Multi-Port Impedance Matching Technique for

2011 IEEE International Symposium on Power Line Communications and Its Applications

Multi-Port Impedance Matching Technique for Power Line Communications Rodolfo Araneo, Salvatore Celozzi, Giampiero Lovat, and Francescaromana Maradei Department of Astronautical, Electrical, and Energetic Engineering, “Sapienza” University of Rome, Italy e-mail: rodolfo.araneo, salvatore.celozzi, giampiero.lovat, [email protected]

Abstract—A computer-aided methodology for designing multiport broadband impedance matching circuits (BIM) to be connected at powerline communication (PLC) networks is presented in order to provide gain equalization and mitigation of the effects of low-impedance loads among transmitter and receivers in a wide frequency range. The design is achieved in successive steps by means of the Vector Fitting method, a rational parametric approximation of the driving impedances, and a nonlinear optimization through a novel Meta Particle Swarm Optimization (MPSO). Special attention is given to the application of such devices on board of yachts, where the use of PLCs could dramatically reduce weights and costs. Preliminary results confirm the efficiency of the proposed approach.

I. I NTRODUCTION Power line communications (PLCs) have received a renewed attention in the last years [1]. Originally used only for protection and telemetering purposes, PLCs have become a viable technology for telecommunications, both at the “last mile” and “last inch” access [2], due to the recent unparalleled growth of Internet, combined with technological advancements. While for the last mile access PLCs could provide another medium alternative to existing technologies (e.g. modems, Digital Subscriber Lines, wireless), as concern the last inch access PLCs are potentially preferred to other available technologies (e.g. cable, wireless and phone-line networking) since power lines are the most pervasive network in a home environment where they could provide an economical physical layer for communication. To support broadband multimedia applications (rate of MB/sec), high speed PLCs are developed working in the frequency range 1.6-30 MHz, i.e., beyond the frequency ranges allowed by available standards (3-148.5 kHz by EN50065-1 and 3-525 kHz by IEC 61000-3-8). The transmission medium that power lines can offer at these frequencies and bands is hostile and signal transmission with high-quality is challenging, since low-voltage power-line networks are primarily designed for an effective energy transportation. Besides several drawbacks (e.g., noise influence, attenuation, echoes, and multi-path reflections) the main enemies are the low-level, the frequency and time-dependence, and the unpredictability of the power-line access impedance [3]–[6]. Moreover, the possible plugging in or switching off of devices connected to the network could dramatically change the network topology and consequently the impedance characteristics. Therefore, the access impedance of the power line fluctuates, resulting in a mismatch with the transmitting/receiving modem impedance

978-1-4244-7750-0/11/$26.00 ©2011 IEEE

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which is always cause of low power gain, transmissiondistance reduction and low reliability. Moreover, several national and international standards limit the power injected by digital signals into power lines by imposing an upper bound to the electromagnetic emission from power lines because of either electromagnetic compatibility (EMC) issues among electronic devices and human exposure. In this framework, it is evident that an efficient solution of the matching problem is of paramount importance. The present work proposes a reliable methodology to design at the same time multiple optimal broadband-impedancematching (BIM) circuits capable of maximizing the power transfer between the transmitting (TX) and receiving (RX) modems connected at the ports of a power line network, while maintaining as low as possible the out-of-band harmonics content (possibly enhanced by nonlinear devices). The matching problems at the ports are clearly coupled, since the insertion of a BIM circuit at the i-th port unavoidably changes the input impedance at the remaining ports. The proposed methodology is particulary suited for closed systems where precise topology and channel’s characteristics are know, while it is a step toward the possible automatic impedance adaption in time-varying systems where switching loads change the requirements and call for a continuous impedance matching. In the past, general-purpose analytical [7] and numerical approaches [8]–[13] have been developed for the design of lossless gain equalizer, in order to solve the so-called singleor double- matching problem, depending on whether the impedances of the generator and the load are purely real or complex. A first analytical theory was developed in [7] which, however, results in explicit formulas very difficult to be used in the case of high-order BIM circuits. Numerical methods have been developed so far based on real frequency techniques [8]– [10], rational parametric approximations [11], [12], and direct stochastic approximations [13]. Variations of these methods have been recently used for the design of couplers for PLCs [3]–[5], [14], [15]. Following our previous studies in [14], [15], where only the matching problem at a single port was addressed and whose theoretical procedure is here extended to multiple ports, the proposed design procedure is organized in successive steps through the application of the Vector Fitting method, rational parametric approximation of the driving impedance, and nonlinear optimization by means of a Meta Particle Swarm Optimization (MPSO). The proposed technique is applied to

