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.6. Probles 593 where T d /c is the one-way travel tie to the fault. Show that the corresponding tie constant τ /a is in the four cases: τ Z C, τ Z C, τ L Z, τ L Z For a resistive fault, show that Γ Z

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.6. Probles 593 where T d /c is the one-way travel tie to the fault. Show that the corresponding tie constant τ /a is in the four cases: τ Z C, τ Z C, τ L Z, τ L Z For a resistive fault, show that Γ Z /R + Z ), or, Γ R/R + Z ), for a shunt or series R. Moreover, show that Γ Z Z )/Z + Z ), where Z is the parallel in the shunt-r case) or series cobination of R with Z and give an intuitive explanation of this fact. For a series C, show that the voltage wave along the two segents is given as follows, and also derive siilar expressions for all the other cases: V ut z/c)+v e at+z/c T ) ut + z/c T ), for z d Vt, z) V e at z/c) ut z/c), for d z d + d Make a plot of V d t) for t 5T, assuing a for the C and L faults, and Γ corresponding to a shorted shunt or an opened series fault. The MATLAB file TDRovie. generates a ovie of the step input as it propagates and gets reflected fro the fault. The lengths were d 6, d 4 in units such that c ), and the input was V.. Coupled Transission Lines Coupled Lines Coupling between two transission lines is introduced by their proxiity to each other. Coupling effects ay be undesirable, such as crosstalk in printed circuits, or they ay be desirable, as in directional couplers where the objective is to transfer power fro one line to the other. In Sections..3, we discuss the equations, and their solutions, describing coupled lines and crosstalk In Sec..4, we discuss directional couplers, as well as fiber Bragg gratings, based on coupled-ode theory Fig... shows an exaple of two coupled icrostrip lines over a coon ground plane, and also shows a generic circuit odel for coupled lines. Fig... Coupled Transission Lines. For siplicity, we assue that the lines are lossless. Let L i,c i, i, bethe distributed inductances and capacitances per unit length when the lines are isolated fro each other. The corresponding propagation velocities and characteristic ipedances are: v i / L i C i, Z i L i /C i, i,. The coupling between the lines is odeled by introducing a utual inductance and capacitance per unit length, L,C. Then, the coupled versions of telegrapher s equations.5.) becoe: C is related to the capacitance to ground C g via C C g + C, so that the total charge per unit length on line- is Q C V C V C gv V g)+c V V ), where V g. .. Coupled Transission Lines Coupled Lines V z L I t L I t, I z C V t + C V t..) V z L I t L I t, I z C V t + C V t When L C, they reduce to the uncoupled equations describing the isolated individual lines. Eqs...) ay be written in the atrix fors: V z L L I L L t..) I z C C V C C t where V, I are the colun vectors: V V V, I I For sinusoidal tie dependence e jωt, the syste..) becoes: dv jω L L I L L di jω C C V C C I..3)..4) It proves convenient to recast these equations in ters of the forward and backward waves that are noralized with respect to the uncoupled ipedances Z,Z : a V + Z I Z, b V Z I Z a V + Z I Z, b V Z I Z a a a, b b b..5) The a, b waves are siilar to the power waves defined in Sec