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2003 SURE Program,Basic Applications of Integral Equations in Electromagnetics,Nathaniel Burt,Kansas State University,Advisor:Professor Chalmers M.Butler,Basic Applications of Integral Equations,Background-What is an Integral Equation?,A Simple Electrostatic Application,Several Electrodynamic Applications,Thin-wire scatterer/antenna problem and associated integro-differential equation,Conclusions and Future Work,Background-What is an Integral Equation?,-limits determined by physical geometry,-unknown function to be determined,-known“kernel(also called a Greens Function),-known forcing function,An Approximation Technique,Difficulty,:possesses infinite number of unknowns,Solution,:,discretize,using series of N weighted pulses,Approximation reduces problem to solving for N unknowns,Further Details of Approximation Technique,xm takes on N unique values within(a,b)interval,yielding N equations,These“match points(xm)lie at centers of pulse functions,N summations of N terms form an NxN“impedance matrix having elements,Recalling that,and using the piecewise constant,approximation of u(x),Further Details of Approximation Technique,Unknown U,N,easily found using Gaussian elimination,Recall piecewise constant approximation of,u(x,),Charged Strip of Infinite Length(electrostatic),If strip is slotted,voltage,v(x,)can vary over the surface,Kernel is analytically,integrable,yielding matrix elements,Solving for column vector ,one easily obtains the approximated charge per unit,lenth,q(x,).,Charge Density per Unit Length on an Infinite Strip,V(x)=1,N=40,V(x)=x,N=20,w=1 m,-w,-w,w,w,Simple Electrodynamic Applications TMExcitation of Bodies Infinite in One Dimension,in which,H,0,(2),is a Hankel function of 0th order,2nd kind,In general,Also,Because must be zero everywhere,Scattered E-field is seen to be,On a PEC,We arrive at a generalized 2-D TM-incidence equation:,TM Excitation of a Conducting Strip,Specializing 2D TM excitation equation to a strip,we have,For 2D,electrodynamic,IEs,utilizing scatterer method,TM Excitation of More Complex Cylinders,Two more complex cases of 2-D TM excitation were investigated,Bent Strip(PEC),Infinite Cavity andInfinite Slot in PEC,Slot must be entirely above cavity,Accomodates,any cavity geometry,Restrictions,Approximation of Geometries for 2-D Cylinders,N=250,Approximating angular geometries using N points is simple,Smoother contours require more innovative geometry approximations,Goal:approximate contour as accurately as possible given N points,N=500,9 m,PEC Plane(L to R),y,Asymmetrical Slotted Cavity Solution,Trough Walls(CCW),-9 m,0.9 m,(0.05,0.9)m,(-0.15,0.9)m,0.9 m,x,y,Thin-Wire Scatterer/Antenna,Rather than using rectangular pulses,we represent I on the PEC in a piecewise linear fashion.,When,I(z,)in the second integral is differentiated,it becomes a series of pulses,Solution of Thin-Wire Integro-Differential Equation,Results of Solution Procedure for Thin Wire,One Wavelength Scatterer,Conclusions and Future Work,Gained a solid understanding of the fundamentals of formulating and numerically solving applied integral equations,Will most likely work with Professor Butler during the Fall semester to investigate effects of a gap capacitance on monopole characteristics like input impedance,radiation pattern,and bandwidth,Special thanks to Professor Butler and to the graduate students in Computational,Electromagnetics,especially Mike,Lockard,and,Jeremy,Rudbeck,
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