T.C. Banwell, S. Galli. {bct, Telcordia Technologies, Inc., 445 South Street, Morristown, NJ 07960, USA

Transcription

1 ON THE SYMMETRY OF THE POWER INE CHANNE T.C. Banwe, S. Gai {bct, Tecordia Technoogies, Inc., 445 South Street, Morristown, NJ 07960, USA Abstract The indoor power ine network is characterized by severa branches and impendence mismatches that cause many refections. In the present paper, we address the probem of determining the exact conditions under which the power ine channe may be considered a symmetric channe, i.e. exhibiting the same frequency transfer function from either side. By using a new way of modeing the power ine channe by means of transmission matrices [3], it is here shown for the first time that the power ine channe is, indeed, a symmetric channe. Moreover, it is aso shown that this property hods regardess of the topoogy of the ink and that the ony condition for symmetry is the requirement that the source and oad impedances used to terminate the ine be the same. The proposed approach impies the vaidity of the TEM approximation. It is possibe to prove that this assumption is vaid assuming correct mode excitation, which differs from conventiona twisted pair. Interestingy, the transmission matrices approach aows us aso to characterize easiy in the frequency domain the effects of the muti-path nature of the power ine channe [3]. Keywords: power ine channe modes, frequency transfer functions, transmission matrices. 1. Introduction Prior contributions propose modeing Power ine Channes (PCs) as a cascade of decouped series resonant circuits, or by using either their scattering parameter matrices or four poe impedance and admittance matrices. In a companion paper [3], it has been proposed for the first time to use the ABCD or transmission matrices to mode the transfer function of the power ine channe. The theoretica and experimenta justification of this mode can be found in [3]. Interestingy, the expoitation of the ABCD matrices in the modeing of the transfer function aows us to prove an interesting property of the PC. In particuar, it is possibe to prove that the PC, regardess of its topoogy, is aways a symmetric channe (same frequency transfer function from either side) provided that the source and oad impedances are the same. An experimenta confirmation of this property can be found in [3]. 2. ABCD Matrix Modeing of a Transmission ine Every power ine cabe can be modeed as a Two-Port Network (2PN) (see [3] for a theoretica as we as experimenta justification of this statement). In transmission ine theory, a common way to represent a 2PN is to use the transmission matrix T, aso known as ABCD matrix [1]. The reationship between current and votage (in the frequency domain) at the two ports is given by the foowing expression (see Figure 1): A B V2 V = = T 2 f. (1) C D I2 I2 To express (V 2, I 2 ) as functions of (V 1, I 1 ), it is necessary to invert the ABCD matrix in (1): 1 V2 A B 1 D B = = = Tb (2) I2 C D AD BC C A In eq.(2), we have changed the signs of B and C according to the convention that considers as positive currents those that fow from the source to the oad. Usuay, the transmission matrices in (1) and (2) are referred to as forward (T f ) and backward (T b ) transmission matrices, respectivey. Given the 2PN in Figure 1, the ABCD parameters aow us to cacuate the foowing usefu quantities (for the sake of simpicity, the expicit dependency on frequency of A, B, C, D, o and γ has been omitted): Input Impedance A B : f + in ( ) = ; C + D (3) Transfer Function V : H ( f ) = 2 = VS A + B + CS + DS ; (4)

2 + S + S Insertion oss : H I ( f ) = H ( f ) = ; (5) A + B + CS + DS The insertion oss is the ratio between the votage on the oad with the cabe present and the votage on the oad without the cabe. If oad and source impedances are the same and are purey resistive, there is a 3 db difference between H(f) and H I (f). Usuay, when deaing with performance cacuations or Signa-to-Noise Ratio (SNR) computations, it is the insertion oss that has to be considered and not the transfer function because any transmit power constraint appies to the power actuay transferred into the cabe and not on the power transferred by the source V S. In fact, the use of insertion oss avoids the signa oss caused by the votage division between source and cabe input impedance whie sti incuding the effect of the source impedance. Moreover, it avoids the division between source and termination impedance when the cabe is very short. As an exampe, H ( f ) = + and H I (f)=1. consider a zero ength cabe. In this case, we woud have: [ ] 2.1. Definition of ABCD Coefficients The ABCD coefficients are compex functions of frequency and fuy characterize the eectrica properties of the 2PN. For a cabe of ength, these coefficients and the corresponding transmission matrix T are: A = D = coshγ coshγ o sinhγ B = o sinhγ T = 1, (6) sinhγ coshγ 1 C = sinhγ o o where γ and o are the propagation constant and the characteristic impedance of the cabe, respectivey. The ABCD coefficients are defined in the foowing way: A = ; B = ; C = ; D =. V2 I2 = I 0 2 V2 = V 0 2 I2 = I 0 2 V2 = 0 The transmission matrix T in (6), that modes a cabe of ength, satisfies the foowing properties: 1. A=D for any frequency; 2. B C for any frequency; (7) 3. Unitary determinant: det(t)=ad-bc=1; 4. Recaing the convention on positive currents, from 1) and 3) it foows that T=T -1, i.e. T f =T b. From property 3), it foows that the 2PN modeing a cabe is a reciproca 2PN, i.e. a 2PN that satisfies the Reciprocity Theorem. The Reciprocity Theorem states that any transfer function with the dimension of impedance or admittance remains unchanged if the points of excitation and response are interchanged. From property 4), it aso foows that a cabe is a symmetrica channe, in the sense that it exhibits the same behavior if driven from either side. Since T f =T b, the transfer function (4) as we as the insertion oss (5) of a cabe are the same from either side, provided that source and oad impedances are switched The Chain Rue In genera, a power ine network is made of severa sections and each section may be constituted by different cabes of different engths. An important property of the transmission matrix is that it easiy aows us to hande tandem connections of 2PNs. For a given network configuration, the overa ABCD matrix of the endto-end circuit is obtained by expoiting the chain rue, i.e. mutipying the ABCD matrices of the singe portions (i) of the network. Therefore, if T f is the forward transmission matrix of the i-th section, the overa forward transmission matrix T f of the end-to-end circuit constituted of N sections is given by the foowing reationship: (1) (2) ( N ) T f = T f f... f. (8) Simiary, the overa backward transmission matrix T b of the end-to-end circuit is given by the foowing reationship: ( ) ( ) ( ) ( ) = 1 ( N ) 1 ( N 1) 1 (1) T =... 1 b T f T f T f T f ). (i) Each forward transmission matrix T f is of the kind in (6) and therefore satisfies the properties (7). In particuar, from property 4) in (7) we can write: 1 ( N ) ( N 1) (1) Tb = ( T f ) = T f f... f. (9) S