The total average power on the line can be expressed conveniently in ters of these: P ReV I ReV I + ReV I P + P a b ) + a b ) a + a ) b + b ) a a b b..6) where the dagger operator denotes the conjugate-transpose, for exaple, a a,a. Thus, the a-waves carry power forward, and the b-waves, backward. After soe algebra, it can be shown that Eqs...4) are equivalent to the syste: da jf a + jg b db jg a + jf b d a j b F G G F a b..7) with the atrices F, G given by: β κ χ F, G κ β χ..8) where β,β are the uncoupled wavenubers β i ω/v i ω L i C i, i, and the coupling paraeters κ, χ are: ) κ ω L C Z Z L β β Z Z L L ) χ ω L + C Z Z L β β + Z Z L L C ) C C C )..9) C C A consequence of the structure of the atrices F, G is that the total power P defined in..6) is conserved along z. This follows by writing the power in the following for, where I is the identity atrix: I a P a a b b a, b I b Using..7), we find: dp ja, b F G G F I I I I F G G F ) a b the latter following fro the conditions F F and G G. Eqs...6) and..7) for the basis of coupled-ode theory. Next, we specialize to the case of two identical lines that have L L L and C C C, so that β β ω L C β and Z Z L /C Z, and speed v / L C. Then, the a, b waves and the atrices F, G take the sipler fors: a V + Z I Z, b V Z I a V + Z I Z, b V Z I..) β κ χ F, G..) κ β χ where, for siplicity, we reoved the coon scale factor Z fro the denoinator of a, b. The paraeters κ, χ are obtained by setting Z Z Z in..9): κ β L L C ), χ C β L L + C ),..) C The atrices F, G coute with each other. In fact, they are both exaples of atrices of the for: a a A a a a I + a J, I, J..3) .. Coupled Transission Lines Coupled Lines where a,a are real such that a a. Such atrices for a coutative subgroup of the group of nonsingular atrices. Their eigenvalues are λ ± a ± a and they can all be diagonalized by a coon unitary atrix: Q e +, e, e +, e..4) so that we have QQ Q Q I and Ae ± λ ± e ±. The eigenvectors e ± are referred to as the even and odd odes. To siplify subsequent expressions, we will denote the eigenvalues of A by A ± a ± a and the diagonalized atrix by Ā. Thus, A+ a + a A QĀQ, Ā..5) A a a Such atrices, as well as any atrix-valued function thereof, ay be diagonalized siultaneously. Three exaples of such functions appear in the solution of Eqs...7): B F + G)F G) Q F + Ḡ) F Ḡ)Q Z Z F + G)F G) Z Q F + Ḡ) F Ḡ) Q..6) Γ Z Z I)Z + Z I) Q Z Z I) Z + Z I) Q Using the property FG GF, and differentiating..7) one ore tie, we obtain the decoupled second-order equations, with B as defined in..6): d a d b B a, B b However, it is better to work with..7) directly. This syste can be decoupled by foring the following linear cobinations of the a, b waves: A a Γ b B b Γa A B I Γ Γ I a b The A, B can be written in ters of V, I and the ipedance atrix Z as follows: A D) V + ZI) B D) V ZI) V DA + B) ZI DA B)..7) D Z + Z I Z..8) Using..7), we find that A, B satisfy the decoupled first-order syste: d A B j B B A B da jba, with solutions expressed in ters of the atrix exponentials e ±jbz : db jbb..9) Az) e jbz A), Bz) e jbz B)..) Using..8), we obtain the solutions for V, I : Vz) D e jbz A)+e jbz B) ZIz) D e jbz A) e jbz B)..) To coplete the solution, we assue that both lines are terinated at coon generator and load ipedances, that is, Z G Z G Z G