A Design Approach for Compact Wideband Transformer WithFrequency-Dependent Complex Loads and Its Application toWilkinson Power Divider
Citation for published version:Liu, L, Liang, X, Jin, R, Fan, H, Bai, X, Zhou, H & Geng, J 2021, 'A Design Approach for Compact WidebandTransformer With Frequency-Dependent Complex Loads and Its Application to Wilkinson Power Divider',IEEE Transactions on Microwave Theory and Techniques, vol. 69, no. 3, pp. 1611-1624.https://doi.org/10.1109/TMTT.2020.3048334
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1
A Design Approach for Compact WidebandTransformer With Frequency-Dependent Complex
Loads and Its Application to WilkinsonPower Divider
Liang Liu , Xianling Liang , Senior Member, IEEE, Ronghong Jin , Fellow, IEEE,
Haijun Fan, Member, IEEE, Xudong Bai , Member, IEEE, Han Zhou ,
and Junping Geng , Senior Member, IEEE
Abstract— In this article, a new design approach for thewideband transformer with frequency-dependent complex loadsis proposed, which is based on the least-square method and thegeneralized modified small reflection theory (MSRT) with properelectrical parameters. Compared with the wideband transformersbased on other approaches such as the generalized MSRTand transmission line theory, the proposed ones feature easydesign, compact size, and wide bandwidth. For validation, severalwideband transformers with frequency-dependent complex loadsare designed, simulated, and measured. The measured results areconsistent with the calculated and simulated results, and a wide-band transformer with a fractional bandwidth of 68.5% (definedby 15-dB return loss) and an overall size of 0.025λ2
g is achieved.In addition, when applied to the design of wideband Wilkinsonpower divider, a fractional bandwidth of 110% ( defined by 25-dBisolation) and an overall size of 0.069λ2
g are attained.
Index Terms— Frequency-dependent complex loads,least-square method, modified small reflection theory(MSRT), wideband transformer, wideband Wilkinson powerdivider (WPD).
I. INTRODUCTION
NOWADAYS, with the rapid development of modernsociety, there is an increasing demand for high-date-rate
wireless communication systems. To fulfill this requirement,much attention has been gained into the research of widebandtransformers, which are widely used in various radio fre-quency components of wireless communication systems, suchas Wilkinson power dividers (WPDs), couplers, transceivers,low-noise amplifiers, and power amplifiers [1]–[6].
Manuscript received July 26, 2020; revised October 25, 2020; acceptedDecember 17, 2020. This work was supported by the National Natural ScienceFoundation under Grant 61671416 and Grant 61801447. (Correspondingauthor: Xianling Liang.)
Liang Liu, Xianling Liang, Ronghong Jin, Han Zhou, and Junping Gengare with the Department of Electronic Engineering, Shanghai Jiao TongUniversity, Shanghai 200240, China (e-mail: [email protected]).
Haijun Fan is with the Institute of Sensors Signals and Systems, Heriot-WattUniversity, Edinburgh EH14 4AS, U.K.
Xudong Bai is with the China Academy of Aerospace Electronics Tech-nology, Shanghai Aerospace Electronics Company Ltd., Shanghai 201821,China.
Color versions of one or more figures in this article are available athttps://doi.org/10.1109/TMTT.2020.3048334.
Digital Object Identifier 10.1109/TMTT.2020.3048334
In [7]–[10], the exact and approximate synthesis methodsto realize the wideband transformers by using multisectionquarter-wavelength transmission lines were proposed. How-ever, they are only applicable to the transformers with fixedand real source and load. In [11], a dual-band transformerbased on the conventional wideband Chebyshev response waspresented. Although the source and load are fixed at realimpedances, the results clearly indicate the feasibility of wide-band transformer using the multifrequency transformer. In par-ticular, the wideband transformer with frequency-dependentcomplex loads can be easily realized by multifrequency trans-former with several adjacent matching frequencies as well asfrequency-dependent complex loads. In [12] and [13], trans-formers consisting of lumped inductors and capacitors wereproposed, which are based on the multiple low-pass–bandpasstransformation sharing one topology for all operating bands.These transformers are characterized by wideband filteringresponses, but they are limited to low-frequency applicationsdue to the parasitic effect of lumped components. In orderto achieve high-frequency applications, various transmission-line transformers were proposed, which are based on themethods such as the transmission line theory [14]–[17] andthe modified small reflection theory (MSRT) [18]. Never-theless, when these methods are applied to the widebandtransformer with two adjacent matching frequencies, theirdominated equations become a quartic equation [16], [17]and a single matrix equation [18]. Therefore, they are notconvenient for practical engineering applications. Furthermore,their cascaded transmission lines are up to three and foursections, which results in bulk size. Also, their bandwidth ofthe single operating band is narrow since the adopted perfectmatching method has a severe impact on the bandwidth oftransformers.
In this article, a new design approach for the widebandtransformer with frequency-dependent complex loads is pre-sented. Its electrical parameters are no longer arbitrarily butproperly selected, and two complex conjugated impedancesdirectly constructed by the least-square method are substi-tuted for the actual impedances of transformers. Consequently,
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2 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
Fig. 1. Structure of wideband transformer realized by dual-frequencytransformer based on (a) generalized MSRT and (b) proposed method.
the dominated equations of transformer are simplified toclosed-form formulas. Meanwhile, the number of transmissionlines for the realization of wideband transformer with dualadjacent matching frequencies is reduced to two or three sec-tions, which leads to a compact size. Moreover, the bandwidthof the transformer is enhanced due to the wideband matchingbased on the least-square method instead of the conventionalperfect matching. In addition, the proposed approach can beeasily applied to the design of wideband WPDs with a compactsize.
II. THEORY AND DESIGN
In order to clearly distinguish the generalized MSRT fromthe proposed method, one first considers the wideband trans-former based on the former method in [18], which is realizedby a dual-frequency transformer with two adjacent matchingfrequencies of f and m f . As shown in Fig. 1(a), the widebandtransformer comprises quad-section transmission lines with thecharacteristic impedance of Zn (n = 1, 2, 3, 4) and electricallength of θ = θn at f (the transmission line number isfour since four unknown characteristic impedances can beobtained from the four independent equations, namely, the realand imaginary parts of two arbitrary complex impedancesat dual matching frequencies). The source and load of thetransformer are assumed as a fixed real impedance of Z0 and afrequency-dependent complex impedance of ZL(θ). Accordingto the perfect matching conditions at f and m f as well asthe restricted condition at the zero frequency, the dominatedequations of the transformer based on the generalized MSRTare given in [18]–[20]
�4(θ = 0) =4∑
n=0
�̃n = Z0−Z
Z0+Z(1a)
�4(θ) =4∑
n=0
�̃n·e− j2nθ = Z∗L(θ)−Z
Z∗L(θ)+Z
(1b)
�4(mθ) =4∑
n=0
�̃n·e− j2n(mθ) = Z∗L(mθ)−Z
Z∗L(mθ)+Z
(1c)
where �4(θ) and �4(mθ) denote the total reflection coefficientsof transformer at f and m f respectively, and the numberof total transmission lines is 4. Meanwhile, the asterisk of* denotes the complex conjugate operation and Z is anarbitrarily selected virtual real load. �̃n is the modified smallreflection coefficient represented in terms of the small reflec-tion coefficient �n , the characteristic impedance of Zn, andZ [19]
�̃0 = �0, �̃n = �n·n−1∏l=0
(1−�2
l
), n = 1, 2, 3, 4 (2a)
�n =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
Zn+1−Z
Zn+1+Z, n = 0
Zn+1−Zn
Zn+1+Zn, n = 1, 2, 3
Z0−Zn
Z0+Zn, n = 4.
