Link_Budget+Magnetic

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 5, MAY 2012 2453

Link Budget and Capacity Performance of InductivelyCoupled Resonant Loops

Umar Azad, Student Member, IEEE, Hengzhen Crystal Jing, Member, IEEE, andYuanxun Ethan Wang, Senior Member, IEEE

AbstractA near field power transfer equation for an induc-tively coupled near field system, analogous to Friis transmissionequation for far field communications, is derived based on theequivalent circuit model of the coupled resonant loops. Experi-mental results show the proposed near field coupling equation istrustworthy as it correctly predicts the transferred power versusdistance relationship for different values of loaded quality factorsat the transmitter and the receiver. Capacity performance of nearfield communication (NFC) links are analyzed based on informa-tion theory, respectively for noise limited and interference limitedscenarios. The analytical results provide guidelines for power andcapacity budget in inductively coupled antenna systems. Examplesof inductively coupled VLF NFC links are evaluated for differentoperating scenarios, demonstrating the efficacy and importance ofthe proposed method for near field link budget.

Index TermsInductive coupling, near field communicationsystem, nonradiative power transfer, undersea communications,very-low frequency/ultra-low frequency (VLF/ULF), wirelesspower transfer.

I. INTRODUCTION

INDUCTIVELY coupled nearfield system is a short-range

wireless technology which allows the devices to communi-cate through the coupling of magnetic field rather than the en-

ergy radiation-interception process in farfield communications.

The technology has been used or proposed in many application

areas such as wireless power transfer [1], contactless power and

information transmission in drilling machines [2], wireless pow-

ering of implantable systems [3], [4], RFID [5], health mon-

itoring [6], real time location system [7], inductively coupled

electric highway system [8] and seamless coverage of littoral

mine warfare operations in shallow water, surf and beach zones

[9]. To evaluate the performance of nearfield energy transfer

links, the power transfer efficiency of strongly coupled loops

has been derived based on the coupled mode theory [1] and

voltage gain expression has been derived based on equivalent

circuit model in [10] for the configuration consisting of coupled

coils with additional driving loops. It is observed that resonance

Manuscript received August 18, 2010; revised June 08, 2011; acceptedNovember 11, 2011. Date of publication March 09, 2012; date of currentversion May 01, 2012. This work was supported by Office of Naval ResearchAward N000140810083.

U. Azad and Y. E. Wang are with the Electrical Engineering Department atUniversity of California at Los Angeles, Los Angeles, CA 90095 USA (e-mail:[email protected]).

H. C. Jing is with the QuinStar Technology Inc., Torrance, CA 90505 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.2012.2189696

plays an essential role in power transfer mechanisms and it im-

proves the efficiency over the case of inductively coupled non-

resonant objects. A nearfield propagation equation is also pro-

posed in [11] in which the path gain concept is introduced to in-

corporate different rate of path loss of the electromagnetic field

in a NF system. However, concerns are often raised regarding

how the properties of antennas and impedance terminations im-

pact on the performance of the nearfield system. This issue is

addressed here by deriving the power transfer relationship of

inductively coupled resonant loops, in a simplified setup con-

sisting of transmitter and receiver coils that connect to variablesource and load impedances respectively. It leads to a concise

formulation called nearfield power transfer equation, which ex-

presses the transferred power in a function of distances between

the loops, dimensions and intrinsic quality factors of the loops

and terminating impedances at both the transmitter and the re-

ceiver. Some preliminary results for weak coupling cases have

been reported in [12]. In thispaper, with insights gained from the

nearfield power transfer equation, a comprehensive discussion

is carried out for both strong coupling and weak coupling cases,

aiming for applications respectively in wireless power transfer

and in nearfield communications (NFC). It shows in strong cou-

pling cases, an optimalload termination condition exists for agiven distance, whichmaximizes the power transfer efficiency

at this distance, while in weak coupling cases the received power

always reaches to its maximum under the conjugate matching

condition. When the coupling is weak, the received power falls

off inversely withthe sixth power of the distance between the

coils but increases with improving quality factors of the trans-

mitting and receiving antennas. However, the benefit brought by

use of high quality factor coils to the capacity of a NFC system

is limited as the increasing quality factor eventually limits the

bandwidth of the communication system. In general, a loaded

quality factor other than the conjugate matching may provide

the best tradeoff between the received power and communica-

tion bandwidth for the maximum capacity.

