1

Hybrid S-Parameters for Transmission LineNetworks with Linear/Nonlinear Load

Terminations Subject to Arbitrary ExcitationsYakup Bayram,Student Member, IEEE,John L. Volakis,Fellow, IEEE,

Abstract— We propose a generalized S-Parameter anal-ysis for transmission lines with linear/nonlinear loadterminations subject to arbitrary plane wave and portexcitations. S-parameters are prevalently used to modeltransmission lines such as cable bundles and interconnectson printed circuit boards subject to port excitations. Theconventional S-parameter approach is well suited to char-acterize interactions among ports. However, non-traditionalport excitations associated with plane wave coupling tophysical ports at transmission line terminals lead to forcedas well as propagating modal waves, necessitating a modi-fication of the standard S-parameter characterization.

In this paper, we consider external plane wave excitationsas well as port (internal) sources, and propose a hybrid S-parameter matrix for characterization of the associated mi-crowave network and systems. A key aspect of the approachis to treat the forced waves at the ports as constant voltagesources and induced propagating modal waves as additionalentries (hybrid S-parameters) in the S-parameter matrix.The resulting hybrid S-matrix and voltage sources can besubsequently exported to any circuit solver such as HSPICEand Advanced Design System for the analysis of combinedlinear and non-linear circuit terminations at ports. Theproposed method is particularly suited for susceptibilityanalysis of cable bundles and printed circuit boards forElectromagnetic interference evaluations. It also exploitsnumerical techniques for structural and circuit domaincharacterization and allows for circuit design optimizationwithout a need to perform any further computationalElectromagnetic analysis.

I. I NTRODUCTION

S-parameters have been widely used in combiningElectromagnetic (EM) analysis of transmission linenetworks with circuits involving of linear/nonlinearloads. They have also been extensively studied forfull wave extraction of parameters to characterize mi-crowave structures ([1], [2], [3], [4]) and for inte-gration of S-parameter networks with linear/nonlinearloads [5], [6], [7], [8], [9], [10], [11]. As opposed to rep-resenting port relations in terms of voltage and currentsvia Z-parameters (Impedance) or Y-parameters (Admit-tance), S-parameters employ modal incident and re-flected electromagnetic fields to establish a physics-

based mathematical relation among the ports. However,the relations among the ports are dependent on the sup-ported modes subject to excitation. Because each modehas distinct characteristic impedance and propagationvelocity, S-parameters are also referred to as modalparameters.

Port analysis is a prevalent approach for EMI/EMCand signal integrity analysis of electronic designs due toits generality and practicality. In addition to analysis ofPCBs with lumped linear circuit elements, it also allowsfor analysis of mixed signal circuits via broad-band char-acterization of the entire Printed Circuit Board (PCB).Depending on whether time or frequency domain analy-sis is considered, various techniques can be employed tointegrate the S-parameter network with circuit solverssuch as HSPICE for time domain [8], [11], [7] andHarmonic Balance Method [12], [13], [14], [15] formixed time-frequency domain characterization.

External field coupling to PCBs and transmission linenetworks is primarily done with Multiconductor Trans-mission Line Theory [16], [17], [18], [19]. However, it isnot suited well for accurate analysis at high frequenciessince it inherently assumes strong quasi-static analysisin its formulation. Time domain techniques such as theFinite Difference Time Domain (FDTD) method havealso been used for concurrent analysis of circuits andEM structures [20]. While time domain techniques alsoyield accurate results, they suffer from computationalinefficiencies due to the meshing of large volumes andsimulation of RF-devices with high quality factors. Theyalso run into convergence problems for circuit elementswith stiff differential equations.

To tackle shortcomings of the aforementioned meth-ods, we have recently introduced hybrid S-parametermatrix that models transmission line networks with lin-ear/nonlinear terminations subject to both plane waveand traditional port excitations (conference papers [21]and [22]). In this paper, we extend the proposed analysisby decomposing external field excitation into forcedand modal waves extracted via Generalized Pencil of

Functions and present a more practical approach tointegrate forced waves with circuit analysis tools such asAdvanced Design System (ADS) and HSPICE. Proposedanalysis also allows for integration of evanescent modesinto circuit analysis for high frequency analysis [2].

