Figure 8 shows the measured radiation patterns including the

copolarization and crosspolarization in the H-plane (y–z plane)

and E-plane (x–y plane). It can be seen that the radiation pat-

terns in both of y–z and x–y planes are nearly omnidirectional

for the two frequencies.

Figure 9 shows the measured and simulated maximum gain

of the proposed antenna and demonstrates a variation similar to

other PIFA antennas. As shown in Figure 9 for the WLAN/

WiMAX frequencies, the measured antenna gain agrees very

well with the simulated results.

4. CONCLUSIONS

In this article, a novel compact printed inverted-F antenna has

been proposed for simultaneously satisfying WLAN and WiMAX

applications. The fabricated antenna has the frequency band of

4.95 to over 5.91. The desired resonant frequencies are obtained

by adjusting the dimension of T-shaped notch. Also, to enhance

the impedance bandwidth characteristic, a rectangular slot is

inserted in the ground plane of the proposed antenna. Prototypes

of the proposed antenna have been constructed and studied exper-

imentally. The measured results showed good agreement with the

numerical prediction and good multiband operation.

ACKNOWLEDGMENT

The authors are thankful to Iran Telecommunication Research Cen-

ter (ITRC) for its financial support of the work.

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York, 2003.

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antenna with U-slotted arms for 2.4/5.2 GHz WLAN operation,

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band center-fed printed dipole antenna, in Proc IEEE Antennas

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der conductor lines, Electron Lett 37 (2001), 933–934.

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Dual wideband printed monopole antenna for WLAN/WiMAX

application, IEEE Antennas Wirel Propag Lett 6 (2007), 149–151.

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nas Propag 54 (2006), 613–619.

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(HFSS), Ver. 10, Ansoft Corporation, Pittsburgh, PA, 2005.

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728–731.

VC 2011 Wiley Periodicals, Inc.

PARTICLE SWARM OPTIMIZATION IN THEDETERMINATION OF THE OPTIMAL BIASCURRENT FOR NOISE PERFORMANCEOF GALLIUM NITRIDE HEMTS

Tyler Ross,1 Gabriel Cormier,1 Khelifa Hettak,2

and Rony E. Amaya21 Faculte d’ingenierie, Universite de Moncton, 57, avenueAntonine-Maillet, Moncton (N.-B), Canada E1A 3E9;Corresponding author: gabriel.cormier@umoncton.ca2Communications Research Centre Canada, 3701 CarlingAvenue, P.O. Box 11490, Station H, Ottawa, ON, CanadaK2H 8S2

Received 1 June 2010

ABSTRACT: Gallium nitride has attracted a great deal of interestin recent years due to its power handling ability. In addition, its

noise performance is known to be good. In this letter, we presenta method for determining the bias current density needed to

obtain optimal noise figure for gallium nitride high-electronmobility transistors (HEMTs). Particle swarm optimization is usedto fit transistor S parameters to a model, enabling the calculation

of the transistor’s two-port noise parameters. This process isperformed for different bias points for different-sized transistors,

leading to the conclusion that a current density of 0.3 mA/lmyields the best minimum noise figure. VC 2011 Wiley Periodicals,

Inc. Microwave Opt Technol Lett 53:652–656, 2011; View this

article online at wileyonlinelibrary.com. DOI 10.1002/mop.25758

Key words: gallium nitride; noise figure; noise modeling; MMIC; lownoise amplifiers

1. INTRODUCTION

In the past few years, gallium nitride has attracted the interest

of microwave circuit designers. This is been principally due to

its high suitability for power amplifiers, including its perform-

ance at high temperatures, its good power handling capacity and

its high breakdown voltage [1, 2].

In addition to its suitability for high heat and high power

applications, previous studies have noted its high resistance to

ionizing radiation, which is much higher than the radiation re-

sistance of gallium arsenide circuits [3]. If a designer would like

to benefit from the advantages that gallium nitride offers, then

one obvious route is to consider its use in a complete integrated

circuit.

In this case, a proper understanding of the noise behavior of

the transistors in the gallium nitride process being considered

for use is required. Different authors have reported on the noise

of gallium nitride circuits and these types of transistors are

reported to have good noise performance (see [4, 5], for

example).

In this work, a predictive noise model was applied to

describe the noise of gallium nitride transistors. Specifically, the

relationship between the minimum noise figure, Fmin, and tran-

sistor bias current was sought. To identify this relationship,

measured device S parameters must be fitted to a small-signal

transistor model. To do so, particle swarm optimization, a global

search algorithm was used.

A previous study by Dickson et al. examined the noise de-

pendence on the current density of various CMOS transistors

having different dimensions, bias conditions, and gate lengths

[6]. In that study, the authors found that regardless of the bias

conditions, transistor dimensions, and fabrication process,

652 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 3, March 2011 DOI 10.1002/mop

optimal Fmin was obtained at 0.15 mA/lm. This letter deals

only with noise performance and will show that a similar

result holds for the gallium nitride high-electron mobility

transistors (HEMTs) used here, albeit at a current density of

0.3 mA/lm.

