This paper presents a distributed model of a lumped-element quadrature coupler that accounts for the distributed nature of parasitic elements. The required capacitors in the constitution of lumped-element quadrature couplers can be properly considered using the proposed distributed model. An integrated low loss lumped-element quadrature coupler is implemented using the proposed model. The modeled responses are compared with the corresponding measured responses; they exhibit good agreement. The measured insertion loss for both direct and coupled ports is 3.23 dB at 5.5 GHz. The variation in the insertion loss is ± 0.3 dB with a phase imbalance of less than 0.95° across a 760-MHz bandwidth

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## I. INTRODUCTION

Quadrature couplers have been widely used in design of balanced amplifiers, phase
shifters, and balanced mixers, as well as other microwave and millimeter-wave components.
Lumped-element quadrature couplers are typically used for on-chip implementations
^{(1-}^{6)}. Fig. 1 shows a schematic of a lumped-element quadrature coupler. The signal power on the
input port is equally divided into direct and coupled ports with a phase difference
of 90° provided that the coupling coefficient, k_{L}, is 0.707. Design values for each element can be founded in ^{(1)}. In this regard, electromagnetic (EM) simulators are widely employed to find design
values of each element. Despite the large applicability of the quadrature coupler,
an accurate model of lumpedelement quadrature couplers has not yet been developed,
to the best of our knowledge. A less accurate model was reported for in ^{(2)}, but the focus was on equation derivation, and modeled responses were not compared
with measured ones. Although EM simulations provide a possible approach for the design
of lumped-element quadrature coupler and for predicting the performance of the coupler,
the simulation time for optimization can be long. Moreover, for getting design insight
and for optimization of quadrature couplers, an accurate model is required.

One of main difficulties for the design of lumpedelement quadrature couplers, whose
generic schematic is depicted in Fig. 1(b), is to achieve a high, well-controlled coupling coefficient ^{(1,}^{2)}. To obtain a high coupling coefficient, the number of turns in inductors typically
needs to be increased in a lateral configuration. It causes the degradation of the
quality factor of inductors, resulting in degradation of insertion loss and phase
balance performance ^{(1,}^{3,}^{4)}.

In this work, we propose an accurate distributed model for a lumped-element quadrature coupler based on physical geometry. In addition, by adopting the proposed distributed model, a low-loss coupler with a relatively low coupling coefficient was implemented. Its results are compared with modeled responses.

## II. COUPLER MODEL AND DESIGN OF LOW LOSS QUADRATURE COUPLER

### 1. Lumped-Element Quadrature Coupler

A lumped-element quadrature coupler, which is shown in Fig. 1(b), is also called Langer coupler because it employs coupled inductors in its geometry.
The equation for the design parameters of the lumped-element quadrature coupler in
Fig. 1(b) can be derived by equating with the parameters of the transmission-line coupler.
In ^{(1)}, the design values of each parameter can be founded by the following equation:

##### (1)

$L=\frac{1.414 \times Z_{0}}{\omega_{0}}$ $C_{C}=\frac{1}{\omega_{O} Z_{0}},$ and $C_{G}=\frac{0.414}{\omega_{O} Z_{0}}$_{C}= 0.58 pF, C

_{G}= 0.24 pF, and k

_{L}= 0.707.

For the design of the coupled inductors, to achieve a large coupling coefficient of
0.707, the number of turns is typically increased in a lateral configuration, resulting
in the decrease of the quality factor of the inductor, $Q$. Fig. 2 shows the plots of simulated S-parameter characteristics with different $Q$ values
using an Advanced Design System (ADS) simulator. Low $Q$ will degrade the insertion
loss as well as the phase balance performance ^{(1,}^{3)}. Although the adjustment for values of C_{C} and C_{G} can be performed for better characteristic ^{(1)}, it can disrupt the symmetry of the coupler and there is no formal analysis for this
optimization process ^{(1)}.