(2b)
From (1), it is noted that for a dual-frequency transformerbased on the generalized MSRT, by selecting any virtualreal load of Z and the reference frequency of r f , a totalof five dominated equations are obtained by separating (1b)and (1c) into real- and imaginary-part equations. Althoughthese dominated equations can be reduced to a single matrixequation with 5 × 5 coefficient matrix, its closed-form solutioninvolves as many as 600 terms according to Cramer’s rule.
A. Proposed Wideband Transformer With Simplified Structure
In order to simplify the computation and structure ofwideband transformer based on the generalized MSRT, oneexpects that part of the dominant equations in (1) can beeliminated by proper selection of the electrical parameters.Subsequently, the unknown parameters of dominated equationsand the required transmission lines are reduced.
For discussion, the operating frequency range of widebandtransformer is assumed to be ( fl , fh). Then, the term of“capacitive and inductive loads” can be defined in the twosubranges of ( fl , fl/2+ fh/2 ) and ( fl/2+ fh/2, fh), whosereactance is almost negative across one of the above twosubranges but positive across the other subrange. Fig. 1(b)shows the proposed wideband transformer with the simpli-fied structure consisting of two parts. The first part, shownin the black dotted box, is the load transformation stagecomprising only one transmission line with the characteristicimpedance of Z1 and electrical length of θ1, which is utilizedto transform an arbitrary frequency-dependent load into theabove-defined capacitive and inductive loads. The second partshown in the red dashed box is the matching stage. It com-prises two-section transmission lines with the characteristicimpedance of Zn (n = 2, 3) and the electrical length ofθn (θ2 = θ3 = θ ), which is used to further transform theobtained capacitive and inductive loads to the source of Z0.Obviously, the matching and load transformation stages areequivalent to the two transformers shown in Fig. 2(a) and 2(b),respectively.
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LIU et al.: DESIGN APPROACH FOR COMPACT WIDEBAND TRANSFORMER 3
1) Closed-Form Algebraic Solution by Properly SelectingElectrical Parameters in Generalized MSRT: For the trans-former shown in Fig. 2(a), its dominated equations are
�2(θ = 0) =2∑
n=0
�̃n = Z0−Z
Z0+Z(3a)
�2(θ) =2∑
n=0
�̃n·e− j2nθ = Z ′L
∗(θ)−Z
Z ′L
∗(θ)+Z(3b)
�2(mθ) =2∑
n=0
�̃n·e− j2n(mθ) = Z ′L
∗(mθ)−Z
Z ′L
∗(mθ)+Z(3c)
where �2(θ) denotes the total reflection coefficient of thematching stage composed of dual-section transmission linesand �̃n(n = 0, 1, 2) is redefined by Z , Z2, Z3, and Z0.Substituting π−θ for θ in (3b), one has
�2(π−θ) =2∑
n=0
�̃n·e− j2n(π−θ) = �∗2(θ) = Z ′
L(θ)−Z
Z ′L(θ)+Z
. (4)
Equalizing (3c) and (4), one obtains the following relation-ship [21]:
mθ = π−θ ⇔ θ = π
1+m, Z ′
L(θ) = Z ′L
∗(mθ). (5)
Thus, by properly selecting the electrical parameters of (5)in the generalized MSRT, the dominated equations are sim-plified to (3a) and (3b). In order to further deduce theclosed-form solution to the simplified dominated equations,substitute Z ′
L(θ) = R′+ j X ′ into (3b), separate it into real-and imaginary-part equations, and then combine it with (3a),one has
2∑n=0
�̃n·cos(2nθ) = R′2+X ′2−Z 2
(R′+Z)2+X ′2 (6a)
2∑n=0
�̃n·sin(2nθ) = 2X ′ Z(R′+Z)2+X ′2 (6b)
2∑n=0
�̃n = Z0−Z
Z0+Z. (6c)
Generally, when Z is an arbitrarily selected virtual load similarto [18], the formula of (6a) is nonzero. Then, applying theproposed method to the simplified structure in Fig. 2(a),the entire dominated equations are reduced to 3 due to the
selected electrical parameter condition of (5). The total termsof the solution to (6) are reduced from 600 to 18 according toCramer’s rule. In particular, by selecting a virtual real load ofZ = (R′2+X ′2)1/2, the formula of (6a) is equal to 0. The totalterms of the solution of (6) are then further reduced from18 to 12 according to Cramer’s rule. After some algebraiccomputations, one obtains the following closed-form algebraicsolution:
From (7), it is observed that when the source, the complexconjugated loads, and m (or θ ) are given, the closed-form alge-braic solution of �̃n can be obtained. Subsequently, the smallreflection coefficient of �n can be derived from (2a), and thecharacteristic impedances of two transmission lines are givenby the following formulas according to (2b):
Z2 =√
R′2+X ′2·1+�0
1−�0, Z3 = Z2·1+�1
1−�1. (8)
It should be pointed out that the electrical length at thecenter frequency of two matching frequencies f and m f isπ/2 according to (5). Therefore, the center frequency of twomatching frequencies is the reference frequency of r f in thegeneralized MSRT, and the condition of (5) is equivalent to
r f = 1+m
2f, Z ′
L(θ) = Z ′L
∗(mθ). (9)
From (9), the parameter of r = (1+m)/2 can be obtainedfrom the two given adjacent matching frequencies, f andm f , which is distinctly different from that in the generalizedMSRT [18]–[20].
2) Realization of Two Complex Conjugated Loads andWideband Matching Based on Least-Square Method: InSection II-B1, although a wideband transformer based on twoadjacent matching frequencies can be realized by two-sectiontransmission lines, its loads at the two matching frequenciesshould be conjugated according to (5) or (9). To fulfill thisrequirement, one widely used approach is the introductionof an additional impedance-transformation transmission line[16], [17]. However, this may lead to a considerable increasein size. In addition, it is not applicable to all dual-frequencytransformers with frequency-dependent complex loads [6].Therefore, it is preferred that the two complex conjugatedloads can be constructed directly.
For analysis, the constructed two complex conjugated loadsare assumed to be R′± j X ′ at f and m f , and the load ofthe matching-stage equivalent transformer is assumed to becapacitive and inductive load of Z ′
L(θn) = R′n+ j X ′
n. Con-sidering that Z ′
L(θn) is frequency-dependent and its reactanceacross the two half frequency-ranges has opposite signs, oneexpands it around the constructed two complex conjugated
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4 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
loads. In addition, the mean impedance R′± j X
′of loads is
defined by a minimum error of Zmin between itself and theloads at N equal-interval frequencies. Namely
Z ′L(θn) = R′
n+ j X ′n
=
⎧⎪⎪⎨⎪⎪⎩R′+ j X ′+�R′
n+ j�X ′n, 1 ≤ n ≤
[N
2
]R′− j X ′+�R′
n+ j�X ′n, N+1−
[N
2
]≤ n ≤ N
(10a)
Zmin = 1
2[
N2
] ·⎧⎨⎩
[ N2 ]∑
n=1
[(R
′−R′n
)2+(
X′−X ′
n
)2]
+N∑
n=N+1−[ N2 ]
[(R
′−R′n
)2+(−X
′−X ′n
)2]⎫⎬⎭.
