Approximate Analytical Calculation of the Skin Effect in Rectangular Wires

Transcription

1 Approimate Analytical Calculation of the Skin Effect in Rectangular Wires Dieter Gerling University of Federal Defense Munich, Neubiberg, Germany Abstract- It is well-known that thick wires energied with highfrequency currents show the effect, that the current mainly flows near the surface of the wire. This effect is quite well described in literature for round wires, for rectangular wires no description of similar simplicity could be found. This effect may be relevant for future electric drives, e.g. in the automotive industry. Therefore, the skin effect of rectangular wires is analyed in this paper for the application of hybrid drives for passenger cars. Inde Terms Analytical Calculation, Rectangular Wire, Skin Effect I. INTRODUCTION In the automotive industry, currently much effort is dedicated to the development of hybrid pure electric drives. As the voltage of the battery is limited (in many cases in the region of 10V), quite thick wires have to be used for a certain power level. In addition, high frequencies are used to reach a high maimum speed. These boundary conditions can lead to the occurrence of the skin effect in such applications (it is well-known that thick wires energied with highfrequency currents show the effect, that the current mainly flows near the surface of the wire). Moreover, sometimes rectangular wires are used to better utilie the slot area of such machines. For round wires the skin effect is well documented, see e.g. [1]. For rectangular wires a description of similar simplicity could not be found in literature. Therefore, in this paper especially the skin effect of rectangular wires is investigated. Even if it can be assumed, that for rectangular wires the skin effect is similar to round wires, a proof of this assumption a detailed calculation method are necessary. The main task of this paper is the elaboration of a simple calculation method for the skin effect in rectangular wires. II. THE SKIN EFFECT IN RECTANGULAR WIRES A. Solution of Mawell s Equations The following figure 1 shows the wire under investigation. We assume that in this wire the current has only a - component (i.e. into the sheet of paper or out of the sheet of paper), but this component depends on the - y- coordinate. This means in terms of the current density: J = J (,y) e (1) Fig. 1. System under investigation. With rot ( H) = J = γe () the following conclusions can be drawn: E = E (,y) e H(,y) = H(,y) e + Hy(,y) ey, where e, e y e are the unit vectors in -, y-, - direction, respectively. Evaluating () in Cartesian Coordinates gives: Hy H = γe (3) y With rot ( E) = μ H (4) we get in Cartesian Coordinates E = μ H (5) y E = μ Hy (6) Differentiating (3) with respect to time inserting (5) (6) gives 1 1 γ E = E + E μ μ y (7) E + E = μγ E y This differential equation shall be solved with the following product set-up:

2 () E,y,t = X Y y T t (8) Inserting this set-up to (7) gives (a prime means the total differential with respect to the relevant variable): X ( ) Y( y) T() t + X( ) Y ( y) T() t =μγx( ) Y( y) T () t (9) T () t X ( ) Y ( y) μγ = + T t X Y y () The left side of (9) depends only on the time, the right side only on the locus. Therefore, both sides must be constant. Evaluating firstly the left side of (9) gives: T () t = c T () t c T() t (10) T() t j t With the set-up for a harmonic wave T() t = e ω we get: jωe ce c = jω (11) Evaluating in a second step the right side of (9) leads to: X ( ) Y ( y) + = γ (1) X Y y As the right side of (1) is constant, both terms of the left side of (1) must be constant (because they depend on different coordinates). It follows: X ( ) = k X( ) X ( ) k X( ) (13) X = C sinh k + C cosh k 1 Analogously we get: Y( y) = C3sinh( y) + C4 cosh( y) (14) In total we get from (8), (1), (13) (14): E,y,t = C sinh k + C cosh k [ ] [ ( ) ( )] C sinh y + C cosh y e, k + = γ B. Calculation of the Variables From (5) (15) we get C sinh k + C cosh k [ ] 1 [ C cosh( y) + C sinh( y) ] e = μ H 1 1 H = [ C1sinh ( k) + C cosh ( k) ] μ jω 3 4 [ ( ) ( )] C cosh y + C sinh y e 3 4 (15) (16) Because of the symmetry, for y the value of H must be ero for any time; therefore it must be valid even for t. It follows: H y,t = 1 1 C3[ C1sinh ( k) + C cosh ( k) ] (17) μ jω C 3 Analogously, we get from (6) (15) C1 (18) Summariing (15), (17) (18), we get: E (, y, t) = C cosh ( k) cosh ( y) e, (19) k + = γ The constant C can be calculated from a b () ˆ I t = Ie = J,y,t dy d (0) a b Solving (19) (0) for the constant C we get: Î k C = (1) 4 γ a b sinh k sinh Equations (19) (1) describe the solution of the field problem inside the rectangular wire, but there is one additional equation missing for the constants k. Possibilities for such an additional equation are given in the following three sections C to E. C. Evaluation of the Magnetic Field Strength In the following we assume that the absolute value of the magnetic field strength for = a,y,y = b are identical. This assumption is valid as long as the frequency is not too high (for DC current the outside circumference of the wire represents a field line) or the wire is (at least approimately) a quadratic one. The evaluation of (5), (6) (19) gives: a a H =,y = Hy =,y 1 a = k Csinh k e () b b H,y = = H,y = 1 b = Csinh e Consequently we get a b ksinh k = sinh (3)

