Analysis of the Skin Effect for Calculating Frequency-Dependent Impedance of the TRTS Power Rail

Transcription

1 Proc. Natl. Sci. Counc. ROC(A) Vol. 23, No. 3, pp Analysis of the Skin Effect for Calculating Frequency-Dependent Impedance of the TRTS Power Rail YAW-JUEN WANG Department of Electrical Engineering National Yun-Lin University of Science & Technology Touliu, Taiwan, R.O.C. (Received June 30, 1998; Accepted August 27, 1998) ABSTRACT Knowledge of frequency-dependent impedance is essential for transient and harmonic analyses of a DC traction power system. This paper deals with the skin effect of the power rail (also known as the conductor rail or third rail) used in the Taipei Rapid Transit System (TRTS). The method of coupledinductance based on subdivision of conductors is employed to calculate the current distribution and the frequency-dependent impedance of the power rail under the skin effect. This paper also proposes a method to reduce the number of subconductors required to achieve a prescribed level of accuracy at high frequency so that computation time and matrix dimensions are reduced. Key Words: skin effect, traction power systems, power rail, frequency-dependent impedance I. Introduction The Taipei Rapid Transit System (TRTS) is a 750 V DC third-rail traction system, where the DC power is supplied by 24-pulse rectifiers installed in Traction Supply Substations (TSS). Intuitively, one may consider that frequency-dependent rail track impedances are important only in AC rail traction systems, and that only DC parameters are required to study DC railway systems. Also, one may expect that a DC railway system will be immune from harmonic disturbances. However, many studies have reported harmonic problems and the need for knowledge of the frequency dependency of track impedances in DC traction power systems. In fact, harmonics and transients do exist in a DC traction power system. While harmonics are attributed to switching operations of power electronics controlled inverters in motor cars and to voltage ripple of rectifiers in the TSS, transients usually stem from acceleration/deceleration of trains or from short-circuit faults. A brief review of the literature reveals that problems concerning harmonics and transients in DC traction power systems are quite well documented. Fracchia et al. (1994) compared the short-circuit faults of a 750 V DC metrorail power system studied using both the fixed-frequency impedance model and frequency-dependent impedance model, and found a significant difference in the initial rate of current-rise for a near-substation fault and in the ripple content of fault current for a fault located far from the substation. Taufiq and Xiaoping (1989) developed a frequency domain method to identify the DC side harmonics produced by a traction pulse-width modulation (PWM) inverter. Coles et al. (1994) identified harmonics produced by power electronics controlled inverters in locomotives and rectifiers in the substations of a 3 kv DC railway system. Steyn and Wyk (1994) studied harmonic currents generated by a 3 kv DC traction locomotive used in the South African Railway and emphasized possible malfunctions of signaling circuits caused by harmonics due to electromagnetic interference. Hill et al. (1995) studied possible harmonic resonance in a 3 kv DC railway system brought about by series inductance of the rail and shunt capacitance of filters in the locomotives. Since the rail inductance encountered by a train varies when the train is moving, the resulting harmonic resonant frequency varies with the train location as well. To deal with this interesting phenomenon, Fracchia et al. (1996) resorted to a probabilistic approach to model random variation of harmonic impedance of that DC railway. The foregoing review of papers has revealed the importance of knowledge of frequency dependency of rail impedance in a DC traction system. A particular difficulty in calculating the impedance of power and running rails as a function of frequency is that they have 419