(10b)
In (10), [N/2] denotes the maximum integer equal or smallerthan N/2, whereas �R′
n and �X ′n represent the resistance and
reactance difference between the nth actual and constructedloads, respectively. Obviously, the minimum error of (10b)can be seen as a least-square equation with two unknownsof R
where the superscript of T denotes the transpose operation ofa matrix and mod (N , 2) is equal to 1 or 0 for the odd or evennumbers N , respectively. After some algebraic computations,R
′and X
′in (11) can be simplified as
R′ = 1
2[
N2
] ·⎛⎝[ N
2 ]∑n=1
R′n+
N∑n=N+1−[ N
2 ]
R′n
⎞⎠ (12a)
X′ = 1
2[
N2
] ·⎛⎝[ N
2 ]∑n=1
X ′n−
N∑n=N+1−[ N
2 ]
X ′n
⎞⎠. (12b)
As the conventional perfect matching condition with zeroreflection at the target frequency usually leads to narrow
matching, in order to achieve a wideband matching, the condi-tion of minimum average power reflection coefficient |� p(θ)|in a given operating frequency range of ( fl , fh) should besatisfied, namely
|� p(θ)| = 1
N·
N∑n=1
|�p,n(θn)| = 1
N·
N∑n=1
|�v,n(θn)|2 (13)
where �p,n(θn) and �v,n(θn) denote the power and voltagereflection coefficients at the nth frequency, respectively, and|�v,n(θn)| is given by [18], [26]
|�v,n(θn)| = | Z in(θ)−Z ′L
∗(θn)
Z in(θ)+Z ′L(θn)
| = | Z in(θ)−R′n+ j X ′
n
Z in(θ)+R′n+ j X ′
n
|. (14)
In (14), Z in(θ) is the input impedance of matching stage,which is complex conjugated to the optimum loads of Z ′
L(θ) =R′± j X ′ for a well-matched transformer. Then, one has
Z in(θ) =
⎧⎪⎪⎨⎪⎪⎩R′− j X ′, 1 ≤ n ≤
[N
2
]R′+ j X ′, N+1−
[N
2
]≤ n ≤ N.
(15)
Substituting (10a), (14), and (15) into (13), after some alge-braic computations, one obtains
|� p(θ)|
= 1
2[
N2
] ·[ N2 ]∑
n=1
[(R′−R′
n
)2+(X ′−X ′n
)2(R′+R′
n
)2+(X ′−X ′n
)2]
+ 1
2[
N2
] · N∑n=N+1−[ N
2 ]
[(R′−R′
n
)2+(X ′+X ′n
)2(R′+R′
n
)2+(X ′+X ′n
)2]
≈ 1
2[
N2
] ·⎧⎪⎨⎪⎩
[ N2 ]∑
n=1
⎡⎢⎣ (R′−R′
n
)2+(X ′−X ′n
)2(R
′+R′n
)2+(
X′−X ′
n
)2
⎤⎥⎦
+N∑
n=N+1−[ N2 ]
⎡⎢⎣ (R′−R′
n
)2+(X ′+X ′n
)2(R
′+R′n
)2+(
X′+X ′
n
)2
⎤⎥⎦⎫⎪⎬⎪⎭ (16a)
= 1
2[
N2
] ·⎧⎨⎩
[ N2 ]∑
n=1
[(bn R′−bn R′
n
)2+(bn X ′−bn X ′n
)2]
+N∑
n=N+1−[ N2 ]
[(bn R′−bn R′
n
)2+(bn X ′+bn X ′n
)2]⎫⎬⎭
bn
=
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
1√(R
′+R′n
)2+(
X′−X ′
n
)2, for 1 ≤ n ≤
[N
2
]1√[
R′+R′
n+mod(N,2)
]2+[
X′+X ′
n+mod(N,2)
]2
for[
N2
]+1 ≤ n ≤ 2
[N
2
](16b)
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LIU et al.: DESIGN APPROACH FOR COMPACT WIDEBAND TRANSFORMER 5
Fig. 3. Relationship among f , m f , fl , and fh .
where, due to the relationship of |R′−R′| � R
′+R′n,
the expressions of (R′+R′n)
2+(X ′−X ′n)
2 and (R′+R′n)
2+(X ′+X ′
n)2 in the denominator are well approximated to (R
′+R′
n)2+(X
′−X ′n)
2 and (R′+R′
n)2+(X
′+X ′n)
2, respectively. Sim-ilar to (10b), the two unknowns of R′ and X ′ in (16b) can alsobe obtained by the least-square method
A = (BT ·B)−1·BT ·Y (17a)
A = (R′ X ′)T
, Y =(
y1 y2 · · · yn · · · y4[ N2 ])T
(17b)
yn =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
b n+12
R′n+1
2, odd n, 1 ≤ n ≤ 2
[N
2
]b n
2X ′
n2, even n, 1 ≤ n ≤ 2
[N
2
]b n+1
2R′
n+12 +mod(N,2)
, odd n, 2
[N
2
]+1 ≤ n ≤ 4
[N
2
]−b n
2X ′
n2 +mod(N,2), even n, 2
[N
2
]+1 ≤ n ≤ 4
[N
2
](17c)
B =(
b1 0 · · · bn 0 bn+1 0 · · · b2[ N2 ] 0
0 b1 · · · 0 bn 0 bn+1 · · · 0 b2[ N2 ]
)T
.
(17d)
In summary, when the capacitive and inductive loads ofZ ′
L(θn) = R′n+ j X ′
n in the matching stage are given in an oper-ating frequency range of ( fl , fh), the two complex conjugatedloads, R′± j X ′, can be directly obtained from (17d). On theother hand, when a wideband transformer in the operatingfrequency range of ( fl , fh) is realized by a dual-frequencytransformer with the matching frequencies of f and m f , it isusually well-matched for the frequencies adjacent to f andm f due to their accurate input impedances of Z in(θ) = R′∓j X ′. Therefore, in order to obtain the small power reflectioncoefficient covering the entire operating frequency range of( fl , fh), the entire operating frequency range is divided intothe two subranges of ( fl , fl/2+ fh/2) and ( fl/2+ fh/2, fh),as shown in Fig. 3. Meanwhile, f and m f are usually selectedto be the center frequencies of the above subranges. Thus,one has f = (3 fl+ fh)/4 and m f = ( fl+3 fh)/4, and thefrequency ratio of m in (5) is then given by
m = fl+3 fh
3 fl+ fh. (18)
C. Load Transformation Stage of Wideband Transformer
In Section II-B, two complex conjugated loads are directlyconstructed based on the least-square method. According tothis, wideband transformer with two well-matched frequenciesis realized by only two transmission lines. However, the loadof matching-stage equivalent transformer should be capacitiveand inductive. Therefore, similar to [27], the load transforma-tion stage shown in Fig. 2(b) should be introduced to obtainthe desired capacitive and inductive loads of matching stage.According to the reactance of loads, the discussion is classifiedinto the following three cases.
1) Wideband Transformer With Capacitive and InductiveLoads: In this case, there is no need for load transforma-tion stage. In other words, the electrical length of the loadtransformation stage and the load of matching stage are givenby
θ1 = 0, Z ′L( f ) = Z L(θ). (19)
2) Wideband Transformer With Capacitive Loads: In thiscase, the reactance of the loads is almost negative in the operat-ing frequency range of ( fl , fh). For analysis, the load of trans-former is assumed to be Z L(θ) = Rn− j Xn(Xn > 0). Then,the transformed impedance of Z ′
L( f ), shown in Fig. 2(b),is given by
Z ′L( f ) = Z1·
ZL(θ)+ j Z1 tan(
ff0θ1
)Z1+ j Z L(θ) tan
(ff0θ1
) (20)
where f0 is the lower matching frequency of widebandtransformer with two matching frequencies and θ1 is theelectrical length at f0. When the characteristic impedance ofZ1 is sufficiently large, one obtains the inequality of Z1 |Z L(θ) tan(θ1 f/ f0)|. Then, the input impedance of Z ′
L( f ) canbe well approximated to
Z ′L( f ) ≈ Z L(θ)+ j Z1 tan
(f θ1
f0
)= Rn+ j
[Z1 tan
(f θ1
f0
)−Xn
]. (21)
From (21), it is seen that Z ′L( f ) is obtained by adding a posi-
tive reactance to Z L(θ). Ideally, one expects Im[Z ′L( f )] = 0 in
the entire operating frequency range. Note that the frequencynumber is often much larger than the only variable of θ1, so therelationship of Im[Z ′
L( f )] = 0 generally is not valid at allfrequencies. However, the solution of θ1 can be obtained withthe least-square method.
