An Accurate Adaptive Method for Drawing 2-D Electric Lines of Force Jie Liu, Jun Zou, Jihuan Tian, Jiansheng Yuan, Xinshan Ma Department of Electrical Engineering, Tsinghua University Beijing, P.R. China
[email protected] Abstract— This paper presents a novel adaptive method for drawing 2-D electric lines of force of electrostatic field. The proposed method is based on high order Taylor expansion technique. By adaptively controlling the drawing step length, the proposed method can meet high drawing precision requirement. The principle, implementation and prerequisites of the proposed method are discussed. Numerical examples are given to validate the proposed method’s accuracy.
I. INTRODUCTION The visualization of the electric lines of force is a very important function of the post-processing module in electromagnetic field analysis software packages. Also, it is necessary to accurately draw electric lines of force in some problems of engineering calculation, such as the numerical analysis of the ion flow field of High Voltage Direct Current (HVDC) power transmission lines [1]-[4]. The existing methods to draw the static electric lines of force can be classified into two categories. The first category is the Euler’s method [3], which plots the electric lines of force ahead by using the tangent direction of the electric field at a certain point to approximate the direction of electric field in the vicinity. The second category is the Complex Potential Function (CPF) method [1], [2], which plots the electric lines of force by utilizing the mathematical property that the equivalue lines of the imaginary part of the complex potential function are the electric lines of force. Euler’s method is inaccurate, because using the tangent direction of electric field to approximate the direction of electric field in the nearby region generally leads to error, especially in the region where the direction of electric field changes rapidly. Furthermore, the error generated in a single drawing step accumulates while plotting electric lines of force step by step, causing further inaccuracy. Two different electric lines of force drawn by Euler’s method sometimes intersect with each other. This obviously deviates from the physical reality. The CPF method is accurate, but the program based on the CPF method to draw electric lines of force in multiconnected domain is difficult to implement and inefficient to run. Because the imaginary part of the complex function is multi-valued in multi-connected domain, it needs to search the entire calculation region to plot electric lines of force corresponding to every single value of the imaginary part of the complex potential function. Such a program is considerably time-consuming to construct and to run. ______________________________________
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In this paper, an accurate and simple-to-implement method is presented to adaptively draw electric lines of force of 2-D electrostatic field. Based on high order Taylor expansion, the proposed method can meet high accuracy requirement by altering drawing step length adaptively. In Section II, the principle and implementation of the proposed adaptive method is discussed; in section III, numerical examples are shown to validate the proposed method. II. PRINCIPLE OF THE PROPOSED METHOD A. General principle The process of the proposed method to draw electric lines of force is similar to that of the Euler’s method. Given the coordinates of a point of an electric line of force, the coordinates of the adjacent next point on the same electric line of force is calculated. Then start again from the newlyobtained point to find its adjacent next point. This process continues until the electric force line reaches the boundary of the drawing region. As shown in Fig. 1, the 2-D locus of every electric line of force can be expressed as y = y ( x) (1)
B : ( x + Δx, y + Δy )
Δy A : ( x, y )
Δx
Fig. 1 Schematic process to draw an electric force line
According to Taylor expansion Δy = y ( x + Δx) − y ( x) = y,x Δx + y,xx Δx 2 2 + y, xxx Δx 3 6 + o(Δx 3 )
(2)
given a point A(x,y) and the drawing step length Δx , the adjacent next point B ( x + Δx, y + Δy ) can be calculated. The direction of the electric field at point A(x,y) is the tangent of the locus of the electric force line, so E y u, y (3) y, x = = Ex u, x
y, xx = y, xxx =
u, xxy u, x
−
u,xy u, x − u, y u, xx u,x
(4)
2
u, y u,xxx + 2u, xy u,xx u, x 2
+
2u, y u,xx u, xxx u, x 3
(5)
where Ex and Ey are x-component and y-component of the electric field at point A(x,y), respectively; u is the electric potential at point A(x,y). When Ex ≤ E y , it is more precise to use Δy as drawing step length and calculate Δx by Taylor expansion to find the adjacent next point B ( x + Δx, y + Δy ) . Firstly, express the locus of the electric line of force as x = x( y ) (6) the Taylor expansion of equation (6) is Δx = x( y + Δy ) − x( y ) = x, y Δy + x, yy Δy 2 2 + x, yyy Δy 3 6 + o(Δy 3 )
(7)
where x, y = x, yy = x, yyy =
u,xyy u, y
−
Ex u, x = E y u, y
(8)
u,xy u, y − u,x u, yy
(9)
u, y 2
u, x u, yyy + 2u,xy u, yy u, y 2
+
2u, x u, yy u, yyy u, y 3
(10)
B. Adaptive Error Control In Euler’ method, the adjacent next point on the locus of an electric line of force is calculated by using Δy = y ( x + Δx) − y ( x) = y,x Δx (11)
Compare equation (2) and equation (11), it is obvious that the Euler’s method neglects the second and higher order items in Taylor series. This is the source of inaccuracy. Therefore, to improve the accuracy, we can calculate Δy by using Δy = y, x Δx + y,xx Δx 2 2
where τ i (i = 1, 2,..., N ) are the values of the simulation charges; ( xi , yi ) are the coordinates of charges τ i . Then, the expressions of derivatives of u can be obtained by differentiating equation (18). For instance, u,x can be calculated by using −( x − xi ) 1 N (19) u,x = ∑τ i 2πε 0 i=1 ( x − xi ) 2 + ( y − yi ) 2 The flowchart of the proposed adaptive method to plot an electric line of force is shown in Fig. 2.