design BIM circuits for a network of a yacht. In fact, yachts are characterized by the presence of highly sophisticated and dense electric and electronic systems so that PLCs become appealing to make these components communicate among them while dramatically reducing the weight of yachts (existing bus networks would be not necessary anymore). II. T HEORETICAL F ORMULATION A. Multiports matching problem In power lines, the line access impedance may vary from few ohms up to 200 ohms [5], depending on the topology of the power line and the value of all the connected loads. In order to improve the power gain between modem and transmission channel and to reduce the partial reflections of the signal caused by changes in the access impedance, coupling units are usually placed between both the transmitting (TX) and receiving (RX) broadband power-line (BPL) modems and the ports of the power-line network, as shown in Fig. 1(a). The coupling units consist of a BIM circuit and a blocking circuit constituted by a radio-frequency (RF) transformer with turn ratio 1 : n (required to provide galvanic isolation for the modem circuitry and to equalize the value of the general powerline access impedance) and a resonant LC-filter (required to block the main voltage at 50/60 Hz present in the power-line network), as shown in Fig. 1(b). By denoting the equalizer impedance (looking toward the modem) and the load impedance (looking toward the power network) as seen from the output port of each BIM circuit as Ze,i (ω) = Re,i (ω) + jXe,i (ω) and Zl,i (ω) = Rl,i (ω) + jXl,i (ω), respectively, the design of each BIM circuit can be achieved by maximizing the power transfer gain Ti given by ∗ Ze,i (ω) − Zl,i (ω) 2 = 4Rl,i (ω) Re,i (ω) , Ti = 1 − 2 Ze,i (ω) + Zl,i (ω) |Ze,i (ω) + Zl,i (ω)| (1) over the prescribed frequency range, where the superscript ∗ denotes complex conjugate. The problem at hand for the single

BIM circuit is the rather classical “broadband single (double) matching problem” [8], which consists in synthesizing a lossless two-port network matching the given generator resistance (or complex impedance) with the arbitrary frequencydependent load impedance, in order to maximize the powertransfer gain. In the considered case, the matching problem is rather more complex since two BIM circuits have to be designed at the same time in order to maximize the overall power gain between the TX and RX modems; the two optimization processes are clearly coupled since the load impedance seen from the output port of one BIM circuit depends on the other BIM circuit (see Fig. 1(a). The solution procedure, as explained in the following, will pass through an optimization process during which all the couplers will be designed at the same time. In particular, at each step, the access impedances of the network at the N ports Znet,i (ω) = Rnet,i (ω) + jXnet,i (ω) (with i = 1, 2, . . . , N ) and the load impedances Zl,i are computed; this is performed by assuming that, at each i-port, the other (N − 1) ports of the power-line network are closed on the impedances Zq,i (ω) = Rq,i (ω) + jXq,i (ω) due to the (N − 1) couplers obtained so far. Hence, the parameters of the couplers can be updated and the optimization process can further proceed.