Considering that the expression of f θ1/ f0 is definitelypositive, one expands tan( f θ1/ f0) around 0.5 rather thanthe normal value of 0 to obtain an approximate expressionapplicable to a wider frequency range
tan
(f θ1
f0
)≈(
f θ1
f0−0.0793
)·sec2 0.5. (22)
Substituting (22) into (21) and solving for θ1 based on thecondition of Im[Z ′
L( f )] = 0 and the least-square method, one
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6 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
has
θ1 = 1∑Nn=1 f 2
n
·N∑
n=1
[f0 fn
(0.0793+0.7702
Xn
Z1
)]. (23)
In (23), the imaginary part of loads −Xn , the frequency of fn ,and the lower matching frequency f0 are known for a giventransformer. Meanwhile, Z1 is selected as a sufficiently largecharacteristic impedance such as 130 or 140 � according tothe used substrate. Therefore, the parameter of θ1 can be easilyobtained. Based on it, the matching-stage loads of Z ′
L(θ) aretransformed into capacitive and inductive impedances, whichcan be derived by substituting θ1 and Z1 into (20). Subse-quently, the remaining electrical parameters of the widebandtransformer can be solved according to Section II-B.
3) Wideband Transformer With Inductive Loads: In thiscase, the reactance of the loads is almost positive in theoperating frequency range of ( fl , fh). For analysis, the load oftransformer at the nth frequency is assumed to be ZL(θn) =Rn+ j Xn = |Z L(θn)|e jθn (Xn > 0, θn > 0). Then, the dis-cussion is classified into the following two cases according towhether the condition of θn � 1 is valid.
In the case of θn � 1, the transformation at a singlefrequency from Z L(θn) = Rn+ j Xn to Z0 is first consid-ered. In the Smith chart with the normalized characteristicimpedance of (Z0|Z L(θn)|)1/2, the reflection coefficients of thesource and load are
| Z0−√Z0|Z L(θn)|
Z0+√Z0|Z L(θn)| | = | |Z L(θn)|−√
Z0|Z L(θn)||Z L(θn)|+√
Z0|Z L(θn)| |
≈ | |Z L(θn)|e jθn−√Z0|Z L(θn)|
|Z L(θn)|e jθn+√Z0|Z L(θn)| |
= | ZL(θn)−√Z0|Z L(θn)|
Z L(θn)+√Z0|Z L(θn)| | (24)
where e jθn ≈ 1+ jθn ≈ 1 is used due to the condition ofθn � 1. From (24), it is seen that the source and load are ona circle of almost equal reflection coefficient. Thus, the loadof Rn+ j Xn can be well matched to Z0 by load-transformationstage with the characteristic impedance Z1 = (Z0|Z L(θn)|)1/2.
Now, one considers the frequency-dependent complex loadZL(θ) in the frequency range of ( fl , fh), which can be wellrepresented by its mean impedance of Z L(θ). Similarly, ZL(θ)also can be well matched to Z0 by the load transformationstage with the characteristic impedance of Z1 = Z 1, whereZ 1 is given by
Z 1 =√
Z0·|Z L(θ)| =√√√√Z0·| 1
N
N∑n=1
(Rn+ j Xn)|. (25)
In order to obtain θ1 of the load transformation stage, substitut-ing Rn+ j Xn and Z 1 for Z L(θ) and Z1 in (20) and separatingit into the real and imaginary parts, one obtains
Z ′L( f )
=Z
21 Rn sec2
(f θ1
f0
)R2
n tan2(
f θc
f0
)+[
Z 1−Xn tan(
f θ1
f0
)]2
+j Z1
[Z 1 Xn+
(Z
21−X2
n−R2n
)tan(
f θ1
f0
)−Z1 Xn tan2
(f θ1
f0
)]R2
n tan2(
f θ1
f0
)+[
Z 1−Xn tan(
f θ1
f0
)]2 .
(26)
For the optimum parameter of θ1, it is expected Im[Z ′L( f )] =
0 across ( fl , fh). Thus, the least-square solution to θ1 is
kn =
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
Z21−R2
n−X2n+√(
Z21−X2
n−R2n
)2+4Z21 X2
n
2Z1 Xn
, Z < Z0
Z21−R2
n−X2n−√(
Z21−X2
n−R2n
)2+4Z21 X2
n
2Z1 Xn
, Z > Z0
(27a)
θ1 =
⎧⎪⎪⎪⎨⎪⎪⎪⎩∑N
n=1 f0 fn·arctan(kn)∑Nn=0 f 2
n
, Z < Z0∑Nn=1 f0 fn·[π+arctan(kn)]∑N
n=0 f 2n
, Z > Z0.
(27b)
In the case that θn � 1 is not valid, the approximate errorin (24) will be introduced. However, Z1 can be expandedaround Z1 of (25) and then is deduced from the conditionof minimum average power reflection coefficient � p(θ) over( fl , fh). After some algebraic computations, � p(θ) is approx-imated to
� p(θ)
= 1
N·
N∑n=1
⎧⎨⎩ Z 1 tan(
fnθ1
f0
)√
cn·Z 2
1
−Z 1 Z 2
0 tan(
fnθ1
f0
)−Xn
[Z
21+Z 2
0 tan2(
fnθ1
f0
)]√
cn
⎫⎬⎭2
+ 1
N
N∑n=1
⎧⎨⎩ Z0 Z 21 sec2
(fnθ1
f0
)√
cn−
Rn
[Z
21+Z 2
0 tan2(
fnθ1
f0
)]√
cn
⎫⎬⎭2
(28)
where, cn is
cn ={
Z0 Z2
sec2
(f θ1
f0
)+Rn
[Z
21+Z 2
0 tan2
(f θ1
f0
)]}2
+{
Z(
Z2−Z 2
0
)tan
(f θ1
f0
)+Xn
[Z
21+Z 2
0 tan2
(f θ1
f0
)]}2
.
Similarly, the formula of (28) can be treated as a least-squareequation with the only unknown of Z 2
1. Then, the optimumcharacteristic impedance of Z1 corresponding to the minimum� p(θ) is given by
b1n =Z1 tan
(fnθ1
f0
)√
cn, b2n =
Z0 sec2(
fnθ1
f0
)√
cn(29a)
y1n =Z1 Z 2
0 tan(
fnθ1
f0
)−Xn
[Z
21+Z 2
0 tan2(
fnθ1
f0
)]√
cn(29b)
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LIU et al.: DESIGN APPROACH FOR COMPACT WIDEBAND TRANSFORMER 7
Fig. 4. Design flowchart of wideband transformer.
y2n =Rn
[Z
21+Z 2
0 tan2(
fnθ1
f0
)]√
cn
Z1 =√√√√∑N
n=1(b1n y1n+b2n y2n)∑Nn=1
(b2
1n+b22n
) . (29c)
When Z1 and θ1 of load transformation stage with inductiveloads are derived, the capacitive and inductive loads of Z ′
L(θ)in the matching stage can be obtained by (20). Subsequently,the electrical parameters of the matching stage can be attainedaccording to Section II-B.
In summary, the load transformation stage can be absent forthe wideband transformer with capacitive and inductive loads,while it can be realized by one-section transmission line forthe transformer with capacitive or inductive loads. Consideringthat the matching stage comprises two-section transmissionlines, the entire wideband transformer can be realized by onlytwo- or three-section transmission lines.
D. Design Flowchart of Wideband Transformer
In order to clearly demonstrate the design process of awideband transformer, Fig. 4 shows the design flowchart basedon the proposed method, which includes eight steps in general.Alternately, in order to obtain a simple but less exact solution,a computer-aided design or optimization procedure with regardto the only two parameters of Z2 and Z3 can be used insteadof those between steps 4 and 7.