2 Y.J. Wang irregular cross sectional shapes. Running rails are made of ferrous material, which makes the calculation even more complicated. In this paper, we will confine ourselves to the problem related to the power rail, the frequency-dependent impedance of which is mainly influenced by the skin effect. The skin effect is a well known physical phenomenon that takes place when alternative current flows through a conductor. The time-varying magnetic field produced by the AC current results in uneven distribution of electric current over the cross section of the conductor. The electric current tends to concentrate near the surface of the conductor, leading to an increase in the resistance and a decrease in the internal inductance. The skin effect is a complicated problem that can be solved analytically only for a very few cases. The most commonly cited examples are infinitely wide flat sheets and long straight cylindrical lines. The former case leads to the concept of the depth of penetration while the latter involves Bessel functions in its analytical closed-form solutions (Meliopoulos, 1988). While the cylindrical conductor model is practically applicable to cases such as electric power transmission lines, it fails to predict the impedances of a conductor with a complicated cross section (e.g., rails and cables) under the influence of the skin effect. An approximate analytical method has been proposed by Ametani and Fuse (1992) for calculating frequency-dependent impedance of a conductor having an arbitrary cross sectional shape. They have tested their method on cable conductors of fan-shaped cross sections with satisfactory accuracy. The case of a power rail is, however, more complicated since its irregular cross section is much more heavily deformed. Therefore, several studies concerning calculation of the rail impedance have resorted directly to a numerical approach (Hill and Carpenter, 1991, 1993; Carpenter and Hill, 1993). The most well known numerical method for solving the skin effect problem of conductors with irregular cross section is perhaps the finite element method (FEM), which can be used to solve partial differential equations governing magnetic fields and electric current in a conductor. The FEM is attractive because it is able to handle the irregular geometric shape by discretizing it into numerous meshes. The difficulties encountered in applying the FEM to rail modeling reside in properly meshing the rail cross section and its surroundings, and in setting boundary conditions to achieve the required accuracy. The FEM software programs for electromagnetic field analysis are expensive, which may in some cases encourage the use of a lower cost method. An alternative to the skin effect problem is a network model based on the coupled-inductance theories. This method was first introduced by Silvester (1966) and has been in widespread use in calculating the current distribution and the frequency-dependent impedance of conductors with irregular cross sectional shapes, where analytical modeling is difficult. Typical examples are cable conductors (De Arizon and Dommel, 1987) and microstrip lines in integrated circuits (Dinh et al., 1990). The coupled-inductance method can be briefly described as follows: The conductor cross section is divided into n identical square (or rectangular) subconductors. By assuming uniform current distribution for each subconductor, one can obtain n magnetically coupled R-L series branches in parallel. Therefore, the actual current distribution over the conductor cross section is approximated by the currents flowing in the R-L branches. As long as n is sufficiently large, an accurate result can be obtained. The coupledinductance method is basically a circuit-oriented model which, though not as powerful as the FEM, is familiar to most electrical engineers and is easy to code with a high level computer language. This paper adopts the coupled-inductance method to analyze the skin effect of the TRTS power rail. The current distribution and the impedance of the power rail as a function of frequency are calculated. This paper begins with a brief introduction to the basic principle of the coupled-inductance method. Subdivision of the rail cross section, which allows the rail to be modeled using magnetically coupled circuits, is illustrated next. We then present a mathematical formulation of the parameters of the coupled circuits. The solution to the circuit matrix equation gives the current distribution and impedance of the power rail at a given frequency. Finally, this paper presents simulation results of the current distribution over the rail cross section at different frequencies using two- and three-dimensional graphs. The resistance and inductance of the power rail as a function of frequency are also shown. A drawback of the coupled-inductance method is that the required number of subconductors increases rapidly when the frequency goes up, leading to an increase in the matrix dimension and in the computation time. To overcome this problem, this paper also proposes a variant method that enables economization of the number of subconductors and, hence, savings in computation time and memory capacity. II. Coupled-Inductance Model 1. Basic Principle The basic concept of the coupled-inductance model 420