As opposed to traditional port excitations, externalplane wave illumination leads to forced waves alongthe transmission line as well as propagating modalwaves. Forced waves stem from enforcing phase match-ing with the incident wave along the structure wallsand propagate with the wave number of the incidentplane wave along the corresponding direction. Suchforced waves are not affected by the loads attachedto the ports. Conversely, backward and forward modalwaves, originated from mismatches at port terminations,travel with corresponding eigenvalues that RF structuresupports at the operating frequency. In our analysis,we consider forced waves as constant voltage sourcesat the ports and characterize the induced propagatingmodes with S-parameter matrix (hybrid S-parameters).The resulting S-matrix and voltage sources can then beexported to any circuit solver such as HSPICE and ADSand analyzed with the corresponding linear and non-linear port terminations. Since transmission line networkis solely treated in EM domain and circuit componentsattached to the ports are handled in the circuit domain,numerical techniques customized for each domain canbe fully exploited in our analysis. This approach alsoallows for circuit design optimization without a need forrepeated analysis of the microwave network.

Below, we first develop the theory for an arbitrarytransmission line network subject to plane wave exci-tation. Next, we validate the proposed concept with apair of transmission lines in free space excited by aplane wave with a current source attached to one ofthe terminals. Similarly, we extend our validation to apair of microstrip lines on a PCB subject to plane waveexcitation. In the final section, we discuss the proposedmethod and remark on future work.

II. T HEORY

In this section, we consider characterization of theinteractions among physical ports within a transmissionline network using modal S-parameters. Subsequently,we propose a hybrid S-parameters matrix to includeexternal plane wave excitations and proceed to describetechniques such as Generalized Pencil of Functions forthe extraction of hybrid S-matrix entries. We then discusshow forced waves are treated at the ports for circuitanalysis.

A. Modal S-Parameters for Coupling Among PhysicalPorts

Fig. 1 displays a typical mixed signal circuit boardwith nonuniform microstrip lines. For the characteri-zation of such a board, we introduce the N-port S-parameter network giving (for thekth mode),

bk1...

bkN

=

Sk1,1 · · · Sk

1,N...

......

SkN,1 · · · Sk

N,N

ak1...

akN

(1)

where Skij = bk

i

akj

with all ports terminated at their

corresponding reference impedanceZrefi , and (aki , bk

i )referring to the incident and reflected waves, respec-tively, at the ith port. We must note that our analysisassume real reference impedance at the ports throughoutour analysis. However, it can be readily extended toaccount for complex reference impedances at the portswith appropriate power relations (see [23], [24]).

We remark that even though a PCB is shown in Fig. 1,our analysis applies to any transmission line configu-ration as is the case with multiconductor transmissionlines and coaxial cable networks. As usual, the scatteringmatrix assumes the following field representation due toport excitation,

E(s) =∑

k

Akeke−γks +∑

k

Bkekeγks (2)

H(s) =∑

k

Ckhke−γks +∑

k

Dkhkeγks

whereek andhk refer to kth modal electric and mag-netic fields, respectively, withγk being the correspond-ing propagation constants whereas,Ak, Bk, Ck andDk

are the coefficients of the expansion.The above modal fields refer to the eigensolutions of

the corresponding Sturm-Liouville problem subject toDirichlet boundary conditions on PEC surfaces

n× ek = 0 (3)

These modal fields must also satisfy the orthogonalitycondition ∮

S

(ep × hk) · ds = δpk (4)

with respect to the cross section of the propagationfront along the transmission line. To relate (ek,hk)with terminal voltages and currents, we introduce thedefinitions

V ki = −

∫Ci

−→Ek ·

−→dl Ik

i = −∮

Ci

−→Hk ·

−→dl (5)

2

(a)

(b)

(c)

