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WIDE DYNAMIC RANGE LOW NOISE AMPLIFIER MODULE FOR Ka-BAND RADAR APPLICATIONS Jinho Jeong, 1 Youngmin Kim, 2 Sangmin Park, 3 Younjin Kim, 4 Changhyun Park, 4 Sangjoo Lee, 4 and Youngwoo Kwon 2 1 Department of Electronic Engineering, Sogang University, Korea; Corresponding author: [email protected] 2 School of Electrical Engineering and Computer Science, Seoul National University, Korea 3 Department of Electronics and Communication Engineering, Kwangwoon University, Korea 4 ISR R&D Lab., LIG Nex1 Co., Ltd., Korea Received 24 June 2011 ABSTRACT: Ka-band low noise amplifier (LNA) module with wide dynamic range is presented using 0.15-lm pseudo-morphic high electron mobility transistors for radar applications. Two of attenuator-combined low-gain LNA microwave monolithic integrated circuits are cascaded to increase the gain without degrading input power handling capability. There exist two operating modes depending on the level of input power, that is, low power and high power modes. The measurement shows a high gain of 21.9 dB and low noise figure of 2.7 dB at 35 GHz in low power mode at which two attenuators are off. At high input power, the attenuators are turned on to handle high power, and the module presents an attenuation of 6.4 dB with a high input 1-dB gain compression point (P 1dB,in ) of 6.5 dBm. The module shows good input and output return losses better than 9.1 dB in both modes. V C 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:1031–1035, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/ mop.26675 Key words: MMIC; low noise amplifier; attenuator; Ka-band; pseudo- morphic high electron mobility transistors 1. INTRODUCTION In general, the receiver for radar and wireless communications should exhibit low noise figure and high conversion gain for high sensitivity. In addition, it should be capable of a wide range of input power for wide dynamic range. When strong sig- nals or interferers are present at the input, they should be suffi- ciently attenuated to prevent the receiver from saturating and to protect the receiver. An low noise amplifier (LNA) plays a criti- cal role in the dynamic range of the receiver, since it is placed right behind an antenna [1]. Several articles have been published on the design of LNAs with wide dynamic range as well as low noise figure. Dual-gain LNA was proposed in Ref. 2, where two gain paths for low gain and high gain were used. They are switched on and off depend- ing on the input power levels. That is, the low gain path is turned on at high input power. In this way, the LNA can handle high input power, resulting in wide dynamic range. In Ref. 1, a large size of transistor was used in the design of LNA to achieve high input power capability at the cost of high DC power consumption. A GaN device is also good choice for wide dynamic range LNA, since it allows low noise performance with high output power capability [3]. For radar applications, attenuators are widely used at the out- put of LNA [4–6] as illustrated in Figure 1(a). At low input power, the attenuator is off and LNA provides low noise figure and high gain. When the strong signal is present at the input, the attenuator is turned on, which reduces the input power to the mixer and IF amplifier following the attenuator. Therefore, the attenuator increases input power capability and dynamic range of the receivers. In this topology, however, the LNA still limits the dynamic range, since the maximum input power is limited by the linearity of the LNA, or P 1dB,in . The LNA in Figure 1(a) suffers from low P 1dB,in due its high gain. In this work, the wide dynamic range LNA module is pre- sented using two low-gain LNAs with attenuators instead of sin- gle high-gain LNA, as shown in Figure 1(b). Each microwave monolithic integrated circuit (MMIC) is designed and fabricated using 0.15-lm pHEMT technology, and two MMICs are cascaded on a printed circuit board (PCB). The operating principle is dis- cussed in Section 2 with a detailed circuit design and simulation results. The measurement results are presented in Section 3. 2. CIRCUIT TOPOLOGY AND DESIGN Figure 1(b) shows the schematic diagram of the LNA module proposed in this work to improve the dynamic range, where two identical low noise MMICs are cascaded. Each MMIC consists of two-stage LNA [Fig. 2(a)] and attenuator [Fig. 3(a)]. The module is designed to provide two operating modes, or low power and high power modes. At low input power less than 23.0 dBm, the module is supposed to provide the total gain higher than 20 dB at Ka-band (35 GHz). Therefore, each LNA is designed to present half the total gain (or 10 dB) with low noise performance. The attenuators are off, showing minimum insertion loss. At high input power greater than 23.0 dBm, the attenuators are turned on, reducing the signal power by more than 25 dB compared with the case of low power mode, which prevents mixer and IF amplifier from saturating. In this mode, the input signal is amplified by IF amplifier and the noise figure perform- ance of LNA can be ignored, since the input power is high enough. Each LNA is always turned on in order for good input and output return losses in both modes. In high power mode, each LNA in Figure 1(b) is expected to saturate at higher input power due to its lower gain, compared with the conventional topology in Figure 1(a). In addition, the input power to the second LNA is dramatically reduced by the first attenuator. As a result, the proposed LNA module can allow a linear gain up to high input power, presenting a high P 1dB,in and wide dynamic range. The module is designed to have a P 1dB,in higher than 5 dBm. Figure 2(a) shows the circuit schematic of the designed LNA MMIC using 0.15 lm pseudo-morphic high electron mobility Figure 1 (a) Conventional LNA module using single high-gain LNA and attenuator. (b) Proposed LNA module using two low-gain LNAs with attenuators DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 4, April 2012 1031