This letter is laid out as follows: section 2 presents the meth-

odology used in this study, section 3 presents the particle swarm

optimization algorithm used to determine the transistor’s small-

signal parameters, section 4 describes the gallium nitride devices

used for this study, and section 5 presents and discusses the

results obtained.

2. METHODOLOGY

Noise models for field-effect transistors, and specialized models

for MESFETs and HEMTs specifically, have been presented and

have been well-known for quite some time. In addition, in more

recent years, noise models for gallium nitride HEMTs in partic-

ular, have been demonstrated [5, 7]. These models typically

have some weaknesses, such as the incorporation of fitting pa-

rameters which must be determined from measurements and

which in some case have no physical origin, noise sources are

occasionally neglected (many gallium nitride devices have gate

leakage currents which becomes a relatively important noise

source [8]), and some models do not enable the determination of

the four noise parameters used in microwave circuit analysis

(Rn, Ropt, Xopt, and Fmin).

A model presented by Sanabria et al. meant for use with gal-

lium nitride HEMTs, does not exhibit these shortcomings [9].

This predictive model uses a typical, small-signal HEMT model

in addition to bias conditions (including gate current) and can

determine all four noise parameters. This model does not depend

on fitting parameters or require noise measurements. The HEMT

model parameters must be determined which could require

measurements, though only S parameters would be necessary,

simplifying the process.

The model presented in [9] was applied to several transistors,

to determine their noise parameters. To validate the results

obtained, noise measurements were taken and compared with

the model’s predictions for several transistors and operating

conditions.

The small-signal model parameters used in the noise calcu-

lations are determined by fitting transistor S parameters to a

13-element small-signal HEMT model, shown in Figure 1.

This model was used because the device parameters used to

calculate the noise parameters cannot be directly obtained

from the measurement-based large-signal model developed by

Gain Microwave to predict the behavior of the measured

devices [10].

Fitting of S parameter data to the small-signal model was

accomplished using particle swarm optimization, a global

search algorithm well-suited to this type of problem. Because

particle swarm optimization is not commonly used in micro-

wave circuit design, the algorithm is presented in the follow-

ing section.

3. PARTICLE SWARM OPTIMIZATION

Particle swarm optimization is a global search algorithm first

proposed Kennedy et al. [11]. This algorithm is based on swarm

intelligence, where a large population collectively seeks a solu-

tion. One example is a swarm of bees seeking flowers to

pollinate.

The behavior of a particle is determined jointly by the col-

lective experience of the swarm and its own ‘‘personal’’ experi-

ence. The particle will wander toward the space between where

it had the most success (the local best) and where the entire

swarm had the most success (the global best). Although the par-

ticle swarm optimization algorithm does use random processes,

it is usually much more effective than a simple random search.

The algorithm can be broken down into a series of individual

steps, shown in Figure 2, which will be discussed further.

3.1. Creation of the SwarmThe first step of the algorithm involves the creation of the

initial swarm of particles that will search for the optimal so-

lution. The particles are placed in an n-dimensional space,

where n is the number of variables in the problem. In the

case of fitting S parameter data to the model shown in Figure

Figure 1 Small-signal transistor model used to calculate noise

parameters

Figure 2 Flowchart illustrating the particle swarm optimization

algorithm

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 3, March 2011 653

1, n would be 13. In this case, each particle is exploring a

13-dimensional space, each of the unknown variables corre-

sponding to one dimension.

The coordinates of each particle in the swarm are initialized

within a predetermined range specific to the problem. For exam-

ple, for the problem at hand, the transconductance gm of the

transistor is likely between 0 S and 4 S. The transconductances

of each particle will therefore fall inside these limits. Care must

be taken when selecting these limits: restricting the variable

space too much could lead to the optimal solution being

excluded, whereas a large distance between the two limits could

lead to convergence issues, requiring more particles and compu-

tational effort and time.

For the particle swarm optimization algorithm, the population

size is dependent on the problem. For the problem of fitting Sparameter data to a model, approximately eight particles per

variable appears to work well, leading to a swarm of 104

particles.

