Chapter 3: Electromagnetic Fields in Simple Devices and Circuits 3.1 Resistors and capacitors 3.1.1 Introduction One important application of electromagnetic field analysis is to simple electronic components such as resistors, capacitors, and inductors, all of which exhibit at higher frequencies characteristics of the others. Such structures can be analyzed in terms of their: 1) static behavior, for which we can set ∂/∂t = 0 in Maxwell’s equations, 2) quasistatic behavior, for which ∂/∂t is non-negligible, but we neglect terms of the order ∂2/∂t2, and 3) dynamic behavior, for which terms on the order of ∂2/∂t2 are not negligible either; in the dynamic case the wavelengths of interest are no longer large compared to the device dimensions. Because most such devices have either cylindrical or planar geometries, as discussed in Sections 1.3 and 1.4, their fields and behavior are generally easily understood. This understanding can be extrapolated to more complex structures. One approach to analyzing simple structures is to review the basic constraints imposed by symmetry, Maxwell’s equations, and boundary conditions, and then to hypothesize the electric and magnetic fields that would result. These hypotheses can then be tested for consistency with any remaining constraints not already invoked. To illustrate this approach resistors, capacitors, and inductors with simple shapes are analyzed in Sections 3.1–2 below. All physical elements exhibit varying degrees of resistance, inductance, and capacitance, depending on frequency. This is because: 1) essentially all conducting materials exhibit some resistance, 2) all currents generate magnetic fields and therefore contribute inductance, and 3) all voltage differences generate electric fields and therefore contribute capacitance. R’s, L’s, and C’s are designed to exhibit only one dominant property at low frequencies. Section 3.3 discusses simple examples of ambivalent device behavior as frequency changes. Most passive electronic components have two or more terminals where voltages can be measured. The voltage difference between any two terminals of a passive device generally depends on the histories of the currents through all the terminals. Common passive linear two-terminal devices include resistors, inductors, and capacitors (R’s, L’s. and C’s, respectively), while transformers are commonly three- or four-terminal devices. Devices with even more terminals are often simply characterized as N-port networks. Connected sets of such passive linear devices form passive linear circuits which can be analyzed using the methods discussed in Section 3.4. RLC resonators and RL and RC relaxation circuits are most relevant here because their physics and behavior resemble those of common electromagnetic systems. RLC resonators are treated in Section 3.5, and RL, RC, and LC circuits are limiting cases when one of the three elements becomes negligible. 3.1.2 Resistors Resistors are two-terminal passive linear devices characterized by their resistance R [ohms]: - 65 -v = iR (3.1.1) where v(t) and i(t) are the associated voltage and current. That is, one volt across a one-ohm resistor induces a one-ampere current through it; this defines the ohm. The resistor illustrated in Figure 3.1.1 is comprised of two parallel perfectly conducting end-plates between which is placed a medium of conductivity σ, permittivity ε, permeability μ, and thickness d; the two end plates and the medium all have a constant cross-sectional area A [m2] in the x-y plane. Let’s assume a static voltage v exists across the resistor R, and that a current i flows through it. (a) σ, ε, μ v = 0 (b) +v ρs x i(t) E + + + + + + + + + + + + z area A d y d - - - - - - - - - - - - +v -ρs v = 0 i(t) Figure 3.1.1 Simple resistor. Boundary conditions require the electric field E at any perfectly conducting plate to be perpendicular to it [see (2.6.16); E ×=0 ], and Faraday’s law requires that any line integral of nˆE from one iso-potential end plate to the other must equal the voltage v regardless of the path of integration (1.3.13). Because the conductivity σ [Siemens/m] is uniform within walls parallel to zˆ , these constraints are satisfied by a static uniform electric field E = zˆEo everywhere within the conducting medium, which would be charge-free since our assumed E is non-divergent. Thus: d∫ o (3.1.2)E • zˆ dz = Ed = v0 where Eo = vd ⎣⎡ Vm -1⎤⎦ . Such an electric field within the conducting medium induces a current density J , where: J =σE ⎣⎡ Am -2 ⎤⎦ (3.1.3) - 66 -The total current i flowing is the integral of J • zˆ over the device cross-section A, so that: i = J • zˆ dxdy = σE • zˆ dxdy = σE dxdy =σE A = vσA d ∫∫(3.1.4)A ∫∫A ∫∫o o ABut i = v/R from (3.1.1), and therefore the static resistance of a simple planar resistor is: Rvi = d = σA o[hms ](3.1.5)The instantaneous power p [W] dissipated in a resistor is i2R = v2/R, and the time-average 2 2power dissipated in the sinusoidal steady state is IR2 = V 2R watts. Alternatively the local instantaneous power density P =•E J W ⎡⎣-3 d m ⎤⎦ can be integrated over the volume of the resistor to yield the total instantaneous power dissipated: 2 22 p =∫∫∫E • J dv =∫∫∫E •σE dv =σE Ad =σAv d = v R []W (3.1.6) V Vwhich is the expected answer, and where we used (2.1.17): J = σE. Surface charges reside on the end plates where the electric field is perpendicular to the perfect conductor. The boundary condition nˆ • D =ρs (2.6.15) suggests that the surface charge density ρs on the positive end-plate face adjacent to the conducting medium is: ρ=εE C ⎡⎣-2 ⎤s om⎦ (3.1.7)The total static charge Q on the positive resistor end plate is therefore ρsA coulombs. By convention, the subscript s distinguishes surface charge density ρs [C m-2] from volume charge density ρ [C m-3]. An equal negative surface charge resides on the other end-plate. The total stored charged Q = ρsA = CV, where C is the device capacitance, as discussed further in Section 3.1.3. The static currents and voltages in this resistor will produce fields outside the resistor, but these produce no additional current or voltage at the device terminals and are not of immediate concern here. Similarly, μ and ε do not affect the static value of R. At higher frequencies, however, this resistance R varies and