Transformation of passive electrical networks with distributed … · 2020. 10. 15. · Every PWL-function describing a waveform of a source in the network is decomposed into PWL-basis-functions

Adv. Radio Sci., 7, 113–118, 2009www.adv-radio-sci.net/7/113/2009/© Author(s) 2009. This work is distributed underthe Creative Commons Attribution 3.0 License.

Advances inRadio Science

Transformation of passive electrical networks with distributedPWL-sources using a Model Order Reduction

S. Ludwig1,2 and W. Mathis1

1Leibniz University Hannover, Institute of Electromagnetic Theory, Hannover, Germany2Fraunhofer Research Institution for Electronic Nano Systems, Advanced System Engineering, Paderborn, Germany

Abstract. In modeling of distributed systems with dis-tributed sources large networks with RLC-elements and in-dependent sources arise. This high complexity leads to ahigh effort in simulations. Therefore model reduction canbe used to reduce these networks, preserving the behaviorat the observed nodes in the networks. For the reduction ofnetworks with a large number of independent sources onlya weak reduction is enabled with standard model reductiontechniques. In this paper an efficient reduction of networkswith a large number of sources with piece-wise-linear wave-forms is presented, using the decomposition of piece-wise-linear functions. With the proposed method a higher reduc-tion of the network and/or a higher accuracy can be achievedwith model reduction. The validity and efficiency of the pro-posed method is shown by reducing a RCI-Grid model.

1 Introduction

Since the early beginnings of circuit theory similarity andequivalence transformation of electrical networks are ofstrong interest. In the modeling of distributed systems, suchas electromagnetic or thermal fields, with distributed sourcesvery large networks are created. For example in electromag-netic fields in ICs the switching currents act like distributedsources or in thermal simulations of substrates the transis-tors act like distributed heat sources. For these very largenetworks a smaller network with approximated port behav-ior is required to allow for fast simulations. With the helpof model order reduction and network synthesis it is possibleto find such a reduced network (Antoulas, 2005; Ionutiu etal., 2007; Ludwig et al., 2008a; Radic et al., 2008). Replac-

Correspondence to:S. Ludwig([email protected])

ing the original network with the reduced network enablesinvestigations with a lower effort for simulations.

In our former work we have developed an efficient reduc-tion method for networks with groups of distributed sourceswhich have equal or proportional amplitudes (Ludwig et al.,2008a,b,c). In this paper we extend this approach to sourceswith piece-wise-linear (PWL) waveforms (PWL-sources),which do not necessarily have equal or proportional ampli-tudes.

In this work the design flow for the approximative trans-formation of large passive electrical networks, which espe-cially contain a high number of PWL-sources, is presentedin Sect. 2. Firstly, the network is transformed with anequivalence-transformation in Sect.2.1 to reduce the num-ber of independent sources and to enhance the reduction pro-cess. Secondly, the network is reduced with an approxima-tive transformation using model order reduction algorithmsin Sect.2.2. The efficiency of the complete reduction pro-cess is shown by reducing an example network in Sect.3 andthe paper is concluded in Sect.4.

2 Transformation of networks with distributed sources

For the transformation of electrical networks the invariant aswell as the approximative properties of the network have tobe defined. Invariant properties are for example stability andpassivity. Properties that can be approximated are the be-havior at the nodes for the connection with other networksand electrical values that should be observed or measured.Also the behavior at the nodes where independent sourcesare connected can be approximated. Other properties, suchas the number of elements or nodes or the behavior at nodeswhich are not of interest can be disregarded.

Distributed systems with distributed sources are mod-eled with passive RLC-elements a high number of inde-pendent sources. In state-of-the-art model reduction these

Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.

brought to you by COREView metadata, citation and similar papers at core.ac.uk

provided by Institutionelles Repositorium der Leibniz Universität Hannover

https://core.ac.uk/display/52920468?utm_source=pdf&utm_medium=banner&utm_campaign=pdf-decoration-v1

http://creativecommons.org/licenses/by/3.0/

114 S. Ludwig and W. Mathis: Transformation of passive electrical networks with PWL-sources using a MOR

2 S. Ludwig and W. Mathis: Transformation of Passive Electrical Networks with PWL-Sources using a MOR

and connected through ports and only the RLC-part is re-duced (left path in Fig. 1). The advantage of this extraction