B. Impedance parametric approximations As also reported in [14] and [15], the first step of the synthesis procedure consists in expressing the Th´evenin impedances of the equalizers Ze,i seen from the output port of the i-th BIM circuit by means of a partial fraction expansion: Ncp,i

Ze,i (s) = r0,i +

X

∗ rk,i rk,i + s − pk,i s − p∗k,i

k=1 Ntot,i

+

X

Znet,1(ω)

C Znet,2(ω)

(a)

Zg,i Eg,i

Coupling unit BIM circuit gain equalizer

Zq,i(ω)

RF filter 1:n

T(ω)

Ze,i(ω) Zl,i(ω)

L

Powerline network

BPL modem

C

(2)

rk,i , s − pk,i

where the subscript i denotes the TX (i = 1) or the RX (i = 2) port, s is the conventional complex-frequency variable in the Laplace domain, Ncp,i is the number of complex-conjugate pole pairs, Nrp,i = Ntot,i − Ncp,i is the number of purely real poles, while rk,i and pk,i are the k-th residue and pole, respectively. The equalizer impedances Ze,i are positive real functions (that is, all the poles are stable < (pk,i ) ≤ 0), the real parts of the impedances are positive semidefinite on the imaginary axis of the complex plane (i.e. < [Ze,i (jω)] ≥ 0), and the residues of the poles located on the imaginary axis (which must be simple poles) are real and positive semidefinite [16], [17]. The driving impedances Ze,i (s) are considered as minimum reactance functions (i.e., zeros are placed on the right halfplane) having only simple poles [8], [11], [12]. Since the matching circuits are assumed to be lossless, the residues rk,i

m RX od em

ou un plin it g

P o ne we tw rli or ne k

ou un plin it g

C

m TX od em

k=Ncp,i +1

!

Znet,i(ω)

(b) Fig. 1. Coupling units connected between the TX-RX modems and the access ports of the power-line network (a) and topological scheme of a coupling unit (b). .

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of poles Ni and a first guess for them, together with a first guess for the residues in (3).

can be computed as rk,i = −

Gi (pk,i ) Gi (−pk,i ) , N Qi  2 pk,i a2i pm,i − p2k,i

(3) B. Optimization process

m=1 m6=k

where ai is a positive real coefficient. The monic polynomial Gi (s) of degree Mi ≤ Ni = (Nrp,i + 2Ncp,i ) (which includes all the zeros of the even part of Ze,i (s)) must be properly chosen in order to synthesize the lossless equalizer as an LC ladder structure [12]. In the general case, zeros are located on the real frequency axis so that Gi (s) assumes the following form: M2,i

Gi (s) = s

M1,i

Y

i=1


s2 − zi2 ,

(4)

where M1 zeros are introduced at dc and M2 zeros are located at some finite frequencies (with M = M1 + M2 ). In order to try to synthetize simple LC ladder structures, zeros can be introduced only at dc or at infinity: in the former case Gi (s) = sM1 , which results in a band-pass behavior of the BIM equalizer, whereas in the latter case Gi (s) = 1, with a resulting low-pass behavior of the equalizer. Anyway, in the case of complex trends of the load impedances Zl,i (s), it is necessary to introduce zeros at finite frequencies, renouncing to synthesize simple LC ladder structures. The optimization procedure of all the Ze,i (s) (with i = 1, 2 . . . N ) consists in the following successive steps: 1) Initialization: the order Ni of the rational approximation of the driving impedance Ze,i (s) and the order Mi of the polynomial Gi (s) are chosen for all the N ports; then the Ni poles pk,i and the Mi zeros zm,i are initialized together with the positive real normalization coefficient ai ; 2) The following cycle is repeated over the N -ports until convergence is achieved: a) the load impedance Zl,i is computed assuming the other couplers fixed; b) the driving impedance is computed through (2) and (3); c) the transfer gain is computed through (1); d) the poles pk,i , the zeros zk,i , and the coefficients ai are updated. Finally, these steps are iterated in an evolutionary stochastic algorithm based on the particle swarm optimization (PSO) to find a near-optimal solution, as described in the following Section. III. N UMERICAL S OLUTION A. Initialization procedure At first, the net impedances Znet,i (s) are computed assuming that the ports, to which the modems have to be connected, are closed on 50Ω-loads. In order to initialize the poles pk,i , the Vector Fitting (VF) approach [15], [18]–[21] has been applied on the load impedances Zq,i (s) to obtain the number

