HW02'sol - Classical Electrodynamics II PHY 506 Spring 2011...

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1 C:\User\Teaching\505-506\S11\HW02'sol.doc PHY 506 Classical Electrodynamics II Spring 2011 Homework 2 with solutions Problem 2.1 (to be graded of 25 points). Calculate impedance Z W of long, straight TEM transmission lines formed by metallic electrodes with the cross-sections shown in Fig. below: (i) two round, parallel wires, separated by distance d >> R , (ii) microstrip line of width w >> d , (iii) stripline with w >> d 1 ~ d 2 , in all cases using the macroscopic boundary conditions on metallic surfaces. Assume that the conductors are embedded into a linear dielectric with constant and . Solutions : (i) From the solution of Problem 2.3 with R 1 = R 2 = R , generalized to dielectric filling by the replacement of 0 with , the mutual capacitance per unit length is ) / ln( 0 R d C  . The inductance L 0 per may be found either from magnetostatics (e.g., using the generalized Ampère law), or directly from Eq. (7.109): R d C L ln 0 0 0 0 0 . Combining these expressions, we get . , ln 2 / 1 2 / 1 0 0 Z R d Z C L Z W This expression is close in structure to that for the coaxial cable – see Eq. (7.113) of the lecture notes, with an extra factor of 2, due to two (rather than one) thin conductors, with distance d between the wires playing the role almost similar to the diameter of the outer tube of the cable. Practical notice: this system is very important for modern information technology. Pairs of parallel copper wires (typically spaced by distance d ~ 4 R and twisted into a spiral to avoid picking up and creating interferences) are broadly used for carrying phone and internet signals (for example, in the Ethernet and DSL technologies) with frequencies up to a few hundred GHz.
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