E. Wideband Transformer in WPD Application
In Sections II-A–II-D, a generalized approach for the wide-band transformer with frequency-dependent complex loadshas been proposed. Based on it, a wideband WPD shownin Fig. 5(a) is developed, which comprises two stages. The firststage named “matching element” is composed of two-sectioncoupled lines (CLs), whereas the second stage named “iso-lation element” comprises a one-section normal transmissionline and an isolation resistor. Due to the symmetry of theWPD, its odd- and even-mode equivalent circuits are obtainedand shown in Fig. 5(b) and (c), respectively. Obviously,the odd- and even-mode equivalent circuits of matching ele-ment act as the matching stage of a transformer discussedin Section II-B, while those of the isolation element work
as the load transformation stage of a transformer. Therefore,the isolation element is used to obtain capacitive and inductiveloads, which are desired for the matching element. Besides,it is also aimed to fulfill the isolation and load conditionsof ZmoL(θm) ≈ ZmeL(θm) = 2Z0, and then, small andfeasible coupling coefficients of matching-element CLs inpractical applications can be obtained. The reason for thisis that under the above condition, the odd- and even-modeequivalent circuits of WPD shown in Fig. 5(b) and (c) can betreated as two transformers with identical source of Z0 andapproximately equal loads of ZmoL(θm) or ZmeL(θm). Hence,the odd- and even-mode characteristic impedances of matchingelement are approximately equal. For analysis, the electricallength of isolation-element transmission line is set to θi atthe center frequency of wideband WPD, and the characteristicimpedance is assumed to be Zi . Meanwhile, the electricallength of matching-element CLs defined as θm at the centerfrequency of wideband WPD, and the odd- and even-modecharacteristic impedances are assumed to be Zmon and Zmen
(n = 1, 2), respectively.1) Design of Isolation Element: Due to the additional
isolation resistor R0 and the zero termination impedance ofodd-mode equivalent circuit in the isolation element, the odd-mode equivalent circuit of the isolation element is differentfrom the load transformation stage in a wideband transformer.Thus, the parameters of the isolation element in WPD are notdetermined by (19), (23), (27), or (29) but can be derived fromthe load condition of ZmoL(θm) ≈ ZmeL(θm) = 2Z0.
In the even-mode equivalent circuit as shown in Fig. 5(c),in order to fulfill the load condition of ZmeL(θm) = 2Z0,the termination impedance of 2Z0 should be transformed to
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8 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
2Z0 by the transmission line of isolation element. Obviously,the characteristic impedance of isolation-element transmissionline can be selected to be
Zi = 2Z0. (30)
In the odd-mode equivalent circuit as shown in Fig. 5(b),the load of ZmoL(θm) is given by
Z ino(θi) = j Zi tan θi (31a)
ZmoL(θm) = Z ino(θi )R0
2Z ino(θi)+R0=
R02
1+ R02Z ino(θi )
. (31b)
From (31b), it is readily concluded that ZmoL(θm) ≈ R0/2under the condition of |Z ino(θi)| R0/2. Therefore, in orderto fulfill the load condition of ZmoL(θm) ≈ 2Z0, Z ino(θi)should be chosen as large as possible, whereas R0 should beapproximate to 4Z0. Moreover, to obtain large |Z ino(θi) =j Zi tan θi | in the entire operating bandwidth of WPD, the fre-quency with maximum |Z ino(θi)| should be selected to be thecenter frequency. Obviously, |Z ino(θi)| increases with θi in therange of (0, π/2) but decreases in the range of (π/2, π).Thus, θi should be selected as π/2 at the center frequency ofwideband WPD. Namely
θi = π
2. (32)
Now, let us consider the parameter of R0. As pointed outabove, in the case of large |Z ino(θi)| in the entire operatingfrequency range of ( fl , fh), R0 should be approximate to 4Z0
to fulfill the load condition of ZmoL(θm) ≈ 2Z0. In the caseof other |Z ino(θi)|, although ZmoL(θm) is also approximate tothe constant R0/2 by selecting R0/2 � |Z ino(θi)|, it resultsin ZmoL(θm) ≈ 0. Therefore, R0 should be sufficiently largeto fulfill the load condition of ZmoL(θm) ≈ 2Z0, and acompromise is made by selecting R0 as R0 = min[|Z ino(θi)|]for this case. Then, for any Z ino(θi), R0 can be given by
R0 = min[|Z ino(θi)|, 4Z0]. (33)
2) Design of Matching Element: In the odd-mode equivalentcircuit, from (31b), it is easily concluded that the imaginaryparts of ZmoL(θm) in the ranges of (0, π/2) and (π/2, π)are positive and negative, respectively. Thus, ZmoL(θm) is acapacitive and inductive load in the operating frequency rangeof WPD. Subsequently, the design of the odd-mode equivalentcircuit is reduced to a wideband real-to-complex transformerwith capacitive and inductive loads discussed in Section II-B,whose source of ZS and load of Z ′
L(θ) are given by
Z S = Z0, Z ′L(θ) = ZmoL(θm) = j Zi R0 tan θi
j2Zi tan θi+R0. (34)
Thus, the electrical length of the matching element and twooptimum complex conjugated loads of R′± j X ′ are derivedfrom (5) and (17d), respectively. Then, the odd-mode char-acteristic impedances of matching-element CLs are easilyobtained by (2a), (7), and (8).
In the even-mode equivalent circuit, obviously, the designis reduced to a real-to-real transformer, whose source of Z S
and load of Z ′L(θ) are given by
Z S = Z0, Z ′L(θ) = R′ = 2Z0. (35)
Fig. 6. Design flowchart of wideband WPD.
Fig. 7. Capacitive and inductive loads of wideband transformer and itsequivalent circuit.
Based on (2a), (7), and (8), all the even-mode characteristicimpedances of matching-element CLs can be easily obtained.
3) Design Flowchart of Wideband WPD: In order to clearlydemonstrate the design process of a wideband WPD, Fig. 6shows its design flowchart. Similarly, in order to obtain asimple but less exact solution, a computer-aided design oroptimization procedure with regard to the only two parametersof Zmo1 and Zmo2 can be used instead of those betweensteps 5 and 6.
III. SIMULATION, COMPARISONS, AND DISCUSSION
To validate the proposed method, several wideband trans-formers with various loads and a wideband WPD are taken asexamples.
A. Wideband Transformers With Various Loads
1) Capacitive and Inductive Loads: The first example isa transformer with source impedance of Z0 = 50 � and afrequency-dependent complex load of ZL(θ), whose equiva-lent circuit consists of an inductor of L = 25 nH, a capacitorof C = 3 pF, and a resistor of R = 180 �, as shownin Fig 7. Based on the exact method [16], [17], the gener-alized MSRT [18], and the proposed method, their electricalparameters listed in Table I can be readily obtained, where allelectrical lengths are referred to 0.71 GHz [for the general-ized MSRT, the structure and parameter definition are shown
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LIU et al.: DESIGN APPROACH FOR COMPACT WIDEBAND TRANSFORMER 9
TABLE I
ELECTRICAL PARAMETERS OF WIDEBAND TRANSFORMER WITHCAPACITIVE AND INDUCTIVE LOADS
Fig. 8. Frequency responses of wideband transformer with capacitive andinductive loads.
in Fig. 1(a), whereas for the exact and proposed methods,they are shown in Fig. 1(b)]. From Table I, the transformerbased on the proposed method comprises only two-sectiontransmission lines, while those based on the exact methodand the generalized MSRT comprise three- and four-sectiontransmission lines, respectively. Accordingly, the size increaseof the latter compared to the former is up to 50.8% and 36.7%.
Fig. 8 shows the frequency responses of transformers basedon various methods. On the one hand, the return loss of trans-former based on the exact method and generalized MSRT attwo aimed frequencies is larger than 35 dB due to their adoptedperfect matching method, while that based on the proposedmethod is limited to 15–20 dB in the entire operating band.On the other hand, as shown in Table I, the bandwidth (definedby 15-dB return loss) of transformer based on the proposedmethod is larger than those based on the exact method andgeneralized MSRT as the least-square method attributes to thewideband matching.