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Page 1: WIDE DYNAMIC RANGE LOW NOISE P AMPLIFIER MODULE FOR …hompi.sogang.ac.kr/rfdesign/paper/28.pdf · WIDE DYNAMIC RANGE LOW NOISE AMPLIFIER MODULE FOR Ka-BAND RADAR APPLICATIONS Jinho

WIDE DYNAMIC RANGE LOW NOISEAMPLIFIER MODULE FOR Ka-BANDRADAR APPLICATIONS

Jinho Jeong,1 Youngmin Kim,2 Sangmin Park,3 Younjin Kim,4

Changhyun Park,4 Sangjoo Lee,4 and Youngwoo Kwon21 Department of Electronic Engineering, Sogang University, Korea;Corresponding author: [email protected] School of Electrical Engineering and Computer Science,Seoul National University, Korea3Department of Electronics and Communication Engineering,Kwangwoon University, Korea4 ISR R&D Lab., LIG Nex1 Co., Ltd., Korea

Received 24 June 2011

ABSTRACT: Ka-band low noise amplifier (LNA) module with widedynamic range is presented using 0.15-lm pseudo-morphic high electron

mobility transistors for radar applications. Two of attenuator-combinedlow-gain LNA microwave monolithic integrated circuits are cascaded to

increase the gain without degrading input power handling capability.There exist two operating modes depending on the level of input power,that is, low power and high power modes. The measurement shows a

high gain of 21.9 dB and low noise figure of 2.7 dB at 35 GHz in lowpower mode at which two attenuators are off. At high input power, the

attenuators are turned on to handle high power, and the modulepresents an attenuation of 6.4 dB with a high input 1-dB gaincompression point (P1dB,in) of 6.5 dBm. The module shows good input

and output return losses better than 9.1 dB in both modes. VC 2012

Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:1031–1035,

2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/

mop.26675

Key words: MMIC; low noise amplifier; attenuator; Ka-band; pseudo-morphic high electron mobility transistors

1. INTRODUCTION

In general, the receiver for radar and wireless communications

should exhibit low noise figure and high conversion gain for

high sensitivity. In addition, it should be capable of a wide

range of input power for wide dynamic range. When strong sig-

nals or interferers are present at the input, they should be suffi-

ciently attenuated to prevent the receiver from saturating and to

protect the receiver. An low noise amplifier (LNA) plays a criti-

cal role in the dynamic range of the receiver, since it is placed

right behind an antenna [1].

Several articles have been published on the design of LNAs

with wide dynamic range as well as low noise figure. Dual-gain

LNA was proposed in Ref. 2, where two gain paths for low gain

and high gain were used. They are switched on and off depend-

ing on the input power levels. That is, the low gain path is

turned on at high input power. In this way, the LNA can handle

high input power, resulting in wide dynamic range. In Ref. 1, a

large size of transistor was used in the design of LNA to

achieve high input power capability at the cost of high DC

power consumption. A GaN device is also good choice for wide

dynamic range LNA, since it allows low noise performance with

high output power capability [3].

For radar applications, attenuators are widely used at the out-

put of LNA [4–6] as illustrated in Figure 1(a). At low input

power, the attenuator is off and LNA provides low noise figure

and high gain. When the strong signal is present at the input,

the attenuator is turned on, which reduces the input power to the

mixer and IF amplifier following the attenuator. Therefore, the

attenuator increases input power capability and dynamic range

of the receivers. In this topology, however, the LNA still limits

the dynamic range, since the maximum input power is limited

by the linearity of the LNA, or P1dB,in. The LNA in Figure 1(a)

suffers from low P1dB,in due its high gain.

