MIT OpenCourseWare http://ocw.mit.edu Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207. Please use the following citation format: Haus, Hermann A., and James R. Melcher, Electromagnetic Fields and Energy. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed [Date]). License: Creative Commons Attribution-NonCommercial-Share Alike. Also available from Prentice-Hall: Englewood Cliffs, NJ, 1989. ISBN: 9780132490207. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms11 ENERGY, POWER FLOW, AND FORCES 11.0 INTRODUCTION One way to decide whether a system is electroquasistatic or magnetoquasistatic is to consider the relative magnitudes of the electric and magnetic energy storages. The subject of this chapter therefore makes a natural transition from the quasistatic laws to the complete set of electrodynamic laws. In the order introduced in Chaps. 1 and 2, but now including polarization and magnetization,1 these are Gauss’ law [(6.2.1) and (6.2.3)] � · (�oE + P) = ρu (1) Amp`ere’s law (6.2.11), ∂ � × H = Ju + ∂t(�oE + P) (2) Faraday’s law (9.2.7), ∂ � × E = −∂tµo(H + M) (3) and the magnetic flux continuity law (9.2.2). � · µo(H + M) = 0 (4) Circuit theory describes the excitation of a two-terminal element in terms of the voltage v applied between the terminals and the current i into and out of the respective terminals. The power supplied through the terminal pair is vi. One objective in this chapter is to extend the concept of power flow in such a way that power is thought to flow throughout space, and is not associated only with 1 For polarized and magnetized media at rest. 12 Energy, Power Flow, and Forces Chapter 11 Fig. 11.0.1 If the border between two states passes between the plates of a capacitor or between the windings of a transformer, is there power flow that should be overseen by the federal government? current flow into and out of terminals. The basis for this extension is the laws of electrodynamics, (1)–(4). Even if a system can be represented by a circuit, the need for the generalization of the circuit-theoretical power flow concept is apparent if we try to understand how electrical energy is transferred within, rather than between, circuit elements. The limitations of the circuit viewpoint would be crucial to testimony of an expert witness in litigation concerning the authority of the Federal Power Commission2 to regulate power flowing between states. If the view is taken that passage of current across a border is a prerequisite for power flow, either of the devices shown in Fig. 11.0.1 might be installed at the border to “launder” the power. In the first, the state line passes through the air gap between capacitor plates, while in the second, it separates the primary from the secondary in a transformer.3 In each case, the current never leaves the state where it is generated. Yet in the examples shown, power generated in one state can surely be consumed in another, and a meaningful discussion of how this takes place must be based on a broadened view of power flow. From the circuit-theoretical viewpoint, energy storage and rate of energy dissi-pation are assigned to circuit elements as a whole. Power flowing through a terminal pair is expressed as the product of a potential difference v between the terminals and the current i in one terminal and out of the other. Thus, the terminal voltage v and current i do provide a meaningful description of power flow into a surface S that encloses the circuit shown in Fig. 11.0.2. The surface S does not pass “inside” one of the elements. Power Flow in a Circuit. For the circuit of Fig. 11.0.2, Kirchhoff’s laws 2 Now the Federal Energy Regulatory Commission. 3 To be practical, the capacitor would be constructed with an enormous number of inter-spersed plates, so that in order to keep the state line in the air gap, a gerrymandered border would be required. Contemplation of the construction of a practical transformer, as described in Sec. 9.7, reveals that the state line would be even more difficult to explain in the MQS case.3 Sec. 11.0 Introduction Fig. 11.0.2 Circuit used to review the derivation of energy conservation statement for circuits. combine with the terminal relations for the capacitor, inductor, and resistor to give dv i = C + iL + Gv (5)dt v = LdiL (6)dt Motivated by the objective to obtain a statement involving vi, we multiply the first of these laws by the terminal voltage v. To eliminate the term viL on the right, we also multiply the second equation by iL. Thus, with the addition of the two relations, we obtain vi = vC dv + iLLdiL + Gv2 (7)dt dt Because L and C are assumed to be constant, we can use the relation udu = d(12u2) to rewrite this expression as vi = dw + Gv2 (8)dt where 1 1 w = Cv2 + Li2 2 2 L With its origins solely in the circuit laws, (8) can be regarded as giving no more information than inherent in the original laws. However, it gives insights into the circuit dynamics that are harbingers of what can be expected from the more general statement to be derived in Sec. 11.1. These come from considering some extremes. If the terminals are open (i = 0), and if the resistor is absent (G = 0), w is • constant. Thus, the energy w is conserved in this limiting case. The solution to the circuit laws must lead to the conclusion that the sum of the electric energy 1Cv2 and the magnetic energy 1Li2 is constant. 2 2 L Again, with G = 0, but now with a current supplied to the terminals, (8) • becomes dw vi = (9)dt� 4 Energy, Power Flow, and Forces
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