2) Inductive Loads: The equivalent circuit in [13], withelectrical parameters of L = 12 nH, C = 0.5 pF, and R =200 �, is used to mimic the inductive frequency-dependentloads in the frequency range of 1.6–2.6 GHz, which coversthe applications of digital cellular system (DCS), personalcommunications service (PCS), long-term evolution (LTE),as well as wireless local area network (WLAN). It is easilyvalidated that there is no exact solution for the dual-frequencytransformer with loads of Z L(θ1) = 89.97 + j33.20 � at1.76 GHz and ZL(θ2) = 65.53 + j78.04 � at 2.28 GHz.The reason for this is that the two frequency-dependent loadscannot be transformed into any complex conjugated loadsthrough a transmission line. However, the capacitive and
Fig. 9. Inductive loads of wideband transformer and its equivalent circuit.
TABLE II
ELECTRICAL PARAMETERS OF WIDEBAND TRANSFORMERS
WITH INDUCTIVE LOADS
Fig. 10. Frequency responses of wideband transformer with inductivefrequency-dependent complex loads.
inductive loads of Z ′L(θ) as shown in Fig. 9 can be achieved by
introducing a load transformation stage. Following the designflowchart shown in Fig. 4, all electrical parameters of thetransformer can be obtained and listed in Table II (θ1, θ , andθtotal are electrical lengths at 1.76 GHz).
Fig. 10 shows the frequency responses of wideband trans-former with inductive loads. It is noted that the transformeris well-matched between 1.8 and 2.4 GHz, which correspondsto a fractional bandwidth of 30%.
3) Capacitive Loads: In the case of capacitive loads,the transformer in [13] is taken as an example, whose sourceand load are Z0 = 50 � and ZL(θ) with the equivalent circuitparameters of L = 2 nH, C = 2 pF, and R = 200 � shownin Fig. 11. On the one hand, based on the exact method [16],[17] and generalized MSRT [18], the electrical parametersof wideband transformers are readily obtained and listed inthe second and third rows of Table III, respectively (θ1, θ ,and θtotal are electrical lengths at 0.6 GHz). On the otherhand, considering the capacitive load of transformer in thefrequency range of 0.3–1.2 GHz, an additional transmissionline with the characteristic impedance of Z1 = 140 � isintroduced for the transformer based on the proposed method.
.
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10 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
Fig. 11. Capacitive loads of wideband transformer and its equivalent circuit.
TABLE III
ELECTRICAL PARAMETERS OF WIDEBAND TRANSFORMERSWITH CAPACITIVE LOADS
Fig. 12. Frequency responses of wideband transformer with capacitive loads.
Then, following the design flowchart shown in Fig. 4, itselectrical parameters are obtained and listed in the fourth rowof Table III. Although it comprises three-section transmissionlines, its size is still smaller than those based on the exactmethod and the generalized MSRT.
Fig. 12 shows the frequency responses of transformers basedon various design methods. It is observed that the bandwidthsof transformers based on the exact method [16], [17] and thegeneralized MSRT [18] are smaller than the one based on theproposed method. The reason for this is that the latter is basedon the wideband-matching method.
Another example with capacitive loads is the transformerwith source of Z0 = 50 � and frequency-dependent loadin [18], which mimics the load of a power amplifier. Thetransformer operates in the frequency range of 0.9–3.0 GHz,which covers various applications such as global positioningsystem (GPS), DCS, PCS, LTE, as well as WLAN. In orderto realize the frequency-dependent loads shown in Fig. 13,
Fig. 13. Frequency-dependent loads of fabricated transformer and itsequivalent circuit.
TABLE IV
ELECTRICAL PARAMETERS OF FREQUENCY-DEPENDENT LOADS
AND PHYSICAL DIMENSIONAL PARAMETERSOF FABRICATED TRANSFORMER
Fig. 14. (a) Dimension of wideband transformer. (b) Photograph of fabricatedtransformer. (c) Photograph of TRL fixture for calibration.
its equivalent circuit parameters are selected as the oneslisted in Table IV. Following the design flowchart shownin Fig. 4, the electrical parameters of wideband transformerbased on the proposed method are easily obtained and listedin Table IV (all electrical lengths are referred to 1.627 GHz).Then, the transformer is simulated and fabricated on aRogers 4350B substrate with the relative dielectric constantof 3.48 and the thickness of 0.508 mm, whose structure andphysical dimensions are shown in Fig. 14(a) and Table IV,respectively. Fig. 14(b) and (c) further shows the photographsof fabricated transformer with an overall size of 0.025λ2
g and
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LIU et al.: DESIGN APPROACH FOR COMPACT WIDEBAND TRANSFORMER 11
Fig. 15. Frequency responses of fabricated wideband transformer withfrequency-dependent complex loads.
TABLE V
COMPARISONS OF FREQUENCY-DEPENDENT TRANSFORMERSBASED ON VARIOUS METHODS
Fig. 16. Physical dimension of wideband WPD.
TABLE VI
ELECTRICAL AND DIMENSION PARAMETERS OF WIDEBAND WPD
its thru-reflect-line (TRL) calibration fixture, respectively (λg
is the guided wavelength at 1.627 GHz).Fig. 15 shows the calculated (MATLAB 2016a), simulated
(HFSS 15.0), and measured (Agilent E8361C) frequencyresponses of the fabricated transformer, which are obtainedon the basis of postprocessing similar to [18]. It is noted thatthe transformer is well-matched between 1.2 and 2.45 GHzwith a return loss larger than 15 dB, which corresponds to afractional bandwidth of 68.5%. Meanwhile, the insertion lossis smaller than 0.3 dB. In addition, good agreements among the
Fig. 17. Photograph of fabricated wideband WPD.
Fig. 18. Frequency responses of wideband WPD. (a) |S11| and |S21|. (b) |S22|.(c) |S32|.
calculated, simulated, and measured results are observed, andthe slight differences are mainly attributed to the meanderingtransmission lines and the manufacture tolerance.
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12 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
TABLE VII
COMPARISONS OF VARIOUS WIDEBAND POWER DIVIDERS
Fig. 19. Imbalance of wideband WPD. (a) Amplitude. (b) Phase.
Table V summarizes various wideband transformers realizedby dual-frequency transformers. It is noted that the widebandtransformer based on the proposed method features easydesign as well as compact size.
B. Wideband Transformer in WPD Application
For validation, a wideband WPD with a center frequencyof 4 GHz is designed, simulated, and fabricated on aRogers 4350B substrate with the relative dielectric constantof 3.48 and the thickness of 0.508 mm. Following thewideband-WPD design flowchart shown in Fig. 6, its electricaland physical dimension parameters in the structure in Fig. 16are easily obtained and listed in Table VI (R0 = 72 � is real-ized by two Yageo’s shunt resistors with resistances of 75 and1800 �, and their part numbers are RE0402DRE0775RLand RE0402DRE071K8L, respectively). Fig. 17 shows thephotograph of the fabricated WPD. It is seen that its overall
size, excluding the connectors, is 0.069 λ2g (λg is the guided
wavelength at the center frequency of 4 GHz).Fig. 18 shows the frequency responses of the wideband
WPD. It is noted that the measured results are almost con-sistent with the calculated and simulated results, and theslight disagreements are mainly attributed to the meanderingtransmission lines, manufacture tolerance, impact of soldering,and parasitic effect of isolation resistor. Furthermore, in theentire operating frequency range of 1.94–6.68 GHz, the inputand output return losses are better than 13 dB, and theinsertion loss is less than 3.6 dB. In addition, a fractionalbandwidth of 110% defined by 25-dB isolation is achieveddue to the proposed wideband-matching method in the designof odd-mode equivalent circuit.
Fig. 19 shows the imbalances of simulated and measuredWPD. It is observed that the amplitude and phase imbalancesbetween the two output ports are less than 0.05 dB and 0.7◦,respectively.
