Electromagnetic Field Theory: A Problem Solving Approach Part 26. 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. | Electrostatic Generators 225 b M Figure 3-38 A modified Van de Graaff generator as an electrostatic induction machine a Here charges are induced onto a segmented belt carrying insulated conductors as the belt passes near an electrode at voltage V. b Now the current source feeding the capacitor equivalent circuit depends on the capacitance Ct between the electrode and the belt. Now the early researchers cleverly placed another induction machine nearby as in Figure 3-39a. Rather than applying a voltage source because one had not been invented yet they electrically connected the dome of each machine to the inducer electrode of the other. The induced charge on one machine was proportional to the voltage on the other dome. Although no voltage is applied any charge imbalance on an inducer electrode due to random noise or stray charge will induce an opposite charge on the moving segmented belt that carries this charge to the dome of which some appears on the other inducer electrode where the process is repeated with opposite polarity charge. The net effect is that the charge on the original inducer has been increased. More quantitatively we use the pair of equivalent circuits in Figure 3-39A to obtain the coupled equations _ du dv i nCiV C nCiV2 C - 2 dt dt where n is the number of segments per second passing through the dome. All voltages are referenced to the lower pulleys that are electrically connected together. Because these 226 Polarization and Conduction M Figure 3-39 a A pair of coupled self-excited electrostatic induction machines generate their own inducing voltage b The system is described by two simple coupled circuits. are linear constant coefficient differential equations the solutions must be exponentials vt V1 I vs -V2es 3 Substituting these assumed solutions into 2 yields the following characteristic roots 2 nC 2 nCi i -F i -7T 4 X o o so that the general solution is _ a Ancjcu a .- ncjcyt Vi Ai e 12 e a AnCJCU a -InCflCU vs Aie A e where At and A2