3 Skin Effect of the TRTS Power Rail geometric mean radius (GMR) of the subconductor if i=j. The GMR of a square can be obtained by the quadruple integral (Aguet and Morf, 1987): ln (g s ) = 1 λ 4 0 λ λ 0 0 λ 0 λ ln (x u) 2 +(y v) 2 dxdydudv, (3) Fig. 1. Rail encased in an imagined thin cylindrical sheath of radius a. The arrows indicate the directions of the currents. was first introduced by Silvester (1966) and is also applicable to analysis of the skin effect of the power rail. Figure 1 shows a rail encased in an imagined thin cylindrical sheath of radius a. The coupled-inductance model assumes that the rail current is returned by the sheath and is uniformly distributed over its surface. It is noted that introduction of the imagined sheath serves not only to provide a defined path for the current, but also to ensure that the current distribution in the rail is not affected by the existence of the return current, i.e., by the proximity effect. Note that the choice of a is mathematically arbitrary. However, to make physical sense, the radius a must be large enough to enclose the rail. The value of a affects only the level of the rail external inductance, not its internal inductance. A cross section of the rail subdivided into n identical slim square subconductors is shown in Fig. 2, in which the dimensions of the rail are indicated. where g s and λ are the GMR and side of the square, respectively, and x, y, u and v are dummy integral variables. Equation (3) has been evaluated using the numerical integration technique and can be rewritten in a simpler form: g s = λ. (4) The GMD between the ith and jth subconductors can be approximated by their center-to-center distance. De Arizon and Dommel (1987) have shown that for the worst case of two adjacent squares, the error of this approximation is only 0.655%. It is noted that in Eq. (2), the GMD between any subconductor and the sheath is equal to the sheath s radius a, and so is the GMR of the sheath. It is this property that allows the inductances to be calculated so easily by Eq. (2), which is another advantage provided by introduction of the imagined sheath. The coupled-circuit model is depicted in Fig. 3, and all the parameters can be obtained by Eqs. (l)-(4). The task remains of determining the relative levels of currents in all subconductors. 3. Analysis of the Coupled-Circuit When steady-state sinusoidal current flows in the 2. Formulation of Circuit Parameters The resistance per unit length of each subconductor in Fig. 2 can be given by r=n/(σa), (1) where σ and A are the conductivity and the cross sectional area of the rail, respectively. The mutual inductance of the ith and jth subconductor can be written as l ij = µ 0 2π ln a d ij, (2) where d ij refers to the geometric mean distance (GMD) between the ith and jth subconductor if i j, and to the Fig. 2. Rail cross section divided into n square subconductors. 421

4 Y.J. Wang distribution in the rail at a specific angular frequency ω, it suffices to solve the linear system in Eq. (5) for [I] by arbitrarily giving a value for the voltage drop V. If one volt per meter is specified as the voltage drop, Eq. (5) becomes a problem of finding the inverse of the impedance matrix [Z], or of finding the admittance matrix [Y]=[Z] 1. (11) Fig. 3. Coupled-circuit model of subconductors. rail, the voltage-current relation in the circuit shown in Fig. 3 can be written in phasor form as where [V]=[R][I]+jω[L][I] [R]= =[Z][I], (5) r r r is the n n diagonal resistance matrix, [L]= l 11 l 12 l 22 l 1n (6) l n1 l nn (7) the symmetric inductance matrix, [I]=[I 1 I 2... I n ] T (8) the current vector, [V]=[V 1 V 2... V n ] T (9) the voltage vector and [Z]=[R]+jω[L] (10) the impedance matrix. The reader is reminded that [R], [L] and [V] are in their respective intrinsic units per unit length. The voltage vector [V] has the same value for all its elements because the voltage drops along all the subconductors must be equal. To evaluate the current 4. Frequency-Dependent Impedance The solution to the skin effect also allows the frequency-dependent resistance and inductance to be evaluated effectively. The current in the rail must be equal to the sum of all the currents in the subconductors. Hence, upon obtaining the current vector [I], the rail current I T can be given by I T = n Σ I i i =1. (12) The impedance of the rail per unit length is, thus, Z r =V/I T =R r (ω)+jωl r (ω) =R r (ω)+jω(l int (ω)+l ext ), (13) where R r (ω) refers to the frequency-dependent rail resistance. The rail inductance L r (ω) includes two components: the internal inductance L int (ω), which is frequency-dependent, and the external inductance L ext, which is independent of the frequency but is related to the geometry of the current return path. Before ending this section, the reader is reminded that Eq. (13) can also be obtained from Eq. (11) by Z r =( n n Σ Σ y i =1 j =1 ij ) 1. (14) 5. Economization of Subdivision When the frequency increases, the gradient of the current near the surface and corners of the rail cross section becomes significant. The result is that the calculation accuracy deteriorates rapidly because the number of subconductors is not large enough to approximate the steep change of the current density at the surface and in the corners. A straightforward way to solve this problem is to increase the number of subconductors. Thus, the dimensions of matrices [R] and [L] increase, and so do the computation time and the memory capacity needed. A technique to improve 422