Table VII compares various reported wideband WPDs.It is noted that the proposed WPD based on the design ofwideband transformer is characterized by only one lumpedelement, compact size, wide bandwidth, high isolation, andsmall amplitude imbalance.
IV. CONCLUSION
In this article, a design approach for the wideband trans-former with frequency-dependent complex loads is presented.By properly selecting the electrical parameters in the gen-eralized MSRT and directly constructing two complex con-jugated loads instead of the actual impedances of trans-former, closed-form design formulas of wideband transformerare deduced. Also, the wideband transformer is realizedby only two- or three-section cascaded transmission lines.Furthermore, the bandwidth is enhanced by introducing theleast-square method instead of the conventional perfect match-ing. In addition, a wideband transformer is readily applied tothe design of WPD with compact size and wide bandwidth.
REFERENCES
[1] M. Chongcheawchamnan, S. Patisang, M. Krairiksh, andI. D. Robertson, “Tri-band wilkinson power divider using a three-section transmission-line transformer,” IEEE Microw. Wireless Compon.Lett., vol. 16, no. 8, pp. 452–454, Aug. 2006.
[2] K.-K.-M. Cheng and C. Law, “A novel approach to the design andimplementation of dual-band power divider,” IEEE Trans. Microw.Theory Techn., vol. 56, no. 2, pp. 487–492, Feb. 2008.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
LIU et al.: DESIGN APPROACH FOR COMPACT WIDEBAND TRANSFORMER 13
[3] Y. Wu, S. Yong Zheng, S.-W. Leung, Y. Liu, and Q. Xue, “An analyticaldesign method for a novel dual-band unequal coupler with four arbitraryterminated resistances,” IEEE Trans. Ind. Electron., vol. 61, no. 10,pp. 5509–5516, Oct. 2014.
[4] S.-F.-R. Chang et al., “A dual-band RF transceiver for multistandardWLAN applications,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 3,pp. 1048–1055, Mar. 2005.
[5] X. Fu, D. T. Bespalko, and S. Boumaiza, “Novel dual-band matchingnetwork for effective design of concurrent dual-band power amplifiers,”IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 61, no. 1, pp. 293–301,Jan. 2014.
[6] K. A. Hsieh, H. S. Wu, K. H. Tsai, and C. K. C. Tzuang, “A dual-band 10/24-GHz amplifier design incorporating dual-frequency complexload matching,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 6,pp. 1649–1657, Jun. 2012.
[7] R. Collin, “Theory and design of wide-band multisection quarter-wavetransformers,” Proc. IRE, vol. 43, no. 2, pp. 179–185, 1955.
[8] H. J. Riblet, “General synthesis of quarter-wave impedance transform-ers,” IRE Trans. Microw. Theory Techn., vol. 5, no. 1, pp. 36–43,Jan. 1957.
[9] C. S. Gledhill and A. M. H. Issa, “Exact solutions of stepped impedancetransformers having maximally flat and Chebyshev characteristics,”IEEE Trans. Microw. Theory Techn., vol. 17, no. 7, pp. 379–386,Jul. 1969.
[10] S. B. Cohn, “Optimum design of stepped transmission-line transform-ers,” IRE Trans. Microw. Theory Techn., vol. 3, no. 3, pp. 16–20,Apr. 1955.
[12] Y. Liu, R. Levy, and Y. Chen, “Optimal dual-band impedance transform-ers with wide bandwidths for frequency dependent complex loads,” inIEEE MTT-S Int. Microw. Symp. Dig., Jun. 2013, pp. 1–4.
[13] Y. Liu, “Synthesis techniques on multiband impedance matching net-works for frequency-dependent complex loads,” IEEE Trans. Microw.Theory Techn., vol. 66, no. 10, pp. 4507–4519, Oct. 2018.
[14] Y. L. Wu, Y. A. Liu, and S. L. Li, “A dual-frequency transformer forcomplex impedances with two unequal sections,” IEEE Microw. WirelessCompon. Lett., vol. 19, no. 2, pp. 77–79, Feb. 2009.
[15] S. C. D. Roy, “Comment on ‘a dual-frequency transformer for com-plex impedances with two unequal sections,”’ IEEE Microw. WirelessCompon. Lett., vol. 19, no. 9, p. 602, Sep. 2009.
[16] X. Liu, Y. A. Liu, S. L. Li, F. Wu, and Y. L. Wu, “A three-section dual-band transformer for frequency-dependent complex load impedance,”IEEE Microw. Wireless Compon. Lett., vol. 19, no. 10, pp. 611–613,Oct. 2009.
[17] Y. Wu, Y. Liu, S. Li, C. Yu, and X. Liu, “A generalized dual-frequency transformer for two arbitrary complex frequency-dependentimpedances,” IEEE Microw. Wireless Compon. Lett., vol. 19, no. 12,pp. 792–794, Dec. 2009.
[18] L. Liu et al., “A generalized approach for multifrequency transmissionline transformer with frequency-dependent complex source and load,”IEEE Trans. Microw. Theory Techn., vol. 67, no. 9, pp. 3603–3616,Sep. 2019.
[19] L. Liu et al., “A novel analytical method for multi-frequency transmis-sion line transformer,” IEEE Microw. Wireless Compon. Lett., vol. 26,no. 8, pp. 556–558, Aug. 2016.
[20] L. Liu et al., “Multifrequency transformer with arbitrary frequency andreal impedance transform ratio,” IEEE Microw. Wireless Compon. Lett.,vol. 27, no. 9, pp. 785–787, Sep. 2017.
[21] C. Monzon, “A small dual-frequency transformer in two sections,” IEEETrans. Microw. Theory Techn., vol. 51, no. 4, pp. 1157–1161, Apr. 2003.
[22] T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation. UpperSaddle River, NJ, USA: Prentice-Hall, 2000.
[23] M. Suzuki, H. Miyashita, A. Kamo, T. Watanabe, and H. Asai, “Asynthesis technique of time-domain interconnect models by MIMOtype of selective orthogonal least-square method,” IEEE Trans. Microw.Theory Techn., vol. 49, no. 10, pp. 1708–1714, Oct. 2001.
[24] K. Seki, “Developing an active filter based on least square method,” inProc. Int. Power Electron. Conf. (ECCE ASIA), Jun. 2010, pp. 191–196.
[25] I. Jeong, B.-G. Gu, J. Kim, K. Nam, and Y. Kim, “Inductance estimationof electrically excited synchronous motor via polynomial approximationsby least square method,” IEEE Trans. Ind. Appl., vol. 51, no. 2,pp. 1526–1537, Mar. 2015.
[26] D. A. Frickey, “Conversions between S, Z, Y, H, ABCD, and T para-meters which are valid for complex source and load impedances,” IEEETrans. Microw. Theory Techn., vol. 42, no. 2, pp. 205–211, Feb. 1994.
[27] M. A. Maktoomi, M. S. Hashmi, and F. M. Ghannouchi, “Improving loadrange of dual-band impedance matching networks using load-healingconcept,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 64, no. 2,pp. 126–130, Feb. 2017.
[28] L. Liu et al., “A compact ultra-wideband power divider with highisolation,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. (APSURSI),Jul. 2014, pp. 641–642.
[29] L. Guo, A. Abbosh, and H. Zhu, “Ultra-wideband in-phase power dividerusing stepped-impedance three-line coupled structure and microstrip-to-slotline transitions,” Electron. Lett., vol. 50, no. 5, pp. 383–384,Feb. 2014.
[30] C.-W. Tang and J.-T. Chen, “A design of 3-dB wideband microstrippower divider with an ultra-wide isolated frequency band,” IEEE Trans.Microw. Theory Techn., vol. 64, no. 6, pp. 1806–1811, Jun. 2016.
[31] M. A. Maktoomi, M. S. Hashmi, and F. M. Ghannouchi, “Theory anddesign of a novel wideband DC isolated wilkinson power divider,” IEEEMicrow. Wireless Compon. Lett., vol. 26, no. 8, pp. 586–588, Aug. 2016.
