2338 ieee transactions on antennas and …theory to account for the parameters [1], [3]–[5]....

2338 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 5, MAY 2014

A Two-Port Model for Antennas ina Reverberation Chamber

Jason B. Coder, Member, IEEE, John M. Ladbury, Member, IEEE, and Mark Gołkowski, Member, IEEE

Abstract—We present an improved model for characterizingfundamental antenna parameters based on the concept of a passiverealizable two-port network. The proposed model is consistentwith circuit theory-based approaches and allows for quantitativedescription and subsequent experimental determination of an-tenna efficiency, loss, and mismatch. A derivation of the model isgiven along with a discussion of the implications of the model. Inaddition, we discuss how the model behaves in a low-loss rever-beration chamber, and give analytical and numerical methods forestimating the parameters of the model in a low-loss environment.An application of the proposed antenna model is shown to yield alower bound for an antenna’s radiation efficiency, although thisbound is useful only when the losses in the chamber are sufficientlylow. Derivation and application of the model is carried out in thecontext of a reverberation chamber, but the results are potentiallyapplicable to an arbitrary environment.

Index Terms—Antenna efficiency, antenna model, environmentcharacterization, parameter estimation, radiation efficiency, rever-beration chambers, two-port networks.

I. INTRODUCTION

I N modern communication systems, the precise quantifica-tion of losses and mismatches of an antenna is a serious

challenge. Even for classical antenna designs with accepted an-alytical models, empirical validation of basic parameters is dif-ficult. With the advent of greater computing resources, numer-ical modeling of antennas has becomemore accessible, but evensophisticated numerical models inherently involve approxima-tions and are prone to error when used for applications for whichthey were not originally intended. The situation of antenna char-acterization is even more cumbersome in the presence of man-ufacturing defects and elaborate designs and in complex en-vironments. Unknown antenna loss, efficiency, and impedancemismatch characteristics propagate into any data acquired witha real antenna and can be the source of significant error. Wepresent an improved model for antennas that accounts for fun-damental antenna characteristics including most imperfections.The model is robust, and can serve as a first order correctionfor many environments. It is composed of a passive, two-port

Manuscript received November 07, 2012; revised July 02, 2013; acceptedDecember 17, 2013. Date of publication February 26, 2014; date of current ver-sion May 01, 2014.J. B. Coder and J. M. Ladbury are with the RF Fields Group, National In-

stitute of Standards and Technology, Boulder, CO 80305 USA (e-mail: [email protected]).M. Gołkowski is with the Department of Electrical Engineering, University

of Colorado Denver, Denver, CO 80204 USA.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2014.2308520

network for which the parameters can be obtained or estimatedfrom experimental data.The modeling of an antenna as a two-port network is not new.

In [1], Rogers, et al. show an antenna as a two-port networkfor use in calculations related to communications channels. Themodel shown here is of similar nature, but approached from adifferent perspective.Similar work has also been done by She, where it is referred

to as “loss evaluation” [2]. In She’s method, certain antenna pa-rameters can be calculated by determining the individual lossesof each antenna component. She’s model works well for an-tennas in which the individual loss components can be easilydetermined. A similar calculation approach can be used to de-termine the parameters for our model.After the parameters of the model network are determined or

estimated in one environment, the results should be able to serveas a first-order correction when applied to the same antenna inan arbitrary environment. The model is designed to be general,and apply to the full range of practical antennas. The modelproposed here is consistent with other models that rely on circuittheory to account for the parameters [1], [3]–[5].Following the basic derivation in Section II, we describe the

characteristics of each of the network’s parameters. In some in-stances we will be able to determine or estimate the parameter,while in others we can only describe its characteristics and es-tablish bounds for its value.Sections III and IV outline the equations governing the two-

port model and each of its parameters. For simplicity, the casewhere there is only one antenna in the environment will be dis-cussed first. Section IV provides details on how the equationschange if two antennas are present.The first application of the model will be in Section V where

we detail one possible way to estimate the model network pa-rameters if the antenna can be placed in a very low-loss envi-ronment. From this estimation, a calculation of the lower boundof transmitting and receiving efficiency is possible. This as-sumption is the same as assuming the walls of the reverberationchamber have a small amount of loss.Section VI shows a validation of this particular solution of

the model network. Antenna efficiency will be calculated frommeasured data, and compared to theoretical calculations. Thesecalculations and comparisons allow us to examine the capabili-ties of the two-port model and identify some of the weaknessesor inaccuracies that exist.

II. ANTENNA MODEL AND NOTATION

The two-port model developed here was first proposed byLadbury and Hill in [6], where a partial derivation was given.

U.S. Government work not protected by U.S. copyright.

CODER et al.: TWO-PORT MODEL FOR ANTENNAS IN A REVERBERATION CHAMBER 2339

Fig. 1. Conceptual model of the proposed unknown two-port model. (a) showsa typical connection between a physical antenna and cable, and (b) shows theunknown two-port network model being inserted and directly connected (con-ceptually) to the antenna.

This approach simulates a real antenna [Fig. 1(a)] as an idealantenna connected to an unknown two-port network [Fig. 1(b)].This ideal antenna has three characteristics: it is lossless as botha transmitter and a receiver, perfectly matched when used as atransmitter in free space, and retains all pattern and scatteringcharacteristics from the original antenna. The imperfections ofthe real antenna (mismatch and loss) are incorporated in the un-known network, while the scattering and pattern characteristicsare not. As with any other two-port network, there are six pa-rameters that need to be considered: the intrinsic impedance ofeach port, the complex reflection coefficient of each port, and thetransmission/coupling coefficient between each port. Obtainingthe value of all six parameters enables one to calculate severalantenna parameters.Though the model we outline is restricted to two ports, a mul-

tiport model would be more complete because it would allow usto account for the scattering and pattern characteristics. In amul-tiport model, one port would represent the connector on the an-tenna, and each remaining port would represent a possible modeor ray path inside the environment or reverberation chamber [7].This approach would be extremely difficult to model mathemat-ically, as the number of possible ray paths inside a highly re-flective environment is very large. Instead of having one portfor each ray path, this model is simplified by accounting forthe overall statistical nature of a highly reflective environment.This simplification is similar to placing an ideal antenna in anideal reverberation chamber [8]. This model also assumes thatthe ideal antenna maintains the same physical size, pattern, andscattering characteristics of the real antenna.The network is attached to the ideal antenna via a virtual port,

denoted in Fig. 1(b) as Port . For simplicity, we will assumethat the intrinsic impedance of Port is the same as Port

Fig. 2. Signal flow graph showing the individual components of the two-portmodel.

