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Dielectric Waveguide 645 Figure 8-30 TE and TM modes can also propagate along dielectric structures. The fields can be essentially confined to the dielectric over a frequency range if the speed of the wave in the dielectric is less than that outside. It is convenient to separate the solutions into even and odd modes. where we choose to write the solution outside the dielectric in the decaying wave form so that the fields are predominantly localized around the dielectric. The wavenumbers and decay rate obey the relations k -2 -a + k. = o 2 +kz = (2) t (2) 2 (0 Eo01_ The z component of the wavenumber must be the same in all regions so that the boundary conditions can be met at each interface. For propagation in the dielectric and evanescence in free space, we must have that &o/e0/w0< k, <-to/ l (3) All the other electric and magnetic field components can be found from (1) in the same fashion as for metal waveguides in Section 8-6-2. However, it is convenient to separately consider each of the solutions for E, within the dielectric. (a) Odd Solutions If E, in each half-plane above and below the centerline are oppositely directed, the field within the dielectric must vary solely as sin kx: A 2 e-a /F= Al sin kxx, A ea(x+d) A,, e" x>d Ixl<-d xj_-d 646 Guided Electromagnetic Waves Then because in the absence of volume charge the electric field has no divergence, jkA22 e- (x-), --ax x jk =, - Alcos kxx, jkax i I xj -d kA 3 e(x+d), x ac while from Faraday's law the magnetic field is k S1 -d x-d (6) jweAl jweoAs (5) aE\ jweoA 2 e-(x-d) H, =cos d kx, IxISd e,(x+d) x -d At the boundaries where x = +d the tangential electric and magnetic fields are continuous: E,(x = Ed_)= E,(x = +d+) A, sin kxd = A 2 -A, H,(x = d-)= H,(x = d+) sin kd = As -joeAi cos kd =--joeoA2 -jwE k, (7) a -jweAI jweoA 3 cos k,d kx a which when simultaneously solved yields A a - 2 = sin kd = cos k•d eoko A'1 As -= AI a=-k.tan = E =a e3a -sin kd k,d (8) e -= cos k,d eokx The allowed values of a and k,are obtained by self-consistently solving (8) and (2), which in general requires a numerical method. The critical condition for a guided wave occurs when a = 0, which requires that k~d = n'r and k. = S2 Eoo.The critical frequency is then obtained from (2) as 2 k = 2 (n/d) l= e8L -EOo eL Eo/o Note that this occurs for real frequencies only if esl> "-t--~ ~ (9) EO00L. 647 Dielectric Waveguide (b) Even Solutions If E, is in the same direction above and below the dielectric, solutions are similarly x d B 2 - a ( - d), ,= BI cos kx, Bs e(x+d), jkzB2 (10) IxI5d xs-d e-a(x-d), k, kBs ea ( a - ,= + >, Be•-B e-a(-d), B sinkx, a SBs ea(x+ d ) , x -5-d x-d Ixl -d (12) X 5 -d Continuity of tangential electric and magnetic fields at x = +d requires B 1 cos kd = B 2 , we BI sin kd = , B 1 cos kd = Bs joeoBs joB jO 1 sin kd = a k, eo B2, a (13) or B2 -=cos kd =Bl sd ea sin kd o°k• -=cos kd = B, 8-7-2 a =-°cot EAk, kd (14) e " Bsa sne kd TE Solutions The same procedure is performed for the TE solutions by first solving for H,. (a) Odd Solutions (A 2 e- a(2 -d), A•,-eAsinkx, As e a ( x + d ), X-d Ixj-d X 5 -d (15) 648 Guided Electromagnetic Waves -k'A2 e-~ ( '-d ) ~, xd io -,=AA coskx,l, jk, As e(x+d ), a x ( -a x-d) d Ix , (16) -- d x-d a E, = A, cos kx, I xl oAs ea (x+d), a x-d d (17) where continuity of tangential E and H across the boundaries requires (18) a = ý k,tan k~d (b) Even Solutions B 2 e-a(-d), ,= B 1 cosh k•,x, x d x d x s -d Bs e'a(+d), -LB2 e - a ( - d), a H,= (19) ikhB-sin k•x, x -d IxJ :d (20) jk a " 'Bs 3 e "O 0 '+ ), ) x --d G-a(x-d), X d a jo04 E,= k,. B--B, sin kx, -]llOBs a ( , +d) IxIsd (21) x: -d where a and A, are related as a =- JA k,cot khd (22) Problems 649 PROBLEMS Section 8-1 1. Find the inductance and capacitance per unit length and the characteristic impedance for the wire above plane and two wire line shown in Figure 8-3. (Hint: See Section 2-6-4c.) 2. The inductance and capacitance per unit length on a lossless transmission line is a weak function of z as the distance between electrodes changes slowly with z. + Re(Voe"' t ) 0 1 (a) For this case write the transmission line equations as single equations in voltage and current. (b) Consider an exponential line, where L(z) = Lo e"', C(z)= Co e -a If the voltage and current vary sinusoidally with time as v(z, t) = Re [i(z) e*"'], i(z, i) = Re [i(z) e""'] find the general form of solution for the spatial distributions of i~(z) and i(z). (c) The transmission line is excited by a voltage source Vo cos wt at z = 0. What are the voltage and current distributions if the line is short or open circuited at z = 1? (d) For what range of frequency do the waves strictly decay with distance? What is the cut-off frequency for wave propagation? (e) What are the resonant frequencies of the short circuited line? (f) What condition determines the resonant frequencies of the open circuited line. 3. Two conductors of length I extending over the radial distance a- r5 b are at a constant angle a apart. (a) What are the electric and magnetic fields in terms of the voltage and current? (b) Find the inductance and capacitance per unit length. What is the characteristic impedance? 