5 Skin Effect of the TRTS Power Rail the calculation accuracy without increasing the number of subconductors is shown in Fig. 4, in which subconductors near the surface are h times slimmer than those more inside. The side ratio h increases with the frequency. In our simulation program, h is an integer and is approximately equal to the exponent of the frequency (i.e., h log(ω)). The area with slimmer subconductors starts from the borders and extends towards the inside by about one depth of penetration. When this technique is used, slight modification of the resistance matrix [R] given by Eq. (6) is needed. Let the rail cross section be divided into m squares of side λ near the border and into k squares of side hλ in the center. Therefore, the total number of subconductors is n=m+k. The resistance of the m slimmer subconductors is given by r s =(m+kh 2 )/(σa), (15) and the resistance of the k grosser subconductors is r s /h 2. The resistance matrix [R] given by Eq. (6) becomes where and R]= [R 1 ] 0 0 [R 2 ] [R 1 ]= [R 2 ]= n n r s r s r s m m r s 0 0 h 2 r 0 s h , (16) r s h 2 k k III. Simulation Results 1. Distribution of the Current (17). (18) A computer program in FORTRAN has been developed based on Eqs. (1)-(5) to study the current distribution of the power rail at various frequencies. The conductivity and permeability of the power rail are σ= S/m and µ=µ 0 =4π 10 7 H/m, respectively. Fig. 4. Subdivision of rail cross sections. Subconductors near the surface are h times slimmer than those in the center. In this figure, h equals 3. The rail cross section has been subdivided into 618 identical squares, and the circuit parameters [R] and [L] have been calculated for the designated frequency. Solving Eq. (5) gives the currents (including the magnitudes and phase angles) in all the subconductors. As only the relative levels of the currents are of interest, the current vector [I] has been normalized so that the maximum current level is equal to one. The simulation results are shown in Fig. 5, in which the current distribution over the rail cross section corresponding to 10, 60, 300 and 1,200 Hz of the exciting frequencies are clearly demonstrated in threedimensional graphs. It is seen that even at 10 Hz, the skin effect is already perceptible. At 60 Hz, the skin effect is more pronounced than can be imagined by most electrical engineers. The current tends to concentrate at the corners of the rail when the frequency exceeds 300 Hz. It is noted that 60, 300 and 1,200 Hz correspond to the fundamental, 5th and 20th harmonic frequencies commonly encountered in electric power systems. The normalized iso-current-density contours corresponding to Fig. 5 are depicted in Fig. 6, which shows the influence of the skin effect on the current distribution in a different manner. It must be noted that only the iso-current-density contours enclosed in the rail cross section profile are meaningful while those outside the profile are the results obtained through interpolation and do not make any physical sense. The iso-current-density contours show that the higher the frequency increases, the more the contours concentrate near the corners and the borders of the rail cross section. Figure 6 thus provides an illustrative demonstration of the skin effect in response to the 423

6 Y.J. Wang Fig. 5. Three-dimensional illustration of the current distribution in the rail under the skin effect at different frequencies. (a) 10 Hz, (b) 60 Hz, (c) 300 Hz, (d) 1,200 Hz. current frequency. 2. Frequency-Dependent Resistance and Inductance An important result obtained through study of the skin effect is the frequency-dependent resistance and inductance of the rail, which play a significant role in evaluating transients and harmonic disturbances of the traction power. With the aid of Eqs. (12) and (13), the program is able to compute, with accuracy, the variation of the resistance and inductance of the power rail. When analyzing the skin effect at a frequency higher than 1,200 Hz, the coupled-inductance model applies the technique of economization of subdivision discussed in the previous section. The calculation was carried out for the case where the rail is encased in an imagined thin cylindrical sheath with a radius of 14 cm. The rail cross section is subdivided into 2,980 inner and 2,682 outer square subconductors, with a side ratio h being equal to 5. To validate the coupled-inductance model, a twodimensional electromagnetic field analysis software program based on the FEM has been used to analyze the same problem for comparison. The geometry designed for the FEM analysis is shown in Fig. 7(a), in which the axis of symmetry bisects the space into two symmetric zones. Because of symmetry, only one 424