Fig. 1. (a) Typical mixed-signal PCB -(b) Port modeling of (a) wherenon-linear component terminals are represented with ports -(c) Circuitrepresentation of (a)

where (V k,Ik ) denote voltages and currents at theith

port due to thekth modal field. From (2), fields to beintegrated areEk = Akeke−γks + Bkekeγks andHk =Ckhke−γks + Dkhkeγks

Mathematical relation between thekth incident andreflected modal amplitudes, in (1) and modal voltagesand currents in (5) is given by

aki =

V ki + ZrefiI

ki

2√

Zrefi

bki =

V ki − ZrefiI

ki

2√

Zrefi

(6)

where Zrefi is the reference impedance for the corre-spondingith port. The expression in (6) can be construedas an interface between the circuit components (ex-pressed in terms of voltages and currents) and the RFstructure treated via full-wave electric and magnetic

Fig. 2. A Typical TL Structure comprised of two conductors

fields. However, we must note that (6) is only applicableto modal excitations at the physical ports. Therefore,one must account for non-modal field contributions atthe ports for non-conventional port excitations. Below,we exploit the relation in (6) and introduce hybrid S-matrix approach to account for non-conventional externalexcitations such as plane waves on transmission linenetworks.

B. Hybrid S-Parameters for External Plane Wave Exci-tation

Let us consider the case of an external plane waveimpinging upon an N-port arbitrary transmission linenetwork shown in Fig. 3. Similar to the lumped port exci-tations, we propose to introduce the external plane waveas a source generated from an additional(N +1)th port.We start our analysis by imposing Dirichlet boundaryconditions along the transmission line walls (see Fig. 2)as follows,

n×Etotal = 0 (7)

n× (Einc + Escat) = 0

where Einc refers to electric field due to the incidentplane wave in absence of the whole transmission linenetwork, andEscat is the electric field radiated by theinduced currents on the transmission line conductors.

For an infinitely long transmission line, (7) impliesthat n×Escat = −n×Einc at transmission line surfaces.However, for a finite transmission line, the reflectedcurrents at the terminals will lead to modal fields whichalready satisfy the boundary conditions given in (3).Therefore, the scattered fields at the transmission linesurfaces satisfy the conditions

n×Escat = n× (−Einc + Emodal) (8)

whereEmodal is given by (2) with the boundary condi-tions along the transmission line surfaces,

n×Emodal = 0 (9)

3

Referring to (8) and [25], we observe that plane waveincidence on a transmission line introduces a forcedwave (having the same wave number as the incidentfield) in addition to the modal fields. Thus, in the case ofa plane wave excitation, we introduce the representation

Etotal = Eforced +∑

k

Akeke−γks +∑

k

Bkekeγks

Etotal = Eforced + Emodal (10)

Comparing the above expression with (2), we deducethat the difference between the plane wave and lumpedport excitation is that the former induces forced wavesin addition to modal fields.

To account for the modal waves coupled to the phys-ical ports at the transmission line terminals, we treat theplane wave as an additional(N + 1)th port and modifythe existing N-port S-matrix accordingly for each modalwave,

bk1...

bkN

bkN+1

= (11)

Sk

1,1 · · · Sk1,N HSk

1,N+1...

......

...Sk

N,1 · · · SkN,N HSk

N,N+1

SkN+1,1 · · · Sk

N+1,N HSkN+1,N+1

ak1...

akN

akN+1

As seen, the(N + 1)th port is characterized by thehybrid S-parameters,HSk

i,N+1, representing plane wavecoupling to ith port for the kth excited mode. Fig. 3clearly demonstrates that plane wave is included asadditional port in the circuit domain and we treat forcedwaves as additional constant voltages at the ports. Inthe subsequent sections, we first describe extraction ofhybrid S-parameters. Next, we explain how we integrateforced waves with circuit analysis.

C. Hybrid S-Parameters via Open Circuit Analysis

To calculate the hybrid S-parameters, we exploit theinherent relation between the incident and reflectedwaves, and voltage and currents at the ports given in (6).We first introduce the corresponding hybrid impedancematrix for the (N+1)-port network representing voltageand current relations at the ports due to modal and plane

(a)

(b)

Fig. 3. (a) Typical mixed-signal PCB subject to plane wave excitation(b) Circuit representation of (a) via hybrid port modeling of plane wave

wave excitations,V1

...VN

=

Z1,1 · · · Z1,N HZ1,N+1

......