This paper is organized as follows. Section II presents the

derivation of the nearfield power transfer equation and the ap-

plications of this equation in both strong and weak coupling

cases. The experimental results in Section III validate the pro-

posed theory. The capacity performance of a NFC link is dis-

cussed in Section IV based on Shannons information theory for

both thermal noise and natural interference limited scenarios.

In Section V, a very-low frequency/ultra-low frequency (VLF/

ULF) NFC link in air is used as an example and its informa-

tion capacity versus distance is analyzed with numerical simu-

lations for different setups of impedance matching. It concludes

that there is an optimum loaded quality factor selection for both

0018-926X/$31.00 2012 IEEE


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2454 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 5, MAY 2012

Fig. 1. (a) Inductively coupled nearfield system. (b) Equivalent circuit modelof nearfield system.

transmitting and receiving loops that results in the maximum

capacity for a certain distance of communication for both noise

and natural interference limited scenarios.

II. THEORETICALANALYSIS

The nearfield system consisting of inductively coupled res-

onant loops is shown in Fig. 1(a). Two circular coils, , and

of radii and , respectively, are centered on a single

axis facing each other in their normal direction. The transmitter

and the receiver are separated by a distance . The coils consist

of and closely wound turns and carry currents and

respectively. As dimensions of coils and distances between the

two coils under consideration are much smaller than the wave-

length of the electromagnetic wave, magnetostatic approxima-

tions can thus be applied, which leads to the equivalent circuit

model in Fig. 1(b). and are the resistances of the coils

at the operating frequency including Ohmic loss resistance, ra-diation resistance and other losses such as the absorption of the

surroundings, , are the self-inductances of the coils and

, are the capacitors to resonate with the transmitters and

receivers coil at an identical frequency in order to create the

maximum coupling sensitivity. and are the source and

load impedances respectively.

A. Self Inductance, Mutual Coupling, and Coupling Coefficient

Under the assumption of infinitesimal thickness of the coil,

the property of homogeneous magnetic field inside a solenoid

is used as a coarse approximation to the field distribution of

the loop antenna. The accuracy of these approximations is to

be examined against COMSOL simulation results in Section V.

More accurate expressions for mutual impedance for various

configurations and shapes of single layer and multi-layer coils

are given in [13][15]. The self-inductance of transmitter and

receiver coils and the mutual inductance between the two coils

in free space (see Appendix) are given by

(1)

(2)

To quantify the strength of the coupling between the coils, the

coupling coefficient is defined as it is in [5]

(3)

where is themutual inductance induced by the inductive cou-

pling between the two coils. Substituting (1) and (2) into (3),

the coupling coefficient between the two coils in free space is

yielded as

(4)

It shows that the coupling coefficient between two conductor

coils in free space is frequency independent and varies with the

inverse cube of the distance, i.e., when the distance be-

tween transmitter and receiver is much larger than the radius of

transmitter and receiver coils i.e., , . This coincides

with the nearfield of an infinitesimal loop, which is in the order

of .

In a homogeneous lossy medium, the attenuation effect of

the lossy material on the coupling coefficient needs to be in-

cluded. The coupling coefficient between the two coils in a lossy

medium is then modified to be

(5)

where is the attenuation constant of the medium. For antennas

other than loops, coupling coefficient may take different forms

but will in general be in the same magnitude unless higher order

resonant modes [16] are used, in which case, a more directive

coupling but a faster attenuation rate versus distance is expected

as predicted by the spatial distribution of the nearfield of those

higher order modes.

