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Electromagnetic Decoupling and ComplexityDing Nie, Bertrand Hochwald, and Erik Stauffer

Abstract—In radio-frequency systems that drive coupled dis-sipative loads, the matching network between the amplifiers andtheir loads needs to account for the coupling. With N amplifiersdriving N loads, a favorite choice is a “decoupling” network,which is a lossless reciprocal network that has N input portsconnected to the sources and N output ports connected to theloads. The decoupling network transforms the coupled impedanceof the loads into the uncoupled characteristic impedance of thesources. Any incident signal at the input ports of the networkis transferred, without reflection, to the loads. Decoupling net-works can be realized by generalized Π-networks of lumpedand distributed impedances, depending on the design frequency.Although the impedance requirements of the network are unique,its realization is not, and networks that involve many impedancescan be complex to lay out on circuit boards. In this paper,we establish that a decoupling network requires a minimum ofN2+N impedances for N arbitrarily coupled loads, and providea systematic method for realizing this lower bound.

Index Terms—RF coupling, decoupling networks, matchingnetworks

I. INTRODUCTION

Impedance matching is the practice of matching a sourceimpedance to a load impedance to maximize the powertransferred to load. When a single-port load is driven by asingle source, the maximum power transfer is obtained whenthe load impedance is the complex conjugate of the sourceimpedance. Sources and loads are often matched in radio-frequency (RF) systems using lossless two-port impedancematching networks [1]. The matching network has the propertythat it presents the complex conjugate of the source impedanceat its input and the complex conjugate of the load impedanceat its output. Transmitters and receivers in radios often usematching networks in the most sensitive portions of the RFchain between the antenna and the RF amplifier stages to avoidlosses in transmitter power when the antenna is acting as aload, and receiver sensitivity when the antenna is acting as asource.

Matching network design techniques for a single-port loadusing two-port lossless networks are well known. Generally,some combination of lumped and distributed reactive compo-nents are employed. Although the impedance requirements fora matching network are unique, the realization of the matchingnetwork to meet these requirements is not. Some criteria usedto judge a given realization include ease of implementationand bandwidth. Ease and cost of implementation are generallygoverned by the number of impedances in the matchingnetwork. The bandwidth of a matching network is its ability to

Ding Nie and Bertrand Hochwald are with the Department of ElectricalEngineering, University of Notre Dame, Notre Dame, IN, 46556 USA. E-mail: nding1@nd.edu, bhochwald@nd.edu.

Erik Stauffer is with the Broadcom Cooperation, Sunnyvale, CA, 94086USA. E-mail: eriks@broadcom.com

maintain a source and load impedance match at frequencies inthe vicinity of the design point. This paper considers ease ofimplementation as having primary importance, and bandwidthas secondary.

When used to connect N sources to N loads, the matchingnetwork is especially complicated if the loads are “coupled”in that the S-matrix of their scattering parameters has non-zero off-diagonal elements. Examples of loads that are cou-pled include closely-spaced antennas. With coupled loads, theprocess of matching to any one load must consider also theremaining loads. A common method to match sources such asradio-frequency amplifiers to loads such as coupled antennasis to insert a multiport decoupling network, which has theproperties that it is lossless, reciprocal and all power incidenton its N input ports is transferred, without reflection, to theN loads.

We assume N independent sources have impedance Z0, thecharacteristic impedance of the system (typically 50 Ω), andN loads are coupled. To match the loads to the sources, thedecoupling network transforms the coupled impedances of theloads to a set of uncoupled ports whose impedances are Z0. Areciprocal lossless decoupling network also works in reverse—power emanating from the loads is transferred to the sourceswithout reflection. In this case, the designation of source andload is interchanged, as when antennas are driving an arrayof receivers. For simplicity, we assume the network is beingused in the forward (transmit) direction, to avoid confusionand issues such as the dependence of the receiver noise-figureon matching conditions.