3 D. Calculation of the Symmetric Case The symmetric case is characteried by a = b (quadratic wire). For this we get because of the symmetry: k = k = γ (4) 1 b k = j δ a + b, δ= 1 a = j δ a + b ωμγ (31) 1 k = = j, δ = δ ωμγ Î k C 4 a sinh k = γ (5) E. Calculation of the (near) DC Case Taylor s series epansion for the hyperbolic sine function with truncation after the first summ gives sinh ( ) (this approimation is valid for small arguments, i.e. for eample for low frequencies). This leads to a a ksinh k k (6) b b sinh For the case of a DC current, i.e. ω, (19) leads to: ω k = (7) E (,y,t) = C Consequently the constant C for the DC case becomes Î C = (8) γ a b For the case of a near DC current, i.e. low frequency, (19), (3) (6) lead to: a k + k = γ b γ b k = a + b Analogously we get: (9) γ a = (30) a + b For the constant C (1) holds true. With the skin depth δ (please refer e.g. to (4) or reference [1]) we get from (9) (30): F. Evaluation of k l for Arbitrary Geometry Frequency Evaluating (31) for the symmetric case ( a = b ) we get the eact result (please refer to (4)). This means: The approimation of the hyperbolic sine function, that leads to k according to (31) is valid for - arbitrary geometry low frequencies - the symmetric geometry ( a = b ) arbitrary frequencies. The solutions (3) (31) for the constants k are based on the same physical constraints (both are valid for low frequency or quadratic wire). Nevertheless, there is a difference when evaluating these equations. To estimate the deviation F between the solutions (3) (31) depending on geometry frequency, the solution (31) will be evaluated the deviation analyed like follows: a b F = ksinh k sinh (3) This results in the following figure for the deviation F (figure shows the real part of the comple value of the deviation F, the imaginary part is qualitatively similar to this with just a slightly lower amplitude). Fig.. Re{ F } as a function of frequency f edge length a. In figure the following conditions were considered: - a frequency of 0H to 1000H; - a copper wire of cross section = 3.75mm ; Awire - a temperature of 0 C (the influence of temperature variation is similar to the influence of frequency variation, please refer to (31));

4 - an edge length variation between 0.1mm 3.0mm (the rectangular copper wire of 3.0mm times 1.5mm is the application described in chapter III). It can be deduced from figure that the deviation F is small, if low frequency or symmetric geometry is regarded (for these cases both solutions deliver the eact result). If one of these conditions is not fulfilled, the deviation because of the used different approimations is increased. G. Numerical Calculation of k l for Arbitrary Geometry Frequency At this time it can not be decided, if the deviation F coming from the different approimate calculations of k for arbitrary frequency geometry has an important influence on the current density distribution inside the wire. Therefore, both solutions to calculate these parameters will be compared. This can be done only numerically, because the set of equations (19) (3) is of transcendent nature. This numerical solution is performed using the software package MathCad. For the same conditions like in the previous section, for varying the frequency between 0H 0kH, for varying the edge length a of the rectangular wire between 0.1mm 10.0mm an absolute value of the failure 10 below 10 for each data point could be reached. The following figure 3 illustrates the difference between the analytical calculation (according to (31)) the numerical calculation according to (3). As an eample the parameter Re{ k } is given. The results for the imaginary part for the parameter are similar. Re{} k frequency in 100H a in 0.1mm Re{} k a in 0.1mm frequency in 100H a) b) Fig. 3. Re{ k } as a function of frequency f edge length a a) analytical calculation according to (31) b) numerical calculation according to (3) H. Evaluation of the Edge Condition Because of the special behavior of the electromagnetic field at the edges of the regarded geometry, a certain edge condition must be fulfilled in addition to the solution of Mawell s equations. This edge condition means that (please refer to []) the electromagnetic energy density must be integrable over any finite domain even if this domain contains singularities of the electromagnetic field. In other words, the electromagnetic energy in any finite domain must be finite. The Poynting s vector S = E H (33) describes the energy, that flows per time unit through the unit area perpendicular to S. As the edge condition must be fulfilled for any finite time interval, this means that Poynting s vector must be finite over any finite domain. The Poynting s vector in the case of the rectangular wire is: S = E H = EHey EHye (34) Inserting (5), (6) (19) into (34) gives: S = E H e E H e y y 1 ( ( ) ) μ y 1 ( ( ) ) μ = C cosh k cosh y e E dt e C cosh k cosh y e E dt e Further we get: 1 S = ( C cosh ( k) cosh ( y) e ) C cosh( k) sinh( y) e ey + C k sinh k cosh y e e jωt ( ( ) ) 1 = C cosh k cosh y e cosh k sinh ( y) ey + ( y) e ] ksinh k cosh y (35) For a certain operating point the parameters k,, C (in addition to ω μ ) are fied values. As the wire has finite dimensions, even the variables y are finite. Therefore, Poynting s vector S is finite over any finite domain. This means that the edge condition is fulfilled for the solution given above; i.e. this solution is valid for rectangular wires (as long as the approimations are valid). III. CURRENT DENSITY DISTRIBUTION From (19) (1) the frequency dependent current density distribution in a rectangular wire can be calculated. For the calculation in this section the time t, the temperature ϑ C, a total current of Î = 10A a rectangular copper wire with a = 3.0mm times b = 1.5mm (0 a,0 y b) cross section is assumed. In a first step, the analytical calculation according to (31) for the parameters k is used to get the result for the current density distribution. Figure 4 shows this current density distribution for 4 different frequencies. The following figure 5 shows the frequency dependent current density distribution for the same boundary conditions like in figure 4. The difference to this former calculation is