B. Near Field Power Transfer Equation

To setup the inductively coupled resonant loops, one must

use capacitors to resonate with the self-inductance of the coils

in both transmitter and receiver at the same resonant frequency

, as shown in Fig. 1(b). In general,

the mutual coupling between two coils affects the impedance

seen from either the transmitter side or the receiver side. As the


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AZADet al.: LINK BUDGET AND CAPACITY PERFORMANCE OF INDUCTIVELY COUPLED RESONANT LOOPS 2455

self-reactance of the coils are cancelled out by those of the res-

onating capacitors, the currents and flowing at the trans-

mitter and receiver coils satisfies the following relationship,

(6)

where is the source voltage. Simultaneously solving the two

equations in (6) yields the current in receiver coil

(7)

Consequently the generalized expression for received power

is

(8)

where is the available power from the source at

the transmitter. In Fig. 1(b), applying the definition of quality

factors to both the transmitting and receiving resonators yields

(9)

where and are the loaded quality factors of the trans-

mitter and the receiver; and arethe intrinsic quality

factors of the transmitting and receiving antennas. Substituting

(9) into (8), the received power can thus be written as a function

of the quality factors

(10)

Equation (10) is so-called near field power transfer equation,

which reveals the impact of impedance terminations and an-

tenna quality factors on nearfield power transfer. Though the

equation was derived based on the equivalent circuit of coupled

resonant loops, one can generalize thisrelationship for other res-

onators coupled through nearfield. The intrinsic quality factors

of antennas are limited by the loss at the coils including both

the radiation and conduction loss. As typically antennas with

extremely small electrical sizes are used in nearfield systems,

their radiation loss can often be ignored and the intrinsic quality

factors are limited by the Ohmic loss of coils, which is primarily

determined by the conductivity and the cross-section of the wire

[4].

C. Power Transfer Under Strong Coupling Assumption

When is close to one, e.g., is comparable

to , it implies the coupling is strong

enough to create a non-negligible effect on the impedance match

in either the transmitter or the receiver. This is so-called strong

coupling region [1] in which wireless power transfer often oper-

ates. It is evident from (10) that a high power transfer efficiencynecessitates use of high coils such that and

. Equation (10) thus reduces to

(11)

It concludes from (11) that the received power is maximized

when , yielding a perfect power transfer efficiency

, e.g.,

(12)

Theoretically 100% effi

cient power transfer for lossless coilscan be obtained for any distance as long as an appropriate

impedance transformation is used so that . This

optimum matching condition requires adjusting the source or

load impedance for different distances, which may be realized

by inserting variable ratio voltage transformers between the

transmitter/receiver and the coils. For coils with finite quality

factors, the maximum power transfer efficiency can be

approximately given by substituting the above condition into

(10)

(13)

D. Power Transfer Under Weak Coupling Assumption

For the case of weak coupling, the effect of the mutual cou-

pling between the two coils on the impedance seen from the

transmitter side can be ignored. The currents and , respec-

tively, at the transmitter and the receiver at the resonant fre-

quency are given by

(14)

The received power under weak coupling assumption is

(15)

On the other hand, the weak coupling case implies

, as and as expected the

generalized power transfer (8) reduces to (15) by applying this

approximation.

Nearfield power transfer equation under weak coupling as-

sumption shows that the received power through inductive cou-

pling in nearfield communication system is proportional to the

square of the coupling coefficient , the loaded quality factors


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Fig. 2. Experimental setup for nearfield power transfer measurements.

, , and it rolls off at the rate of , in contrast to the far

field power rolling off inthe order of . Thisrapid rolling off

behavior provides nearfield system more advantages for com-

munications in short ranges as less likely a near field system

interferes with other systems outside a certain range [11].

The termination efficiency at the transmitter and receiver

is characterized by the coupling factor and, respectively. To maximize the received power

through the coupling, the critical coupling condition [17]

should be selected both at the transmitter and at the receiver,

e.g., and and the received power

under this condition is thus

(16)

It is evident that and should be made as high as

possible to maximize the power coupled through under both the

strong and the weak coupling assumption.