An early example of a decoupling network includes [2]for multiple-antenna systems, where lengths of transmissionlines are used to decouple the antennas when their mutualimpedances (off-diagonal elements of the antenna impedancematrix) are reactive. Decoupling networks are shown toachieve the maximum channel capacity among all losslessmatching networks when they are utilized in multiple-antennacommunication systems [3]. When used in multiple-antennasystems, decoupling network designs fall into two main cat-egories: those based on microwave couplers include [4]–[8];those based on reactive components and transmission lines in-clude [2], [9]–[16]. Examples for two antennas include [4], [5],[9]–[12]; three antennas include [6], [13]–[15]; four antennasinclude [7], [8]. Example papers using reactive componentsfor N antennas include [2], [16], where particular antennastructures are considered, and [8], where imperfect decouplingis considered.

In many of these references, the analyses and designs aretailored to specific load structures and a small number ofantennas, and performance metrics of network complexity aretailored to the structure. Hence, these analyses are of limiteduse in the design of large-scale networks that work with arbi-

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trary load structures and coupling. We seek systematic, unified,design methods that work for any N and load structure, and weseek standardized limits against which network performancein complexity can be measured. We require the loads to bedissipative and reciprocal, but otherwise unconstrained.

Next-generation 5G and WiGig wireless networks are con-sidering using “massive MIMO,” especially at millimeter-wave frequencies, where large bandwidths of spectrum areavailable. Massive MIMO involves many antennas in basestations and portable devices to assist with poor wirelesspropagation and building penetration at these frequencies.These antennas would be in close proximity and interactelectromagnetically with each other, especially on portabledevices. For example, Samsung has recently demonstrated [17]a prototype of a system that delivers 256 megabits/sec of datain the 28 GHz band with N = 64 antennas at the transmitterand receiver. Large-scale decoupling networks are needed tomaximize output power and capacity for such systems withcoupled antennas.

A decoupling network can be realized by a generalizedΠ-network using lumped impedances at frequencies belowmillimeter-wave bands, and combinations of lumped anddistributed impedances and transmission lines at millimeter-wave frequencies. However, the number of impedances in arealization is generally quadratic in the number of sourcesand loads. For N = 64 loads, the large number of resultingimpedances can be onerous in cost and layout difficulty ona circuit board. Especially troublesome are crossovers, wherelumped or distributed components would need to cross eachother to reach an intended connection on a board.

The following results are presented:• The lower bound for the number of impedances needed

in a decoupling network is established as N2 + N forarbitrarily coupled loads.

• Systematic designs of decoupling networks are presentedthat achieve the lower bound.

Section II introduces notation and defines a decouplingnetwork. Section III shows the properties and a realization ofthe decoupling network. Section IV presents one of the mainresults: minimizing the number of impedances for generalcoupled loads. The proofs and notes for the design methodgo in Section V, and the conclusion goes in Section VI.

II. PROPERTIES OF DECOUPLING NETWORKS

We introduce the properties of the decoupling networks inS-matrix representation. In the RF system shown in Figure1, we assume SL is the N × N complex S-matrix of a setof N dissipative reciprocal loads that we wish to match to Ndecoupled sources, each with characteristic impedance Z0. Theijth element of this matrix represents the complex scatteringparameter for an incident wave on port j and exiting porti [1]. The matrix SL is symmetric, STL = SL, because theloads are reciprocal, and the singular values of SL are lessthan one because the loads are dissipative. We note that thereis an implicit assumption that the scattering parameters arebeing measured at a particular design frequency fd since theseparameters can change with fd.

0Z

0Z

2N-port matching network

.

.

.

LSLMS

N-port loads

1a

1b

2a

2b

.

.

.

Output ports N

+1~2N

Input ports 1~N

Fig. 1. An RF system where N independent sources drive N loads. SL

is the S-matrix of the loads, and SLM is the S-matrix as presented to thesources.