7 Skin Effect of the TRTS Power Rail Fig. 6. Normalized iso-current-density contours over the cross section of the rail at different frequencies. (a) 10 Hz, (b) 60 Hz, (c) 300 Hz, (d) 1,200 Hz. Note that contours outside the rail profiles are the results of interpolation and have no physical meaning. zone needs to be modeled by the FEM. Also shown in the figure are boundary conditions: the Dirichlet condition for the circle of l-m-radius and the Neumann condition for the axis of symmetry. The meshing of the rail cross section and part of its surroundings is shown in Fig. 7(b), in which elements near the corners are made sufficiently small so as to increase the computation accuracy. The FEM allows the stored magnetic energy and power dissipation in a specific area to be calculated, which in turn permits the inductance and resistance to be obtained with the very fundamental formulae E=L r I T 2 /2 and P=I T 2 R r, where E and P are the stored magnetic energy and dissipated electric power in that area, respectively. To enable comparison with the coupled-inductance model, the stored magnetic energy and power dissipation in the area enclosed by the circle of 14-cm-radius were calculated at various frequencies. Figure 8 presents the calculation results obtained using the coupled-inductance model and the FEM for frequencies ranging from 1 to 10 5 Hz. For the rail inductance (dashed line and triangular dots), the two 425

8 Y.J. Wang as harmonic analysis and resonance prediction, the frequency range up to 10 5 Hz is largely sufficient. If, however, rail resistance and inductance at higher frequencies are required, it suffices to adjust the side ratio h or to increase the number of subconductors in the coupled-inductance model. Figure 8 also conveys an advantage of the coupled-inductance model over the FEM in that it is able to give continuous variations of resistance and inductance versus frequency. On the other hand, each simulation scenario of the FEM gives values corresponding only to a specific frequency. To have a global view of how the resistance and inductance vary with the frequency, many simulation scenarios must be run using the FEM. It can be observed that the rail resistance increases rapidly when the frequency exceeds 1,000 Hz. When the frequency increases, the inductance of the rail drops as an inverse s curve as expected. Note that the inductance declines and gradually attains its minimum value. It is informative to point out that the quantity by which the inductance decreases is the internal inductance of the rail while the minimum level of the inductance accounts for its external inductance. As we mentioned earlier, the internal inductance varies with the frequency while the external inductance does not. We note finally that if the radius of the sheath a is chosen differently, the external inductance will change, but the internal inductance will remain the same. IV. Conclusions The influence of the skin effect on the current Fig. 7. Calculation of the rail resistance and inductance using the FEM to validate the coupled-inductance model. (a) Schematic diagram showing the geometry designed for finite element analysis and the boundary conditions. (b) Meshing of the rail and its surrounding boundary for finite element modeling. methods agree with each other over the whole frequency range. Discrepancy between the rail resistances (solid line and square dots) obtained using the two methods is not observed until the frequency exceeds 10 4 Hz, beyond which the coupled-inductance model slightly underestimates the resistance. At 10 5 Hz, the error increases to about 5.3%. Although it is expected that this error will increase rapidly when the frequency increases further, for most practical applications, such Fig. 8. Rail resistance (solid line) and inductance (dashed line) calculated using the coupled-inductance model, compared with those (represented by square and triangular dots, respectively) obtained using the FEM. 426