In this work, the wide dynamic range LNA module is pre-

sented using two low-gain LNAs with attenuators instead of sin-

gle high-gain LNA, as shown in Figure 1(b). Each microwave

monolithic integrated circuit (MMIC) is designed and fabricated

using 0.15-lm pHEMT technology, and two MMICs are cascaded

on a printed circuit board (PCB). The operating principle is dis-

cussed in Section 2 with a detailed circuit design and simulation

results. The measurement results are presented in Section 3.

2. CIRCUIT TOPOLOGY AND DESIGN

Figure 1(b) shows the schematic diagram of the LNA module

proposed in this work to improve the dynamic range, where two

identical low noise MMICs are cascaded. Each MMIC consists

of two-stage LNA [Fig. 2(a)] and attenuator [Fig. 3(a)]. The

module is designed to provide two operating modes, or low

power and high power modes. At low input power less than

�23.0 dBm, the module is supposed to provide the total gain

higher than 20 dB at Ka-band (35 GHz). Therefore, each LNA

is designed to present half the total gain (or 10 dB) with low

noise performance. The attenuators are off, showing minimum

insertion loss.

At high input power greater than �23.0 dBm, the attenuators

are turned on, reducing the signal power by more than 25 dB

compared with the case of low power mode, which prevents

mixer and IF amplifier from saturating. In this mode, the input

signal is amplified by IF amplifier and the noise figure perform-

ance of LNA can be ignored, since the input power is high

enough. Each LNA is always turned on in order for good input

and output return losses in both modes.

In high power mode, each LNA in Figure 1(b) is expected to

saturate at higher input power due to its lower gain, compared

with the conventional topology in Figure 1(a). In addition, the

input power to the second LNA is dramatically reduced by the

first attenuator. As a result, the proposed LNA module can allow

a linear gain up to high input power, presenting a high P1dB,in

and wide dynamic range. The module is designed to have a

P1dB,in higher than 5 dBm.

Figure 2(a) shows the circuit schematic of the designed LNA

MMIC using 0.15 lm pseudo-morphic high electron mobility

Figure 1 (a) Conventional LNA module using single high-gain LNA

and attenuator. (b) Proposed LNA module using two low-gain LNAs

with attenuators

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 4, April 2012 1031

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transistors. Two-stage common source FETs are used to achieve

a gain greater than 10 dB which is high enough to minimize the

noise contribution of the insertion loss of the attenuator. The

transistor with a gate width of 100 lm is used for the common-

source FETs. According to the measured noise data up to 26.5

GHz, it also shows the best noise figure performance among the

transistors provided by the foundry. The bias voltages are care-

fully selected, since they strongly affect the gain, noise figure

and output power performance. In low power mode, both gate

bias voltages Vgg1 and Vgg2 are set to �0.8 V at which the tran-

sistor shows a best noise performance with a moderate gain.

The pinch-off voltage is �1.2 V. The noise figure is degraded

by increasing the drain bias voltage as shown in Figure 2(b). In

this figure, the maximum available gain of 100 lm pHEMT is

presented as a function of frequency, together with the minimum

noise figure performance. The measured noise data were extrap-

olated to predict the performance at frequencies higher than

26.5 GHz. Based on this graph, the drain bias voltage is selected

as 2.0 V for low noise figure. However, the low drain bias volt-

age limits the available output power from the transistor, which

results in low P1dB,in. To solve this problem, the drain bias is

increased to 3.0 V in high power mode.

As shown in Figure 2(a), inductive line feedback (L1) is usedat the source of the first FET for simultaneous noise and input

impedance matching. The second stage is designed to provide a

sufficient gain without using source feedback. Impedance match-

ing is performed using distributed elements only such as micro-

strip lines. The bias circuits are designed using quarter-wave

long lines and capacitors as shown in Figure 2(a). The resistors

(Rb2, Rb3, and Rb4) are inserted between the quarter-wave long

lines and capacitors to improve the stability of the amplifier,

even though they may slightly degrade the in-band gain and

noise figure. However, the noise contribution of these resistors

is greatly reduced by the gain of the first FET. The first gate

bias resistor Rb1 is differently connected from others so that it

can’t degrade the in-band noise figure performance. Figure 2(c)

shows the simulated performance of LNA. At a design fre-

quency of 35 GHz, the gain is as high as 13.7 dB with a low

noise figure of 2.4 dB, and input and output return losses are

better than 10 dB.

The attenuator is integrated at the output of two-stage LNA.