5 that during this evaluation the numerical calculation for the parameters k according to (3) is used to obtain the result for the current density distribution. Some slight differences to figure 4 can be noticed at high frequency. The time dependent current density distribution is illustrated in the following figure 6. As there are some slight deviations (for high frequency) between both calculation methods, only the results of the numerical calculation according to (3) are given. The calculation was performed for a frequency of f kh. Fig. 4. Current density distribution in a rectangular copper wire (cross section 3.0mm times 1.5mm, current Î = 10A, time t, temperature ϑ C ): analytical calculation according to (31). Fig. 6. Current density distribution in a rectangular copper wire (cross section 3.0mm times 1.5mm, frequency f kh, current Î = 10A, temperature ϑ C ): numerical calculation according to (3). Fig. 5. Current density distribution in a rectangular copper wire (cross section 3.0mm times 1.5mm, current Î = 10A, time t, temperature ϑ C ): numerical calculation according to (3). IV. ENTIRE SET OF MAXWELL S EQUATIONS Fulfilling the entire set of Mawell s equations, the following equation (36) has to be checked: div( E) = E + Ey + E (36) y E = E,y is true, the condition (36) is fulfilled. As Additionally, div B H H,y e H,y e must be fulfilled. With = + B = μh we get y y (37)

6 div( B) H + Hy y With (5) (6) this leads to 1 1 H + Hy = Edt + Edt y μ y y μ 1 = Edt Edt μ y y 1 = Edt Edt μ y y Therefore, even condition (37) is fulfilled. V. COMPARISON TO LITERATURE DATA There are only very few papers dealing with the skin effect in rectangular wires. In addition, not the time dependent current density distribution is given in [3, 4], but the absolute value of current density or electrical field strength versus the wire cross section. Therefore, a qualitative (but no quantitative) comparison is possible illustrated in the following figures 7 8. It can be deduced that the results obtained with the proposed calculation method correspond quite well with the FEM results published in literature. Nevertheless, there is some deviation along the edges of the wires. Most probably this comes from the approimative assumption mentioned in section II. Fig. 7. Comparison to literature data (left: data published in [3], right: solution according to this paper for the same boundary conditions. VI. CONCLUSION In this paper the skin effect of rectangular wires is investigated. In addition to the well-known effect of frequency beside the influence of the wire geometry (edge lengths) even the influence of temperature material (e.g. copper or aluminum) can be deduced very easily from the formulae given. For the rectangular wire an analytical solution could be found that is eact for the symmetric case (i.e. quadratic wire) nearly true for low frequency arbitrary geometry. For arbitrary geometry frequency, the solution given is just an approimation. Two different mathematical descriptions for the approimation are given, but physically both are based on the same assumption. The solutions deviate from each other depending on frequency geometry, but there is no criterion to decide which one fits better to reality. Evaluating Poynting s vector, it could be proven that the necessary edge condition according to [] is fulfilled for the solution given. The analytical calculation of the current density distribution deduced in this paper shows (in accordance to the FEM results known from literature) that with increasing frequency the current is displaced more more to the corners of the wire. This is different to round wires, where the entire circumference is conducting the current. For the eample of winding characteristics of nowadays hybrid electric drives for the automotive industry (typical rectangular copper wire with a = 3.0mm times b = 1.5mm cross section), it could be shown that even for the critical case of high frequency up to f = 1000H, the approimate analytical solution is fairly good enough to describe the current density distribution. Therefore, even the AC loss calculation will be fairly good using the approimate analytical solution. Consequently, for practical AC loss calculation this simple time efficient approimate analytical calculation method may be used to evaluate the application envisaged. REFERENCES [1] K. Simonyi, Theoretische Elektrotechnik, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980 (in German). [] J. Meiner, The behaviour of electromagnetic fields at edges, IEEE Transactions on Antennas Propagation, 0(197)4, pp [3] Fichte, L. O.: Berechnung der Stromverteilung in einem System rechteckiger Massivleiter bei Wechselstrom durch Kombination der Separations- mit der Rintegralgleichungsmethode, Ph.D. dissertation, Helmut-Schmidt-University, Hamburg, 007 (in German) [4] Faraji-Dana, R; Chow, Y.: Edge condition of the field a.c. resistance of a rectangular strip conductor, IEE Proceedings 17(1990) Fig. 8. Comparison to literature data (left: data published in [4], right: solution according to this paper for the same boundary conditions.