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Electromagnetic Field Theory: A Problem Solving Approach Part 22

Electromagnetic Field Theory: A Problem Solving Approach Part 22. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. Heavy reliance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked problems. To keep the subject exciting, many of these problems are based on physical processes, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering. | Lossy Media 185 M Figure 3-22 Two different lossy dielectric materials in series between parallel plate electrodes have permittivities and Ohmic conductivities that change abruptly across the interface a At t 0 right after a step voltage is applied the interface is uncharged so that the displacement held is continuous with the solution the same as for two lossless dielectrics in series b Since the current is discontinuous across the boundary between the materials the interface will charge up. In the de steady state the current is continuous c Each region is equivalent to a resistor and capacitor in parallel. so that the displacement field is f 0 --- 13 e2a T6io The total current from the battery is due to both conduction and displacement currents. At t 0 the displacement current 186 Polarization and Conduction is infinite an impulse as the displacement field instantaneously changes from zero to 13 to produce the surface charge on each electrode afa 0 Tf x a b DX 14 After the voltage has been on a long time the fields approach their steady-state values as in Figure 3-226. Because there are no more time variations the current density must be continuous across the interface just the same as for two series resistors independent of the permittivities Jx t oo o-iEi-o-2E2 ai 2V 15 T2a T D where we again used 12 . The interfacial surface charge is now x r- r- 2O-l- ia-2 V 1Z.X rz x a 2E2- i i ------- ---- 16 cr2a -ro iO What we have shown is that for early times the system is purely capacitive while for long times the system is purely resistive. The inbetween transient interval is found by using 12 with charge conservation applied at the interface I d n j2-Ji r D2-Di -0 d a2E2-aiEi e2E2-e1Ei 0 17 at With 12 to relate E2 to E we obtain a single ordinary differential equation in E - h I8 at r 2a E b where the relaxation time is a weighted average of relaxation times of each material 19 a b r2a Using the initial condition of 13 the solutions for the fields are E -- 20 e2--- .