1, or . For most applications this is 50 . We willalso make the simplifying assumption that the real antenna, andtherefore the two-port network, is reciprocal.Fig. 2 depicts a signal flow graph representation of the two-

port model. Here, represent the transmission from Port 1 intothe environment, and represents the signal received fromthe environment by antenna 1. is the portion of the receivedsignal from the environment that is reflected back into the envi-ronment, and is the input reflection looking into port 1. The 1subscripts indicate the port or antenna number being referencedby the equation. represents the amount of energy reflectedback to the antenna from the environment. In this case, the su-perscript is used to denote “environment.” The parametersof the environment (reverberation chamber or otherwise) willbe discussed in greater detail in Section IV. Since port isa virtual port, we are free to define the “location” of the refer-ence port in a way that is mathematically convenient. In otherwords, we can attach a lossless length of matched transmissionline so that we have control of the phase of either the transmis-sion coefficient or the reflection coefficient (but not bothsimultaneously). We choose to set the phase of to zero. This,combined with the assumptions of reciprocity and equal portimpedances give us , and both parameters are real.To begin the analysis of the model network, we can examine

how such a network would represent an antenna in free space.If the ideal antenna from Fig. 1(b) is placed in free space, thenthere will be no reflection from the environment, and thereforethe will be zero. This would be equivalent to replacing theideal antenna in Fig. 1(b) with a non-reflecting load. This givesa reflection coefficient at port 1 of , or the reflection coeffi-cient of the network when terminated by a non-reflecting termi-nation. For an incident power of , the reflected power willbe , the net power into the antenna will be

where is the transmissionmismatch factor. The power into the ideal antenna, which is thetransmitted power, will be .From these basic definitions, equations for the efficiency of

the real antenna can be developed. According to IEEE 145-1993, radiation efficiency is defined as: “The ratio of the totalpower radiated by an antenna to the net power accepted by theantenna from the connected transmitter” [9]. We now introducetwo terms. The transmission efficiency of an antenna is simplythe radiation efficiency for an antenna in free space [9]. Sim-ilarly, we define the transmission efficiency of a two-port net-work depicted in Fig. 2 as the ratio of the power dissipated by a


reflection-less load attached to the virtual port of the network topower accepted by the network at port 1, which is analogous tothe radiation efficiency outlined in [9]. This efficiency (denotedas ) is equivalent to the radiation efficiency of the real an-tenna, or

(1)

An optional subscript has been added to to denote theproper antenna of interest. In most cases, we will omit this sub-script.When used as a receiving antenna, we will assume that the

receiving instrument attached to Port 1 also represents a non-reflecting load. This assumptions simplifies the description, butimperfect instruments can be included with a few modificationsto the equations. Receiving characteristics of an antenna aremore complicated than the corresponding transmitting charac-teristics. In the receiving case, every incident plane wave willresult in scattering in all directions. The portion of the inci-dent energy that is received by the antenna and reflected by thetwo-port network will also be scattered (retransmitted) in all di-rections. As a result, discussing the receiving reflection coef-ficient of the antenna is somewhat ambiguous, and phase an-gles have very little meaning. However, referring to the modeldepicted in Figs. 1 and 2, we can define as the powerreceived by an ideal antenna for a given plane-wave excita-tion, the power reflected by the two-port network and the an-tenna is , net power received by the antenna will be

where is thereceiving mismatch factor. The power into the measurement in-strument, which is the measured power, will be .Conceptually, modeling the receiving antenna inside an ideal

reverberation chamber is equivalent to connecting Port of thetwo-port network to a random source that has the same statisticalproperties as the reverberation chamber. In this case, the signalmeasured at Port 1 will have the same statistical properties, butwill be scaled by . This result is consistent with [8] and [10].These definitions lead to an equation for the receiving ef-

ficiency. Unfortunately, IEEE 145-1993 does not define “re-ceiving efficiency” [9]. We define the receiving efficiency ofthe two-port network to be the ratio of the power dissipated bya reflection-less load attached to Port 1 of the network [refer toFig. 1(b)] to the amount of power accepted by the network froma source (of impedance ) connected to port . This can beexpressed algebraically as

(2)

The receiving efficiency of the antenna being modeled can beexpressed as a ratio of the total amount of power available tothe antenna (over steradians) to the power accepted by theantenna. The efficiency expressed in (2) is equivalent to the ef-ficiency of the real antenna.Because the model network is bound by the strict realizability

constraints of a passive two-port network, the transmission effi-ciency, receiving efficiency, and the transmission and receivingmismatch factors must all fall between 0 and 1 (inclusive).

The definition of transmission efficiency (and thus radiationefficiency) used here is consistent with the IEEE 145-1993 stan-dard definition of radiation efficiency. This definition does notconsider power reflected from the antenna due to an impedancemismatch in the efficiency calculation. Therefore, if 1 W is in-cident on the input port of an antenna, 0.5 W is reflected, andthe remaining 0.5 W is radiated, the antenna is considered to be100 % efficient.From the above analysis, a reasonable first step to model a

real antenna is to assume that for our two-port network isthe same as the free-space reflection coefficient for a realantenna in free space. If the efficiency is known, we can usethe efficiency relationship in (1) or (2) and the antenna’s reflec-tion coefficient to determine , but efficiency is not generallyknown or provided by antenna manufacturers, and can be dif-ficult to measure. One possibility is to estimate the efficiencywith a reverberation chamber [8], [10].For a reciprocal network, we know that . However,

we cannot be assume that . We will show in Section IIIthat for small losses, will be similar to , but they may besignificantly different in higher loss networks. In the case wherethe network is lossless, .With the model defined, the challenge of estimating each net-

work parameter in the two-port network is apparent. Using thelimits of physically realizable networks to bound the problem,we can analytically determine the allowable ranges for theparameters.

III. SINGLE ANTENNA EQUATIONS

With the basic definitions outlined in Section II, a series ofequations can be developed that will allow bounds to be estab-lished for the individual parameters.For simplicity, these equations will be shown in the context

of a single antenna inside a reverberation chamber. Therefore,and network parameters with a “1” subscript will be used.