Fig. 1. Transformation flow for networks with distributed sources

is that only passive RLC-elements are used in the reduciblemodel and the passivity of the whole model can be preservedduring a model order reduction (Odabasioglu, 1998). Thisleads to a passive reduced model and prevents unstable time-domain simulations. For networks with a large number ofindependent sources, this extraction produces a large num-ber of ports. A large number of ports is a strong limitationfor the model reduction (Silva, 2007b). In many distributedsystems the waveforms of the sources are not completely in-dependent of each other. For equal or proportional wave-forms of the sources, which are widely used in modeling,we have developed a method to describe these sources effi-ciently for model reduction (Ludwig, 2008a,b,c). The ideais to reduce the number of independent sources in order toenable stronger reduction of the reducible part (dotted linesin Fig. 1) of the network. For these networks we propose anequivalence transformation (right path in Fig. 1) instead ofthe extraction of the sources. The basis-waveforms are givenby one representative of each group and are realized withadditional independent sources. The waveforms are copiedinto the network by replacing the distributed sources withcontrolled sources with a specific gain. As only the inde-pendent sources, representing a group of sources, have to beextracted and connected through ports, a network with a re-duced number of ports is generated. The resulting networkcan be more efficiently reduced than the network with ex-tracted sources (Ludwig, 2008a). The network generated in

this way is generally not passive. The connection of non-passive systems can lead to unstable time-domain simula-tions and therefore a passive system is preferable. Controlledsources connected with the additional independent sourcesare added. These sources act like dissipative elements thatdissipate exactly the amount of energy generated by the con-trolled sources in the network and vice a versa. Thereby theequivalent network is passive (Ludwig, 2008a,c) and passiv-ity can be preserved in the reduced model with the descrip-tion proposed in (Ludwig, 2008c).

In models that are generated from measurements of dis-tributed systems the sources in the model have PWL-waveforms, which are not necessarily equal or propor-tional. In the next section the described approach for anequivalence-transformation of the network is extended tonetworks with PWL-sources.

2.1 Equivalence Transformation for the Reduction of theNumber of PWL-Sources

In this section we propose a method to describe PWL-sourcesin a network for an efficient reduction. We consider net-works with sources, where the waveforms of the sources aredescribed with continuous PWL-functions fq(t) on a closedinterval [0, T ] for which holds

fq(t) = mit + oi for all ai−1 ≤ t ≤ ai (1)

for pairs of real numbers (mi, oi), i = 1, ..., k with

0 < a0 < a1 < ... < ak < T (2)

according to the definition in (Aliprantis, 2005).Every PWL-function describing a waveform of a source

in the network is decomposed into PWL-basis-functions.The union of all basis-functions gives the complete basis-functions space of the size b. The waveforms of all sourcesare now the linear combination of all b basis-functions mul-tiplied with a corresponding weighting factor wi.

fq(t) =b∑

i=1

wq,ifi(t). (3)

The basis-functions can be found for example with themethod proposed in (Aliprantis, 2005).

After their decomposition the PWL-functions have to berealized in the network. In the network every basis-functionis realized with an independent PWL-source. The distributedsources in the network are replaced with controlled sources,with the controlling values and gains corresponding to thelinear combination of the basis-functions as in Eqn. 3. Withthis method a network with equivalent behavior at the pinsof the network is generated, hence this modification is anequivalence-transformation as shown in the right path inFig. 1

If the number of basis-functions is lower than the numberof distributed PWL-sources in the original network, with this

Fig. 1. Transformation flow for networks with distributed sources.

independent sources are extracted from the RLC-part of thenetwork and connected through ports and only the RLC-partis reduced (left path in Fig.1). The advantage of this ex-traction is that only passive RLC-elements are used in thereducible model and the passivity of the whole model can bepreserved during a model order reduction (Odabasioglu et al.,1998). This leads to a passive reduced model and preventsunstable time-domain simulations. For networks with a largenumber of independent sources, this extraction produces alarge number of ports. A large number of ports is a stronglimitation for the model reduction (Silva et al., 2007b; Feld-mann, 2004). In many distributed systems the waveformsof the sources are not completely independent of each other.For equal or proportional waveforms of the sources, whichare widely used in modeling, we have developed a method todescribe these sources efficiently for model reduction (Lud-wig et al., 2008a,b,c). The idea is to reduce the number ofindependent sources in order to enable stronger reduction ofthe reducible part (dotted lines in Fig.1) of the network.For these networks we propose an equivalence transforma-tion (right path in Fig.1) instead of the extraction of thesources. The basis-waveforms are given by one representa-tive of each group and are realized with additional indepen-dent sources. The waveforms are copied into the network by

replacing the distributed sources with controlled sources witha specific gain. As only the independent sources, represent-ing a group of sources, have to be extracted and connectedthrough ports, a network with a reduced number of ports isgenerated. The resulting network can be more efficiently re-duced than the network with extracted sources (Ludwig etal., 2008a). The network generated in this way is generallynot passive. The connection of non-passive systems can leadto unstable time-domain simulations and therefore a passivesystem is preferable. Controlled sources connected with theadditional independent sources are added. These sources actlike dissipative elements that dissipate exactly the amount ofenergy generated by the controlled sources in the networkand vice a versa. Thereby the equivalent network is pas-sive (Ludwig et al., 2008a,c) and passivity can be preservedin the reduced model with the description proposed inLud-wig et al.(2008c).