......

ZN,1 · · · ZN,N HZN,N+1

I1

...IN

aN+1

= [Z]I+ V modal

oc (12)

where V modal

oc = HZaN+1 refers to open circuitvoltage at the ports due to modal fields excited bythe external plane wave. Coupling to the(N + 1)th

port is of no interest in our analysis. Therefore, it isexcluded in the Z-matrix (namely, the(N + 1)th rowof the Z-matrix is set to zero). This also helps uscircumvent mismatch problems at the EMI port. The lastelement added to theI column,aN+1, represents thenormalized plane wave coefficient. Thus, the column,HZ can be construed as that relating the open circuitmodal voltages at the ports to the incident plane waveexcitation.

To associate impedance matrix entries with S-

4

parameters, we employ (6) to update (12) giving√[Zref ](a+ b) = (13)

[Z](√

[Zref ])−1(a − b) + V modal

oc

where

aT = a1 · · · aN bT = b1 · · · bN (14)

[Z] =

Z1,1 · · · Z1,N

......

...ZN,1 · · · ZN,N

√

[Zref ] =√

Zref1

... √ZrefN

in which [Z] and

√[Zref ] are already known matrices.

Rearranging the terms, we find that the coefficients ofthe incident and reflected waves at the physical ports aregiven by

b = [Z](√

[Zref ])−1 +√

[Zref ]−1

[Z](√

[Zref ])−1 −√

[Zref ]a

+[Z](√

[Zref ])−1 +√

[Zref ]−1V modaloc (15)

Comparing (15) with (11) and settingaN+1 =|E0|√

ZrefN+1to normalize the incident plane wave, we

readily identify that

HS = (16)√ZrefN+1

|E0|[Z](

√[Zref ])−1 +

√[Zref ]−1V modal

oc

where|E0| is the magnitude of the incident plane waveandHST = HS1 · · ·HSN.

The evaluation of[Zij ] in (11) is done in the usualmanner via open circuit analysis. However, the eval-uation of V

modal

oc requires more attention. Once theopen circuit modal voltages are obtained,HS canbe calculated via (16). Since the forced voltages do notdepend on the terminations, they can be directly exportedto the circuit solver shown in Fig. 3.

D. Generalized Pencil of Functions for Extraction ofHybrid S-Parameters

As noted above, the hybrid scattering matrix assumesthe propagation of a discrete set of modes within thenetwork. Knowledge of these modes and their associated

parameters (eg.γk, Ak and Bk as in (2)) is neces-sary for the extraction of the hybrid S-matrix entries.Generalized Pencil of Functions [26], [27] method canbe employed for the extraction of these parameters.Such an analysis has been successfully employed inthe literature [28], [29]. Specifically, in [28] and [29],the current induced on a microstrip line is decomposedinto the bound (dominant)and higher order modes andauthors employed Generalized Pencil of Functions tofind the corresponding mode amplitude and propagationconstants to achieve the best fit. Similarly, in [30], FDTDwas used in conjunction with Generalized Pencil ofFunctions to extract the S-Parameters of a waveguidestructure via a full wave analysis. Further, in [31],Generalized Pencil of Functions was used to extractthe parameters of current induced on large scatterersrepresented with sum of complex exponentials.

Once the parameters of the exponential terms in (10)are attained, one can readily distinguish forced anddominant modal waves by examining the propagationconstants such that the dominant modal terms appearas a pair of backward and forward travelling voltageswith negligible decay/attenuation constant. A more rig-orous comparison can be also made by computing thepropagation constants of the transmission line networkby invoking the eigenfunction representation with theappropriate boundary conditions. Since the computedeigenvalues correspond to the propagation constants, onecan then extract the modal propagation constants fromGeneralized Pencil of Functions Method results.