However, we allow for the possibility of different or multipleantennas which would require changing the subscript from “1”to “ .” The details of modeling multiple antennas is discussedin Section IV.In an ideal reverberation chamber, the electric field is statis-

tically uniform throughout the working volume of the chamber.This uniformity is achieved by changing the boundary condi-tions using a rotating metal paddle. In a fundamental sense, stir-ring creates a new electromagnetic environment that contains adistribution of energy physically different than the previous en-vironment. A single chamber can be stirred to create hundredsof unique but statistically equivalent electromagnetic environ-ments. Models indicate that the ensemble of such environmentsyields a statistically uniform environment wherein electromag-netic parameters are independent of location and/or orientationin the working volume of the chamber [8].If is measured inside an ideal reverberation chamber, it

will have a uniformly distributed phase (0 to ), its magnitudewill be independent of the phase, and it will have an unknowndistribution bounded between 0 and 1. If the losses are high,such that the average reflection coefficient is much less than 1,a Rayleigh distribution can be used to approximate the distribu-tion of the magnitude. When expressed as a complex quantity,


the real and imaginary components are independent and nor-mally distributed with zero mean and identical variances [10].This concept of placing an antenna into a reverberation

chamber which is described by the random chamber reflectioncoefficient is analogous to terminating the unknowntwo-port network with a load that has a random reflectioncoefficient , which will have the statistical properties ofthe reverberation chamber. If the reflection coefficient of Port1, is examined while the random load is still attached toport , the result will also be random.With an idea of how the model performs in an ideal reverber-

ation chamber, a series of equations can be developed that ex-press these characteristics. These equations are rooted in wave-guide junction theory [11], and will be centered around the re-flection coefficients and .Waveguide theory [11] already provides an expression for, given :

(3)

If we write in polar form as we can rewrite(3) as

(4)

Utilizing the characteristics of an ideal reverberationchamber, we can assume that the magnitude and phase of

are independent of each other. Given this, it is usefulto examine the effects of changing the magnitude and phaseof separately. If the magnitude is held constant whilethe phase is varied, this would be conceptually equivalent toattaching a sliding mismatch to the two-port network. Thisproblem has been well characterized in [11]. By use of thissliding mismatch concept and (3), a circle (of radius ,centered at the origin) is transformed to a new circle with thefollowing radius and center :

(5)

and

(6)

where represents the phase component of the indicated pa-rameter. Applying our simplifying assumptions of reciprocity,matched port impedance, and zero phase for , (6) simplifiesto

(7)

where the bar over denotes the complex conjugate. If ,, , and can be measured or estimated, then it is possible

to determine (both magnitude and phase) as

(8)

We will make use of this in Section III-C. For the very specialcase of complete reflection , these equationscan be further simplified. From (5) we get

(9)

and (8) simplifies to

(10)

These simplifications give some hope that we may be able toestimate from measured data in a realistic environment.Fig. 3 shows the transformation of several circles from theplane (a) to the plane (b). In these examples, the mag-

nitude of was stepped from 0.0 to 1.0 in 0.1 increments.The network parameters are set to at a phase angleof 45 , with a 0 phase angle, andat a phase angle of 315 . These parameters result in both thetransmitting and receiving efficiencies being 1.

A. Estimating

Of the parameters in the two-port antenna model, is theeasiest to estimate. It is also the parameter we can determinewith the lowest uncertainty.We proceed by evaluating the expected value of . For this

analysis, we assume the phase of is uniformly dis-tributed. To keep this analysis general, we assume thathas an arbitrary distribution of , where . Forsimplicity, we will let and let . Because

and are independent, the joint distribution can bewritten as

(11)

The expected value of the measured value of , which isa function of the complex random variable [as given in(4)] and the joint distribution given in (7), can be obtained byintegration [12]:

(12)

This integral can be simplified:

(13)

The bracketed term in (13) integrates to 0 forand , which implies that

(14)


Fig. 3. (a) shows a series of circles representing data. (b) shows the samedata transformed to represent .

This validates the long held assumption that the free-space re-flection coefficient of an antenna is equivalent to the ensembleaverage of observed in a reverberation chamber [10]. Thisvalidation is a powerful result that will allowmeasured data(assuming a Raleigh distributed magnitude and uniformly dis-tributed phase) to be used in the calculation of the reflectioncoefficient ( ) of the unknown two-port network. With a re-verberation chamber, data can be measured at each stirrerposition (e.g., 100 stirrer positions), then averaged to obtain anestimate of .There are some subtle implications of the results here that re-

quire some additional comments. Since the bracketed term in(13) integrates to 0 for all between 0 and 1, the mean of eachcircle in Fig. 3(b), (or any other circle generated using asliding mismatch with uniformly spaced phase samples), is ,although (6) clearly indicates that the center of each circle isoffset from unless , , and/or are zero. For example,

Fig. 4. Data from Fig. 3(b) are re-graphed as individual points to show theirdistribution.

in the case of Fig. 3(b), the circle generated for resultsin an circle centered at the origin, but with a mean of .This seems non-intuitive, but can be explained if we takeuniform phase samples in the plane, and transform thosesamples into the plane, as shown in Fig. 4. The higher den-sity of points near is sufficient to shift the mean of each circleaway from its center. In fact, the offset between and pro-vides information about the magnitude and phase of , as givenby (10).

B. Characteristics of and

In Section II, we stated that the intrinsic impedance of bothports of the model network are assumed to be equal. This as-sumption coupled with the reciprocity of the network leads to

. The identical port impedances and reciprocal net-work do not mean the functions of the model network (e.g.,transmitting and receiving efficiency) are equal. Differences inthe and terms can cause the transmitting and receivingcharacteristics of the model network to be different.In Section V we will show one example of how the and

parameters can be determined. If the efficiency of the antennabeing modeled is known and can be determined using (14),then can be determined using a version of (1):

(15)

C. Characteristics of

The parameter is the most difficult to quantify because itrepresents a process that is not readily accessible theoretically orexperimentally. Instead of solving for a unique value, a rangeof possible values can be obtained. Utilizing information aboutthe other three network parameters, possible values can bebounded. Since the random and statistical nature of reverbera-tion chambers prevents us from determining an exact value, wewill show a series of constraints that can be placed on the valueof .