### 2. Quadrature Coupler using Distributed Model

The key to achieve an accurate modeling of couplers is the identification of relevant parasitic components. Unlike models of inductors or transformers, the composition of parasitic capacitance in lumped-element couplers plays a crucial role for the normal operation of quadrature couplers. To accurately model the composition of parasitic capacitance, distributed effects should be carefully considered.

Fig. 3(a) shows a layout of a symmetrical quadrature coupler. Two metal traces are inter-winded
together. Each winding is referred to as a 1.5-turn inductor. A topmost thickest metal
layer is used for most inductor winding to minimize its loss. A lower metal layer
is used only for underpass elements. Each metal winding is divided into three segments,
as shown in Fig. 3(a). The proposed equivalent circuit model for the quadrature coupler is depicted in
Fig. 3(b). Each segment is generated independently, but then linked together with additional
mutual capacitive and magnetic coupling components. Each segment includes self-inductance,
$L$, with ZS. The series resistance, R_{sm}, in Z_{S} represents a series metal loss. Z_{S} also includes L-R branches, R_{pm}, and L_{pm}, to model skin and proximity effects. At starting and ending points of each segment,
parasitic capacitances are considered. External capacitors for C_{C} and C_{G} in Fig. 1(b) can be added. However, because parasitic capacitances from two metal traces can be
absorbed for a large portion of the required C_{C} and C_{G} in Fig. 1(b), they should be carefully analyzed. Considering coupler design using GaAs technology,
C_{G} is a parasitic capacitance of dielectric and substrate owing to the absence of resistance
in a semiinsulating GaAs substrate. Fig. 3(a) also shows the metal stack-up information of a GaAs technology. The mutual magnetic
coupling between two adjacent inductors is indicated with the coupling coefficient,
k_{L}. Note that C_{C} in Fig. 3(a) describes the coupling capacitance between two adjacent segments. Given that the
spacing is typically required to be small to increase k_{L} and the metal thickness is usually larger than 3 μm in most CMOS and GaAs technology,
the parasitic C_{C} can constitute a large portion of the required total C_{C}. Note that K_{p} is the additional coupling coefficient between segments in the same winding.

### 3. Design of Low Loss Quadrature Coupler

Lumped models in light yellow and light green blocks in Fig. 3(b) are identical for an ideal quadrature coupler if C_{G} and C_{C} at middle point are neglected. It seems that two ideal lumped-element couplers are
combined in series. In ^{(3,}^{4)}, the required coupling coefficient in a two-stage cascade design of quadrature coupler
can be much lower than that of an ideal single-stage coupler. A coupler design with
high $Q$ will improve its loss performance. Therefore, in this study, a low-loss quadrature
coupler is developed with a relatively low coupling coefficient using the proposed
model. The proposed quadrature coupler structure using a distributed model includes
loss resistors, as shown in Fig. 3(b). Thus, unlike coupler designs using (1), the design method using the proposed model can be optimized with finite $Q$ of inductors.
In addition, because a cascade structure has a relatively large bandwidth performance,
the model in Fig. 3(b) with a structure similar to a cascade can improve the bandwidth performance.

## III. IMPLEMENTATION AND MEASUREMENTS

The proposed low-loss quadrature coupler was fabricated using GaAs technology. The
thicknesses of the top and lower metal layers are 4 μm and 1 μm, respectively. A metal
layer with a large width is used to improve $Q$. Specifically, this width is 30 μm.
A spacing of 10 μm is chosen for k_{L} and C_{C}. The size of the coupler is mainly determined to provide the necessary inductance
$L$. However, increasing the size also increases the parasitic capacitances, C_{C} and C_{G}. Its size is 1250 μm × 650 μm. In this design, no external capacitor is added. The
values for each component in Fig. 3(b) are obtained from optimization based on the initial values from physical properties.
A slight adjustment within the physical range is done for better fitting with measurement
results. The value of k_{L} in Fig. 3(b) is 0.51, which is lower than the value from an ideal lumped- element quadrature coupler.
Fig. 4 shows the microphotograph of the fabricated quadrature coupler.