and Z L Z L Z L. The generator voltages V G,V G are assued to be different. We define the generator voltage vector and source and load atrix reflection coefficients: V G VG V G, The terinal conditions for the line are at z and z l : Γ G Z G I Z)Z G I + Z) Γ L Z L I Z)Z L I + Z)..) V G V)+Z G I), Vl) Z L Il)..3) They ay be re-expressed in ters of A, B with the help of..8): A) Γ G B) D ZZ + Z G I) V G, Bl) Γ L Al)..4) But fro..9), we have: e jbl B) Bl) Γ L Al) Γ L e jbl A) B) Γ L e jbl A)..5) Inserting this into..4), we ay solve for A) in ters of the generator voltage: A) D I Γ G Γ L e jbl ZZ + ZG I) V G..6) Using..6) into..), we finally obtain the voltage and current at an arbitrary position z along the lines: Vz) e jbz + Γ L e jbl e jbz I Γ G Γ L e jbl ZZ + ZG I) V G Iz) e jbz Γ L e jbl e jbz I Γ G Γ L e jbl Z + ZG I) V G..7) These are the coupled-line generalizations of Eqs..9.7). Resolving V G and Vz) into their even and odd odes, that is, expressing the as linear cobinations of the eigenvectors e ±, we have: V G V G+ e + + V G e, where V G± V G ± V G Vz) V + z)e + + V z)e, V ± z) V z)±v z)..8) In this basis, the atrices in..7) are diagonal resulting in the equivalent solution: Vz) V + z)e + + V z)e e jβ+z + Γ L+ e jβ+l e jβ+z Z + Γ G+ Γ L+ e jβ+l V G+ e + Z + + Z G The atrices D, Z, Γ G,Γ L,Γ,B all coute with each other. + e jβ z + Γ L e jβ l e jβ z Γ G Γ L e jβ l Z Z + Z G V G e..9) .. Coupled Transission Lines 599 where β ± are the eigenvalues of B, Z ± the eigenvalues of Z, and Γ G±,Γ L± are: Γ G± Z G Z ±, Γ L± Z L Z ±..3) Z G + Z ± Z L + Z ± The voltages V z), V z) are obtained by extracting the top and botto coponents of..9), that is, V, z) V + z)±v z) / : V z) e jβ+z + Γ L+ e jβ+l e jβ+z Γ G+ Γ L+ e jβ+l V + + e jβ z + Γ L e jβ l e jβ z Γ G Γ L e jβ l V V z) e jβ+z + Γ L+ e jβ+l e jβ+z Γ G+ Γ L+ e jβ+l V + e jβ z + Γ L e jβ l e jβ z Γ G Γ L e jβ l V..3) where we defined: ) Z ± VG± V ± Z ± + Z G 4 Γ G±)V G ± V G )..3) The paraeters β ±,Z ± are obtained using the rules of Eq...5). Fro Eq...), we find the eigenvalues of the atrices F ± G: F + G) ± β ± κ + χ) β ± L ) ω L ± L ) Z F G) ± β ± κ χ) β C ) ωz C C ) C Then, it follows that: β + F + G) + F G) + ω L + L )C C ) β F + G) F G) ω L L )C + C ) F + G) + L + L Z + Z F G) + Z Z F + G) F G) L C C L L C + C..33)..34) Thus, the coupled syste acts as two uncoupled lines with wavenubers and characteristic ipedances β ±,Z ±, propagation speeds v ± / L ± L )C C ), and propagation delays T ± l/v ±. The even ode is energized when V G V G, or, V G+,V G, and the odd ode, when V G V G, or, V G+,V G. When the coupled lines are iersed in a hoogeneous ediu, such as two parallel wires in air over a ground plane, then the propagation speeds ust be equal to the speed of light within this ediu 65, that is, v + v / μɛ. This requires: L + L )C C ) μɛ L L )C + C ) μɛ L μɛc C C L μɛc C C..35) Therefore, L /L C /C, or, equivalently, κ. On the other hand, in an inhoogeneous ediu, such as for the case of the icrostrip lines shown in Fig..., the propagation speeds ay be different, v + v, and hence T + T. 6. Coupled Lines. Crosstalk Between Lines When only line- is energized, that is, V G,V G, the coupling between the lines induces a propagating wave in line-, referred to as crosstalk, which also has soe