E. Integration of Forced Waves with Circuit Analysis

As described above, forced and modal waves can beextracted via Generalized Pencil of Functions Method.Also, we have shown that modal waves can be combinedwith circuit analysis through the hybrid S-parametermatrix. Next, we describe incorporation of forced wavesinto the circuit analysis.

We start our analysis with the following impedanceboundary condition that must be satisfied regardless oflinear/nonlinear loads attached to the ports. Specifically,we have

Etan = ZsHtan (17)

where Zs is measured inΩ per square unit cell. Theimpedance boundary conditions for EM analysis at theports can be translated into Ohm’s law to relate thevoltage and currents at the terminals using the generalform,

Vtotal = f(ZL, Itotal) (18)

5

Fig. 4. A Pair of Transmission Lines Subject to Plane WaveIllumination

where ZL is the complex linear/nonlinear loadimpedance at the ports. We must remark that the surfaceimpedanceZs in (5) can be expressed in terms of thetotal port impedanceZL and the port dimensions. Forinstance, such a relation for the microstrip lines can bewritten as

ZL = Zsh

w(19)

where h and w correspond to the height of the trans-mission line from the ground plane and the width of thestrip line, respectively.

For plane wave excitation, the total voltage at the portis expressed in terms of modal and forced waves via(10) and (5), viz.

Vtotal = Vforced + Vmodal (20)

Vmodal = Vtotal − Vforced

whereVtotal represents the total voltage at the terminalsof any linear or nonlinear loads. As stated,Vforced is aconstant term, not associated with the loads at the ports.Thus, the forced voltage can be added to the ports asa constant source term to enforce Ohm’s law in circuitdomain or equivalently the surface impedance boundarycondition (see Fig. 3).

III. VALIDATION STUDIES

To demonstrate the validity of the hybrid S-parameters, we first consider a pair of transmissionlines subject to concurrent plane wave and a direct portexcitation. Subsequently, we consider a more complexconfiguration consisting of a pair of microstrip lines ona PCB illuminated by an obliquely incident plane wave.

IV. A PAIR OF TRANSMISSIONL INES SUBJECT TO

PLANE WAVE EXCITATION

Consider the Transmission Lines (TL) shown in Fig. 4excited by a current source at the left and terminated bya loadZL = 100 located 250mm from the source. TheTL is comprised of two wires of radius 0.125mm and

separated by 2mm having a characteristic impedance ofZ0 = 332.24Ω [32]. We are interested in computing thevoltage induced at the load when the TL is concurrentlyilluminated by a plane wave operating at 2 GHz (thesame as the port source). This problem is therefore atypical EMI/EMC coupling analysis.

We consider the plane wave excitationE =Eince

−jk.r with Einc = (x1000 + y500 + z1500)V/mand k = k0

x+y−z√3

. Further, we assume that the currentonly flows in the y direction since the wire radiusis much smaller than the wavelength at the operatingfrequency.

We break down our analysis into two sections(1) Current source excitation(2) Plane wave excitationSuch an approach implicitly assumes linear circuit

components attached to the ports. However, we mustnote that the proposed solution can be applied to thecases where non-linear loads are included by employingbroad band S-parameter characterization.

A. Current Source Excitation

To compute the total voltage induced at the load,we employ S-parameter matrix defined for two portswhere the current source and lumped ports are attached,respectively.

Since the current source supports quasi-TEM modesalong the transmission line, one can establish a 2-PortS-Parameter network based on a quasi-TEM mode prop-agation. The resulting 2-Port S-Parameter network canbe exported to any circuit simulator such as ADS (seeFig. 5) and connect the current source and the load atthe corresponding ports.

After performing a full wave analysis (HFSS)1, weextracted2× 2 S-parameter matrix

S2×2 =[

0.909∠8.5 0.415∠97.70.415∠97.7 0.909∠8.5

](21)

Subsequently, we exported the resulting S-matrix to ADSwith the connected the current source at port 1 and theload at port 2. Using ADS (see Fig. 5) we can then findthe load voltage as a function of the current source.