One such set of constraints are the strict realizability con-straints. Any passive, linear, time-invariant network must sat-isfy all of these conditions. From waveguide theory [11], theseconstraints are simplified to account for identical intrinsic im-pedances. The three stipulations for a realizable network are

(16a)

and (16b)

(16c)

Note that the first two constraints are based on the transmittingand receiving efficiencies of the model network.The realizability constraints combined with the equations for

the other three parameters can considerably limit the possiblevalues of . To further describe the value of , we can take ad-vantage of reciprocity and obtain a range of possible valuesthrough algebraic manipulation. The algebraic result will be alocus of solutions for , all located within a circle, and subjectto the following constraints:1) The distance from the center of the circle to the originof the complex plane is

(17a)

2) The radius of the circle will be

(17b)

3) The phase angle to the center of the circle will be

(17c)

The term in the (17c) becomes zero as a result of ourassumption that the phase of is zero.Given these three sets of constraints, can only be defini-

tively determined in a few special cases. From (17b), the pos-sible solutions for can be reduced to a small circle if the effi-ciency is high. In cases where the efficiencies are low, there willbe a large range of possible values. In Section V-C, we willdetail an iterative method whereby the parameter can be ap-proximated when the antenna is in a low-loss/highly reflectiveenvironment.If the model network is lossless (or assumed to be), then the

efficiencies shown in (17a) and (17b) will be equal to 1, and thethird constraint, (17c), becomes , or

, and since , we end up with .When the set of three bounds are applied with and equalto 1, the range of possible values shrinks to a single point.Assuming the network is reciprocal and the phase of the andparameters is zero, this becomes one of the few cases wherecan be uniquely determined.In cases where the model network is nearly lossless, the effi-

ciencies will be high, causing the radius of possible values tobe very small. For these cases, it may be reasonable to assumea value for . If the network has more loss, the radiusof solutions will grow. Assuming a value for in these casescould lead to large errors in subsequent applications.

Fig. 5. Series of two-port networks that describe a common reverberationchamber measurement. Within this series of networks are two networks thataccount for the imperfections of each antenna being used.

IV. TWO-ANTENNA CASE

Up to this point, discussion of the two-port antenna modelhas focused on a single antenna inside an arbitrary environ-ment. However, many common reverberation chamber appli-cations require two antennas be used. If we wanted to modelthe case where there are two antennas present in a reverberationchamber, the previous discussions could be expanded. Now, in-stead of a single two-port network representing the imperfec-tions of a single antenna, two additional networks are required:one for the reverberation chamber and one for the second an-tenna. This set of networks can be thought of as three cascadedtwo-port networks, as shown in Fig. 5. Though we specificallyspeak about using a reverberation chamber (denoted as ),this method could be used with any environment (denoted as

). The two antennas present in the environment need not besimilar because each will be represented by its own network.The notation for two antennas in an arbitrary environment

is shown in Fig. 5. The notation remains similar to the singleantenna case, but the subscript of each parameter is adjusted toindicate the appropriate port.Equations for the two-antenna case become significantly

more complicated because three networks are being cascaded:antenna-environment-antenna. In this configuration, the goalof the two-port model is the same: to remove the effects of theantennas, allowing a better characterization of the environment.This becomes more difficult when two antennas are present inthe chamber, as the effects of both antennas must be removed.To describe of a series of cascaded networks, a modified

version of (3) can be used. From (3), additional terms are in-cluded to account for the environment, the scattering due to thesecond antenna, and the interactions between the two antennas.The full expression for is shown as

(18)

(19)

and is the result of algebraically solving for from the setof three cascaded network matrices [11]. An examination of (3)shows that it is a special case of (18) where .


An equation for the signal that propagates through the envi-ronment can also be developed. Similar to (18), it is possibleto compute (note the lack of superscripts) by solving forit from the set of cascaded network matrices. The result of thisis (19), which accounts for the imperfections and scattering ofboth antennas, and represents the signal propagating from testport to test port.These two equations can also be used to provide expressions

for and , by simply changing the terms to apply to theopposite ports. The assumptions outlined in Section III for eachindividual two-port network still apply when they are cascaded.From (18) and (19), we can see that the only four directly-

measurable parameters are , , , and . From these,the only parameters we can directly estimate from the measureddata are and . Depending on the type of environment be-tween the two antennas, we may be able to estimate the charac-teristics of the environment ( , , , and ).If the two antennas are separated, such that they have no sig-

nificant scattering or multiple interactions between them, theparameters of the two-port network would be bounded by theequations in Section III. With each antenna properly character-ized by the two-port network, the characteristics of the environ-ment can be obtained.However, if the two antennas are strongly interacting with

each other, the single antenna solutions cannot be applied. An-tenna-to-antenna interactions can alter their respective radiatingand receiving properties. If some assumptions are made aboutthe environment (e.g., high- or low-loss), other methods couldbe used to find some of the antenna parameters. For example, ifantenna efficiency is obtained using another method [13], [14],it could be used to help identify some of the parameters of themodel network, and of the cascaded networks. This would thenlead to approximations about the parameters of the environment.

V. CALCULATING A LOWER BOUND FOR ANTENNA EFFICIENCY

One method for estimating the four parameters of the modelnetwork can be derived if we assume a very low-loss environ-ment. This derivation begins with the efficiency equations. Bymanipulating these equations, the four network parameters canbe estimated. In Section III, equations were developed assumingthe reverberation chamber was ideal. If we want to relax thoseassumptions, and estimate the network in a low-loss environ-ment, we need to understand how the change in environmentwill affect the results of the two-port model.We define a low-loss environment to be one where the power

absorbed by the walls is approximately the same or less thanthe power absorbed by the antennas. A reverberation chamberis well suited for this application because it can be operated atfrequencies where the losses are low and the amount of reflectedenergy [due to the walls and stirrer(s)] is high. The degree towhich this assumption is true in practice is what makes the fol-lowing calculation a lower bound of antenna efficiency.In cases where the environment has sufficient loss ( 70 %),

the statistical distribution of reverberation chamber has beenstudied in detail [8], [10]. These showed that, under some gen-eral conditions (well-matched antennas, over-moded chamber,and sufficient loss), that the phase of any measured -param-

eter will be uniformly distributed over all possible phases (0to ) and the magnitude will have an approximately Rayleighdistribution. The magnitude cannot have a true Rayleigh distri-bution, since a Rayleigh distribution has a nonzero probabilitythat a magnitude sample can exceed the unit circle. Even so, theRayleighmodel proved to be useful even for predicting extrema,as shown by close agreement between theoretical and measuredpeak-to-average ratios. This appears to be consistent as long asthe power absorbed by the chamber is much greater than thepower absorbed by antennas (or anything connected to the an-tenna) in the chamber, because under high-loss conditions, mea-sured values near the unit circle are very unlikely.As chamber losses decrease (which generally occurs as fre-

quency decreases), the Rayleigh model becomes less and lessappropriate. This is illustrated by a much lower peak-to-averageratio. So, even though average coupling continues to increaseas loss decreases, the peak is more constrained. However, thephase still appears to be uniformly distributed. In the most ex-treme case of a lossless chamber containing a single transmittingantenna, the reflection coefficient must be 1 (all power radiatedby the antenna must also be received by the antenna in a loss-less chamber). In this case, we also assume that the phase isuniformly distributed.Based on these observations, we will make some general

predictions regarding the behavior of any of the reverbera-tion chamber -parameters . For allconditions we will assume a uniform phase distribution. Forhigher chamber loss conditions, we will assume that the mag-nitude of the chamber -parameters has a truncated Rayleighdistribution, constrained between 0 and 1. As loss decreases,the probability density near 1 increases, which means that theprobability of observing a random sample near 1 increases. Asloss decreases further, the probability density shifts even moretowards 1. Regardless of the nature of the true distributions, wewill simplify the assumptions to the following for the low-losscase:1) The phase is uniformly distributed.2) There is a good chance that, given several hundred samplesof the magnitude, that at least three will be “close” to 1.