In models that are generated from measurements of dis-tributed systems the sources in the model have PWL-waveforms, which are not necessarily equal or propor-tional. In the next section the described approach for anequivalence-transformation of the network is extended tonetworks with PWL-sources.

2.1 Equivalence transformation for the reduction of thenumber of PWL-sources

In this section we propose a method to describe PWL-sourcesin a network for an efficient reduction. We consider net-works with sources, where the waveforms of the sources aredescribed with continuous PWL-functionsfq(t) on a closedinterval[0, T ] for which holds

fq(t) = mi t + oi for all ai−1 ≤ t ≤ ai (1)

for pairs of real numbers(mi, oi), i=1, ..., k with

0 < a0 < a1 < ... < ak < T (2)

according to the definition inAliprantis et al.(2005).Every PWL-function describing a waveform of a source

in the network is decomposed into PWL-basis-functions.The union of all basis-functions gives the complete basis-functions space of the sizeb. The waveforms of all sourcesare now the linear combination of allb basis-functions mul-tiplied with a corresponding weighting factorwi .

fq(t) =

b∑i=1

wq,ifi(t). (3)

The basis-functions can be found for example with themethod proposed inAliprantis et al.(2005).

After their decomposition the PWL-functions have to berealized in the network. In the network every basis-functionis realized with an independent PWL-source. The distributedsources in the network are replaced with controlled sources,with the controlling values and gains corresponding to the

Adv. Radio Sci., 7, 113–118, 2009 www.adv-radio-sci.net/7/113/2009/

S. Ludwig and W. Mathis: Transformation of passive electrical networks with PWL-sources using a MOR 115

linear combination of the basis-functions as in Eq. (3). Withthis method a network with equivalent behavior at the pinsof the network is generated, hence this modification is anequivalence-transformation as shown in the right path inFig. 1.

If the number of basis-functions is lower than the numberof distributed PWL-sources in the original network, with thismethod a network with a lower number of independent PWL-sources can be generated. By extracting the lower number ofindependent PWL-sources a reducible network with a lowernumber of ports can be generated, which can be more effi-ciently reduced. Additionally, the reducible part of the net-work is still passive and reciprocal by adding appropriatecontrolled sources to the additional independent sources forthe basis-functions, as described earlier in this section.

The difference compared to the former method for groupsof waveforms with equal or proportional amplitude is thatnot for every group of waveforms an independent source isadded but every basis-function is realized with an indepen-dent source. The distributed sources inside the network areagain replaced by controlled sources, which are now not onlycontrolled by the representative source for each group but byall additional independent sources.

2.2 Approximative transformation using a Model Or-der Reduction (MOR)

Model order reduction algorithms based on projection arecommonly used for the reduction of large systems of elec-trical networks (Ionutiu et al., 2007). The advantage of theprojection methods is the simple implementation and thelow computational effort for the construction of the reducedmodel.

For MOR the electrical network of the model has to bedescribed as a system of differential algebraic equations inthe form of

(sC + G)x = Bu, y = LT x (4)

using modified nodal analysis, where the number of equa-tionsN is defined as the order of the system.p is the numberof ports. The positive definite system matricesC, G∈RN×N

contain the stamps for capacitors, inductors and resistors.The input and output vectorsu, y are related with the sys-tem vectorx∈RN through ports with the help of the matricesB, L∈RN×p, B=L . The matrix transfer functionH∈Cp×p

of the system is given by

H(s) = LT(sC + G)−1B. (5)

Circuit equations of typical networks have system matricesthat are very sparse, even though the orderN is quite large.For the reduction we use the approach proposed inPhillipsand Silveira(2005). Firstly a number of real frequency pointssi is defined and for every frequency point

zk = (skC + G)−1 B (6)

S. Ludwig and W. Mathis: Transformation of Passive Electrical Networks with PWL-Sources using a MOR 3

method a network with a lower number of independent PWL-sources can be generated. By extracting the lower number ofindependent PWL-sources a reducible network with a lowernumber of ports can be generated, which can be more effi-ciently reduced. Additionally, the reducible part of the net-work is still passive and reciprocal by adding appropriatecontrolled sources to the additional independent sources forthe basis-functions, as described earlier in this section.