3 2.3. Modeing Series and Shunt Impedances Aong the ine Particuar attention must be given to bridged taps or, in genera, to series and shunt impedances aong the ine. In fact, in these cases the ABCD matrix takes a particuar form that differs from that of (6). However, as a genera resut, any ine discontinuity can be embedded in the mode in the form of cascaded 2PNs. A bridged tap occurs whenever branching connection is spiced onto the cabe (see Figure 2a), a situation that is very common both in indoor and outdoor wiring. The end of the branch can be either open or terminated. A bridged tap can be viewed as a three-port section, but one of the ports appears as a oad impedance to the ine between the two sections on each side of the bridged tap. The branching connection can sti be modeed as a 2PN with a given ABCD matrix as shown in Figure 2b. Such a situation can sti be modeed as a cascade of 2PNs since bridged tap can be modeed as a shunt impedance across the ine with the impedance equa to the input impedance of the bridged tap (see Figure 2c). In this case, the ABCD matrix taking into account the effects of the branching connection is: 1 0 T bt =, (10) 1/ inbt 1 where the input impedance of the bridged tap inbt can be cacuated as in (3). If the branching connection is unterminated (i.e., infinite oad), the input impedance in (3) bois down to A + B A im in ( f ) = im =, C + D C so that we have the foowing particuar case: 1 0 T bt = C. 1 A The same approach can be used to mode simpe shunt impedances aong the ine. This occurs in correspondence to power pugs where a working appiance may be present. In this case, inbt is repaced by the input impedance of the appiance. Simiary, it is straightforward to compute the ABCD matrix of series impedance se aong the ine: 1 se T bt = The Proof of Symmetry Matrix mutipication does not usuay satisfy the commutative property, therefore T f in (8) is different from T b in (9). The mutipication of two matrices may be commutative if and ony if the two matrices share a common eigenvectors. However, this is not the case because the eigenvectors of a matrix of the kind in (4) are given by: u1 = [ α o α] and u2 = [ αo α ]. These eigenvectors depend on the characteristic impedance of the cabe and, therefore, the transmission matrices pertaining to two different kinds of cabes (i.e. with two different characteristic impedances) having different eigenvectors. Athough it may seem counterintuitive, this means that even a simpe connection constituted by two different cabes spiced together has different overa forward and backward matrices. Interestingy, this property has never been pointed out in the iterature. Athough the overa forward and backward transmission matrices of a power ine network are different, the reciprocity property 3) in (7) for T f and T b is sti maintained. In fact, we have: det (1) (2) ( N ) (1) ( N ) ( T f ) = det( T f f... f ) = det( T f )... det( T f ) = 1, ( N ) ( N 1) (1) ( N ) (1) ( ) = det( T... ) = det( T )... det( T ) = 1 det T b f f f f f. Therefore, the cascade of reciproca 2PNs is sti a reciproca 2PN. However, the fact that the whoe network may be modeed as a reciproca 2PN does not mean that the power ine channe is a symmetric channe (i.e., has the same frequency response in both directions) as it is sometimes stated. In fact, the Reciprocity Theorem hods for transfer functions with the dimension of an impedance or an admittance and does not hod for dimensioness transfer functions, such as the votage gain (4) (see [2, Sect. 1.8] for an eegant proof of this statement). Moreover, the symmetry property is satisfied if and ony if both properties 1) and 3) in (7) are satisfied. In the case of a network made of severa sections, property 1) in (7) is not satisfied anymore by the overa transmission matrix. In fact, we have: a b x y ax + bz ay + bx =, c a z x az + cx ax + cy confirming that the product of two matrices of the kind of (6) is no onger of the kind of (6).