The decoupling network for the loads is lossless, reciprocaland has a total of 2N ports, where ports 1 through N are inputports that connect to the sources, and ports N + 1 through2N are output ports that connect to the loads. We define itsS-matrix as a 2N × 2N matrix S; S satisfies the reciprocalcondition ST = S and lossless condition SHS = I , where thesuperscript H denotes conjugate-transpose. When the loads areconnected to the output ports of the decoupling network, thedecoupling network should present an uncoupled characteristicimpedance to the sources at its input ports.

We express this matching condition mathematically as fol-lows. We partition the 2N × 2N S-matrix of the decouplingnetwork as

S =

[S11 S12

S21 S22

], (1)

where each Sij is an N × N submatrix. If we use ~a1,~b1 asN ×1 vectors of incident and reflected voltage waves at inputports, and ~a2,~b2 as vectors at the output ports (see Figure 1),then

~b1 = S11~a1 + S12~a2,~b2 = S21~a1 + S22~a2,

~a2 = SL~b2.

(2)

Let SLM denote the N ×N complex S-matrix of the cascadeof the matching network connected to the loads. Then ~b1 =SLM~a1, and using (2) yields

SLM = S11 + S12SL(I − S22SL)−1S21. (3)

We define a decoupling network as follows.Definition 1: A decoupling network for N dissipative recip-

rocal loads with S-matrix SL is a lossless, reciprocal, 2N -portnetwork S that satisfies SLM = 0.

The decoupling network, by definition, ensures that ~b1 = 0independently of the incident signal ~a1. Hence, the aggre-gate source power is delivered in its entirety to the loads.The network does not, however, necessarily maintain pairingassociations between sources and loads. Such associationsare generally meaningless when there is coupling, since anyattempts to stimulate one load by one source will potentiallystimulate all the loads, dependening on the nature of thecoupling. On average, over all possible incident signals, everyload receives 1/N of the total power. However, for anygiven incident signal on the decoupling network, some loads

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could be delivered more power than others. This is of noconsequence in a wireless system where the loads are antennasand are treated as approximately equal in their ability tocommunicate with a receiver. Hence, the decoupling networkdoes not isolate the loads from each other, but does providea set of isolated input ports at the characteristic impedancewhen its output ports are connected to the loads.

By solving SLM = 0 in (3), we conclude that it is necessaryand sufficient for a decoupling S to be an element of the set

S := S ∈ C2N×2N : S22 = SHL , SHS = I, ST = S, (4)

(see, for example, [18]). The number of real degrees offreedom in the set S can be computed by noting that a2N × 2N complex, symmetric, unitary matrix has 2N2 + Nreal degrees of freedom. Because (4) requires S22 = SHL ,which is symmetric, we have an additional N2+N constraintsimposed, leaving N2 degrees of freedom for S ∈ S. Thus, Shas N2 degrees of freedom to search for decoupling networkrealizations that have desirable properties.

III. REALIZATION OF DECOUPLING NETWORKS

We design decoupling networks using ideal, pure imaginaryadmittances (positive for capacitance, negative for inductance).For a lossless network, there is a one-to-one correspondencebetween its S-matrix, its admittance matrix, and its realizationusing a generalized multiport Π-network. These concepts arenow defined and explained.

A. Network realization

There is a one-to-one correspondence between the 2N×2Nscattering matrix S and 2N × 2N admittance matrix Y of thematching network using the Cayley transform [1]

Y = Y0(I − S)(I + S)−1, (5)

where Y0 = 1Z0

is the arbitrary characteristic admittance. Fordecoupling networks, S is symmetric and unitary, implyingthat Y is symmetric and skew-Hermitian (Y H = −Y ) andhence must be purely imaginary. We use yij to denote theijth element of Y . We then have yij = yji and Reyij = 0.

Network synthesis is the process of realizing the networkcircuit from the admittance matrix Y using components withthe prescribed admittances. For example, the two-port Π-network is shown in Figure 2(a), where c12 = −y12 is theadmittance of the component connecting ports 1 and 2, andcii = yii + y12 is the admittance of the component thatconnects port i to ground [1]. Although we have drawn thecij as lumped components, their actual realization may be in adistributed manner, depending on fd. The Y matrix is therefore

Y =

[c11 + c12 −c12

−c12 c12 + c22

]. (6)

We are interested in networks that have N single-ended inputsand N single-ended outputs, relative to ground, and hence havea total of 2N ports (not including ground) where ground isthe common return port for all inputs and outputs. Given anadmittance matrix of any dimension, we define the generalized2N -port Π-network as follows.