In this work, the absorptive type attenuator is designed as shown

in Figure 3(a), since it allows excellent impedance matching

performance in both modes [7]. In low power mode, all the tran-

sistors in the attenuator are off so that the attenuator forms a

conventional 50-X transmission line with half-wave length. The

simulation in Figure 3(b) shows an insertion loss of 1.46 dB at

35 GHz. The transistors are all turned on in high power mode.

The short-ended quarter-wave long transmission lines by on-

transistor Q2 presents very high impedance to input and output.

Thus, the input power is bypassed to the resistor R1 (45 X) andon-transistor Q1. The simulation shows a high attenuation of

15.2 dB at 35 GHz when the attenuator is on. The simulated

return loss at both modes is better than 14.4 dB at 35 GHz. The

Figure 2 LNA design. (a) Circuit schematic of LNA; (b) Maximum

available gain and minimum noise figure of 100 lm pHEMT at various

drain bias voltages (gate bias voltage is fixed to �0.8 V); and (c) Simu-

lated performance of LNA. [Color figure can be viewed in the online

issue, which is available at wileyonlinelibrary.com]

Figure 3 Attenuator design. (a) Circuit schematic of attenuator. (b)

Simulated performance of attenuator when it is off (solid) and on (dot-

ted). [Color figure can be viewed in the online issue, which is available

at wileyonlinelibrary.com]

1032 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 4, April 2012 DOI 10.1002/mop

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length and impedance of the transmission lines are carefully

determined considering the parasitic inductances and capacitan-

ces of the transistors (all the transistors have a gate width of

100 lm).

3. MEASUREMENT RESULTS

The designed MMIC was fabricated using 0.15-lm GaAs

pHEMT technology. The substrate thickness is 100 lm. The

LNA and attenuator were integrated on a single chip, and its

photograph is presented in Figure 4. The chip size is 2.0 mm �1.0 mm. The basic operation of the fabricated MMIC was veri-

fied by on-wafer measurement. Figure 5(a) shows the measured

small-signal performance (solid lines) of the attenuator-com-

bined LNA MMIC in low power mode, where the attenuator is

off and bias condition is as follows : Vgg1 ¼ Vgg2 ¼ �0.8 V,

Vdd ¼ 2.0 V, and Vatt ¼ �2.0 V. The bias current is 22 mA. At

35 GHz, the gain is as high as 12.5 dB with good noise figure

of 2.5 dB and good output return loss of 22.8 dB. The measured

input return loss is 6.3 dB at 35 GHz, indicating somewhat big

discrepancy with the simulation. In high power mode, the atten-

uator is on and bias condition is changed as follows: Vgg1 ¼�0.9 V, Vgg2 ¼ �1.1 V, Vdd ¼ 3.0 V, and Vatt ¼ 0.4 V. The

gate biases were slightly adjusted to improve the input return

loss and to increase attenuation level. The measured attenuation

or insertion loss is 2.9 dB at 35 GHz, which corresponds to an

Figure 4 Photograph of the fabricated Ka-band attenuator-combined

LNA MMIC. [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com]

Figure 5 Measured (solid) and simulated (dotted) small-signal performance of attenuator-combined LNA MMIC: (a) Low power mode; (b) High

power mode. Measured (solid) and simulated (dotted) large-signal performance of attenuator-combined LNA MMIC; (c) Low power mode; and (d) High

power mode. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]

Figure 6 Photograph of the Ka-band LNA module. [Color figure can be

viewed in the online issue, which is available at wileyonlinelibrary.com]

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 4, April 2012 1033

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attenuation of 15.4 dB compared with the gain of 12.5 dB in

low power mode. The large-signal performance was also meas-

ured as shown in Figures 5(c) and 5(d). The P1dB,in is �4.0 and

6.0 dBm in low power and high mode, respectively. These

measured performances agree well with the simulation and are

good enough to be used in implementing the LNA module of

Figure 1(b).

Two of these MMICs were cascaded on a PCB to increase

the gain without lowering P1dB,in. Figure 6 shows the photo-

graph of the fabricated LNA module. Simple off-chip matching

using transmission lines was performed to compensate for para-

sitic inductances of bond wires. K-connectors were mounted on

the input and output of the module for the measurement (They are excluded in Figure 6 to show the PCB layout and MMICs

with better quality).

