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ee256-08-lecture02 - 1 EE256 Numerical Electromagnetics...

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1 EE256 Numerical Electromagnetics H. O. #4 Marshall 24 June 2008 Summer 2008 LECTURE 2 2.1 PARTIAL DIFFERENTIAL EQUATIONS AND PHYSICAL SYSTEMS Most physical systems are described by one or more partial differential equations, which can be derived by direct application of the governing physical laws. For electromagnetic phenomena, the governing physical laws are experimentally based and are Faraday’s law, Ampere’s law, Coulomb’s law, and the conversation of electric charge. Application of these laws lead to Maxwell’s equations. For voltage and current waves on transmission lines, application of Kirchoff’s voltage and current laws (which are contained in Maxwell’s equations) leads to the Telegrapher’s equations : V ∂z = - R I - L I ∂t [2.1 a ] I ∂z = - G V - C V ∂t [2.1 b ] where V ( z, t ) and I ( z, t ) are respectively the line voltage and current, while R , L , G , and C are re- spectively the distributed resistance, inductance, conductance, and capacitance of the line. Manipulation of [2.1 a ] and [2.1 b ] leads to the general wave equation for a transmission line: 2 V ∂z 2 = RG V + ( RC + LG ) V ∂t + LC 2 V ∂t 2 [2.2] Note that making the substitution V ↔I in [2.2] gives the wave equation in terms of current I ( z, t ) . A similar wave equation can be derived in terms of the electric field E by manipulating [1.1] and [1.3], for simple media for which , μ , σ , and σ m are simple constants: 2 E = σσ m E + ( μσ + σ m ) E ∂t + μ 2 E ∂t 2 [2.3] The corresponding equation for the magnetic field H can be obtained by making the substitution E ↔ H . General solutions of equations [2.2] and [2.3] are in general quite complex, except in special cases. Solutions for [2.2] take a simple and analytically tractable form when we assume that the variations in time are sinusoidal (time-harmonic), or when we assume that the loss terms (i.e., those involving R , G in [2.2]) are zero. Solutions for [2.3] also become tractable for the time-harmonic and lossless (assuming that terms involving σ , σ m in [2.3] are zero) cases, if we make additional simplifying assumptions about the direction and spatial dependency of E ( x, y, z, t ) .
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2 Partial differential equations arise in a variety of ways and in the context of a wide range of different physical problems, remarkably having forms quite similar to equations [2.1] and [1.1] through [1.5]. The basic reason for the commonality of the form of governing equations for different physical systems is the fact that most of classical physics is based on principles of conservation, whether it be principle of conservation of mass, energy, momentum, or electric charge. For example, application of the principle of conservation of momentum to an incremental volume of a compressible fluid (e.g., a gas such as air) gives the ‘telegrapher’s equations’ for acoustic waves: Transmission - line Analog Acoustic waves V ∂z = - L I ∂t ∂u x ∂x = - 1 γ g p a ∂p ∂t I ∂z = - C V ∂t ∂p ∂x = - ρ v ∂u x ∂t [2.4] where u x is the velocity in the x direction, p is the variation in pressure (above an ambient level), ρ v is the mass per unit volume, γ
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