Opt2'sol - Classical Electrodynamics II PHY 506 Spring 2011...

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1 C:\User\Teaching\506\S11\Opt2'sol.doc PHY 506 Classical Electrodynamics II Spring 2011 Optional Problems Set 2 with solutions Problem O.8 . What lumped ac circuit is equivalent to the system shown in Fig. 7.19b of the lecture notes, with monochromatic incident wave of power P i ? Assume that the wave reflected from the load circuit does not return to it. Solution : It is intuitively clear that the circuit has to include: (i) the lumped load with impedance Z L ( ), (ii) a passive lumped element with impedance Z W , presenting passive properties of the transmission line, and (iii) some signal generator, presenting the incident wave. Following these arguments, let us try to use the circuit shown on the right, where E ( t ) is an e.m.f. presenting the incident wave. An elementary calculation of the complex amplitude of the current in the load and voltage drop across it yields I = E /[ Z L ( ) + Z W ( )], V = I Z L ( ) = E Z L ( )/[ Z L ( ) + Z W ( )], so that the calculation similar to the derivation of Eq. (7.40) yields the following expression for the average power absorbed in the load: ) ( Re ) ( ) ( 2 1 2 1 2 2 * L W L L Z Z Z I V E P . (*) On the other hand, we can calculate this power in the genuine circuit (Fig. 7.20) by subtracting from the incident power P i the reflected wave power P R = (1/2)P i R 2 , where R is given by Eq. (7.115):  2 2 2 2 2 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 W L W L W L i W L W L i i L Z Z Z Z Z Z Z Z Z Z R P P P P . In the most important case of a loss-free line, its impedance Z W is real, and this expression is reduced to   ) ( Re ) ( 4 2 L W L W i L Z Z Z Z P P . (**) Comparing Eqs. (*) and (**), we see that they give similar results (and hence the lumped circuit is equivalent to the wave system shown in Fig. 20), if the effective e.m.f. has amplitude     2 / 1 8 W i Z P E . Such presentation of a transmission line by its impedance and e.m.f. is valid even if the lumped load is nonlinear and/or time-dependent, and is broadly used for the analysis of not only linear, but also nonlinear and parametric microwave devices. 1 1 For a brief discussion of the nonlinear and parametric interactions, see, e.g., CM Sec. 5.5. ) ( t E ) ( W Z ) ( L Z ) ( t I
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2 C:\User\Teaching\506\S11\Opt2'sol.doc Problem O.9 . Calculate the skin-effect contribution to the attenuation coefficient of (i) the basic ( H 11 ) mode, and ( i i ) t h e H 01 mode of waves in a metallic waveguide with the circular cross-section (Fig. 7.22a of the lecture notes), and analyze the low-frequency ( c ) and high-frequency ( >> c ) behavior of for these two modes. Solutions : (i) For the H 11 mode, the longitudinal component H z of the magnetic field (or rather its complex amplitude) is described by Eq. (7.140) of the lecture notes:  84 . 1 with , , 11 11 1 ' ' l z i e R J H H .
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Opt2'sol - Classical Electrodynamics II PHY 506 Spring 2011...

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