The measured S-parameters and noise figure of the module

in low power mode are shown in Figure 7(a). The module shows

the gain higher than 21.9 dB and good noise figure of 2.7 dB at

35 GHz, which is an excellent performance considering the loss

of off-chip matching circuits. The input off-chip matching was

optimized for the noise figure and input return loss. The input

return loss was slightly improved to 9.1 dB at 35 GHz. The out-

put return loss is better than 10 dB over wide bandwidth. Figure

7(b) shows the measured S-parameters in high power mode. The

measured attenuation is 6.4 dB with good input and output loss

better than 20 dB at 35 GHz.

The output power and power gain were also measured as a

function of input power as shown in Figure 7(c). In low power

mode at which the input power is less than �23 dBm, the devel-

oped LNA module exhibits almost constant power gain of 21.9

dB. At input power higher than �23 dBm, two attenuators are

turned on and the module shows an attenuation of 6.4 dB with a

high P1 dB,in of 6.5 dBm. These results indicates that the devel-

oped LNA module can handle high power and provide wide

dynamic range, so that it can be successfully applied for the

millimeter-wave radar receivers.

Comparison with the reported LNA module is presented in

Table 1. It can be found that the developed LNA module in this

work exhibits higher gain and attenuation with comparable noise

figure compared with the X-band LNA module in Ref. 5.

4. CONCLUSIONS

In this article, the Ka-band LNA module with wide dynamic

range was presented using 0.15-lm pHEMT technology for ra-

dar receiver applications. Two low-gain LNAs with attenuators

are cascaded to enhance power handling capability. Attenuators

are placed after each LNA and are normally off at low input

power. Therefore, the module provides high gain and low noise

figure performance. The attenuators are turned on at high input

power to provide signal attenuation, so that the first LNA is sat-

urated at high input power due to its low gain, and the second

LNA doesn’t suffer from power-saturating since its input power

is dramatically reduced by the first attenuator. The second atten-

uator further increases the signal attenuation. In this way, the

proposed LNA module can allow high input power handling

capability or wide dynamic range with high gain and low noise

figure. Therefore, the developed LNA module can be effectively

integrated with the mixer and IF amplifier for high performance

Ka-band radar receivers.

ACKNOWLEDGMENTS

This work was supported by the Acceleration Research Program of

the Ministry of Education, Science and Technology of the Republic

of Korea and the Korea Science and Engineering Foundation.

Figure 7 Measured performance of the fabricated LNA module. (a)

S-parameters and noise figure in low power mode; (b) S-parameters in

high power mode; and (c) Output power and gain as a function of input

power at 35 GHz. [Color figure can be viewed in the online issue, which

is available at wileyonlinelibrary.com]

TABLE 1 Comparison of the Reported LNA Modules

[5] This Work

Technology 0.25 lm pHEMT 0.15 lm pHEMT

Frequency 7–11 GHz 35 GHz

Gain (attenuator off) 17 dB 21.9 dB

Gain (attenuator on) 8 dB �6.4 dB

Noise figure 2.0�2.5 dB 2.7 dB

Input/output return loss > 8.5 dB > 9.1 dB

P1dB,in – 6.5 dBm

1034 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 4, April 2012 DOI 10.1002/mop

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REFERENCES

1. T.K. Thrivikraman, C.M. Grens, W.-M.L. Kuo, J.M. Andrews, and

J.D. Cressler, A High-linearity, X-band, SiGe low-noise amplifier

for improved dynamic range in next-generation radar and wireless

systems, IEEE Topical Meeting on Silicon Monolithic Integrated

Circuits on RF systems, 2009, 1–4.

2. Y. Aoki, N. Hayama, M. Fujii, and H. Hida, A 23/3-dB dual-gain

low-noise amplifier for 5-GHz-band wireless applications, IEEE

GaAs IC Symposium, 2002, 197–200.

3. H. Xu, C. Sanabria, A. Chini, S. Keller, U.K. Mishra, and R.A.

York, A C-band high-dynamic range GaN HEMT low-noise ampli-

fier, IEEE Microwave Wirel Compon Lett 14 (2004), 262–264.

4. S.T. Janesch, K.H.G. Duh, P. Ho, S.C. Wang, and S.M.J. Liu, 0.25

lm PHEMT X band multifunction LNA MMIC with T/R switch

and attenuator achieves 1.85 dB noise figure, IEEE Microwave

Symposium, Albuquerque, NM, 1992, 1179–1182.

5. W. Yau, H. Kanber, C.S. Wu, B.M. Paine, S. Bar, and Z. Bardai,

Design translation of an X-Band multifunction PHEMT MMIC,

IEEE Microwave Symposium, San Diego, CA, 1994, 1155–1158.