Table I shows a comparison of the full wave resultswith the proposed S-matrix/ADS simulation. As seen, anexcellent agreement is achieved.

B. Plane Wave Excitation

We now proceed to include the plane wave couplingin terms of travelling wave components [25]. Referring

1For the full wave analysis, we further use strips in place of wiresby employing the standard equivalencea = w

4where thea is radius

of the wire andw is the width of the equivalent strip.

6

Fig. 5. 2-Port circuit representation of transmission lines

I=10mA andZL = 100ΩMag(VL)(Full Wave-HFSS) 1.87

Mag(VL) (2-Port Network-ADS) 1.89Angle(VL)(Full Wave-HFSS) 2.31

Angle(VL)(2-Port Network-ADS) 2.51

TABLE I

VOLTAGE INDUCED AT THE ZL = 100Ω DUE TO A CURRENT

SOURCE OF10MA

to Fig. 6, theC1 and D1 coefficients correspond tocoupling onto infinite transmission line and have thesame phase as the incident field to force phase matchingalong the wires. The remaining terms represent forwardand backward travelling (modal) currents and the totalvoltage along the TL can be also expressed as,

Vtotal = Vforced + Vmodal (22)

Vtotal = Vince−j

k0√3y + V1e

−jk0√

3y + V2e

jk0y + V3e−jk0y

Vforced = (Vinc + V1)e−j

k0√3y

Vmodal = V2ejk0y + V3e

−jk0y

where Vinc is the voltage due to the plane wavein the absence of the wires andV1, V2, and V3 areassociated with the induced currents (C1, D1), (C2, D2)and (C3, D3), respectively.

TheVinc andV1 terms depend only on the polarizationand magnitude of the plane wave excitation and inde-pendent of the loads connected to the wire terminals.Therefore, they are forced waves and independent of theattached loads. Further, their presence disappears at themoment the incident field stops to exist, even the modalfields continue their presence.

To substantiate the above expansion, we carried out afull wave analysis of the TL in Fig. 6 with plane waveexcitation and extracted the resulting travelling waverepresentation via the Generalized Pencil of Functionsmethod. Two sets of loads (ZL1,ZL2) were used at theeach port.

Fig. 6. Physical current decomposition of the modes travelling alongthe transmission line

For the loadsZL1 = 250Ω, ZL2 = 800Ω, we foundthat

Vscat1 = (0.57 + j0.04)e−j24.69y (23)

+(−0.37− j1.29)e+j42.76y + (−0.74− j0.61)e−j42.03y

and for the loadsZL1 = 400Ω, ZL2 = 400Ω, weobtained

Vscat1 = (0.56 + j0.05)e−j24.46y (24)

+(−0.77− j1.29)e+j42.77y + (−0.16− j0.37)e−j41.84y

Considering thatk0 = 2πλ = 2π

0.15 = 41.88 andk0√3

= 24.14, it is clear that (23) and (24) are inagreement with (22). In other words, forced voltageterms did not alter with changing loads while modalwaves responded to the attached loads. Based on thisclaim, one can establish a hybrid S-parameters networkbased on the quasi-TEM travelling voltages (V2 andV3) in conjunction with the 2-port S-parameter networkconstructed in the previous section. Similarly, the forcedvoltage terms can be included in the analysis as constantvoltage sources at the ports.

C. Hybrid 3-Port Quasi-TEM S-Parameter NetworkConstruction

In this section, we proceeded to construct a 3-Port Hy-brid S-Parameter network such that Port 1 and Port 2 arephysical ports at the terminals of the transmission linewith Port-3 representing the plane wave source leadingto the quasi-TEM wave induced along the transmissionlines. We first computed the open circuit modal voltagesat the ports and employed (16) to compute the hybridS-parameters.The resulting3× 3 hybrid S-matrix is

7

Fig. 7. Circuit representation of transmission line pair subject toconcurrent plane wave and current source excitation

S3×3 = (25) 0.909∠8.5 0.415∠97.7 0.187∠− 11.810.415∠97.7 0.909∠8.5 0.190∠170.0

0.0∠0.0 0.0∠0.0 0.0∠0.0

where we set the last row to zero because onlyHS1,3

andHS2,3 are non-zero since they represent the couplingof the incident plane wave onto the physical ports at thetransmission line terminals.