Unfortunately, we cannot directly measure the reverberationchamber -parameters, which means the distributions cannotbe directly observed. However, we can take measurements at afew frequencies where the antennas appear to be well matched.Fig. 6(a) shows an example of 1000 samples taken at 265.0MHzwith a dual-ridged horn antenna. Fig. 6(b) shows a histogram ofthe magnitude from Fig. 6(a).If we reexamine the case of a lossless chamber containing a

single transmitting antenna, we know that , and as-sume a uniformly distributed phase, so all samples will fall onthe unit circle, and the analysis above in Section III is appro-priate. We can now relax the assumptions of losslessness andassume instead that a few (at least 3) samples have a magnitudenear one. Now, only a few of the samples fall near the unit circle,and the analysis from Section III is more complicated. However,if we take the smallest circle that contains all of the samples, thisshould be very close to the unit circle. If we transform all of thepoints and the enclosing circle to using (3), then the trans-formed circle will still contain all of the transformed points, and


Fig. 6. Sample set of measured data, taken in a reverberation chamber witha small amount of loss. (b) Shows the histogram of the magnitude.

ideally the transformed circle should be the smallest circle thatcontains all of the transformed points.Unfortunately, if the antenna (modeled network) is poorly

matched, random samples tend to cluster near , which in-creases the probability that the transformed extremes will alsobe clustered near .

A. Receiving Efficiency

The two-port antenna model allows for calculating the re-ceiving efficiency as well as the transmission efficiency. Bothefficiencies are usually presumed to be equal, but this is not nec-essarily true. As the model two-port network is defined, the re-flections on either port may be different ( may not equal ).Even if the transmitting and receiving efficiencies of a givenantenna are found to be different, the transmitting and receivingpatterns, as well as and will remain equal.The equation for receiving efficiency was outlined in (9). To

calculate the receiving efficiency in a lossless environment, wemeasure the complex (e.g., using a vector network analyzer)and compute the radius of the resulting circle. This result is con-sistent with [11].Though the receiving efficiency of the model two-port net-

work can be calculated, verifying it empirically is very difficult.

There currently is no way to experimentally verify the receivingefficiency of an antenna. Consider a plane wave incident on anantenna under test (AUT). As the plane wave reaches the an-tenna, part of the signal will be received by the antenna, and partof it will be scattered by the antenna. We can measure the por-tion of the signal that is received by the antenna, but cannot yetmeasure the portion of the signal that is scattered by the antenna.

B. Transmission Efficiency

Once the receiving efficiency is known, calculating the trans-mission efficiency requires a few more steps.Mathematically, transmission efficiency is defined by (1). As

with the receiving efficiency, not all of the required parametersare known. However, it is possible to write the transmissionefficiency in terms of receiving efficiency. Beginning with (1),the denominator can be replaced with (due to reciprocity).The denominator can then be expressed in a version of (2),solved for

(20)

In previous sections, methods for estimating each of the pa-rameters in (20) have been described.Recall that in Section III, we discussed that the true value ofis very difficult to calculate. Usually, can be estimated

because it is dependent on the radius of the smallest circle thatbounds all of the measured data.

C. Calculating Efficiency—An Iterative Approach

From (2) and (20), we see that in order to calculate the trans-mission efficiency, we first need to find the receiving efficiency.Receiving efficiency is determined by , which is the radiusof a circle that bounds all data points in the plane. In thecase of a reverberation chamber, each data point could repre-sent the complex reflection coefficient measured at a dif-ferent paddle position.When the chamber is lossless , measured data

values will be within the unit circle, and calculated valueswill be on the unit circle. Thus, the radius of the minimumcircle will be 1. If the reverberation chamber is not lossless, and

, the data points will not extend to the unit circle. Themore loss present in the chamber, the further from the unit circlethe data may be. Similarly, as the loss increases, the radius ofthe minimum circle will decrease. This decrease in the radiuswill result in a lower estimate of receiving and transmission ef-ficiency simply due to a change in the environment.The other potential issue with the use of a minimum ra-

dius bounding circle arises when the AUT has a significantimpedance mismatch. In this case, may be appropriatelydistributed, but the impedance mismatch will cause thepoints to cluster close to the edge of the unit circle, as in Fig. 7.When a minimum radius circle is applied, the circle could crossthe unit circle, resulting in a set of equations that cannot besolved.One possible approach to circumvent this problem would be

to determine the smallest bounding circle that encompasses allpoints and uses the unit circle as one of the three points in de-termining the minimum bounding circle. This is what we hope


Fig. 7. Example of data observed that cannot be processed to obtain ef-ficiency information. The solid green circle denotes the unit circle and the redcircle is the minimum radius bounding circle.

to accomplish using an iterative method outlined below. Thisapproach begins with a set of measured data. The easiestway to describe this iterative method is to show a sample cal-culation. The same set of data shown in Fig. 7 will be used forthe sample calculation of efficiency. This data came from themeasurement of a single dual ridged horn antenna inside a re-verberation chamber. For Fig. 7 and the efficiency calculations,there are 1000 stirrer positions at a frequency of 468.8 MHz.The iterative method begins by assuming the antenna is 100

% efficient. This assumption, albeit incorrect, allows us to cal-culate an initial estimate of all four of the parameters in the two-port network. These four parameters will be used as a startingpoint for the iterative process. As the iterations progress, thesefour parameters will change to more reasonable values. Derivedfrom the strict realizability constraints, the following three equa-tions can be used to calculate the initial network parameters

(21a)

(21b)

(21c)

Once the four network parameters are calculated, assumingthe antenna is 100 % efficient, the data are transformed to the

plane by use of (3), solved for :

(22)

The goal with this transformation is to redistribute the pointsto have a more uniform phase. After this transformation iscomplete, the minimum radius bounding circle is found. Fig. 8shows the original data transformed to the plane(notice the more uniform phase distribution). The solid redcircle around the data is the minimum radius bounding circle.Now that the minimum radius circle bounding all of the

points has been found, the circle (not the data) must be trans-formed back to with (3).