11y

22y

33y

44y

12y

13y

24y

34y

14y

23y

1

2

3

4

. . .

. . .

1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N(a) (b)

. . .

. . .1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N

11c

22c

33c

44c

12c

13c

24c

34c

14c

23c

1

2

3

4

(b)

22c21

12c

(a)

(c)

11c

Fig. 2. (a) A two-port Π-network that corresponds to a 2 × 2 admittancematrix. (b) A four-port Π-network that corresponds to a 4 × 4 admittancematrix. (c) A generalized Π-network has 2N ports, where the connectionsbetween ports through a (lumped or distributed) component is indicated by aline; each of the 2N ports is also connected to ground through a component,which is not shown. The generalized Π-network has 2N2 + N components.

Definition 2: A generalized 2N -port Π-network has 2Nsingle-ended ports and a ground. Every port i is connectedto port j through a component with admittance cij = cji, andevery port i is grounded through cii, where i 6= j = 1, . . . , 2N .

The 2N × 2N admittance matrix Y of the generalized 2N -port Π-network is

Y =

∑2Ni=1 c1i −c12 · · · −c1(2N)

−c12

∑2Ni=1 c2i · · · −c2(2N)

......

. . ....

−c1(2N) −c2(2N) · · ·∑2Ni=1 ci(2N)

. (7)

The resulting 4-port network is illustrated in Figure 2(b),and the 2N -port network in Figure 2(c). Because Figure 2(c)is dense, only lines are used to indicate components, andconnections to ground are not shown. In general, a fully-populated Π network requires 2N2 +N components to realizea 2N×2N admittance matrix Y ; 2N of the components are toground, while the remaining 2N2 −N interconnect the portsin direct correspondence with the off-diagonal elements of Y .

As seen in (7), there is a simple relation between yij andcij , the components connecting two distinct ports. In general,

cij =

−yij i 6= j,∑2Nk=1 yik i = j.

(8)

Observe that yij = yji = 0 (i 6= j) means that ports iand j have zero admittance between them, which impliescij = 0 (no component needed). Thus, the more off-diagonalentries of Y we can make zero, the simpler the networkrealization is. However, making yii = 0 does not eliminate thecomponent between node i and ground since this componenthas admittance value given by the sum of the ith row of Y ,or cii =

∑2Nk=1 yik.

B. Network complexity

Let 1(x) be an indicator such that 1(0) = 0 and 1(x) = 1for x 6= 0. The decoupling network simplification problem is

Y ? = arg minY :(Y0I−Y )(Y0I+Y )−1∈S

I(Y ), (9)

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where I(Y ) =∑i>j 1(yij) +

∑Ni=1 1(

∑Nj=1 yij) represents

the number of components with non-zero admittance in therealization of Y ; S is defined in (4). Hence, (9) represents asearch over all Y that are decoupling networks such that thenumber of components is minimized.

For a general dense Y , I(Y ) = 2N2 +N . However, since Shas N2 degrees of freedom, and the mapping between S andY is one-to-one, we have the potential to eliminate up to N2

components, and hence there is a lower bound I(Y ?) ≥ I?,where

I? = 2N2 +N −N2 = N2 +N. (10)

Method 1 in the next section achieves this lower bound in asystematic manner.

IV. DECOUPLING NETWORKS THAT MINIMIZE I(Y )

One way to solve (9) and achieve (10) is to create 2N2 zeroson the off-diagonal entries of Y , because zero off-diagonalentries directly eliminate components in the realization of thematching network. Recall that Y is symmetric and hence onlyN2 of these zeros are distinct. Similarly to (1), we define

Y =

[Y11 Y12

Y21 Y22

]. (11)

The method we now present makes Y22 diagonal and makesY12 upper triangular. Moreover, we also create specific zeros inY12. The total of 2N2 zeros is thereby created since Y21 = Y T12.