The S-parameter on-wafer measurement was carried out using a four-port Agilent N5242A
network analyzer and Cascade S300 probe station. During the procedure in full four-port
calibration, calibration data was measured by connecting an open standard, a short
standard, a load standard, and a through standard to the four test ports. After the
calibration process, the integrated coupler was measured. The measured insertion losses
between input and direct ports (S_{21}) and between input and coupled ports (S_{31}) are compared with the modeled responses. As shown in Fig. 5, the modeled responses are well matched with the measured responses. The measured
insertion losses for both S_{21} and S_{31} are 3.23 dB at 5.5 GHz. Across a 760-MHz bandwidth, the insertion loss varies only
± 0.3 dB. Fig. 6 shows the phase differences between S_{21} and S_{31} in the measured and modeled responses. The measured phase difference is less than
0.95° over a 760MHz bandwidth. Fig. 7 and 8 show the responses of isolations and return losses of the implemented coupler,
respectively. The measured isolation and return losses are less than 30 dB and 23
dB across a 760-MHz bandwidth, respectively. There is a slight gap between modeled
and measured responses. The values for inductors and capacitors of each segment are
slightly different each other owing to the different length of each element, which
can give rise to discrepancies between modeled and measured responses. However, the
same values for the components in each segment are applied to provide a design insight.
The results of the proposed quadrature coupler are compared in Table 1 with previously published results. The proposed coupler achieves lower insertion
loss with low phase error over a wide bandwidth compared with other works.

Table 1. Comparison table with integrated quadrature coupler

## IV. CONCLUSIONS

An accurate distributed model for a quadrature coupler has been proposed. The proposed approach is validated with measured results. The model agrees well with measured results. This work shows that couplers with the proposed accurate model can achieve low insertion loss across a wide bandwidth. The proposed distributed model can be directly applied to the design of any lumpedelement quadrature couplers.

### ACKNOWLEDGMENTS

This work was supported by Korea Institute for Advancement of Technology(KIAT) grant funded by the Korea Government(MOTIE) (P078200021, The Competency Development Program for Industry Specialist). The CAD tools were supported by the IDEC.

### REFERENCES

## Author

received the B.S. degree in electrical engineering from Pusan National University, Busan, Korea, in 2015, and is currently pursing the Ph.D. degree in electrical engineering at Pusan National University, Busan, Korea.

His interests include high-frequency integrated circuits and system design for wireless communications.

received the B.S. degree in EE from Yonsei University, Korea, in 1999, and the M.S. and Ph.D. degrees in EECS from the KAIST, Korea, in 2001 and 2005, respectively. From 2005 to 2007, he was a Senior Engineer with Samsung Electronics, Gyeonggi, Korea, where he was involved in the development of mobile digital TV tuner IC.

In 2007, he joined the School of Electrical Engineering, Pusan National University, Busan, Korea, and is now a Professor.

received the B.S., M.S., and Ph.D. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2000, 2002, and 2007, respectively.

From 2007 to 2009, he was in Georgia Institute of Technology, where he developed CMOS power amplifiers.

In 2009, he joined Skyworks Solutions, Inc., Cedar Rapids, IA, where he was involved with the design of power amplifiers and front end modules.

In 2010, he joined the faculty of Hanbat National University, Daejeon, Korea.

His research interests include RF power amplifier design, microwave module design, and ultrasonic circuit design.

received the B.S. degree in electrical engineering from Sungkyunkwan University, Korea, in 2001, the M.S. degree in electrical engineering from the KAIST, Korea, in 2005, and the Ph.D. degree in electrical and computer engineering from the Georgia Institute of Technology, USA, in 2009.

Upon completion of the doctoral degree, he joined Qualcomm Inc., USA, as a Senior Engineer, where he was involved in the development of transmitters and integrated passive circuits on mobile applications.

He is currently a faculty member with the Department of Electrical Engineering, Pusan National University, Korea.

His research interests include high-frequency integrated circuits and system design for wireless communications.