4 On the basis of the previous considerations, the forward and backward transmission matrices of a power ine ink are indeed different. However, it can be proved that the forward and backward transfer functions of a power ine ink are the same provided that source and oad impedances are equa. As previousy mentioned, this cannot be proved by means of the Reciprocity Theorem but has to be proved in some other way. et us consider the genera expressions of T f and T b given in (8) and (9): A1 B1 (1)... ( N ) A2 B2 ( N ) (1) T f = = T f f, Tb = = T f... C1 D1 C2 D2 f. (11) It is easy to prove that the forward and backward chain matrices of a ink coincide with the forward matrices of the origina ink and the one that exhibits a symmetric topoogy, respectivey. The matrices in (11) are the chain matrices of the whoe connection and they satisfy ony property 3) in (7), i.e. they have unitary determinant; on (i) the other hand, matrices T f satisfy a the properties in (7), for any i. On the basis of matrix mutipication properties and of induction, it is possibe to show that the foowing reationships aways hod: A 1 A2 = ( D1 D2 ), B1 B2 = C1 C2 = 0. (12) Now, the forward and backward transfer functions H f (f) and H b (f) of the whoe ink can be cacuated on the basis of the chain matrices (11): H f ( f ) = A1 + B1 + C1 S + D1 ; S H b ( f ) = A2 + B2 + C2S + D2. S The difference between the reciprocas of the previous quantities is: 1 1 ( A1 A2 ) + ( B1 B2 ) + ( C1 C2 ) S ( D1 D2 ) S H = =. H f ( f ) Hb ( f ) Expoiting the reationships in (12), we finay obtain: ( A1 A2 )( S ) H =. Now, if = S, we can write: 1 1 H = = 0 H f ( f ) = Hb ( f ). H f ( f ) Hb ( f ) Since A 1 and A 2 are aways non-zero quantities and are aways different for different cabe configurations, we can state that if and ony if and S are equa, the forward and backward transfer functions are the same and, consequenty, the channe is symmetric. 4. Experimenta confirmation and Concusions An experimenta confirmation of the theoretica resuts discussed here is aso presented. The topoogy of the power ine ink considered is the one shown in Figure 3 of [3]. Experiments were conducted with various vaues chosen for R s, R 1, R 2, R 3 and R Y. Specific refections and resonant modes coud be isoated by seectivey disconnecting the bonding shunt, 15ft, 25ft and 60ft branches, which resuts in 12 distinct topoogy variations. Transmission experiments were conducted between nodes (X) and (Y) in this mode for the case of R 1 =0, R S =2 Ω, and R 2 =R 3 =R Y =. The transfer function between X and Y (H XY ) and between Y and X (H YX ) are shown here in Figure 3. The response is neary identica in both directions with matched sending and receiving impedance of 140Ω. Severa resonant modes are apparent in Figure 3. Using seective path isoation, the ground bonding shunt was found to create dips at 3.3, 9.9, 16.9 and 23.3 MHz; R 2 = creates dips at 7.0 and 21 MHz, the 15ft branch creates a 11.4 MHz dip, and the mains feed creates dips at 4.8, 9.8, 14.9, 19.7 and 25.0 MHz. SPICE simuations based on the equivaent circuit in Figure 5 of [3] are consistent with the measured resuts for a 12 topoogy variations investigated. In the present paper, we addressed the probem of determining the exact conditions under which a power ine ink may be considered a symmetric channe. In fact, to the best of the authors knowedge, no expicit proof has ever been given to prove this symmetry. In particuar, it was shown that the symmetry property hods regardess of the topoogy of the ink and that the ony condition for the symmetry requires that the source and oad impedances used to terminate the ine be the same. An experimenta confirmation of this property has aso been shown here. As far as the vaidity of the ABCD modeing of a power cabe is concerned, we refer to the companion paper [3] where the vaidity of the ABCD modeing is theoreticay and experimentay justified.

5 5. References [1] Transmission Systems for Communications, Be aboratories, Fifth edition, [2]. Weinberg, Network Anaysis and Synthesis, R. E. Krieger Pubishing Company, New York, [3] T.C. Banwe, S. Gai, A New Approach to the Modeing of the Transfer Function of the Power ine Channe, IEEE Internationa Symposium on Power ine Communications and its Appications, ISPC 01, Mamo, Sweden, Apri 4-6, S I 1 I 2 Two-Port Network V S V 1 A B V 2 C D Fig. 1 - A generic two-port network. (a) (b) [T] inbt [T i] [T i+2] (c) [T i+1 ]=[T bt ] [T i ] inbt [T i+2 ] Fig. 2 (a) Bridged tap: branching connection spiced onto a ine; (b) equivaent circuit of (a) in terms of 2PNs; (c) fina circuit modeing the bridged tap as a cascaded 2PN.

6 0 TRANSMISSION GAIN ( db ) H XY H YX -30 FREQUENCY ( MHz) Figure 3 Pot of the transfer function from X to Y (H XY ) and from Y to X (H YX ) for the topoogy in Figure 3 of [3].