The difference compared to the former method for groupsof waveforms with equal or proportional amplitude is thatnot for every group of waveforms an independent source isadded but every basis-function is realized with an indepen-dent source. The distributed sources inside the network areagain replaced by controlled sources, which are now not onlycontrolled by the representative source for each group but byall additional independent sources.

2.2 Approximative Transformation using a Model OrderReduction (MOR)

Model order reduction algorithms based on projection arecommonly used for the reduction of large systems of elec-trical networks (Ionutiu, 2007). The advantage of the projec-tion methods is the simple implementation and the low com-putational effort for the construction of the reduced model.

For MOR the electrical network of the model has to bedescribed as a system of differential algebraic equations inthe form of

(sC + G)x = Bu, y = LT x (4)

using modified nodal analysis, where the number of equa-tions N is defined as the order of the system. p is the num-ber of ports. The positive definite system matrices C,G ∈RN×N contain the stamps for capacitors, inductors and re-sistors. The input and output vectors u,y are related withthe system vector x ∈ RN through ports with the help of thematrices B,L ∈ RN×p,B = L. The matrix transfer functionH ∈ Cp×p of the system is given by

H(s) = LT(sC + G)−1B. (5)

Circuit equations of typical networks have system matricesthat are very sparse, even though the order N is quite large.For the reduction we use the approach proposed in (Phillips,2005). Firstly a number of real frequency points si is definedand for every frequency point

zk = (skC + G)−1 B (6)

is calculated. By combining all zi in Z = [z1, z2, ..., zi] andusing a singular value decomposition

[U,S,V] = SVD(Z) (7)

the projection matrix T is the orthogonal column span of U(:, 1 : q) and the singular values in S can be used for choosing

the reduced order and for error estimation (Phillips, 2005),as only the dominant singular vectors corresponding to theq largest singular values are used for the projection matrix.With the projection matrix T the system vector is changedwith x = Tx. The reduced system matrices are calculatedwith

C = TT CT G = TT GT

B = TT B L = TT L. (8)

If the projection matrix is real, this method preserves thepassivity of suitably described systems (Odabasioglu, 1998).The order reduced transfer function

H(s) = LT(sC + G)−1B (9)

with H ∈ Cp×p is generated, with an order n ¿ N ofthe system matrices. The reduced system closely matchesthe input-output-behavior of the original model, H ≈H (Phillips, 2005).

In the last step for the approximative transformation of thenetwork, the reduced system is synthesized as an electricalnetwork, for example with a method described in (Palenius,2004; Yang, 2007; Ludwig, 2008a).

With this transformation a smaller network can be found.In the resulting network, properties like passivity are pre-served while the behavior at the pins is approximately thesame as in the original network.

3 Results for an example RCI-Grid

As an example a network grid as shown in Fig. 2 is used. Theexample grid contains resistors, capacitors and independentcurrent PWL-sources. The 4900 resistors of our example areconnected as a two-dimensional grid with 50x50 nodes andpins at two opposite corners of the network. Between ev-

Fig. 2. Example of an 3x3 RCI-Grid model with 4 PWL-sources

ery node in the 50x50 grid and the ground node a capac-itor is connected, which amounts to 2500 capacitors. Thevalues of the elements are randomly chosen in the ranges1 ± 0.5ohm and 0.1 ± 0.05F for the resistors and ca-pacitors, respectively. Between 55 arbitrary chosen nodesin the network and the ground node independent currentsources with PWL-waveforms are connected. Some exam-ples for the PWL-waveforms used in the example are shown

Fig. 2. Example of an 3×3 RCI-Grid model with 4 PWL-sources.

is calculated. By combining allzi in Z = [z1, z2, ..., zi] andusing a singular value decomposition

[U, S, V] = SVD(Z) (7)

the projection matrixT is the orthogonal column span ofU(:, 1 : q) and the singular values inScan be used for choos-ing the reduced order and for error estimation (Phillips andSilveira, 2005), as only the dominant singular vectors cor-responding to theq largest singular values are used for theprojection matrix. With the projection matrixT the systemvector is changed withx=Tx. The reduced system matricesare calculated with

C = TT CT G = TT GT

B = TT B L = TT L . (8)

If the projection matrix is real, this method preserves thepassivity of suitably described systems (Odabasioglu et al.,1998). The order reduced transfer function

H(s) = LT(sC + G)−1B (9)

with H∈Cp×p is generated, with an ordern�N of the sys-tem matrices. The reduced system closely matches the input-output-behavior of the original model,H≈H (Phillips andSilveira, 2005).