6. K. Li, J.-Y. Huang, and J.-F. Teng, Research on receiver dynamic

range extension with adjustable attenuator, International Confer-

ence on Wireless Communications, Networking and Mobile Com-

puting, 2009, 1–4.

7. T. Buber, F. Kolak, N. Kinayman, and J. Bennett, A low-loss high-

isolation absorptive GaAs SPDT PIN switch for 24 GHz automo-

tive applications, IEEE Proceedings of Radio and Wireless Confer-

ence, Philadelphia, PA, 2003, 349–352.

VC 2012 Wiley Periodicals, Inc.

AN ANALYTICAL TECHNIQUE TO FASTEVALUATE MUTUAL COUPLINGINTEGRALS IN SPECTRAL DOMAINANALYSIS OF MULTILAYEREDCOPLANAR COUPLED STRIPLINES

M. LucidoUniversity of Cassino, via G. Di Biasio 43, Cassino 03043, Italy;Corresponding author: [email protected]

Received 24 June 2011

ABSTRACT: The analysis of propagation in multilayered coplanarcoupled striplines by means of Galerkin’s method in the spectral domainwith expansion functions factorizing the edge behavior of the surface

current density on each strip leads to the evaluation of slowlyconverging integrals. In this work a new analytical technique to express

the mutual coupling integrals as rapidly converging series is presented.VC 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:1035–

1039, 2012; View this article online at wileyonlinelibrary.com.

DOI 10.1002/mop.26674

Key words: analytical technique; multilayered coplanar coupledstriplines; Galerkin’s method; spectral domain analysis

1. INTRODUCTION

In the past decades and recently, many researchers have devoted

attention to the analysis of propagation in multiple coupled

microstrip transmission lines and striplines, to simultaneously

obtain low computational cost and high accuracy in the evalua-

tion of the dispersion characteristics of the modes (see [1–4]

and the references therein for an overview).

It is well-known that fast convergence can be achieved by means

of Galerkin’s method performed in the spectral domain with analyti-

cally Fourier transformable expansion functions factorising the edge

behavior of the surface current densities on the strips [5, 6].

Unfortunately, the elements of the impedance matrix are

improper integrals of oscillating functions that can have a slow

asymptotic decay. Among these, the mutual coupling integrals

between coplanar strips are particularly troublesome because the

greater the distance between the involved strips is, the much

more rapidly the integrands oscillate.

To guarantee an accurate and efficient evaluation of such

integrals acceleration techniques are typically used. A common

approach consists in the extraction of the asymptotic behavior of

the kernel [7–9]. The slowly converging integrals of the

extracted contributions are independent of frequency and propa-

gation constant and can be expressed in closed form in some

special cases (coefficients of self-contribution or mutual-contri-

bution between two identical coplanar strips). A new very effi-

cient acceleration technique has been introduced in [10], consist-

ing in the extraction of a different contribution from the kernel

so as to obtain exponentially decaying integrands. Moreover, the

slowly converging integrals of the extracted contributions are

expressed as combinations of proper integrals and/or improper

integrals of nonoscillating exponentially decaying functions by

using suitable integration procedures in the complex plane.

In this work, a different approach specially designed to effi-

ciently evaluate the mutual coupling integrals that come out in

the analysis of propagation in multilayered coplanar coupled

striplines is introduced. Observing that the spectral domain

dyadic Green’s functions for such problems are single-valued

meromorphic functions, and the Fourier transforms of the expan-

sion functions are entire functions, such integrals are reduced to

series by means of Jordan’s lemma and residue theorem. The

obtained series are quickly convergent and, instead of what hap-

pens to the mutual coupling integrals, the required computa-

tional time rapidly decreases as the distance between the

involved strips increases. This goal makes the method particu-

larly suited to analyze structures with a large number of strips.

2. FORMULATION OF THE PROBLEM

In Figure 1 L perfectly conducting strips, of dimensions 2ai andcentred at the abscissas �xi with i [ {1,...L}, are located at the

interface between two lossless homogeneous and isotropic

dielectric slabs, of dimensions d1, d2 and relative dielectric per-

mittivity er1, er2, delimited by two perfectly conducting ground

planes.

It is assumed for the fields a behavior with z (longitudinal

direction) of the kind e�jkzz where kz is the propagation constant.

The following homogeneous system of integral equations in

the spectral domain can be obtained by imposing the tangential

components of the electric field to be vanishing on the strips

surfaces [5]

Figure 1 Geometry of the problem

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 54, No. 4, April 2012 1035