementary cell taking into account the skin and proximity effects, and dielectric losses.
In the following section to validate the proposed shielded cable model, a simulation of a 10-m cable length is carried out. C. Application of the Model for a 10-m Cable Length The elementary cell of the model used to simulate the shielded cable with various electrical parameter values is presented in Fig. 22. Simulation results in the frequency domain of the 10-m cable length in both common mode and differential mode configurations are presented and compared with experimental measurements in Figs. 23 and 24.
Fig. 25. Current waveforms in the 10-m cable length in common mode configuration.
These results show good agreement between simulation and measurement until 20 MHz. As before, we carried out a test in the time domain. The simulation and experimental waveforms of the current in the cable are shown in Fig. 25 (common mode) and in Fig. 26 (differential mode). These results show good concordance between the simulation and the measurements, with a slight delay of 5 ns. IV. CONCLUSION To analyze and reduce the EMI generated by the power converter used with a long energy cable, it is necessary to build a
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[4] N. Idir, J. J. Franchaud, and R. Bausiere, “Evaluation and reduction of common mode currents in adjustable speed drives,” in Power Electronics Intelligent Motion (PCIM), Nuremberg, Germany, 2004. [5] E. Gubia, P. Sanchis, O. Alonso, J. Lopez, A. Lumbreras, and L. Marroyo, “Cable modeling for electrical drives,” in Eur. Conf. Power Electronics Conference (EPE), Toulouse, France, 2003. [6] A. F. Moreira and T. H. Lipo, “High-frequency modeling for cable and induction motor over voltage studies in long cable drives,” IEEE Trans. Ind. Appl., vol. 38, no. 5, pp. 1297–1306, Sep.–Oct. 2002. [7] H. G. Brachtendorf and R. Laur, “Simulation of skin effects and hysteresis phenomena in the time domain,” IEEE Trans. Magn., vol. 37, no. 5, pp. 3781–3789, Sep. 2001. [8] S. Kim and D. P. Neikirk, “Compact equivalent circuit model for the skin effect,” in Microwave Symp. Dig. IEEE MTT-S Int., 1996, vol. 3, pp. 1815–1818. [9] S. Ogasawara and H. Agaki, “Modeling and damping of high frequency leakage currents in PWM inverter-fed AC motor drive system,” IEEE Trans. Ind. Appl., vol. 32, no. 5, pp. 1105–1114, Sep.–Oct. 1996. Fig. 26. Current waveforms in the 10-m cable length in differential mode configuration.
satisfactory model of the cable, which is valid in both the time and the frequency domains. In a first approach, the study of an unshielded two-wire cable allows us to eliminate the problems created by the asymmetry of the cable and by the nonexistence of the common mode currents. The proposed model of unshielded and shielded energy cables takes into account the skin and proximity effects as well as the dielectric losses. In this study, a finite-element method is used to calculate most of the electrical parameter values of the energy cable model. The proposed unshielded and shielded models were validated in both the frequency and time domains. The resulting cable model can thus be used to test various solutions making it possible to reduce output overvoltages at the cable terminal (under the motor input winding when it is supplied by the power converter). The proposed method will be used as a basis to develop more advanced models, which will be necessary for the study of multiwire cables. REFERENCES [1] C. R. Clayton, Introduction to Electromagnetic Compatibility. New York: Wiley, 1992. [2] R. J. Kerkman, D. Leggate, and G. L. Skibinski, “Interaction of drive modulation and cable parameters on AC motor transients,” IEEE Trans. Ind. Appl., vol. 33, no. 3, pp. 722–731, May–Jun. 1997. [3] O. A. Mohammed, S. Ganu, N. Abed, S. Liu, and Z. Liu, “High frequency PM synchronous motor model determined by FE analysis,” IEEE Trans. Magn., vol. 42, no. 4, pp. 1291–1294, Apr. 2006.
Manuscript received December 14, 2005; revised March 23, 2006. Corresponding author: N. Idir (e-mail:
[email protected]).
Y. Weens received the Dipl-Ing. degree from Ecole Polytech’ Lille in 2003. He is currently pursuing the Ph.D. degree in electromagnetic compatibility (EMC) in power converters at Université des Sciences et Technologies de Lille in the L2EP Lille (Laboratory of Electrical Engineering of Lille), France. His main research interest is EMC in power converters.
N. Idir (M’03) received the Ph.D. degree from Université des Sciences et Technologies de Lille, France, in 1993. Since 1994, he has been an Associate Professor at the Institut Universitaire de Technologie de Lille and at the Laboratory of Electrical Engineering of Lille (L2EP Lille). His main research interest is EMC in power converters.
R. Bausière (M’93) received the Ph.D. degree in physical sciences from the Université des Sciences et Technologies de Lille, France, in 1982. He has been a Professor of power electronics at the Université des Sciences et Technologies de Lille since 1989. He is now in the Laboratoire d’Electrotechnique et d’Electronique de Puissance de Lille (L2EP Lille), where his main research interests are control of switching devices and multilevel power converters. He is co-author of three power electronic books.
J. J. Franchaud is currently a Research Engineer at the Université des Sciences et Technologies de Lille in the Laboratory of Electrical Engineering of Lille (L2EP Lille), France. His research interests include power electronics and EMC in power converters.