In the last step for the approximative transformation of thenetwork, the reduced system is synthesized as an electricalnetwork, for example with a method described inPaleniusand Roos(2004); Yang et al.(2007); Ludwig et al.(2008a).

With this transformation a smaller network can be found.In the resulting network, properties like passivity are pre-served while the behavior at the pins is approximately thesame as in the original network.

3 Results for an example RCI-grid

As an example a network grid as shown in Fig.2 is used.The example grid contains resistors, capacitors and indepen-dent current PWL-sources. The 4900 resistors of our ex-ample are connected as a two-dimensional grid with 50×50

www.adv-radio-sci.net/7/113/2009/ Adv. Radio Sci., 7, 113–118, 2009



in Fig. 3. All waveforms can be decomposed into the threebasis-waveforms shown in Fig. 4.

Fig. 3. Some waveforms of the current sources of the 50x50 RCI-Grid model

Fig. 4. Basis-waveforms for the 50x50 RCI-Grid model using de-composition of the original PWL-waveforms

With the standard method of extracting all independentsources the system has 57 ports. If the waveforms of theindependent sources are decomposed into the three basis-waveforms and the equivalence-transformation in Sect. 2.1is used, only three independent sources are necessary. Thisresults in only five ports of the reducible part of the network.With this description the behavior at the two pins of themodel is not changed. Notably the system of the equivalence-transformed network can be, already before the reductionwith MOR, calculated almost four times faster due to thelower number of ports (Tab. 1). Both system descriptions arereduced with the PMTBR algorithm as described in Sect. 2.2.One hundred expansion points si are equally distributed inthe frequency range range of interest 10−5 ≤ f ≤ 105 andused for the generation of the projection matrix. In Fig. 5the singular values S of the system with extracted sourcesand the equivalence-transformed system are shown. It can beseen that the singular values of the system after equivalence-transformation decay faster, which means that the reducedsystems will be smaller and more accurate. For comparisonboth systems are first reduced to the same order of n = 40.Secondly both systems are reduced to a maximal singularvalue of 10−1, resulting in an order of n = 40 for the sys-tem of the equivalence-transformed network and n = 165 forthe system of the network with extracted sources. The pos-sible MATLAB-speed-up of the reduction of both systems isshown in Tab. 1. The transfer functions of the reduced sys-tems with the maximal singular value of 10−1 show a goodagreement with the original system, as shown for the transferfunction Z11 in Fig. 6. However, since the reduced systemof the equivalence-transformed network is smaller, it can befaster simulated than the reduced system of the network with

Fig. 5. Hankel singular values of the 50x50 RCI-Grid

extracted sources (52x in comparison to 9.4x, Tab. 1). Forthe same order, the speed-up of the system with extractedsources is almost as high as for the equivalence-transformedsystem (42x and 52x, Tab. 1). Nevertheless, the reduced sys-tem of the equivalence-transformed network is more accu-rate (Fig. 6 and Fig. 7). With this example it is shown, that

Table 1. Comparison of Reduction Results of the SystemOriginal Order Ports Reduced MATLAB Accuracy

N = 2500 p Order n Speed-up(Reduction)

Extracted 57 2500 1x Exact atsources - the pins

Extracted reduced 57 165 9.4x Verywith PMTBR (93.4%) Good

Extracted reduced 57 40 42x Badwith PMTBR (98.4%)Equivalence- 5 2500 3.8x Exact attransformed - the pins

Equivalent reduced 5 40 52x Verywith PMTBR (98.4%) Good

Fig. 6. Magnitude and phase of transfer function Z11 of the 50x50RCI-Grid (MATLAB)

the proposed method of equivalence-transformation with thedecomposition of the waveforms as an additional step be-fore the reduction leads to smaller and therefore faster mod-els than the standard method of extracting all PWL-sources.Also it was shown, that for the same size of the reduced sys-tem our equivalence-transformation before reduction leads tomore accurate models.

For investigations with SPICE-like simulators the smallestreduced system that is close enough to the original, which is

Fig. 3. Some waveforms of the current sources of the 50×50 RCI-Grid model.

















Fig. 4. Basis-waveforms for the 50×50 RCI-Grid model using de-composition of the original PWL-waveforms.

nodes and pins at two opposite corners of the network. Be-tween every node in the 50×50 grid and the ground nodea capacitor is connected, which amounts to 2500 capaci-tors. The values of the elements are randomly chosen in theranges 1±0.5 ohm and 0.1±0.05 F for the resistors and ca-pacitors, respectively. Between 55 arbitrary chosen nodesin the network and the ground node independent currentsources with PWL-waveforms are connected. Some exam-ples for the PWL-waveforms used in the example are shownin Fig. 3. All waveforms can be decomposed into the threebasis-waveforms shown in Fig.4.

With the standard method of extracting all independentsources the system has 57 ports. If the waveforms of theindependent sources are decomposed into the three basis-waveforms and the equivalence-transformation in Sect.2.1is used, only three independent sources are necessary. Thisresults in only five ports of the reducible part of the net-work. With this description the behavior at the two pinsof the model is not changed. Notably the system of theequivalence-transformed network can be, already before thereduction with MOR, calculated almost four times faster dueto the lower number of ports (Table1). Both system descrip-tions are reduced with the PMTBR algorithm as described inSect.2.2. One hundred expansion pointssi are equally dis-tributed in the frequency range range of interest 10−5

≤f≤105

and used for the generation of the projection matrix. In Fig.5the singular valuesS of the system with extracted sourcesand the equivalence-transformed system are shown. It can beseen that the singular values of the system after equivalence-transformation decay faster, which means that the reducedsystems will be smaller and more accurate. For comparisonboth systems are first reduced to the same order ofn=40.Secondly both systems are reduced to a maximal singular

















Fig. 5. Hankel singular values of the 50×50 RCI-grid.

Table 1. Comparison of reduction results of the system.

Original order Ports Reduced MATLAB AccuracyN=2500 p ordern Speed-up

(Reduction)

Extracted 57 2500 1× Exact atsources – the pins

Extracted reduced 57 165 9.4× Verywith PMTBR (93.4%) Good

Extracted reduced 57 40 42× Badwith PMTBR (98.4%)

Equivalence- 5 2500 3.8× Exact attransformed – the pins

Equivalent reduced 5 40 52× Verywith PMTBR (98.4%) Good

value of 10−1, resulting in an order ofn=40 for the systemof the equivalence-transformed network andn=165 for thesystem of the network with extracted sources. The possi-ble MATLAB-speed-up of the reduction of both systems isshown in Table1. The transfer functions of the reduced sys-tems with the maximal singular value of 10−1 show a goodagreement with the original system, as shown for the transferfunction Z11 in Fig. 6. However, since the reduced systemof the equivalence-transformed network is smaller, it can befaster simulated than the reduced system of the network withextracted sources (52× in comparison to 9.4×, Table1). Forthe same order, the speed-up of the system with extractedsources is almost as high as for the equivalence-transformedsystem (42× and 52×, Table1). Nevertheless, the reducedsystem of the equivalence-transformed network is more ac-curate (Figs.6 and7). With this example it is shown, thatthe proposed method of equivalence-transformation with thedecomposition of the waveforms as an additional step be-fore the reduction leads to smaller and therefore faster mod-els than the standard method of extracting all PWL-sources.Also it was shown, that for the same size of the reduced sys-tem our equivalence-transformation before reduction leads tomore accurate models.



S. Ludwig and W. Mathis: Transformation of passive electrical networks with PWL-sources using a MOR 117

















Fig. 6. Magnitude and phase of transfer functionZ11 of the 50×50RCI-grid (MATLAB).S. Ludwig and W. Mathis: Transformation of Passive Electrical Networks with PWL-Sources using a MOR 5

Fig. 7. Magnitude- and phase-error of transfer function Z11 50x50of the RCI-Grid (MATLAB)

the equivalence-transformed network reduced with PMTBR(last line in Tab. 1) is synthesized with the method describedin (Ludwig, 2008a). The resulting network is smaller (Tab. 2)than the original network and simulations are around tentimes faster. As an example the good agreement of the net-

Table 2. Comparison of Reduction Results of the NetworkOriginal ReducedNetwork Network

Network elements 7450 1685Network nodes 2500 45

SPICE parsing time 18s 0.9sSPICE simulation time 27s 2.6s

SPICE memory 16.7mbytes 2.1mbytes

work behavior at the pins can be seen for a simulation of thevoltage fluctuation at pin1 in Fig. 8.