Fig. 8. Original measured data after transformation to . The solid redcircle is the minimum radius bounding circle. The unit circle is shown in solidgreen.

Fig. 9. Minimum radius bounding circle from the plane after it is trans-formed back to the plane. The blue dot represents the center of the circle,and the solid green line is the unit circle.

Note that (22) contains several operations: reciprocal, scale,and offset. Each of these operations must be done on the circle asa whole. By use of a bilinear transformation technique, (3) canbe computed using only the radius and center of the circle [11].The key to this transformation is that if the circle is transformed(as well as all of the points), the transformed points will still allbe within the transformed circle.Fig. 9 shows the circle transformed back into the plane.

From this point, the center and radius of the circle in theplane can be used to calculate a new

(23)


Fig. 10. Data and minimum radius bounding circle (red) in the plane. Theunit circle is shown in solid green.

where is the new parameter calculated from the pointsof the circle, and are the radius and centerof the circle used to calculate the new , respectively.Unless the antenna being used is truly 100 % efficient, this

process may need to be repeated at least once more. This meanscalculating a set of new network parameters and repeating the

and transformations again, as described above. To re-peat the process, a new must be calculated:

(24)

As the process is repeated a second time, Fig. 10 shows theoriginal measured data transformed into the plane.To compare the results from iteration to iteration, compareFigs. 8–10.A careful examination of Fig. 10 reveals that the minimum

radius bounding circle has points that fall outside of the unitcircle. In this example, the difference is very difficult to see, butit can be more pronounced if the antenna’s mismatch is moresevere. While an exact mathematical explanation for this hasnot yet been determined, this appears to be an artifact of theprocessing that occurs only in intermediate iterations, and onlyfor a small fraction of the data points. A probable cause forthis artifact is that we are attempting to find parameters thatwould map points directly to the unit circle, even though allnetwork parameters are not yet estimated. Further investigationis required to determine the cause of this anomaly. We wouldbe more concerned if any of the redistributed points or center ofthe circle exceed the unit circle.Similar to Fig. 9, Fig. 11 shows the second iteration of data

after the minimum radius bounding circle has been transformedto the plane. The calculation ends when the lower boundof radiation efficiency stops significantly changing from itera-tion-to-iteration. In practice, this is usually after three iterations.The iterative method remains a lower bound calculation be-

cause the reverberation chamber and paddle are not ideal, and

Fig. 11. Minimum radius bounding circle from the plane after it is trans-formed back to the plane. The blue dot represents the center of the circle,and the solid green line is the unit circle.

do contain some loss. As the amount of loss increases, the lowerthe lower-bound calculation of efficiency will be. This methodyields results that are consistent with those in [15].

VI. ANALYTICAL EFFICIENCY CALCULATIONSAND MEASUREMENTS

As a first step toward validating the two-port antenna model,we will examine its specific application to antenna efficiency.If the model and calculations are shown to be consistent withanalytical calculations, we can have some confidence that themodel is functioning properly.In order to provide a basic analytical validation, we consider

a monopole antenna with finite conductivity over an infiniteground plane for which the transmission efficiency is analyti-cally tractable under standard antenna theory assumptions. Wedefine a dipole of radius , length and conductivity . By def-inition, the radiation efficiency is

(25)

where is the radiated power and is the Ohmic powerloss [16]. By use of image theory, the current distribution on theantenna (along the axis) is

(26)

where is the wavenumber and is the maximum current.With the given current distribution, the radiated power is givenby an integral over the space above the conducting groundplane:

(27)


Fig. 12. Photograph of the 750-MHz monopole used to compare the analyticalcalculation to the measured data.

where is the impedance of free space. Equation (27) canbe readily evaluated numerically. To evaluate ohmic losses, webegin with an expression for total resistance for any conductorof uniform conductivity:

(28)

where is the conductor length and is the cross-sectional areaof current flow. In the antenna, the current will flow primarilyon the surface, penetrating the conductor to a depth of the skindepth . Under the conditions of the crosssectional area of current flow in the monopole is ,allowing us to write the Ohmic loss as

(29)

The integral in (29) can be evaluated as

(30)

The above analysis is based on a perfectly conducting infiniteground plane. If we assume that the ground plane has a finiteconductivity, it is not unreasonable to also assume that losses inthe ground plane are equal to that in the antenna. The expressionfor efficiency then becomes

(31)

We can estimate the efficiency of a given monopole using thisset of equations. This calculation can then be compared tomeasurements made with a precision constructed monopoleinside a reverberation chamber. For this comparison, threedifferent monopoles were constructed. Each monopole wasdesigned to have a different resonant frequency: 750 MHz,1 GHz, and 1.5 GHz, respectively. Fig. 12 shows one of themonopoles constructed for this measurement.The diameter of each of the three monopoles is 6.4 mm. The

lengths are 100.4 mm, 62.7 mm, and 41.4 mm, respectively. Theground plane of each monopole has a diameter of 298.4 mm.The monopoles are constructed from Copper 101, which has a

Fig. 13. Comparison of the analytical efficiency calculation and the measuredlower bound of transmission efficiency for each monopole. To reduce noise, an11-point moving averaging is applied to all measured data.

nominal conductivity of S/m. Given these dimensions,the efficiency we expect for eachmonopole is shown as a dashedline in Fig. 13.The calculation indicates that the transmission efficiency of

each monopole is nearly flat (within 10 %) across the frequencyrange 500 MHz to 1.5 GHz. This is the result of the definitionof efficiency we are using; energy reflected by the monopole isnot considered to be a loss.Also shown in Fig. 13 is the lower bound of transmission effi-

ciency as estimated by the iterative method shown in Section V.In the case of the 750 MHz monopole, the measured data arewithin 5 % of the analytical model at the lower frequencies. Asthe frequency increases, the lower bound diverges from the an-alytical calculation.This divergence has twomain contributors. First, the assump-

tion that the chamber is lossless for at least a few paddle posi-tions becomes weaker as frequency increases. At lower frequen-cies, this assumption is close to being correct. As the frequencyincreases, losses in the stirrer and chamber begin to increase.This increase in loss shrinks the radius of the measured data,and thus results in a reduced lower-bound efficiency calculation.Second, the analytical calculation assumes that the ground