3.2. Evaluation of the Objective FunctionThe objective function determines the quality of a solution. In

fitting S parameters, the objective function measures the error

between the already-known S parameter data and the S parame-

ters of the small-signal model with parameters whose perform-

ance is to be evaluated. Equation (1) shows our objective func-

tion; it is based on the commonly used statistical measure of

goodness-of-fit for sum-of-squares minimization. In this equa-

tion, N is the number of (frequency) measurements, the sub-

scripts m and c indicate the measured and calculated values,

respectively, and e is the uncertainty on the S parameters. The

notation of the inner summation is simply to indicate that all

four S parameters are used in the evaluation of the objective

function.

v2 ¼ 1

4N

XNk¼1

XSi;j

ReðSi;jkm � Si;jkc Þ� �2þ ImðSi;jkm � Si;jkc Þ

� �2e

" #

(1)

The use of this biased objective function offers some

advantages over an unbiased objective function. First, it pro-

vides for weighting of the error between measurements and

model results, depending on the uncertainty of the measure-

ments (less reliable measurements are accorded less weight).

Second, it gives an indication of the error of the measure-

ments. When v2 � 1, the error is similar to the error of the

measured data, and the fit is good. If v2 >> 1, the error

between measurements and calculations is significantly differ-

ent from the uncertainty of the measurements. However, when

v2 >> 1, the uncertainty of the measurements may have

been overestimated.

3.3. Local and Global BestsThe local best of any given particle is found by comparing the cur-

rent value of the objective function with that particle’s previous

local best. If it is better, it will become the new local best. This

results in each particle having a limited ‘‘memory’’ of its past.

Similarly, the swarm as a whole keeps track of its global

best. The global best is the best of the individual particles’ local

bests, past or present.

3.4. Velocity and Position UpdatesOnce the local and global bests have been determined, they are

used to update the velocities of the particles. This is done to

guide each particle toward a stronger result. The velocity of par-

ticle k for the time step t þ 1 is given by

~vk½tþ 1� ¼ w~vk½t� þ c1~r1ðxg �~xk½t�Þ þ c2~r2ðxk �~xk½t�Þ; (2)

where ~v is the velocity of a particle, ~x is its position, and x sig-

nifies a best, whereas the subscript g indicates a global best and

the subscript k references the local particle. ~r1 and ~r2 are vectors

of random numbers with the number of elements of xk, in the

range (0,1). c1 and c2 decide how quickly the particle should

move toward its local best and the global best. Finally, w is the

inertia weight, a parameter that balances global and local explo-

ration of the search space [12].

Initial suggestions for these weights are w [ (0.9, 1.2) [12]

and c1 ¼ c2 ¼ 2 [11]. These values typically need to be hand-

tuned if they do not lead to convergence or if the algorithm con-

verges slowly.

Figure 3 Microphotograph of a wafer with the measured transistors.

The measured dimensions of the wafer are 1.2 mm � 0.76 mm. The

inset shows a 2 � 10 lm transistor (gate on right, drain on left, source

access on top and bottom)

Figure 4 Measured and modeled S11 and S21 for a 2 � 40 lm HEMT

654 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 3, March 2011 DOI 10.1002/mop

Lastly, the new position of each particle, k, is updated using

the following equation,

~xk½tþ 1� ¼~xk½t� þ~vk½tþ 1�: (3)

4. GALLIUM NITRIDE TRANSISTORS

Different gallium nitride devices were studied. The transistors

are AlGaN/GaN HEMTs, manufactured using the National

Research Council Canada’s gallium nitride foundry process, and

were supplied by Gain Microwave Corporation [10].

The AlGaN/GaN structure was grown on a semi-insulating,

500-lm-thick 4H-SiC substrate using ammonia molecular beam

epitaxy. Contact optical lithography was used to define transis-

tors having gate lengths of 0.8 lm. The devices have an ft of �15 GHz and an fmax of 45 GHz. All metal traces are 1-lm-thick

gold for this process. The fabrication process is more fully

described in [10]. Figure 3 shows a microphotograph of one of

the two identical wafers used for measurements.

As shown in Figure 3, the devices can be probed directly

using coplanar ground-signal-ground probes having a 100 lmpitch. The devices measured are all two-fingered transistors,

with gate finger widths of 10, 20, 40, 100, or 200 lm.

Noise measurements were taken from 3 to 6 GHz under vari-

ous bias conditions. The bias points were selected to provide the

best noise performance. This is typically just in the saturation

region of the I-V characteristic. For these gallium nitride devi-

ces, this usually occurs for drain voltages of � 5–7 V. The gate

voltage in this case was varied from �2 to �4 V, which yields

a fairly large range of drain current densities.

5. RESULTS AND DISCUSSION

The transistor S parameters were first measured on-wafer using

an Agilent E8361C PNA network analyzer. For noise measure-

ments, the transistors were probed using an Agilent N5242A

PNA-X network analyzer with a noise figure measurement sys-

tem. Ground-signal-ground probes with 100 lm pitch and short-

open-load-through de-embedding were used to remove the effect

of the measurement cables and probes.

Small-signal S parameters were measured first and compared

to a large-signal device model. As can be seen in Figure 4,

model simulations and measurements agree very well. Only S11and S21 are shown in this figure to avoid cluttering the image;

measurements of S12 and S22 also agree well with simulation

results.