Method 1 (Systematic Design Method): For N dissipative,reciprocal loads, with N ×N S-matrix SL, and source admit-tance Y0, follow the steps below:

1) Calculate N ×N complex matrices P and Q using:

P = (I − SLSHL )−1,Q = SHL (I − SLSHL )−1.

(12)

and use pij , qij to denote the ijth element of P and Q.2) If N ≥ 3, solve the following quadratic equation for

real θ1:

d1 tan2 θ1 + d2 tan θ1 + d3 = 0, (13)

where

d1=γ21(δN1γN2 − γN1δN2)− δ21(δN1αN2 − γN1βN2)d2=α21(δN1γN2 − γN1δN2)− δ21(βN1αN2 − αN1βN2)

+γ21(βN1γN2 − αN1δN2)− β21(δN1αN2 − γN1βN2)d3=α21(βN1γN2 − αN1δN2)− β21(βN1αN2 − αN1βN2)

and

αij = Repij + qij, βij = Im−pij + qij,γij = Impij + qij, δij = Repij − qij.

If a real solution for θ1 does not exist or N = 2, setθ1 = π/2.

3) Calculate real θ2, . . . , θN using

tan θi =|pi1| cos(∠pi1 − θ1) + |qi1| cos(∠qi1 − θ1)

|pi1| sin(∠pi1 − θ1)− |qi1| sin(∠qi1 − θ1),

(14)and let

Θ = diag(θ1, . . . , θN ) (15)

be an N ×N diagonal matrix.

. . .

. . .

. . .

. . .

1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N

1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N(a) (b)

. . .

. . .

1

2

3

1N −

N

1N +

2N +

3N +

2 1N −

2N

Fig. 3. The 2N -port decoupling network generated by Method 1. The dashedline represents a component that is present when ? in (19) is non-zero. Notethat each of the 2N ports is also connected to the ground through a componentthat is not shown.

4) Use the Cholesky factorization to find an N × N reallower triangular matrix Lc that is non-negative along thediagonal and satisfies

LcLTc = 2ReejΘPe−jΘ+ 2Ree−jΘQe−jΘ − I.

(16)5) Realize the Π-network using (8) applied to

Y ? =

[Y ?11 Y ?12

Y ?21 Y ?22

], (17)

where

Y ?11 = jY0L−1c Re2ReejΘP+ 2Ree−jΘQ − e−jΘ

× (csc Θ)Lc (18a)

Y ?12 = (Y ?21)T = −jY0LTc csc Θ (18b)

Y ?22 = jY0 cot Θ. (18c)

Theorem 1: For N ≥ 3 and any real θ1 that solves (13),Method 1 creates Y ? such that I(Y ?) = I?. OtherwiseI(Y ?) = I? + 1. Furthermore, the structure of Y ? is

Y ? =

× × × · · · × × × 0 0 · · · 0 0× × × · · · × × 0 × × · · · × ?× × × · · · × × 0 0 × · · · × ×...

......

. . ....

......

......

. . ....

...× × × · · · × × 0 0 0 · · · × ×× × × · · · × × 0 0 0 · · · 0 ×× 0 0 · · · 0 0 × 0 0 · · · 0 00 × 0 · · · 0 0 0 × 0 · · · 0 00 × × · · · 0 0 0 0 × · · · 0 0...

......

. . ....

......

......

. . ....

...0 × × · · · × 0 0 0 0 · · · × 00 ? × · · · × × 0 0 0 · · · 0 ×

,

(19)where × represents non-zeros elements; ? is zero if N ≥ 3and a real solution to (13) exists. The design corresponding tothis Y ? appears in Figure 3, where the dashed line representsa component if and only if ? is non-zero.

Method 1 is systematic, constructive, and works for any Nand SL. The next section shows the proofs and some notesfor Method 1 and Theorem 1.