III. EXPERIMENTALVALIDATION OF NEARFIELD

POWERTRANSFEREQUATION

In order to validate the near field power transfer equation,

two coils of 5-cm radius and 24 tightly packed turns are built

using a copper wire of 1-mm radius. The self-inductance of the

coil computed using (1) is 58.4 H while the measured value

of the self-inductance using the 4342A Q-meter is 60 H. With

330 pf capacitors attached to both coils, the resonant frequency

is observed at 1.06 MHz. The quality factor of both coils at

1.06 MHz measured using 4342A Q-meter is 59. Therefore the

transmitter and receiver coil resistance calculated from (9) is6.75 . The transmitter is an arbitrary waveform generator with

the standard 50- output impedance and the receiver is a digital

oscilloscope with 50- input impedance. The source voltage

is 10 volts peak-to-peak and therefore the maximum available

power from the transmitter is 0.25 watts. The experimental setup

of the nearfield power transfer is shown in Fig. 2. Two coils are

placed normal to each other with centers aligned in one line,

which are connected to both the transmitter and the receiver

through voltage transformers.

By selecting the turn ratio of the voltage transformer among

8:3, 1:1, and 3:8, one can obtain source and load terminations

with three different loaded quality factors of 29.5, 7, and 1.1

alternatively. The received power is measured for different

distances and compared with the calculated received power in

Fig. 3. Measured received power (dBm) and calculated received power plotted(dBm) using generalized power transfer equation and power transfer equation

derived under weak coupling assumption against Distance between same Coils(cm) on a log scale for different values of loaded Q of transmitter and receiver.

Fig. 3. The three groups of curves plotted in Fig. 3 in different

colors correspond to high-Q, medium-Q, and low-Q termina-

tions, respectively.

Within each group, solid, dashed, and dotted lines are repre-

senting the measured result, the result calculated with the near

field power transfer (10), and theresult calculated with the equa-

tion under weak coupling assumption (15). In general, all the

curves within each group converge at far distances when cou-

pling is weak. The discrepancy between the dotted curves andother curves at close distances can be attributed to the weak cou-

pling assumption used in deriving (15), as the measured result

agrees well with the calculated result without such an assump-

tion even at close distances. The coupled power increases mono-

tonically when distance draws closer until the coupling is strong

enough to affect the impedance matching so that the condition

of no longer holds. It is observed that the cou-

pled power indeed reaches to the peak at the proximity of the

distance satisfying for all cases except the low-Q

case where the distance satisfying such a condition is out of the

measured range.

Among the three groups of curves, lower power is observed

for lower Q cases at far distances, while the high-Q case ex-

hibits the greatest transferred power as it operates at the critical

coupling condition. However, the medium-Q case has greater

measured power transfer efficiency than that in the high-Q case

at close distances as the strong coupling condition kicks in. The

measured maximum power transfer efficiency for any distance

is 72.1% for the medium-Q case and 21.1% for the high-Q case

versus the theoretical predictions of 77.7% and 25% respec-

tively given by (13).

IV. CAPACITYPERFORMANCE OFNFC LINK

To discuss the capacity performance of a NFC system under

the weak coupling assumption, the ShannonHartley theorem


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defining the capacity of a digital communication system is

introduced

(17)

where is the bandwidth in Hz, is the received power and

is total noise power at the receiver over the bandwidth .

Therefore, when a digital communication link is built upon near

field coupling mechanism, not only the transferred power, but

also the bandwidth of the communication system is of impor-

tance to the capacity performance of such a link. The fractional

bandwidth of a NFC system can be estimated from the loaded

quality factors of the transmitter and the receiver through the

following:

(18)

where is thecenter frequency. A termination based on criticalcoupling increases the signal to noise ratio by improving the

received power, yet it may not offer the optimal system capacity

as the signal bandwidth may be sacrificed. It suggests that the

transferred power and the bandwidth of a NFC system must be

traded-off for the optimal capacity.

The source of the noises in the receiver can be either natural

interference dominated or thermal noise dominated,and

the total receiver noise is the sum of natural interference

and thermal noise. The capacity performance of such a system

is further discussed as follows.

A. Capacity Performance in Thermal Noise Limited Scenario

According to Plancks blackbody radiation law, the thermal

noise power is approximately given by

(19)

where is the Boltzmanns constant having value

and is the system noise temperature mea-

sured in Kelvins. By substituting (19) and (18) into (17), the

capacity of a NFC system in a situation where the thermal noise

is the dominant source of noise is expressed entirely as a func-

tion of loaded quality factors of the transmitter and receiver as

shown in (20) at the bottom of this page. Hence, one needs to

search through all the possible values of loaded quality factorsin both the transmitter and the receiver for an optimum pair that

maximizes the system capacity given by (20).