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During the last years, the PSO [22] has become increasingly popular as an efficient optimization method for solving singleobjective [23] and multi-objective optimization problems [24]; it shares many features with Genetic Algorithms (GAs) previously adopted by the authors [18]–[20], [27], but presents several advantages. In the PSO algorithm, the social behavior is modeled to guide a population of particles towards the most promising region of the search space. By assuming an N -dimensional problem characterized by a cost function F and a swarm composed  m m m T of M particles, the position xm t = x1,t , x2,t , . . . , xN,t (with m = 1, 2, . . . , M ) of each particle at the time t in the space domain D represents a potential solution. Each particle moves inside the space domain with a velocity vector  m m m T vtm = v1,t , v2,t , . . . vN,t , searching for food, i.e., the best solution which gives rise to the smallest value of the cost function. The core idea at the base of the PSO algorithm is the exchange of information among the particles of the swarm population: in the algorithm implementation, the m-th particle not only knows its personal best position bm L (local optimum solution), that is the best position this particle has visited so far that yields the highest fitness value, but also the global best position bm G (global optimum solution), that is the best position ever attained by the particle swarm that gives the best fitness value in the entire population. The exchange of these information allows for determining the position and the velocity of the m-th particle at the next iteration through the following two updating equations: m m m vt+1 = wvtm + c1 ϕ1 (bm L − xt ) + c2 ϕ2 (bG − xt ) (5a) m n xm (5b) t+1 = xt + vt+1 .

The entire optimization relies on the correct manipulation of the particles’ velocities: w is the inertia factor, which keeps the particle in its current trajectory; the last two terms inject deviations according to the distances from the personal bm L and global bG best locations, through the cognitive factor c1 and the social factor c2 , respectively (also called acceleration coefficients). The two numbers ϕ1 and ϕ2 are two random variables distributed in the range [0,1], which inject the unpredictability of the particles’ movement. The convergence of the algorithm depends on the proper tuning of the acceleration coefficients [25] and on the boundary conditions used to prevent the explosion of the particles [26]. In the present work, a suitable variation of the original PSO (referred to as Meta Particle Swarm Optimization, MPSO [28], [29]) is adopted. It simply consists in using more than a single swarm moving through the space domain D. By denoting the position vector of the m-th particle of the j-th swarm at the time t as xm,j , and its velocity vector as vtm,j , the updating t

L2(2×10m) L2(6×6m)

Load5

L2(8m)

RX2

L3(2×60m)

L1(31m)

L1(30m) L1(3×40m)

L3(15m)

L7(60m)

L2(3×15m) L2(8×10m)

Load3

L2(10m)

RX3

L1(40m)

L2(10×15m)

L2(6m)

L2(11×10m)

L2(20m)

L4(40m)

L2(2×10m) L2(18m)

L3(2×10m) L2(20m)

Load1

L1(40m)

L2(15m)

RX1

  = + c1 ϕ1 bm,j − xm,j t L     + c3 ϕ3 bjS − xm,j + c2 ϕ2 bG − xm,j , t t

(6)

k=1

where L is the number of sampling frequencies ωk distributed over the frequency band of interest, T0 is the desired flat gain to be approximated (ideally 1, but practically set in the range 0.7 − 0.8 in the following), Ti (ωk ) is the computed gain at the i-th port, and N is the number of ports to which the BPL modems have to be connected.

TABLE I F REQUENCY- DEPENDENT LINE PARAMETERS (R : Ω/m, L : µH/m, C : pF/m)

L1

3 × 16 mm2

L2

3 × 1.5mm2

L3

3 × 2.5 mm2

L4

3 × 35 mm2

L5

3 × 140 mm2

L6

3 × 120 mm2

L7

3 × 95 mm2

Load7

chosen according to [24], [25] and the reflecting boundary conditions in [26] have been used to relocate the particles that fly outside the allowed solution space. The cost function F that needs to be minimized is defined as "N #2 L 1 X Y Ti (ωk ) − T0 , (7) F= 2L i=1

where bjS is the global best position even attained by the j-th swarm and c3 is a social factor local to the j-th swarm. In the considered problem, each particle xm,j consists of t the possible values of the lumped components of the BIM circuits. The inertia and acceleration coefficients have been