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V. PROOFS AND NOTES

A. Proof of Method 1

We begin with a lemma that is used in the proof.Lemma 1: Let N loads have S-matrix SL, with SVD SL =

ULΛLVHL . For any N × N unitary matrix V , define the

partitioned Y in (11) as

Y11 = Y0(V ∗AT + V AH)(V ∗AT − V AH)−1 (20a)

Y12 = Y T21 = −2Y0(AV H −A∗V T )−1 (20b)

Y22 = Y0(BV H +B∗V T )(AV H −A∗V T )−1, (20c)

where

A = UL(I − Λ2L)−

12 + VL(I − Λ2

L)−12 ΛL (21)

B = UL(I − Λ2L)−

12 − VL(I − Λ2

L)−12 ΛL. (22)

Then the admittance matrix Y of any decoupling network hasthe form (20) for some unitary matrix V . Conversely, the Π-network with component admittances given in (8) applied toY in (20) is a decoupling network for any V .

Proof: The proof proceeds by relating S ∈ S in (4) to Yin (11) using the Cayley transform.

Using the SVD of SL, we are able to write S ∈ S that ispartitioned as in (1) as

S11 = −V V HL U∗LΛLVT (23a)

S12 = V (I − Λ2L)

12UHL (23b)

S21 = U∗L(I − Λ2L)

12V T (23c)

S22 = VLΛLUHL . (23d)

for some unitary V . The converse is also true, that theseequations yield S ∈ S; see, for example, [22].

To obtain Y , the Cayley transform in (5) is used:

Y = Y0

[I − S11 −S12

−S21 I − S22

] [I + S11 S12

S21 I + S22

]−1

.

The Cayley transform is reversible and every Y yields adistinct S. The partitioned Y is then

Y11=Y0(I − S11 + S12(I + S22)−1S21)××(I + S11 − S12(I + S22)−1S21)−1

Y12=−2Y0××(S−1

12 + S−112 S11 + S22S

−112 + S22S

−112 S11 − S21)−1

Y21=−2Y0××(S−1

21 + S−121 S22 + S11S

−121 + S11S

−121 S22 − S12)−1

Y22=Y0(I − S22 + S21(I + S11)−1S12)××(I + S22 − S21(I + S11)−1S12)−1.

Substituting (23) into the above and simplifying yields (20).These steps are readily reversible, thus proving the converseas well.

The following matrix will be also used in the proof:

C = ejΘUL(I − Λ2L)−

12 + e−jΘVL(I − Λ2

L)−12 ΛL, (24)

where Θ = diag(θ1, · · · , θN ) and θ1, · · · , θN are given in(13), (14). Moreover, we will use the fact that when doingthe LQ factorization A = LaQa and C = LcQc, the lowertriangular matrices La and Lc are real. This result is due tothe fact that AAH and CCH are real symmetric matrices.

Proof of Method 1: We use Lemma 1 by finding asuitable unitary matrix V that is used in (20) to obtain (18).This shows that (18) is the admittance matrix of a decouplingnetwork for SL.

We apply the LQ factorization C = LcQc so that Lc is reallower triangular and has real positive diagonal elements. ThenV = Qc. The calculation using Lemma 1 is as follows. Westart with Y ?11. Equation (20a) yields

Y ?11 = jY0ReV AH(ImV AH)−1

= jY0ReQcAH(ImQcAH)−1

= jY0L−1c ReCAH(ImCAH)−1Lc

= jY0L−1c Re2ReejΘP+ 2Ree−jΘQ − e−jΘ(csc Θ)Lc,

which is (18a). Here the following relation is used:

ACH = 2RePe−jΘ+ 2ReQe−jΘ − ejΘ. (25)

We now consider Y ?12. Equation (20b) yields

Y ?12 = jY0(ImAV H)−1 = jY0(ImAQHc )−1

= jY0LTc (ImACH)−1 = −jY0L

Tc csc Θ,

where we use (25) again. This completes (18b).Last, we examine Y ?22. Equation (20c) yields

Y ?22 = −jY0ReBV H(ImAV H)−1

= −jY0ReBQHc (ImAQHc )−1

= −jY0ReBCHL−Tc LTc (ImACH)−1

= jY0ReejΘ(ImejΘ)−1 = jY0 cot Θ,

where (25) and

BCH = 2jImPe−jΘ − 2jImQe−jΘ+ ejΘ (26)

are used in the calculation. This completes (18c).