(12)
Because the neglected high order items in equation (12) decrease as Δx becomes smaller, the error can be controlled by adaptively decreasing Δx . Here, Δx is chosen to satisfy the precision criterion y, xxx Δx 3 6 < ε (13) where ε is a small positive constant specified in advance to control drawing precision. When Ex ≤ E y , the error can be controlled in a similar way according to the criterion x, yyy Δy 3 6 < ε
Therefore, the proposed method is accurate only when these derivatives can be computed precisely. Numerical methods such as the boundary element method (BEM) and the charge simulation method (CSM) can be used to precisely calculate the derivatives of the potential u. If BEM [5] is applied, potential u can be calculated from the boundary potential u0 and the boundary potential’s directional derivative q0 ∂G u ( x, y ) = ∫ u0 dΓ − ∫ Gq0 dΓ (15) Γ ∂n Γ where Γ( xΓ , yΓ ) is the boundary of BEM calculation domain; n is the unit vector normal to the boundary; G is the Green function of 2-D Laplace problems 1 2 2 G ( x, y ) = ln ( x − xΓ ) + ( y − yΓ ) (16) 2π Then, derivatives u,x, u,y, u,xx, u,xy, u,yy, u,xxx, u,xxy, u,xyy, u,yyx can be calculated by differentiating equation (15). For example, u,x can be calculated by using ⎛ ∂G ⎞ u,x = ∫ ⎜ (17) ⎟ u0 d Γ − ∫ G,x q0 dΓ Γ ⎝ ∂n ⎠ , x Γ If CSM [6] is applied, u is expressed as yi 1 N τ i ln (18) u ( x, y ) = ∑ 2 2πε 0 i=1 ( x − xi ) + ( y − yi ) 2
Δy
Δx y,xxx Δx 3 6 < ε
x, yyy Δy 3 6 < ε
Δy
Δx
x = x + Δx y = y + Δy
( x + Δx, y + Δy )
(14)
C. Calculation of Potential Derivatives As shown in equations (2)-(5) and equations (7)-(10), the derivatives of u are the prerequisites to calculate Δy or Δx . Fig. 2 Flowchart of the proposed adaptive method
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III. EXAMPLES A. Calculation Models The electric lines of force of two 2-D models shown in Fig. 3 are plotted by using both the Euler’s method and the proposed adaptive method. The model (i) in Fig. 3 is two parallel circular conductors placed near to each other, with potential U1 and U2 respectively; the model (ii) is a ± 400kV bipolar HVDC power transmission line [7].
(a)
(b) Fig. 4. Electric lines of force of model (i) with U1=U2=1000V by two methods: (a) the proposed adaptive method; (b) the Euler’s method.
Fig. 3. Schematic models for plotting 2-D static electric lines of force
B. Numerical Results The electric lines of force of model (i) with U1=U2 are plotted in Fig. 4 by using both the proposed adaptive method and the Euler’s method. Because the Euler’s method can not effectively control the error, electric lines of force sometimes intersect with each other in the region where the direction of the electric field changes rapidly, as shown in Fig. 4 (b). In such cases, the visualization of the electric lines of force by using the Euler’s method obviously deviates from the physical reality. However, from Fig. 4 (a), it can be seen that the electric lines of force plotted by the proposed adaptive method are more reasonable and precise, because the error of every drawing step is controlled adaptively. The electric lines of force of model (i) with U1=－U2 are plotted in Fig. 5 by using both the proposed adaptive method and the Euler’s method. Due to the symmetry property about y-axis of the model (i) in Fig. 3, an electric line of force starting from one marked point on the left circular conductor should end at the symmetric point marked on the right circular conductor. So, it is obvious in Fig. 5 that the proposed adaptive method is more precise than the Euler’s method.
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(a)
(b) Fig. 5. Electric lines of force of model (i) with U1=1000V, U2=－1000V by two methods: (a) the proposed adaptive method; (b) the Euler’s method.
(a)
(c)
(b) (d) Fig. 6. Electric lines of force of model (ii) plotted by two methods. (a) and (b): the proposed adaptive method; (c) and (d): the Euler’s method.
In Fig. 6, the electric lines of force of model (ii) are plotted by both the proposed adaptive method and the Euler’s method. The electric lines of force are plotted to start from the +400kV conductor bundle at left and end at the -400kV conductor bundle at right. It can be seen in Fig. 6 that the proposed adaptive method is more accurate than the Euler’s method. As contrasted in Fig. 6 (b) and Fig. 6 (d), in plotting regions where the direction of the electric field changes rapidly, the proposed adaptive method’s superiority in precision is especially obvious. IV. CONCLUSIONS This paper presents a new method to draw electric lines of force of 2-D electrostatic fields. By adaptively controlling the plotting error, the proposed method can draw electric lines of force more accurately than the existing Euler’s method. The superiority of the proposed method’s accuracy is especially obvious in the region where the direction of the electric field changes rapidly.
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