R L C R L C) R L C R L C R L C R L C R L C

Load8

Simplified one-line diagram of the yacht network.

wvtm,j

Cable

L3(3×6m)

L5(6m)

rule (5a) can be written as

Line

L3(6m)

TX Fig. 2.

m,j vt+1

L2(3×15m)

L1(53m)

Load2

L6(8m)

Switch board

L2(30m) L2(5×15m)

Load9

L3(5×20m)

Service panel

Load4

Switch board

L3(2×20m)

Switch board

Load6

Switch board

L2(2×6m) L2(4×15m)

1 MHz 0.06413 33.389 0.14188 67.285 0.11661 61.213 0.056107 36.527 0.025321 32.296 0.029611 30.582 0.033779 30.714

5 MHz 10 MHz 0.15862 0.19833 32.987 32.936 99.079 0.36544 0.49605 66.194 65.978 50.365 0.33062 0.43555 60.356 60.204 54.766 0.12281 0.19556 36.326 36.345 89.328 0.045191 0.077261 32.286 32.303 100.76 0.053539 0.092191 30.551 30.562 106.43 0.064531 0.10734 30.677 30.688 106.3

C. Synthesis

15 MHz 0.20987 32.914

Finally, the synthesis of Zq (s) as a lossless LC ladder network with a resistive termination can be performed by means of well-established methods [16], [17].

0.55379 65.886

IV. R ESULTS

0.47481 60.153

The methodology has been applied to synthesize coupling units for the TX and RX modems to be used in the network of a real yacht. In particular, the simplified one-line diagram of the network is reported in Fig. 2. The measured frequencydependent per-unit-length (p.u.l.) parameters [30] of the seven types of lines that appear in Fig. 2 are reported in Table I only at the frequencies of 1, 5, 10 and 15 MHz, for the sake of coinciseness. In fact, the frequency window where the power-transfer gain between modem and network needs to be maximized lies between 1 and 15 MHz, because this

0.22471 36.372 0.11962 32.315 0.14118 30.578 0.15674 30.707

99

TABLE II L OAD IMPEDANCE VALUES

Resistance (Ω) 75 22.89 26.56 75 14.3397 75 32.0395 0.95255 0.64

Magnitude of input impedance Znet [Ω]

1 2 3 4 5 6 7 8 9

Inductance (H) 0 0.26 0 0 0.1629 0 0.0755 0.0296 0.0015

is the frequency range where the network presents the best environment for an high-speed wide-band communication. Finally, the measured values of the load impedances are reported in Table II. In Fig. 2 one TX and three RX modems are considered since our goal was to investigate the performance of the proposed procedure to optimize the overall power gain among several pairs of ports. However, in the following, results between the transmitting modem TX and the receiving modem RX3 are reported. In Fig. 3(a), the magnitude of the initial input impedances Znet,1 (ω) and Znet,3 (ω) at the TX and RX3 ports are reported as functions of frequency; in Fig. 3(b) the relevant phases are also shown. In particular, the initial input impedances have been computed by means of an equivalent model of the network based on the connection of transmission-line lengths. Moreover, as explained before, the initial impedances have been computed assuming that the other ports where a modem has to be connected are closed on 50 Ω loads. The initial approximations, obtained applying the vector fitting procedure with N = 25 poles, are also reported in Fig. 3. As can be observed, in the high-frequency range the input impedances to be matched show a quite rapidly oscillating behavior and, in particular, the amplitude can vary also by one order of magnitude. The proposed design methodology based on the parametric approximation and the MPSO described in the previous Section is then applied to obtain the parametric approximation of the two BIMs to be connected at the ports TX and RX3 . The MPSO requires 1734 iterations to converge to a near-optimum solution. Then the two BIMs are synthesized as lossless 16th order T-ladders. The frequency behavior of the optimized final power gain T (ω) at the TX port is reported in Fig. 4. From this plot, the improvement brought by the BIM circuits in the frequency band of interest is clearly visible. In fact, it can be observed that the initial power gain between the two modems is low, below 0.5 in almost the whole range of interest 1 − 15 MHz. Despite the oscillating behavior of the net impedances, the insertion of only one BIM circuit at the TX port (optimized assuming the RX3 port closed on a 50 Ω load) improves the power gain (around 7 dB gain) that essentially remains between 0.8 and 0.9 over the whole considered frequency range, presenting only few bumps at isolated frequencies.