The small-signal measurements show that the model is accu-

rate. However, it is difficult to extract accurate values for some

of the small-signal model parameters from this type of model.

For this reason, a 13-parameter small-signal model was fitted to

S parameters, to apply the predictive noise model described in

[9], using particle swarm optimization, as described in section 3.

Note that some of the parameters in the small-signal model vary

with bias, so this procedure must be repeated for each bias

point. Some of the parameters which do not vary with bias can

be held constant.

Noise parameters were calculated for several operating points

for each transistor. Figure 5 shows Fmin for the five devices, as

a function of drain current density. The drain voltage was set to

5 V in all cases. This places the transistors in their saturation

Figure 5 Minimum noise figure, Fmin, as a function of current density,

for transistors having different gate widths, at 5 GHz, and VD ¼ 5 V

Figure 6 Noise figure, F, for VG ¼ �2 V and VD ¼ 5 V, for the 2 �40 lm device, with a 50 X source impedance

Figure 7 Noise figure, F, calculated for VG ¼ �2 V and VD ¼ 5 V,

for the 2 � 40 lm device after fitting measured S parameters to a small-

signal HEMT model, with a 50 X source impedance

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 3, March 2011 655

regions. The noise figures shown were determined for transistor

operation at 5 GHz.

The variability in the noise figure is due to the small-signal

transistor model not fitting the S parameters predictions of the

large-signal model. However, the error is fairly small, and the

tendencies of the noise figure as a function of current density

are clear.

As can be seen from Figure 5, the best Fmin is obtained for

current densities of � 0.3 mA/lm, even for transistors of differ-

ent sizes. This indicates that optimal noise performance can be

obtained from an amplifier if it is biased using a constant-cur-

rent configuration.

To validate the results of the model, noise figure measure-

ments were taken. The noise figure measured and simulated at

50 X for a 2 � 40 lm device is shown in Figure 6. These

curves fit quite well, indicating that the model accurately repre-

sents reality.

Instead of relying on a large-signal model to be able to deter-

mine the small-signal parameters and then calculate noise pa-

rameters, it is possible to simply extract the HEMTs small-sig-

nal parameters from the S parameter measurements. In this case,

any additional terminal resistance and inductance will be

included directly in the fitted parameters, which should lead to

improved accuracy of the noise predictions.

Figure 7 shows the result of this procedure. Again, it should

be noted that no noise figure measurements were included in

this fitting process; the noise figure is determined using only

measured scattering parameters. The fit is good and the models

used were able to accurately predict noise figure, lending more

confidence to the results of Figure 5.

6. CONCLUSIONS

This letter has shown the dependence of the optimal minimum

noise figure (Fmin) on the drain current density of gallium nitride

HEMTs. It was found that the best noise performance was

obtained for current densities of � 0.3 mA/lm. This result was

obtained by using particle swarm optimization to determine a

transistor’s small-signal parameters, which were then used to

predict the noise parameters.

This result is useful to those designing gallium nitride low-

noise amplifiers, who must attain the lowest possible noise fig-

ure. Because the best noise figure is found at a given current

density, an effort should be made when designing an amplifier

to use constant-current biasing. This ensures that an optimal

noise figure is obtained and will not be unnecessarily degraded

by foundry process variations and tolerances.

The results discussed in this letter are similar to previous

results regarding CMOS transistors, although the optimal cur-

rent density for noise figure is found to be twice the current

density required for CMOS transistors. Gallium arsenide

designers have also been aware of a similar rule, where 0.15

mA/lm, or � 0.2IDSS, has traditionally yielded optimal noise

performance. During our work, we have found that gallium

nitride devices work best at even higher drain current

densities.

ACKNOWLEDGMENTS

The authors express their appreciation to the Natural Sciences

and Engineering Research Council of Canada (NSERC) for

their financial support. The authors thank Agilent Technologies,

Inc., for the loan of measurement equipment. Finally, the

authors appreciate Gain Microwave Corporation having pro-

vided the gallium nitride transistors studied and presented in

this article.

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A NOVEL COMPACT DUAL-MODEBANDPASS FILTER

M. Keshvari and M. TayaraniDepartment of Electrical Engineering, Iran University of Scienceand Technology, Tehran, Iran; Corresponding author:keshvari.mohammad@gmail.com

Received 2 June 2010

ABSTRACT: Using slow-wave transmission line ring resonator and

square-patch element, a novel compact dual-mode microstrip bandpassfilter is developed in this article. A filter having a 3% bandwidth at 2

GHz is designed, fabricated, and measured. The design procedure isgiven, and the mode-splitting characteristic and impact of the patchelement on the performance of the filter is investigated. The proposed

656 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 3, March 2011 DOI 10.1002/mop