[32] X. Yu and S. Sun, “A novel wideband filtering power dividerwith embedding three-line coupled structures,” IEEE Access, vol. 6,pp. 41280–41290, 2018.
[33] H. Zhu, Z. Cheng, and Y. J. Guo, “Design of wideband in-phase andout-of-Phase power dividers using microstrip-to-slotline transitions andslotline resonators,” IEEE Trans. Microw. Theory Techn., vol. 67, no. 4,pp. 1412–1424, Apr. 2019.
Liang Liu received the B.S. degree in electronicsscience and technology from Southwest JiaotongUniversity, Chengdu, China, in 2005, and the M.S.degree in electronics science and technology fromZhejiang University, Hangzhou, China, in 2008.He is currently pursuing the Ph.D. degree at Shang-hai Jiao Tong University, Shanghai, China.
From 2008 to 2009, he was an RF Engineer withWinhap Communications Inc., Shenzhen, China.From 2009 to 2010, he was a Research Assistantwith the City University of Hong Kong, Hong Kong.
From 2010 to 2013, he was an RF Engineer with SED Wireless Com-munications Inc., Shenzhen. His current research interests include multifre-quency/wideband power dividers, filters, and power amplifiers.
Xianling Liang (Senior Member, IEEE) received theB.S. degree in electronic engineering from XidianUniversity, Xi’an, China, in 2002, and the Ph.D.degree in electric engineering from Shanghai Uni-versity, Shanghai, China, in 2007.
From 2007 to 2008, he was a Post-DoctoralResearch Fellow with the Institute National de laRecherche Scientfique, University of Quebec, Mon-treal, QC, Canada. In 2008, he joined the Depart-ment of Electronic Engineering, Shanghai Jiao TongUniversity (SJTU), Shanghai, as a Lecturer where
he became an Associate Professor in 2012. He has authored or coauthoredover 250 papers, including 141 journal articles and 113 conference papers,and coauthored one book and three chapters in microwave and antenna fields.He holds 15 patents in antenna and wireless technologies. His current researchinterests include OAM-EM wave propagation and antenna design, time-modulated/4-D array and applications, anti-interference antenna and array,integrated active antenna and array, and ultrawideband wide-angle scanningphased array.
Dr. Liang was a recipient of the Award of Shanghai Municipal ExcellentDoctoral Dissertation in 2008, the Nomination of the National ExcellentDoctoral Dissertation in 2009, the Best Paper Award presented at theInternational Workshop on Antenna Technology: Small Antennas, InnovativeStructures, and Materials in 2010, the SMC Excellent Young Faculty and theExcellent Teacher Award of SJTU in 2012, the Shanghai Natural ScienceAward in 2013, the Best Paper Award presented at the IEEE InternationalSymposium on Microwave, Antenna, Propagation, and EMC Technologiesin 2015, the 4th China Publishing Government Book Award, and the OkawaFoundation Research Grant in 2017.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
14 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
Ronghong Jin (Fellow, IEEE) received the B.S.degree in electronic engineering, the M.S. degree inelectromagnetic and microwave technology, and thePh.D. degree in communication and electronic sys-tems from Shanghai Jiao Tong University, Shanghai,China, in 1983, 1986, and 1993, respectively.
In 1986, he joined the Department of ElectronicEngineering, Shanghai Jiao Tong University, as aFaculty Member, where he has been an Assistant,a Lecturer, and an Associate Professor and is cur-rently a Professor. From 1997 to 1999, he was a
Visiting Scholar with the Department of Electrical and Electronic Engineer-ing, Tokyo Institute of Technology, Meguro, Japan. From 2001 to 2002,he was a Special Invited Research Fellow with the Communication ResearchLaboratory, Tokyo, Japan. From 2006 to 2009, he was a Guest Professorwith the University of Wollongong, Wollongong, NSW, Australia. He isalso a Distinguish Guest Scientist with the Commonwealth Scientific andIndustrial Research Organization, Sydney, NSW, Australia. He has authored orcoauthored over 300 papers in refereed journals and conference proceedingsand coauthored four books. He holds approximately 60 patents in antennaand wireless technologies. His current research interests include antennas,electromagnetic theory, numerical techniques for solving field problems, andwireless communication.
Dr. Jin is a Committee Member of the Antenna Branch of the ChineseInstitute of Electronics, Beijing, China. He was a recipient of theNational Technology Innovation Award, the National Nature Science Award,the 2012 Nomination of the National Excellent Doctoral Dissertation (Super-visor), the 2017 Excellent Doctoral Dissertation (Supervisor) of the ChinaInstitute of Communications, the Shanghai Nature Science Award, and theShanghai Science and Technology Progress Award.
Haijun Fan (Member, IEEE) received the B.S.degree in applied physics from the Universityof Electronic Science and Technology of China,Chengdu, China, in 2009, and the M.S. degreein electromagnetic and microwave technology andthe Ph.D. degree in electronic science and technol-ogy from Shanghai Jiao Tong University, Shanghai,China, in 2011 and 2016, respectively.
From 2016 to 2017, he was a Post-Doctoral Fellowwith the Department of Electronic Engineering, TheChinese University of Hong Kong, Hong Kong,
SAR, China. From 2018 to 2019, he was a Research Associate with theInstitute of Sensors, Signals and Systems, Heriot-Watt University, Edinburgh,U.K. From 2019 to 2020, he was a Senior Design Engineer with Ampleon,Nijmegen, The Netherlands. He is currently an Assistant Professor with theInstitute of Sensors, Signals and Systems, Heriot-Watt University, Edinburgh,U.K. His current research interests include monolithic microwave integratedcircuit (MMIC) power amplifier, filter synthesis, antenna array, and 5G/6Grelated areas.
Dr. Fan was a recipient of the Outstanding Ph.D. Thesis Award of ChinaInstitute of Communications in 2017.
Xudong Bai (Member, IEEE) received the B.S.degree in electronic engineering, the M.S. degree inelectromagnetic and microwave technology, and thePh.D. degree in electronic science and technologyfrom Shanghai Jiao Tong University (SJTU), Shang-hai, China, in 2009, 2012, and 2016, respectively.
In 2016, he joined the Research and DevelopmentCenter, Shanghai Aerospace Electronics CompanyLtd., Shanghai, where he is currently an AdvancedEngineer. His current research interests includephased arrays, electromagnetic metamaterials, and
OAM-EM wave propagation and antenna design.
Han Zhou received the B.S. degree from ShanghaiJiao Tong University, Shanghai, China, in 2014,where he is currently pursuing the Ph.D. degree inelectromagnetic and microwave technology.
His current research interests include multi-in–multiout antenna, metamaterials, microwave devices,and phased arrays.
Junping Geng (Senior Member, IEEE) received theB.S. degree in plastic working of metals, the M.S.degree in corrosion and protection of equipment,and the Ph.D. degree in circuit and system fromNorthwestern Polytechnic University, Xi’an, China,in 1996, 1999, and 2003, respectively.
From 2003 to 2005, he was a Post-DoctoralResearcher with Shanghai Jiao Tong University(SJTU), Shanghai, China. In 2005, he joined theDepartment of Electronic Engineering, SJTU, as aFaculty Member, where he is currently an Asso-
ciate Professor. He has been involved in electromagnetic compatibility forhigh-altitude platform stations, multiantennas for terminals, smart antennas,and nanoantennas. He has authored or coauthored over 260 refereed journaland conference papers, two book chapters, and one book. He holds 50 patentswith over 50 pending. His current research interests include antennas, elec-tromagnetic theory, and computational techniques of electromagnetic andnanoantennas.
Dr. Geng is a member of the Chinese Institute of Electronics, Beijing,China. He was a recipient of the Technology Innovation Award of the ChineseMinistry of Education in 2007 and the Technology Innovation Award of theChinese Government in 2008.
