MIT OpenCourseWare http://ocw.mit.edu Electromagnetic Field Theory: A Problem Solving Approach For any use or distribution of this textbook, please cite as follows: Markus Zahn, Electromagnetic Field Theory: A Problem Solving Approach. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.chapter 8guided electromagneticwaves568 Guided Electromagnetic WavesThe uniform plane wave solutions developed in Chapter 7cannot in actuality exist throughout all space, as an infiniteamount of energy would be required from the sources.However, TEM waves can also propagate in the region offinite volume between electrodes. Such electrode structures,known as transmission lines, are used for electromagneticenergy flow from power (60 Hz) to microwave frequencies, asdelay lines due to the finite speed c of electromagnetic waves,and in pulse forming networks due to reflections at the end ofthe line. Because of the electrode boundaries, more generalwave solutions are also permitted where the electric andmagnetic fields are no longer perpendicular. These newsolutions also allow electromagnetic power flow in closedsingle conductor structures known as waveguides.8-1 THE TRANSMISSION LINE EQUATIONS8-1-1 The Parallel Plate Transmission LineThe general properties of transmission lines are illustratedin Figure 8-1 by the parallel plate electrodes a small distance dapart enclosing linear media with permittivity e andpermeability Cj. Because this spacing d is much less than thewidth w or length i, we neglect fringing field effects andassume that the fields only depend on the z coordinate.The perfectly conducting electrodes impose the boundaryconditions:(i) The tangential component of E is zero.(ii) The normal component of B (and thus H in the linearmedia) is zero.With these constraints and the, neglect of fringing near theelectrode edges, the fields cannot depend on x or y and thusare of the following form:E = E,(z, t)i,(1)H = H,(z, t)i,which when substituted into Maxwell's equations yieldIThe TransmissionLine Equations569Figure 8-1 The simplest transmission line consists of two parallel perfectly conduct-ing plates a small distance d apart.= aE, aH,ata• Z atVxH= -E ' =-Eat az atWe recognize these equations as the same ones developedfor plane waves in Section 7-3-1. The wave solutions foundthere are also valid here. However, now it is more convenientto introduce the circuit variables of voltage and current alongthe transmission line, which will depend on z and t.Kirchoff's voltage and current laws will not hold along thetransmission line as the electric field in (2) has nonzero curland the current along the electrodes will have a divergencedue to the time varying surface charge distribution, o-, =±eE,(z, t). Because E has a curl, the voltage differencemeasured between any two points is not unique, as illustratedin Figure 8-2, where we see time varying magnetic flux pass-ing through the contour LI. However, no magnetic fluxpasses through the path L2, where the potential difference ismeasured between the two electrodes at the same value of z,as the magnetic flux is parallel to the surface. Thus, thevoltage can be uniquely defined between the two electrodes atthe same value of z:v(z,t)= Jz =constE -dl = E,(z, t)d570Guided Electromagnetic WavesAL2' 3E dl=0E 22 all L2fE -di = -podf a-dsL1 stFigure 8-2 The potential difference measured between any two arbitrary points atdifferent positions z, and zg on the transmission line is not unique-the line integral L,of the electric field is nonzero since the contour has magnetic flux passing through it. Ifthe contour L2 lies within a plane of constant z such as at z,, no magnetic flux passesthrough it so that the voltage difference between the two electrodes at the same valueof z is unique.Similarly, the tangential component of H is discontinuousat each plate by a surface current +K. Thus, the total currenti(z, t) flowing in the z direction on the lower plate isi(z, t)= K,w = H,wSubstituting (3) and (4) back into (2) results in the trans-mission line equations:av ai-. = -L-Oz at(5)ai av-= -c-z -atwhere L and C are the inductance and capacitance per unitlength of the parallel plate structure:IldL = -henry/m,wC=w-farad/mdIf both quantities are multiplied by the length of the line 1,we obtain the inductance of a single turn plane loop if the linewere short circuited, and the capacitance of a parallel platecapacitor if the line were open circuited.It is no accident that the LC productLC= ejA = 1/c2is related to the speed of light in the medium.8-1-2 General Transmission Line StructuresThe transmission line equations of (5) are valid for anytwo-conductor structure of arbitrary shape in the transverseThe Transmission Line Equations 571xy plane but whose cross-sectional area does not change alongits axis in the z direction. L and C are the inductance andcapacitance per unit length as would be calculated in thequasi-static limits. Various simple types of transmission linesare shown in Figure 8-3. Note that, in general, the fieldequations of (2) must be extended to allow for x and ycomponents but still no z components:E = ET(x, y, z, t) = E.i + E,i,, E,=0(8)H= HT(x, y, z, t)= Hi.+H+i,, H = 0We use the subscript T in (8) to remind ourselves that thefields lie purely in the transverse xy plane. We can then alsodistinguish between spatial derivatives along the z axis (a/az)from those in the transverse plane (a/ax, alay):aV= T_+iz (9)ix+iy-We may then write Maxwell's equations asVTXET+ (i. XET)= -a--Ta aETVTXHT+-(i xHT)= e-az atVT-ET=O (10)VT-HT=OThe following vector properties for the terms in (10) apply:(i) VTX HT and VTX ET lie purely in the z direction.(ii) i, xET and i, x HT lie purely in the xy plane.D D€,/a /-•----D ------nCoaxial cable Wire above planeFigure 8-3 Various types of simple transmission lines.572 Guided Electromagnetic WavesThus, the equations in (10) may be separated by equatingvector components:VTET=O, VrXHr=0(11)VT'ET=0, VT HT)=08 , HT 8DET ,.(i, x ET)= -AT -= L-(i, x HT)az at az at (12)8 OET-(i. HT)= -az atwhere the Faraday's law equalities are obtained by crossingwith i, and expanding the double cross producti, X (iZXET)=i(i ET)-ET(i i,)=-ET (13)and remembering that i, *Er= 0.The set of 'equations in (11) tell us
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