650 Guided Electromagnetic Waves b r a 4. A parallel plate transmission line is filled with a conducting plasma with constitutive law J=oPeE at + Re(Voe "t) itjY 0Z 0 (a) How are the electric and magnetic fields related? (b) What are the transmission line equations for the voltage and current? (c) For sinusoidal signals of the form ei ( "' ) , how are w and k related? Over what frequency range do we have propagation or decay? (d) The transmission line is short circuited at z = 0 and excited by a voltage source Vo cos wt at z = -1. What are the voltage and current distributions? (e) What are the resonant frequencies of the system? 5. An unusual type of distributed system is formed by series capacitors and shunt inductors. VIZ - •z, I z - Az '')I i (z + Az, t) I z 1 z + Az I1 (a) What are the governing partial differential equations relating the voltage and current? Problems 651 (b) What is the dispersion relation between w and k for signals of the form ei<'-k)? (c) What are the group and phase velocities of the waves? Why are such systems called "backward wave"? cos wt is applied at z = -1 with the z = 0 (d) A voltage Vo end short circuited. What are the voltage and current distributions along the line? (e) What are the resonant frequencies of the system? Section 8-2 6. An infinitely long transmission line is excited at its center by a step voltage Vo turned on at t = 0. The line is initially at rest. Zo V(t) Zo 0 (a) Plot the voltage and current distributions at time T. (b) At this time T the voltage is set to zero. Plot the voltage and current everywhere at time 2 T. 7. A transmission line of length I excited by a step voltage source has its ends connected together. Plot the voltage and current at z = 1/4, 1/2, and 31/4 as a function of time. 0 + V0 8. The dc steady state is reached for a transmission line loaded at z = 1with a resistor RL and excited at z = 0 by a dc voltage Vo applied through a source resistor R,. The voltage source is suddenly set to zero at t = 0. (a) What is the initial voltage and current along the line? 652 Guided Electromagnetic Waves V) R, Z "1 Vt) R I (b) Find the voltage at the z = I end as a function of time. (Hint: Use difference equations.) 9. A step current source turned on at t= 0 is connected to the z = 0 end of a transmission line in parallel with a source resistance R,. A load resistor RL is connected at z = i. (a) What is the load voltage and current as a function of time? (Hint: Use a Thevenin equivalent network at z = 0 with the results of Section 8-2-3.) (b) With R, = co plot versus time the load voltage when RL = co and the load current when RL = 0. (c) If R, = co and Rt = co, solve for the load voltage in the quasi-static limit assuming the transmission line is a capacitor. Compare with (b). (d) If R, is finite but RL = 0,what is the time dependence of the load current? (e) Repeat (d) in the quasi-static limit where the transmission line behaves as an inductor. When are the results of (d) and (e) approximately equal? 10. Switched transmission line systems with an initial dc voltage can be used to generate high voltage pulses of short time duration. The line shown is charged up to a dc voltage Vo when at t = 0 the load switch is closed and the source switch is opened. Opens at t = 0 Closes at t = 0 = Zo 653 Problems (a) What are the initial line voltage and current? What are V+ and V_? (b) Sketch the time dependence of the load voltage. 11. For the trapezoidal voltage excitation shown, plot versus time the current waveforms at z = 0 and z = L for RL = 2Zo and RL = jZo. Rs =Z 0 T 3T I 2Zo 22o 4T 0 i 12. A step voltage is applied to a loaded transmission line with RL = 2Zo through a matching source resistor. R, =Z v(t) +R V0 V(t) Zu, T = C :RL = 2Zo 2Vo -> - T 2T T 2T _t VC (a) Sketch the source current i,(t). (b) Using superposition of delayed step voltages find the time dependence of i,(t) for the various pulse voltages shown. (c) By integrating the appropriate solution of (b), find i,(t) if the applied voltage is the triangle wave shown. 13. A dc voltage has been applied for a long time to the .transmission line circuit shown with switches S1 and S2 open 654 Guided ElctromagnticWaves when at t = 0: (a) S2 is suddenly closed with SI kept open; (b) S, is suddenly closed with S 2 kept open; (c) Both S, and S2 are closed. S2 For each of these cases plot the source current i,(t) versus time. 14. For each of the transmission line circuits shown, the switch opens at t = 0 after the dc voltage has been applied for a long time. Opens at t = 0 I 0 I 1 Opens at t = 0 _3 (a) What are the transmission line voltages and currents right before the switches open? What are V+ and V_ at t = 0? (b) Plot the voltage and current as a function of time at z=1/2. 15. A transmission line is connected to another transmission line with double the characteristic impedance. (a) With switch S 2 open, switch S, is suddenly closed at t = 0. Plot the voltage and current as a function of time halfway down each line at points a and b. (b) Repeat (a) if S 2 is closed.