HIGH VOLTAGE TECHNIQUES electrostatic field analysis methods Assistant Professor Suna BOLAT Eastern Mediterranean University Department of Electric & Electronic Engineering Electrostatic field analysis methods 1. Analytical calculations 2. Analog methods 3. Numerical methods Analytical calculations • • • • Analytical solution of differential equations (Laplace, Poisson) Conform transform Schwarz – Christoffel transform ... Analog methods • Graphical methods • Experimental methods – On the model – On a real system Numerical methods • • • • • • Finite difference method Finite element method Boundary element method Charge simulation method Monte – Carlo method Moment method Experimental methods • • • • Electrolytic tank experiment Semi-conductor paper method Resistance simulation method Grass seed method Electrolytic tank experiment Principle: static electric field has an analogy with current field. Application: • Create a scaled model of electrode system • Replace the dielectric with a conductive material • Determine the current field lines on conductive media • Draw the electric field lines perdendicular to them Current lines analogy Flux lines To voltage source Model electrodes Electrolitic liquid (?) Experimental setup Measurement bridge • 𝐼= 𝑈 𝑅1 +𝑅2 • 𝑈2 = 𝐼𝑅2 = 𝑅2 𝑈 𝑅1 +𝑅2 = 𝑈𝐴 Numerical methods • Finite difference method Principle: it leans on finite difference operations All the derivatives are substituted by numerical representations. 𝜕2𝑉 𝜕2𝑉 + 2=0 2 𝜕𝑥 𝜕𝑦 𝑑2 𝑉 𝑉 𝑥 + ℎ, 𝑦 − 2𝑉 𝑥, 𝑦 + 𝑉 𝑥 − ℎ, 𝑦 ≅ 2 𝑑𝑥 ℎ2 𝑑 2 𝑉 𝑉 𝑥, 𝑦 + 𝑘 − 2𝑉 𝑥, 𝑦 + 𝑉(𝑥, 𝑦 − 𝑘) ≅ 2 𝑑𝑦 𝑘2 𝑉 𝑥 + ℎ, 𝑦 − 2𝑉 𝑥, 𝑦 + 𝑉 𝑥 − ℎ, 𝑦 ℎ2 𝑉 𝑥, 𝑦 + 𝑘 − 2𝑉 𝑥, 𝑦 + 𝑉(𝑥, 𝑦 − 𝑘) + =0 2 𝑘 Letting k = h, (square grids) 𝑉 𝑥 + ℎ, 𝑦 − 2𝑉 𝑥, 𝑦 + 𝑉 𝑥 − ℎ, 𝑦 𝑉 𝑥, 𝑦 + ℎ − 2𝑉 𝑥, 𝑦 + 𝑉(𝑥, 𝑦 − ℎ) + =0 ℎ2 ℎ2 1 𝑉 𝑥, 𝑦 = 𝑉 𝑥 + ℎ, 𝑦 + 𝑉 𝑥 − ℎ, 𝑦 + 𝑉 𝑥, 𝑦 + ℎ + 𝑉(𝑥, 𝑦 − ℎ) 4 Example Numerical methods • Charge Simulation method Principle: simulating the field between condutors by using simulation charges Q1 + X B1 + Q4 Q2 X B2 + + Q3 XB3 + XB4 + + + V1 Ṽ - - - - - - V2 - Steps • Place simulation charges outside of the region to be analyzed • Determine boundary points • Solve potential equation to calculate simulation charges for boundary points • Control the value of charges • Calculate potential and electric field values for the desired point using determined simulation charges Accuracy of this method depends on 1. 2. 3. 4. 5. Type of the simulation charges Number of simulation charges Location of simulation charges Number of boundary points Location of boundary points Types of simulation charges • Point charge • Line charge • ... Point charge For spherical systems 𝑉= 𝑞 4𝜋ε𝑟 𝑟= potential factor: 𝑃 = 𝑥𝑞 − 𝑥𝑝 2 + 𝑦𝑞 − 𝑦𝑝 1 4𝜋ε𝑟 (𝑋𝑞 , 𝑌𝑞 ) q P (𝑋𝑝 , 𝑌𝑝 ) 2 q1 q2 electrode V X B1 X B2 q3 X B1 Boundary points Voltages at the boundaries 𝑉𝐵1 𝑞1 𝑞2 𝑞3 = + + 4𝜋 ε 𝑟11 4𝜋 ε 𝑟12 4𝜋 ε 𝑟13 𝑉𝐵2 𝑞1 𝑞2 𝑞3 = + + 4𝜋 ε 𝑟21 4𝜋 ε 𝑟22 4𝜋 ε 𝑟23 𝑉𝐵3 𝑞1 𝑞2 𝑞3 = + + 4𝜋 ε 𝑟31 4𝜋 ε 𝑟32 4𝜋 ε 𝑟33 In general... [P] [q] = [V] Simulation charges vector Potential factor matrix Potential vector • After finding simulation charges, the value of the charges should be controlled • Choose control points on known potentials& Potential of any point 𝑃11 𝑃21 𝑃31 𝑃12 𝑃22 𝑃32 𝑃13 𝑃23 𝑃33 𝑞1 𝑉1 𝑞2 = 𝑉2 𝑞3 𝑉3 Potential of any K point in the region: 𝑞1 𝑞2 𝑞3 𝑉𝐾 = + + 4𝜋 ε 𝑟1𝐾 4𝜋 ε 𝑟1𝐾 4𝜋 ε 𝑟1𝐾 𝑟𝑖𝑘 = (𝑥𝑖 − 𝑥𝑘 )2 −(𝑦𝑖 − 𝑦𝑘 )2 Electric field at any point 𝑑𝑉 𝐸=− =− 𝑑𝑟 𝑑 𝑞 4𝜋ε𝑟 𝑑𝑟 𝑞 1 𝐸= . 2 4𝜋ε𝑟 𝑟 Infinite line charge For cylindrical systems 𝜆= 𝑉𝑃 = l r0 V=0 P 𝑄 𝑙 𝜆 𝑟0 ln 2𝜋𝜀 𝑟 r0: the distance between line charge and the point with 0 potential r: the distance between charge and the point P Potential at my heart if I stand under a high voltage line r q rHP VL h 𝑉ℎ𝑒𝑎𝑟𝑡 Suna V=0 Ground (earth) 𝑞 ℎ = ln 2𝜋𝜀 𝑟𝐻𝑃 𝑞 ℎ 𝑉𝐿 = ln 2𝜋𝜀 𝑟 Chapter 1 is over...