plane is infinite in size. This allows for simple assumptionsabout the current structure on the monopole. On an infiniteground plane, the monopole will appear to be a perfect dipoledue to imaging currents [5]. If the ground plane is not infinite,the image currents are not perfect, and the current structureis less favorable. This leads us to underestimate the lossesin the monopole and ground plane, causing a slightly highertheoretical efficiency.The data for the 1 GHz and 1.5 GHz monopoles follow

the same fall off as for the 750 MHz monopole. This gives avery clear indication that the loss of the reverberation chamberand stirrers is dominating the calculation. For comparison, abroadband (200 MHz–2 GHz) dual ridged horn was measuredin the same reverberation chamber at the same frequencies asthe three monopole antennas. These antennas are designed to


be reasonably well matched, low-loss, and efficient at all fre-quencies in this band. The similar fall off between the horn andthree monopole antennas is performing equally poorly for allantennas in cases where the antenna mismatch may be severe.Each of the three monopole antennas has a severe mismatchwhen it is operating outside of its resonant frequency.Despite this drop-off at higher frequencies, this comparison

provides some support for the two-port model being valid. Ifthe lower bound of efficiency were higher than the analyticalsolution, it would indicate that the radius of measured wasartificially large.Future research may indicate that this large difference can be

corrected. If the loss of the chamber is known from other means(e.g., a measurement of insertion loss), it may be possible to re-move the effects of chamber loss on the lower bound efficiencymeasurement.The next step towards validating this antenna model would be

to perform a finite element analysis. This analysis would requirea detailed model of a monopole be created and simulated tocalculate the transmission efficiency.

A. Measurement Setup and Uncertainties

For the measured data in Fig. 13, the antenna under test wasthe only antenna inside the reverberation chamber during themeasurement. Complex was measured by use of a vectornetwork analyzer (VNA). Data were acquired at 31.25-Hzintervals.The reverberation chamber measures 4.2 m in length, 3.6 m

width, and 2.9 m in height. There are two stirrers inside thechamber. One runs side to side, and the other from floor toceiling. Each paddle was stepped 23 times to create 529 differentenvironments. These data were then used to calculate the lowerbound of antenna efficiency. Data for the horn and 750 MHzmonopole antennas began at 500 MHz, and ended at 2 GHz.Data for the 1 GHz and 1.5 GHz monopole antennas began at750 MHz and 1 GHz, respectively.Uncertainties associated with themeasurement of include

uncertainties of the VNA (e.g., receiver nonlinearity), effects ofimpedance mismatch, and connector repeatability. These uncer-tainties total 3.3 % (coverage factor ). In addition to theseuncertainties, additional consideration is given to how the mea-surement uncertainty propagates through the model.To assess these uncertainties, a bootstrap analysis was per-

formed [17]. For this analysis, 100 random data sets were takenfrom each set of 529 stirrer positions. For each random dataset, transmission efficiency was calculated. The standard devia-tion of the results are used as an indication of the uncertainty ofthat parameter. The VNA uncertainty and standard deviation arecombined (via the root of the sum squared of the components)and a coverage factor of is applied. Fig. 14 shows an en-velope of the total expanded uncertainty of the lower bound ofradiation efficiency.Fig. 14 indicates the uncertainty of the lower bound estimate.

Although the uncertainty of the lower bound is low, this doesnot imply that the lower bound is actually a good estimate ofthe efficiency of the antenna. In some cases, the true efficiencymay be significantly higher.

Fig. 14. Uncertainty envelope of the lower bound of radiation efficiency, ex-pressed in terms of percentage points of radiation efficiency.

VII. APPLICATIONS AND FUTURE WORK

The proposed antenna model has the potential to isolate andremove the effects that an antenna has on the measurement of anenvironment. This would have a significant impact on our abilityto measure an environment utilizing an antenna. If all four pa-rameters of the model could be determined, and an antenna’seffects removed, the uncertainty of a variety of measurementswould decrease significantly.Here, we have shown one way in which the model network

parameters can be estimated for a special set of conditions: whenthe reverberation chamber is lossless, or has very little loss. Forone of the parameters, , the estimation is weak. Additionalwork is required to develop a deterministic solution for .The lower bound on antenna efficiency discussed in

Section V is only one example of an application. Once es-timates for the model’s parameters have been determined, thereare sure to be a wide variety of other applications. This willbe the focus of future work: what other characteristics can becalculated from the two-port model? Can we gain a better un-derstanding of the antenna’s mismatch or loss characteristics?Revisions or supplements to the model may also be possible.

By supplying supplemental information about the environment(e.g., loss characteristics) it may be possible to revise the esti-mations of the network parameters to include this information.Developing better estimations of the two-port model param-

eters when there is only a single antenna in an arbitrary en-vironment would result in significant progress toward solvingthe case where two antennas are in an arbitrary environment(Section IV). If the two-antenna case was solved, we could thor-oughly describe how signals propagate through an environment,while utilizing the two-port model to remove the effects of bothantennas. This has the potential to significantly improve mea-surements done to characterize wireless channels [18].Another important future research topic would be to focus

on the difference between the measured data and the analyticcalculation. This difference may yield some information aboutthe environment the antenna is operating in. Consider the casewhere an antenna is characterized in a lossless reverberation


chamber. In this environment the two-port model parameters areestimated or determined, and the parameters of the antenna be-come known. If the same antenna is then put in an unknown en-vironment, any difference between the expected efficiency andthe lower bound could be considered an aspect of the environ-ment. This could provide a useful method for characterizing anunknown environment.

VIII. CONCLUSION

An improved model for antennas in a reverberation chamberhas been presented. The proposed model works by accountingfor antenna imperfections (e.g., impedance mismatch and loss)with an unknown two-port network. One method for estimatingthe parameters of the two-port network under special conditionswas detailed. In this case, the network was estimated by in-serting the antenna into a reverberation chamber, data weremeasured, and a lower bound of antenna efficiency was calcu-lated. As a preliminary validation of the model, three nearlyideal monopole antennas were modeled and constructed. Theefficiency of each monopole was calculated frommeasured dataand compared to the numerical efficiency of the model. Theoverall model performance and applications were discussed.Areas for future work were identified, including a brief exami-nation of the case where two antennas are present in an arbitraryenvironment.

ACKNOWLEDGMENT

The authors would like to thank Dr. Ryan J. Pirkl for hishelpful input and ideas concerning the revised network notation.They would also like to thank Masih Hajihabib for his efforts inconstructing the monopole antennas.