inor influence back on line-. The near-end and far-end crosstalk are the values of V z) at z and z l, respectively. Setting V G in..3), we have fro..3): V ) Γ G+ ) + Γ L+ ζ+ ) Γ G+ Γ L+ ζ+ V V l) ζ + Γ G+) + Γ L+ ) Γ G+ Γ L+ ζ + V Γ G ) + Γ L ζ ) Γ G Γ L ζ V ζ Γ G ) + Γ L ) Γ G Γ L ζ V..) where we defined V V G / and introduced the z-transfor delay variables ζ ± e jωt± e jβ±l. Assuing purely resistive terination ipedances Z G,Z L, we ay use Eq..5.5) to obtain the corresponding tie-doain responses: V,t) Γ G+) Vt)+ + ) Γ G+ Γ L+ ) Vt T + ) Γ G+ Γ G Γ G ) Vt)+ + ) Γ G Γ L ) Vt T ) V l, t) Γ G+) + Γ L+ ) Γ G+ Γ L+ ) Vt T + T + ) Γ G ) + Γ L ) Γ G Γ L ) Vt T T )..) where Vt) V G t)/. Because Z ± Z, there will be ultiple reflections even when the lines are atched to Z at both ends. Setting Z G Z L Z, gives for the reflection coefficients..3): Γ G± Γ L± Z Z ± Γ ±..3) Z + Z ± In this case, we find for the crosstalk signals: V,t) + Γ +) Vt) Γ+ ) Γ + Vt T + ) + Γ ) Vt) Γ ) Γ Vt T )..4) V l, t) Γ +) Γ ) Γ + Vt T + T + ) Γ Vt T T ) Vt) is the signal that would exist on a atched line- in the absence of line-, V Z V G /Z +Z G ) V G /, provided Z G Z. .. Crosstalk Between Lines 6 6. Coupled Lines Siilarly, the near-end and far-end signals on the driven line are found by adding, instead of subtracting, the even- and odd-ode ters: V,t) + Γ +) Vt) Γ+ ) Γ + Vt T + ) + + Γ ) Vt) Γ ) Γ Vt T )..5) V l, t) Γ + ) Γ+ Vt T + T + ) + Γ ) Γ Vt T T ) These expressions siplify drastically if we assue weak coupling. It is straightforward to verify that to first-order in the paraeters L /L,C /C, or equivalently, to first-order in κ, χ, we have the approxiations: β ± β ± Δβ β ± κ, Γ ± ± ΔΓ ± χ β, Z ± Z ± ΔZ Z ± Z χ β, T ± T ± ΔT T ± T κ β v ± v v κ β..6) where T l/v. Because the Γ ± s are already first-order, the ultiple reflection ters in the above suations are a second-order effect, and only the lowest ters will contribute, that is, the ter for the near-end, and for the far end. Then, V,t) Γ + Γ )Vt) V l, t) Vt T+ ) Vt T ) Γ+ Vt T + ) Γ Vt T ) Using a Taylor series expansion and..6), we have to first-order: Vt T ± ) Vt T ΔT) Vt T) ΔT) Vt T), Vt T ± ) Vt T ΔT) Vt T) ΔT) Vt T) V dv dt Therefore, Γ ± Vt T ± ) Γ ± Vt T) ΔT) V Γ ± Vt T), where we ignored the second-order ters Γ ± ΔT) V. It follows that: V,t) Γ + Γ ) Vt) Vt T) ΔΓ) Vt) Vt T) V l, t) Vt T) ΔT) V Vt T) ΔT) V dvt T) ΔT) dt These can be written in the coonly used for: V,t) K b Vt) Vt T) V l, t) K f dvt T) dt near- and far-end crosstalk)..7) where K b,k f are known as the backward and forward crosstalk coefficients: K b χ β v ) L + C Z, K f T κ ) 4 Z β v T L C Z Z..8) where we ay replace l v T. The sae approxiations give for line-, V,t) Vt) and V l, t) Vt T). Thus, to first-order, line- does not act back to disturb line-. Exaple..: Fig... shows the signals V, t), V l, t), V, t), V l, t) for a pair of coupled lines atched at both ends. The uncoupled line ipedance was Z 5 Ω L /L.4, C /C.3 line near end line far end line near end line far end t/t L /L.8, C /C.7 line near end line far end line near end line far end t/t Fig... Near- and far-end crosstalk signals on lines and. For the left graph, we chose L /L.4, C /C.3, which results in the even and odd ode paraeters using the exact forulas): Z Ω, Z Ω, v +.v, v.3v Γ +.7, Γ.9, T +.99T, T.88T, K b.75, K f.5 The right graph corresponds to L /L.8, C /C.7, with paraeters: Z +.47 Ω, Z 7.5 Ω, v +.36v, v.7v Γ +.4, Γ.49, T +.73T, T.58T, K b.375, K f.5 The generator input to line- was a rising step with rise-tie t r T/4, that is, Vt) V Gt) t t r ut) ut tr ) + ut t r ) The weak-coupling approxiations are ore closely satisfied for the left case. Eqs...7) predict for V,t) a trapezoidal pulse of duration T and height K b, and for V l, t), a rectangular pulse of width t r and height K f /t r. starting at t T: V l, t) K f dvt T) dt K f t r ut T) ut T tr ) These predictions are approxiately correct as can be seen in the figure. The approxiation predicts also that V,t) Vt) and V l, t) Vt T), which are not quite true the effect of line- on line- cannot be ignored copletely. .3. Weakly Coupled Lines with Arbitrary Terinations Coupled Lines The interaction between the two lines is seen better in the MATLAB ovie xtalkovie., which plots the waves V z, t) and V z, t) as they propagate to and get reflected fro their respective loads, and copares the to the uncoupled case V z, t) Vt z/v ). The waves V, z, t) are coputed by the sae ethod as for the ovie pulseovie. of Exaple.5., applied separately to the even and odd odes..3 Weakly Coupled Lines with Arbitrary Terinations The even-odd ode decoposition can be carried out only in the case of identical lines both of which have the sae load and generator ipedances. The case of arbitrary terinations has been solved in closed for only for hoogeneous edia 6,65. It has also been solved for arbitrary edia under the weak coupling assuption 7. Following 7, we solve the general equations..7)..9) for weakly coupled lines assuing arbitrary terinating ipedances Z Li,Z Gi, with reflection coefficients: Γ Li Z Li Z i, Γ Gi Z Gi Z i, i,.3.) Z Li + Z i Z Gi + Z i Working with the forward and backward waves, we write Eq...7) as the 4 4 atrix equation: a β κ χ dc jmc, c a b, M κ β χ χ β κ b χ κ β The weak coupling assuption consists of ignoring the coupling of a,b on a,b. This aounts to approxiating the above linear syste by: β dc j κ β χ ˆMc, ˆM.3.) β χ κ β Its solution is given by cz) e j ˆMz c), where the transition atrix e j ˆMz can be expressed in closed for as follows: e jβz κ e j ˆMz ˆκe jβz e jβz ) e jβz ˆχe jβz e jβz ) ˆκ β β, e jβz χ ˆχ ˆχe jβz e jβz ) ˆκe jβz e jβz ) e jβz β + β The transition atrix e j ˆMl ay be written in ters of the z-doain delay variables ζ i e jβil e iωti, i,, where T i are the one-way travel ties along the lines, that is, T i l/v i. Then, we find: a l) a l) b l) b l) ζ ˆκζ ζ ) ζ ˆχζ ζ ) ζ ˆχζ ζ ) ˆκζ ζ ) ζ a ) a ) b ) b ).3.3) These ust be appended by the appropriate terinating conditions. Assuing that only line- is driven, we have: V )+Z G I ) V G, V )+Z G I ), which can be written in ters of the a, b waves: V l) Z L I l) V l) Z L I l) a ) Γ G b ) U, b l) Γ L a l) a ) Γ G b ), b l) Γ L a l), U Γ G ) V G Z.3.4) Eqs..3.3) and.3.4) provide a set of eight equations in eight unknowns. Once these are solved, the near- and far-end voltages ay be deterined. For line-, we find: Z V ) a )+b ) + Γ Lζ Γ G Γ L ζ V Z V l) a l)+b l) ζ + Γ L ) Γ G Γ L ζ V where V Γ G )V G / Z V G /Z + Z G ). For line-, we have: V ) κζ ζ )Γ L ζ + Γ L ζ )+ χ ζ ζ ) + Γ L Γ L ζ ζ Γ G Γ L ζ ) Γ G Γ L ζ ) V l) κζ ζ ) + Γ L Γ G ζ ζ )+ χ ζ ζ )Γ L ζ + Γ G ζ Γ G Γ L ζ ) Γ G Γ L ζ ).3.5) ) V ) V l.3.6) where V + Γ G )V + Γ G ) Γ G )V G / and V l + Γ L )V, and we defined κ, χ by: κ χ Z ˆκ Z Z ˆχ Z Z Z Z Z κ β β χ β + β ) ω L C Z β β Z ) ω L + C Z β + β Z.3.7) In the case of identical lines with Z Z Z and β β β ω/v, we ust take the liit: e jβl e jβl li d e jβl jle jβl β β β β dβ Then, we obtain: κζ ζ ) jωk f e jβl jω l ) L C Z e jβl Z χ K b v ).3.8) L + C Z 4 Z where K f,k b were defined in..8). Setting ζ ζ ζ e jβl e jωt, we obtain the crosstalk signals: .4. Coupled-Mode Theory Coupled Lines V ) jωk f Γ L + Γ L )ζ + K b ζ ) + Γ L Γ L ζ ) V Γ G Γ L ζ ) Γ G Γ L ζ ) V l) jωk f + Γ L Γ G ζ )ζ + K b ζ )Γ L + Γ G )ζ Γ G Γ L ζ ) Γ G Γ L ζ ) V l.3.9) The corresponding tie-doain signals will involve the double ultiple reflections arising fro the denoinators. However, if we assue the each line is atched in at least one of its ends, so that Γ G Γ L Γ G Γ L, then the denoinators can be eliinated. Replacing jω by the tie-derivative d/dt and each factor ζ by a delay by T, we obtain: V,t) K f Γ L + Γ L + Γ L Γ G ) Vt T) + K b + Γ G ) Vt) Vt T) + K b Γ L Γ L Vt T) Vt 4T) V l, t) K f + ΓL ) Vt T)+Γ L Γ G Vt 3T) + K b Γ L + Γ G + Γ L Γ L ) Vt T) Vt 3T).3.) where Vt) Γ G )V G t)/, and we used the property Γ G Γ L to siplify the expressions. Eqs..3.) reduce to..7) when the lines are atched at both ends..4 Coupled-Mode Theory In its siplest for, coupled-ode or coupled-wave theory provides a paradig for the interaction between two waves and the exchange of energy fro one to the other as they propagate. Reviews and earlier literature ay be found in Refs , see also for the relationship to fiber Bragg gratings and distributed feedback lasers. There are several echanical and electrical analogs of coupled-ode theory, such as a pair of coupled pendula, or two asses at the ends of two springs with a third spring connecting the two, or two LC circuits with a coupling capacitor between the. In these exaples, the exchange of energy is taking place over tie instead of over space. Coupled-wave theory is inherently directional. If two forward-oving waves are strongly coupled, then their interactions with the corresponding backward waves ay be ignored. Siilarly, if a forward- and a backward-oving wave are strongly coupled, then their interactions with the corresponding oppositely oving waves ay be ignored. Fig..4. depicts these two cases of co-directional and contra-directional coupling. Fig..4. Directional Couplers. Eqs...7) for the basis of coupled-ode theory. In the co-directional case, if we assue that there are only forward waves at z, that is, a) and b), then it ay shown that the effect of the backward waves on the forward ones becoes a second-order effect in the coupling constants, and therefore, it ay be ignored. To see this, we solve the second of Eqs...7) for b in ters of a, assuing zero initial conditions, and substitute it in the first: z bz) j e jfz z ) G az ) da z jf a + Ge jfz z ) G az ) The second ter is second-order in G, or in the coupling constant χ. Ignoring this ter, we obtain the standard equations describing a co-directional coupler: da jf a d a β κ a j.4.) a κ β a For the contra-directional

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