Electromagnetic Field Theory: A Problem Solving Approach Part 64. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. Heavy reliance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked problems. To keep the subject exciting, many of these problems are based on physical processes, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering. | Sinusoidal Time Variations 605 dL P 2 Figure 8-16 A transmission line with lossy walls and dielectric results in waves that decay as they propagate. The spatial decay rate a of the fields is approximately proportional to the ratio of time average dissipated power per unit length Pu to the total time average electromagnetic power flow P down the line. 44 can be rewritten as P z Az P z I Pd dxdydz 46 Dividing through by dz Az we have in the infinitesimal limit P z Az P z d P hrn---------------------- ------ Az- o Az dz Pd dx dy 47 where P u is the power dissipated per unit length. Since the fields vary as e- the power flow that is proportional to the square of the fields must vary as e-2 1 so that d P ---- -2a P PdL dz 48 which when solved for the spatial decay rate is proportional to the ratio of dissipated power per unit length to the total 606 Guided Electromagnetic Waves electromagnetic power flowing down the transmission line 1 PdL 2 P 49 For our lossy transmission line the power is dissipated both in the walls and in the dielectric. Fortunately it is not necessary to solve the complicated field problem within the walls because we already approximately know the magnetic field at the walls from 42 . Since the wall current is effectively confined to the skin depth 8 the cross-sectional area through which the current flows is essentially w8 so that we can define the surface conductivity as rwS where the electric field at the wall is related to the lossless surface current as crw5E w 50 The surface current in the wall is approximately found from the magnetic field in 42 as -Hy -EJr 51 The time-average power dissipated in the wall is then P u. w wall Re Ew Kw 1 Kw 2 1w 1 2w 2 crw5 2 52 The total time-average dissipated power in the walls and dielectric per unit length for a transmission line system of depth w and plate spacing d is then PdL 2 Pdz. wal 2 l P 2w - 53 where we multiply 52 by two because of the losses in both electrodes. The time-average electromagnetic