Fig. 8. Voltage drop at pin1 of the 50x50 RCI-Grid (HSPICE)

4 Conclusions

In this paper, an efficient reduction process of networks witha large number of PWL-sources is presented. The former de-veloped method of an equivalence-transformation instead ofextracting sources is extended to PWL-waveforms which aredecomposed into basis-functions. The validity and efficiencyof the proposed method is shown by reducing a RCI-Gridmodel. With the proposed reduction process a higher reduc-tion and a higher accuracy can be achieved. This results insmaller and / or more accurate networks generated with thereduction process.

Acknowledgements. S. Ludwig wants to acknowledge Lj. Radic-Weissenfeld and R. Kazemzadeh of the Institute of ElectromagneticTheory at the Leibniz University of Hannover for many useful dis-cussions. Represented research and development work is carried outin the frame of the MEDEA+ project A701 PARACHUTE projectThis particular research was supported by the BMBF under 01M3169 A, 01M 3169 D and 01M 3169 E. The responsibility for thispublication is held by the authors only.

References

Aliprantis, C.D., D. Harris and R. Tourkey: Continous PiecewiseLinear Functions, Macroeconomic Dynamics, Vol. 10, No. 1,2005.

Antoulas, A.C.: Approximation of Large-Scale Dynamical Sys-tems, Advances in Design and Control, SIAM, 2005.

Feldmann, P.: Model order reduction techniques for linear systemswith large number of terminals, DATE2004 - Design, Automa-tion and Test in Europe, Exhibition Conference, Vol. 2, Paris,France, 2004.

Ionutiu, R., S. Lefteriu and A.C. Antoulas: Comparison of ModelReduction Methods with Applications to Circuit Simulation, Sci-entific Computing in Electrical Engineering SCEE 2006, Mathe-matics in Industry, Springer, Vol. 11, 2007

Ludwig, S., Lj. Radic-Weissenfeld, W. Mathis and W. John: Ef-ficient Model Reduction of Passive Electrical Networks with aLarge Number of Independent Sources, IEEE International Sym-posium on Circuits and Systems ISCAS, Seattle, USA, 2008a.

Ludwig, S., Lj. Radic-Weissenfeld, W. Mathis and W. John: Ef-ficient Passive Network Description of IC Conducted Emis-sion Models for Model Reduction, Advances in Radio Sciences,2008b.

Ludwig, S., Lj. Radic-Weissenfeld, W. Mathis and W. John: Passiveand Reciprocal Network Description of Independent Sources forEfficient Model Reduction. International Conference on Signalsand Electronic Systems - ICSES, Krakow, Poland, 2008c.

Odabasioglu, A., M. Celik and L.T. Pileggi: PRIMA: PassiveReduced- Order Interconnect Macromodeling Algorithm, IEEETransactions on Computer-Aided Design of Integrated Circuitsand Systems, Vol. 17, No. 8, August 1998.

Palenius, T. and J. Roos: Comparison of reduced-order interconnectmacromodels for time-domain simulation, IEEE Transactions onMicrowave Theory and Techniques, Vol. 52, 2004.

Phillips, J.R. and L.M. Silveira: Poor Mans TBR: A Simple ModelReduction Scheme, IEEE Transactions on Computer-Aided De-sign of Integrated Circuits and Systems, Vol. 24, No. 1, 2005.

Radic-Weissenfeld, Lj., S. Ludwig, W. Mathis and W. John: Modelorder reduction of linear time-invariant systems, Advances in Ra-dio Sciences, 2008a.

Silva, J.M.S., J.F. Villena, P. Flores and L.M. Silveira: OutstandingIssues in Model Order Reduction, Scientific Computing in Elec-trical Engineering, Springer, Berlin Heidelberg, Germany, 2007.

Yang, F., X. Zeng, Y. Su and D. Zhou: RLCSYN: RLC EquivalentCircuit Synthesis for Structure-Preserved Reduced-order Modelof Interconnect, IEEE International Symposium on Circuits andSystems ISCAS, New Orleans, USA, 2007.

Fig. 7. Magnitude- and phase-error of transfer functionZ11 50x50of the RCI-grid (MATLAB).

the equivalence-transformed network reduced with PMTBR(last line in Table1) is synthesized with the method de-scribed in (Ludwig et al., 2008a). The resulting network issmaller (Table2) than the original network and simulationsare around ten times faster. As an example the good agree-ment of the network behavior at the pins can be seen for asimulation of the voltage fluctuation at pin1 in Fig.8.

4 Conclusions

In this paper, an efficient reduction process of networks witha large number of PWL-sources is presented. The former de-veloped method of an equivalence-transformation instead ofextracting sources is extended to PWL-waveforms which aredecomposed into basis-functions. The validity and efficiencyof the proposed method is shown by reducing a RCI-Gridmodel. With the proposed reduction process a higher reduc-tion and a higher accuracy can be achieved. This results in

Table 2. Comparison of reduction results of the network.