9 Skin Effect of the TRTS Power Rail distribution and the frequency-dependent resistance and inductance of the power rail used in the Taipei Rapid Transit System have been studied in this paper using the coupled-inductance method. A comparative study using the FEM has also been conducted to validate the proposed coupled-inductance model. The accuracy of the coupled-inductance method deteriorates rapidly at high frequency if the number of subconductors does not increase with frequency. To solve this problem, a simple technique that uses subconductors of different sizes has been introduced. The proposed technique is found to be able to economize on the number of subconductors and, hence, to reduce the computation and memory requirements. Two- and three-dimensional illustrations of the current distribution over the rail cross section at different frequencies have been given in this paper, which allows the skin effect to be clearly visualized. The resistance and inductance of the power rail as functions of frequency have been obtained through skin effect analysis. Knowledge of the frequency-dependent characteristics of the power rail has many potential applications. It is of immediate interest with regard to transients and harmonic analysis of the traction power system. It can aid synthesis of an equivalent circuit that has frequency response similar to that of the rail impedance for time-domain dynamic simulation of the railway circuit. If it is incorporated into the frequency-dependent impedance model of running rails and ground, suspicious harmonic resonance and electromagnetic interference with signaling circuits can be investigated in more depth and detail. Acknowledgment The author appreciates the assistance given by Mr. Wen-Jin Lin of the Rapid Transit System department, Taipei municipal government. This work was carried out with financial support provided by the National Science Council, R.O.C., under research grant (NSC E ). References Aguet, M. and J. J. Morf (1987) Énergie Électrique, pp Dunod, Paris, France. Ametani, A. and I. Fuse (1992) Approximate method for calculating the impedances of multiconductors with cross sections of arbitrary shapes. Electrical Engineering in Japan, 111, Carpenter, D. C. and R. J. Hill (1993) Railroad track electrical impedance and adjacent track crosstalk modeling using the finiteelement method of electromagnetic systems analysis. IEEE Trans. on Vehicular Technology, 42, Coles, P. C., M. Fracchia, R. J. Hill, P. Pozzobon, and A. Szelag (1994) Identification of a catenary harmonics in 3 kv DC railway traction systems. 7th Mediterranean Electrotechnical Conference, Antalya, Turkey. De Arizon, P. and H. W. Dommel (1987) Computation of cable impedance based on subdivision of conductors. IEEE Trans. on Power Delivery, 2, Dinh, T. V., B. Cabon, and J. Chilo (1990) Time domain analysis of skin effect on lossy interconnections. Electronics Letters, 26, Fracchia, M., R. J. Hill, P. Pozzobon, and G. Sciutto (1994) Accurate track modeling for fault current studies on third-rail metro railways. The 1994 ASME/IEEE Joint Railroad Conference in Conjunction with Area 1994 Annual Technical Conference, Chicago, IL, U.S.A. Fracchia, M., P. Pozzobon, and L. Pierrat (1996) Probabilistic determination of harmonic impedance in distributed dc traction systems based on the invariance of resonance frequencies. 7th International Conference on Harmonics and Quality of Power, Las Vegas, NV, U.S.A. Hill, R. J. and D. C. Carpenter (1991) Determination of rail internal impedance for electric railway traction system simulation. IEE Proceedings-B, 138, Hill, R. J. and D. C. Carpenter (1993) Rail track distributed transmission impedance and admittance: theoretical modeling and experimental results. IEEE Trans. on Vehicular Technology, 42, Hill, R. J., M. Fracchia, P. Pozzobon, and G. Sciutto (1995) A frequency domain model for 3 kv dc traction dc-side resonance identification. IEEE Transactions on Power Systems, 10, Meliopoulos, A. P. S. (1988) Power System Grounding and Transients: An Introduction, pp Marcel Dekker, NewYork, NY, U.S.A. Silvester, P. (1966) Modal network theory of skin effect in flat conductors. Proceedings of the IEEE, 54, Steyn, B. M. and J. D. V. Wyk (1994) An experimentally calibrated locomotive simulation model for the determination of harmonic content in the traction current. 6th International Conference on Harmonics in Power Systems, Bologna, Italy. Taufiq, J. A. and J. Xiaoping (1989) Fast accurate computation of the DC-side harmonics in a traction VSI drive. IEE Proceedings- B, 136,

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