REFERENCES[1] S. D. Rogers, J. T. Aberle, and D. T. Auckland, “Two-port model of an

antenna for use in characterizing wireless communications systems,obtained using energy efficiency measurements,” IEEE AntennasPropag. Mag., vol. 45, no. 3, pp. 115–118, Jun. 2003.

[2] Y. She, J. Hirokawa, andM.Ando, “Loss evaluation of post-wall wave-guide and its effect on antenna efficiency in millimeter-wave bands,”in Proc. IEEE Microw. Conf., Asia Pacific, 2009, pp. 2041–2044.

[3] L. J. Chu, “Physical limitations of omnidirectional antennas,” Mass.Inst. of Technol., Tech. Rep. 64, May 1948.

[4] C. T. A. Johnk, Engineering Electromagnetic Fields andWaves. NewYork, NY, USA: Wiley, 1988.

[5] R. S. Elliott, Antenna Theory and Design, D. G. Dudley, Ed. NewYork, NY, USA: IEEE Press/Wiley-Interscience, 2003.

[6] J. M. Ladbury and D. Hill, “An improved model for antennas in rever-beration chambers,” in Proc. IEEE Int. Symp. EMC, Ft. Lauderdale,FL, USA, Jul. 2010.

[7] R. J. Pirkl, “Spherical wave scattering matrix description of antennacoupling in arbitrary environments,” IEEE Trans. Antennas Propag.,vol. 60, no. 12, pp. 5654–5662, Dec. 2012.

[8] D. A. Hill, Electromagnetic Fields in Cavities—Deterministic and Sta-tistical Theories, 1st ed. Piscataway, NY, USA: IEEE Press, 2009.

[9] IEEE Standard Definitions of Terms for Antennas, Standard 145, TheInstitute of Electrical and Electronics Engineers Std. 145, Rev. 145-1993, Antenna Standards Committee, 1993.

[10] J. Ladbury, G. Koepke, and D. Camell, “Evaluation of the Nasa Lan-gley Research Center Mode-Stirred Chamber Facility,” National Insti-tute of Standards and Technology, Boulder, CO, USA, Tech. Rep. 1508,Jan. 1999.

[11] D. M. Kerns and R. W. Beatty, Basic Theory of Waveguide Junctionsand Introductory Microwave Network Analysis, 1st ed. New York,NY, USA: Pergamon, 1967.

[12] A. Papoulis and S. U. Pillai, Probability, Random Variables and Sto-chastic Processes, 4th ed. New York, NY, USA:McGraw-Hill, 2002.

[13] E. H. Newman, P. Bohley, and C. H. Walter, “Two methods for themeasurement of antenna efficiency,” IEEE Trans. Antennas Propag.,vol. AP-23, no. 4, pp. 457–461, Jul. 1975.

[14] H. A. Wheeler, “The radiansphere around a small antenna,” in Proc.IRE, Aug. 1959, pp. 1325–1331.

[15] J. B. Coder, J. M. Ladbury, and M. Golkowski, “On lower bound an-tenna efficiency measurements in a reverberation chamber,” in Proc.IEEE Int. Symp. Electromagn. Compatibility, 2012, pp. 216–221.

[16] C. Balanis, Antenna Theory: Analysis and Design, 3rd ed. New York,NY, USA: Wiley, Apr. 2005.

[17] B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap. BocaRaton, FL, USA: Chapman & Hall/CRC, 1998.

[18] K. Remley, G. Koepke, C. Grosvenor, R. T. Johnk, J. M. Ladbury,D. Camell, and J. Coder, “NIST tests of the wireless environment inautomobile manufacturing facilities,” National Inst. of Standards andTechonol., Boulder, CO, USA, Tech. Rep. 1550, Oct. 2008.

Jason B. Coder (S’06–M’11) received the B.S. andM.S. degrees in electrical engineering from the Uni-versity of Colorado, Denver, CO, USA, in 2008 and2010, respectively.As a graduate student his research focused on

signal processing and electromagnetics. He hasworked in the RF Fields Group of NIST, Boulder,CO, since 2005. His research interests include appli-cations of reverberation chambers, EMC metrology,standards development, and unusual applications ofEMC. In his short career at NIST, he has published

nearly two dozen papers. Outside of NIST, he serves as a Consultant for thedesign and implementation of electronic systems and scientific instrumentation.His consulting achievements include a U.S. patent and several new devicesthat have been deployed across the globe. He also continues his work at theUniversity of Colorado Denver as an adjunct faculty member, and collaborateswith faculty to create undergraduate and graduate curricula that develop andencourage interest in electromagnetics.

John M. Ladbury (M’09) was born in Denver, CO, in 1965. He received theB.S.E.E. andM.S.E.E. degrees (specializing in signal processing) from the Uni-versity of Colorado, Boulder, CO, USA, in 1987 and 1992, respectively.Since 1987 he has worked on EMC metrology and facilities with the

Radio Frequency Technology Division, National Institute of Standards andTechnology (NIST), Boulder. His principal focus has been on reverberationchambers, with some investigations into other EMC-related topics such astime-domain measurements and probe calibrations. He is a member of theInternational Electrotechnical Commission (IEC) joint task force on reverber-ation chambers.Mr. Ladbury received three Best Paper Awards at IEEE International EMC

symposia over the last six years.

Mark Gołkowski (M’10) received the B.S. degreein electrical engineering from Cornell University,Ithaca, NY, USA, in 2002 and the M.S. and Ph.D.degrees in electrical engineering from StanfordUniversity, Stanford, CA, USA, in 2004 and 2009,respectively.He served as a Postdoctoral Research Fellow with

the Space, Telecommunications, and RadioscienceLaboratory, Department of Electrical Engineering,Stanford University, from 2009 to 2010. He iscurrently an Assistant Professor at the University of

Colorado, Denver, CO, USA, in the Department of Electrical Engineering andalso Bioengineering. He actively conducts research on electromagnetic wavesin plasmas, ionospheric physics, near-Earth space physics, characterizationof antennas in reverberation chambers, hybrid imaging technologies, andbiomedical applications of gas discharge plasmas. He is associate editor of thejournal Earth, Moon, Planets.Dr. Gołkowski was recipient of the International Association of Geomag-

netism and Aeronomy (IAGA) Young Scientist Award for Excellence in 2008and American Geophysical Union Outstanding Student Paper Award in Fall2005. He is a member of the American Geophysical Union and the InternationalUnion of Radio Science (URSI)—Commission H (Waves in Plasmas).

2338 ieee transactions on antennas and …theory to account for the parameters [1], [3]–[5]....

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