B. Proof of Theorem 1

We show that Y ? in (18) has zero elements at the locationsindicated in (19). In other words, we want to show that: (i)Y ?12 in (18b) is a lower triangular matrix and has zeros at 1ith(i = 2, . . . , N ) and 2N th elements (if real θ1 exists); and (ii)Y ?22 in (18c) is diagonal. The latter part is obvious since Θ isdiagonal by its definition, so we focus on the first part.

Equation (18b) shows that the zeros in Y ?12 mirror those inthe real lower triangular matrix Lc. Hence, (i) is proved if wecan show that the symmetric matrix LcL

Tc in (16) has zeros

at the i1th (i = 2, . . . , N ) and N2th elements. According to(16), the ijth element of LcLTc can be made zero by choosingθi, θj such that

Reejθipije−jθj+ Ree−jθiqije−jθj = 0. (27)

The i1th (i = 2, . . . , N ) and N2th elements of LcLTc are madezero by the solutions of (13) and (14).

First we explain how θi in (14) makes the i1th element ofLcL

Tc zero. Given any θ1, (14) is equivalent to

|pi1| cos(∠pi1 − θ1) cos θi + |qi1| cos(∠qi1 − θ1) cos θi= |pi1| sin(∠pi1 − θ1) sin θi − |qi1| sin(∠qi1 − θ1) sin θi

⇔|pi1| cos(∠pi1 − θ1 + θi) + |qi1| cos(∠qi1 − θ1 − θi) = 0⇔Repi1ej(θi−θ1)+ Reqi1ej(−θi−θ1) = 0.

Combining this result with (27) indicates the i1th element ofLcL

Tc is zero.

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Next we show that a solution to (13) makes the N2thelement of LcLTc equal to zero. In fact, (13) is obtained bysolving three equations jointly: (27) for (i, j) = (2, 1), (N, 1),and (N, 2). After some straightforward calculation, we find θ1

must satisfy (13). The detailed calculation is omitted.

C. Notes for Method 1

We mention some comments about Method 1:1) θ1 is computed differently from θ2, . . . , θN because the

solution for θ1 makes y2(2N) zero, while θi makesy1(N+i) zero for i = 2, . . . , N .

2) For N = 2, y24 cannot be made zero by choosing θ1

and hence we only get I(Y ?) = N2 +N + 1 = 7.3) The properties S must have so that θ1 has a real solution

are not yet known, but θ1 has a real solution in allexamples given in this paper.

4) The solutions to θ1, . . . , θN in (13) and (14) are notunique. Hence, there are multiple realizations of de-coupling networks with I(Y ?) = N2 + N . Thisnon-uniqueness is useful when searching among allminimum-complexity decoupling networks for realiza-tions with component admittance values within somedesired range.

5) If θ1 has no real solution or N = 2, θ1 can be any valueexcept 0,±π,±2π, . . .. Hence, multiple realizations ofa decoupling networks with I(Y ?) = N2 + N + 1 arepossible. We choose θ1 = π/2 for convenience.

6) The lower-triangular Cholesky factorization in step (4)can be replaced by an upper-triangular factorization.This results in a decoupling network whose topologyis “flipped” relative to Figure 3.

VI. CONCLUSION

We provided a systematic method to construct minimum-complexity decoupling networks for arbitrarily coupled loads.We utilized the properties of the decoupling network toestablish N2 + N as the lower bound for the number ofimpedances needed in a decoupling network. Then Method1 was introduced to realize this lower bound for any load sizeand structure.

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