Computed Approximated

Znet,3 Znet,1

100

10

10k

100k

1M Frequency [Hz]

10M

(a) 1.5 Znet,1 Phase of Input impedance Znet [deg]

Load Load Load Load Load Load Load Load Load Load

1000

1 0.5 Znet,3 0 -0.5 -1

Computed Approximated

-1.5 10k

100k

1M Frequency [Hz]

10M

(b) Fig. 3. Input impedance of the yacht network in Fig. 2 computed at the TX and RX3 ports: magnitude (a) and phase (b).

Finally, the insertion of two BIM circuits (optimized together by means of the proposed procedure) further improves the power-transfer gain, especially for frequencies above 10 MHz, in addition to smooth some dips (around 5 and 7 MHz). This is a result of practical interest that confirms the validity of the proposed approach and hopefully opens the way to a suitable use of PLCs in telecommunication systems in vehicular and naval environments. V. C ONCLUSION In recent years, an increasing interest has raised around PLCs due to the fact that they can provide an economical medium for modern broadband communications. The naval industry also looks with great attention at the potentialities of PLCs since they could help to reduce the weight of yachts, improve their performance, and reduce construnction costs. However, power lines present an extremely harsh environment for the transmission of high-speed wideband signals. In particular, the frequency-dependent and time-varying character of the channel impedance, which is affected by the loads possibly

100

With both TX-RX BIMs

0.9

Power gain TTX

0.8

0.7 With single TX BIM 0.6 Without BIM

0.5 0.4

1M

10M Frequency [Hz]

Fig. 4. Power gain T between the transmitting modem and the yacht network as a function of frequency.

connected to the network, is one of the major problems. Basically, it leads to a mismatch between the modem and the network load impedance, which is the cause of a low reliability of the whole system. The paper has presented a design optimization procedure for synthesizing impedance matching circuits for the equalization of the power-transfer gain in PLC structures over a wide frequency band. The methodology is based on a parametric representation of the driving impedances, which are optimized by means of a novel Meta Particle Swarm Optimization method. The presented results, obtained for networks already existing in yachts, show that the proposed methodology is efficient and robust, leading to coupling units capable of maximizing the level of the transit signals in the prescribed frequency range. ACKNOWLEDGMENT This work has been partially supported by the Italian Ministry of University (MIUR) under a Program for the Development of Research of National Interest (2007 PRIN grant ] 20072347AY-004). The author would like to thank T. Zheng, M. Raugi and M. Tucci for having provided the measured data reported in Tables I and II. R EFERENCES [1] A. Majumder and J. Caffery Jr, “Power lines communications: an overview,” IEEE Potentials, vol. 23, pp. 4–8, Oct.-Nov. 2004. [2] N. Pavlidou, A. J. Han Vinck, and J. Yazdani, “Power line communications: State of the art and future trends,” IEEE Communications Magazine, pp. 34–40, April 2003. [3] S. N. Yang, H.Y. Li, M. Goldberg, X. Carcelle, F. Onado, and S. M. Rowland, “Broadband impedance matching circuit design using numerical optimization techniques and field measurements,” 2007 Proc. IEEE ISPLC Symp., 26-28 March 2007, Pisa, Italy, pp. 425–430. [4] W. H. Choi and C. Y. Park, “A simple line coupler with adaptive impedance matching for power line communication,” 2007 Proc. IEEE ISPLC Symp., 26-28 March 2007, Pisa, Italy, pp. 187–191. [5] C. Y. Park, K. H. Jung, and W. H. Choi, “Coupling circuitry for impedance adaptation in power line communications using VCGIC,” 2008 Proc. IEEE ISPLC Symp., 2-4 April 2008, Jeju Island, Korea, pp. 293–298.

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