Next, we proceeded to employ ADS in conjunctionwith (25) to find the port voltages (see Fig. 7).

We performed three studies in which current sourceis set to zero, 5mA and 10mA respectively. Next, wecompared the proposed method solution with full wavesolution for the voltage induced at the loadZL for eachcurrent source and plane wave excitation. It is clearlyobserved in Fig. 8 that hybrid S-parameters agree verywell with full wave results.

D. A Pair of Coupled Microstrip Lines Subject ToConcurrent Plane Wave and On-Board Current SourceExcitation

We now consider the geometry in Fig. 9, displaying apair of coupled microstrip lines residing on a RT/Duroid5880 board with 2.2 dielectric constant and thickness of31mils (0.7874mm).

The microstrip lines are terminated with compleximpedances and an on-board current source was placedin port 1 (P1) given in Table II. Additionally, a planewave operating at 2.5GHz (the same as the current

(a)

(b)

Fig. 8. Comparison of hybrid S-parameter method results with fullwave results for the validation problem Fig. 4 - (a) Amplitude of thevoltage induced at the loadZL = 100Ω for various current sourcevalues - (b) Phase of the voltage induced at the loadZL = 100Ω forvarious current source values

Fig. 9. A pair of coupled microstrip lines residing on a PCB andsubject to oblique plane wave illumination

source) also impinged on the microstrip lines.E = Eince

−jk.r with Einc = (x2000 + y2000 +z3000)V/m andk = k0

x0.5+y−z1.5

In our analysis, we aim to find the total voltageinduced at the ports. To do so, we first extracted thestandard4 × 4 S-parameter matrix for the given portconfiguration with respect to50Ω reference impedance.Afterwards, we performed full wave analysis on thestructure (with open ports) subject to only plane waveexcitation. Subsequently, we conducted Generalized Pen-cil of Functions analysis to extract the forced voltageand propagating modes along the lines. Upon obtainingmodal and forced voltages at the ports, we then em-

8

P1 I=50mAP2 Z2 = 10− j250ΩP3 Z3 = 100− j100ΩP4 Z4 = 400 + j50Ω

TABLE II

PORT TERMINATIONS IN FIG. 9

ployed (16) to extract the hybrid S-parameters. Resulting5×5 hybrid S-parameter matrix is exported to ADS andcorresponding forced voltages and port terminations areconnected to the respective ports (see Fig. 10).

Fig. 10. Circuit representation of the coupled microstrip lines subjectto concurrent plane wave and current source excitation

In this configuration, portsP1 − P4 correspond tophysical ports at the terminals of the coupled microstriplines and port P5 represents the plane wave coupling.Circuit analysis was run at 2.5 GHz and performanceof the proposed method is compared with full waveresults (see Fig. 11). It is clearly observed that proposedmethod results agree well with full wave results.

V. CONCLUSION

A novel approach for the analysis of transmissionline networks subject to a plane wave excitations wasproposed. It was shown that plane wave coupling leadsto both forced and modal waves. The former is constantat the ports regardless of the attached load whereas, thelatter travels along the transmission lines. While modalwaves were taken into account by treating the plane waveas an additional port, forced waves appear as constantvoltage sources at the terminals to enforce the Ohm’s lawin circuit, or equivalently, surface impedance boundary

(a)

(b)

Fig. 11. Comparison of the proposed method with full wave resultsfor the voltage induced at the ports shown in Fig. 9 - (a) Total inducedvoltage amplitude at each port - (b) Total induced voltage phase ateach port

condition in EM domain. Two validation studies werecarried out with a pair of transmission lines in freespace and also a coupled microstrip lines residing ona PCB. It was shown that the proposed method agreesvery well with full wave results. The key advantage ofthe proposed method is the treatment of EM structure inthe EM domain whereas circuit components are treatedin circuit domain. Therefore, this analysis can address alarge variety of circuit components.

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