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 8 guided electromagnetic waves 568 Guided Electromagnetic Waves The uniform plane wave solutions developed in Chapter 7 cannot in actuality exist throughout all space as an infinite amount of energy would be required from the sources However TEM waves can also propagate in the region of finite volume between electrodes Such electrode structures known as transmission lines are used for electromagnetic energy flow from power 60 Hz to microwave frequencies as delay lines due to the finite speed c of electromagnetic waves and in pulse forming networks due to reflections at the end of the line Because of the electrode boundaries more general wave solutions are also permitted where the electric and magnetic fields are no longer perpendicular These new solutions also allow electromagnetic power flow in closed single conductor structures known as waveguides 8 1 8 1 1 THE TRANSMISSION LINE EQUATIONS The Parallel Plate Transmission Line The general properties of transmission lines are illustrated in Figure 8 1 by the parallel plate electrodes a small distance d apart enclosing linear media with permittivity e and permeability Cj Because this spacing d is much less than the width w or length i we neglect fringing field effects and assume that the fields only depend on the z coordinate The perfectly conducting electrodes impose the boundary conditions i The tangential component of E is zero ii The normal component of B and thus H in the linear media is zero With these constraints and the neglect of fringing near the electrode edges the fields cannot depend on x or y and thus are of the following form E E z t i H H z t i which when substituted into Maxwell s equations yield I 1 The TransmissionLine Equations Figure 8 1 569 The simplest transmission line consists of two parallel perfectly conduct ing plates a small distance d apart aE ata Z VxH E at az aH at E at We recognize these equations as the same ones developed for plane waves in Section 7 3 1 The wave solutions found there are also valid here However now it is more convenient to introduce the circuit variables of voltage and current along the transmission line which will depend on z and t Kirchoff s voltage and current laws will not hold along the transmission line as the electric field in 2 has nonzero curl and the current along the electrodes will have a divergence due to the time varying surface charge distribution o eE z t Because E has a curl the voltage difference measured between any two points is not unique as illustrated in Figure 8 2 where we see time varying magnetic flux passing through the contour LI However no magnetic flux passes through the path L 2 where the potential difference is measured between the two electrodes at the same value of z as the magnetic flux is parallel to the surface Thus the voltage can be uniquely defined between the two electrodes at the same value of z v z t J z const E dl E z t d 570 Guided Electromagnetic Waves AL2 3E E fE L1 22 all di podf a ds dl 0 L2 st Figure 8 2 The potential difference measured between any two arbitrary points at different 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 If the contour L2 lies within a plane of constant z such as at z no magnetic flux passes through it so that the voltage difference between the two electrodes at the same value of z is unique Similarly the tangential component of H is discontinuous at each plate by a surface current K Thus the total current i z t flowing in the z direction on the lower plate is i z t K w H w Substituting 3 and 4 back into 2 results in the transmission line equations av ai LOz at ai z av 5 c at where L and C are the inductance and capacitance per unit length of the parallel plate structure Ild L henry m w C w farad m d If both quantities are multiplied by the length of the line 1 we obtain the inductance of a single turn plane loop if the line were short circuited and the capacitance of a parallel plate capacitor if the line were open circuited It is no accident that the LC product LC ejA 1 c2 is related to the speed of light in the medium 8 1 2 General Transmission Line Structures The transmission line equations of 5 are valid for any two conductor structure of arbitrary shape in the transverse The Transmission Line Equations 571 xy plane but whose cross sectional area does not change along its axis in the z direction L and C are the inductance and capacitance per unit length as would be calculated in the quasi static limits Various simple types of transmission lines are shown in Figure 8 3 Note that in general the field equations of 2 must be extended to allow for x and y components but still no z components E ET x y z t E i E i E 0 8 H 0 H HT x y z t Hi H i We use the subscript T in 8 to remind ourselves that the fields lie purely in the transverse xy plane We can then also distinguish between spatial derivatives along the z axis a az from those in the transverse plane a ax alay V T iz a 9 ix iy We may then write Maxwell s equations as VTXET i XET a VTXHT i xHT e az a T aET at 10 VT ET O VT HT O The following vector properties for the terms in 10 apply i ii VTX HT and VTX ET lie purely in the z direction i xET and i x HT lie purely in the xy plane D a Coaxial cable Figure 8 3 Wire above plane Various types of simple transmission lines D n D 572 Guided Electromagnetic Waves Thus the equations in 10 may be separated by equating vector components 8 VTET O VrXHr 0 VT ET 0 VT HT 0 8DET L i HT az i x ET AT at 8 i HT az at 11 x HT OET az 12 at where the Faraday s law equalities are obtained by crossing with i and expanding the double cross product i X iZ XET i i i ET ET ET i 13 and remembering that i Er 0 The set of equations in 11 tell us that the …
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