Original Reducednetwork network

Network elements 7450 1685Network nodes 2500 45

SPICE parsing time 18 s 0.9 sSPICE simulation time 27 s 2.6 s

SPICE memory 16.7 mbytes 2.1 mbytes

Fig. 8. Voltage drop at pin1 of the 50×50 RCI-grid (HSPICE).

smaller and/or more accurate networks generated with thereduction process.

Acknowledgements.S. Ludwig wants to acknowledge Lj. Radic-Weissenfeld and R. Kazemzadeh of the Institute of ElectromagneticTheory at the Leibniz University of Hannover for many useful dis-cussions. Represented research and development work is carried outin the frame of the MEDEA+ project A701 PARACHUTE projectThis particular research was supported by the BMBF under 01M3169 A, 01M 3169 D and 01M 3169 E. The responsibility for thispublication is held by the authors only.

References

Aliprantis, C. D., Harris, D., and Tourkey, R.: Continous Piece-wise Linear Functions, Macroeconomic Dynamics, 10(1), 77–99, 2005.

Antoulas, A. C.: Approximation of Large-Scale Dynamical Sys-tems, Adv. Design Control, SIAM, 2005.

Feldmann, P.: Model order reduction techniques for linear systemswith large number of terminals, DATE’2004 – Design, Automa-tion and Test in Europe, Exhibition Conference, vol. 2, Paris,France, 2004.

Ionutiu, R., Lefteriu, S., and Antoulas, A. C.: Comparison of ModelReduction Methods with Applications to Circuit Simulation, Sci-entific Computing in Electrical Engineering SCEE 2006, Mathe-matics in Industry, Springer, vol. 11, 2007.

Ludwig, S., Radic-Weissenfeld, Lj., Mathis, W., and John, W.: Ef-ficient Model Reduction of Passive Electrical Networks with aLarge Number of Independent Sources, IEEE International Sym-posium on Circuits and Systems ISCAS, Seattle, USA, 2008a.

Ludwig, S., Radic-Weissenfeld, Lj., Mathis, W., and John, W.: Effi-cient passive network description of IC conducted emission mod-els for model reduction, Adv. Radio Sci., 6, 133–137, 2008b,http://www.adv-radio-sci.net/6/133/2008/.

Ludwig, S., Radic-Weissenfeld, Lj., Mathis, W., and John, W.:Passive and Reciprocal Network Description of Independent

www.adv-radio-sci.net/7/113/2009/ Adv. Radio Sci., 7, 113–118, 2009

http://www.adv-radio-sci.net/6/133/2008/


Sources for Efficient Model Reduction, International Conferenceon Signals and Electronic Systems – ICSES, Krakow, Poland,2008c.

Odabasioglu, A., Celik, M., and Pileggi, L. T.: PRIMA: PassiveReduced- Order Interconnect Macromodeling Algorithm, IEEETransactions on Computer-Aided Design of Integrated Circuitsand Systems, 17(8), 645–654, 1998.

Palenius, T. and Roos, J.: Comparison of reduced-order intercon-nect macromodels for time-domain simulation, IEEE Transac-tions on Microwave Theory and Techniques, 52, 2240–2250,2004.

Phillips, J. R. and Silveira, L. M.: Poor Mans TBR: A Simple ModelReduction Scheme, IEEE Transactions on Computer-Aided De-sign of Integrated Circuits and Systems, 24(1), 43–55, 2005.

Radic-Weissenfeld, Lj., Ludwig, S., Mathis, W., and John, W.:Model order reduction of linear time invariant systems, Adv. Ra-dio Sci., 6, 129–132, 2008a,http://www.adv-radio-sci.net/6/129/2008/.

Silva, J. M. S., Villena, J. F., Flores, P., and Silveira, L. M.: Out-standing Issues in Model Order Reduction, Scientific Computingin Electrical Engineering, Springer, Berlin Heidelberg, Germany,2007.

Yang, F., Zeng, X., Su, Y., and Zhou, D.: RLCSYN: RLC Equiv-alent Circuit Synthesis for Structure-Preserved Reduced-orderModel of Interconnect, IEEE International Symposium on Cir-cuits and Systems ISCAS, New Orleans, USA, 2007.


http://www.adv-radio-sci.net/6/129/2008/

Documents

Transformation of passive electrical networks with distributed … · 2020. 10. 15. · Every PWL-function describing a waveform of a source in the network is decomposed into PWL-basis-functions