B. Capacity Performance in Natural Interference Limited

Scenario

In many situations, natural interference caused by lightning

in the ionosphere may become the main source of the receiver

noise. This is particularthe case when the system operates at low

frequencies such as ELF/VLF bands [18]. The received interfer-

ence power from ELF/VLF noise is derived (see Appendix)

(21)

where

The capacity performance of such a system is thus given by

(22)

(23)

The first equation in (23) shows that in the case that the trans-

mitter bandwidth limits the system bandwidth, the capacity per-

formance becomes independent of the size and the quality factor

of the receiver antenna.

C. Comparison of Thermal Noise and Natural InterferenceThe ratio of natural interference to thermal noise in the re-

ceiver is obtained by dividing the natural interference noise

picked up by receiver in (21) by the thermal noise power in (19)

(24)

The noise temperature of a receiver can vary between 100 and

400 K. Termination efficiency factor in gen-

eral varies between 0.5 for high-Q receiver when

and 1 for low-Q receiver when . The natural in-

terference can be comparable to the thermal noise in power

depending on the size of coil, loaded quality factor, termina-tion efficiency and temperature at receiver. In general, for re-

ceivers with low-loaded Q, the thermal noise is comparable or

if

if

(20)


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Fig. 4. Variation of (a) mutual inductance and (b) coupling coefficientwith the communication distance in free space.

greater than the received natural interference since a large re-

ceiver bandwidth leads to more thermal noise, while for re-

ceivers with high-loaded Q, the received natural interference

can be dominant since a higher loaded Q at the receiver inter-

cepts a greater amount of natural interference.

V. SIMULATION RESULTS

An inductively coupled NFC air-air link operating at VLF

frequencies is examined in this paper to demonstrate the im-

pact of quality factors to the capacity performance of a weakly

coupled nearfield communication system. The simulations are

carried out in the following steps. First, the commercial soft-ware COMSOL Multiphysics is used to extract the equivalent

, parameters and the coupling coefficient of the coils by

performing quasi-static electromagnetic simulations. One can

calculate the quality factors from the extracted and param-

eters, which are then substituted into (15) to lead to the received

power versus different distances. Finally, (17) is used to com-

pute the capacity of the link for a certain assumed noise level.

On the other hand, those parameters can also be analytically de-

rived using (1)(3). The capacity performance can thus be cal-

culated in the same fashion for comparison. Due to the axial

symmetry of the coils, the simulation is performed in thetwo-di-

mensional space of the wire cross section. In all the simulation

scenarios, two identical circular coils made of copper with loop

radii m, the wire radius cm are chosen

Fig. 5. Capacity versus transmitter/receiver Q of the near field link at threediffe rence distances ( a) at kHz (b) at kHz.

as the antennas at the transmitter and receiver. The coils consist

of 50 wound turns and the spacing between two windings next

to each other is at cm.Two identical circular coils centered on a single axis and

separated by distance in free space form a communication

link in the air. The simulation is performed when the coils are

operating at two resonant frequency points, KHz and

KHz, respectively. The obtained resistance of the two

coils from simulations is at 1 KHz

and at 3 KHz. The self-inductance of

the two coils from simulation is mH at both 1

and 3 KHz. In contrast, the analytic value of the self-inductance

is mH at both 1 and 3 KHz obtained from (1).

This is because that the uniformity assumption of the magnetic

field inside the solenoids is made for the loops, leading overesti-

mations of self-inductance for loops than the simulated results.

The intrinsic quality factor of the coils from The simulation is


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Fig. 6. Effect of the available power from the transmitter on the capacity per-formance of the nearfield link.

at 1 KHz and

at 3 KHz.

Fig. 4(a) and (b) show the variations of the simulated mutual

inductance and coupling coefficient versus the separation

distance between the transmitter and receiver in free space in

the range of 1 to 10 km, in comparison to analytical results ob-

tained from (1)(3). The solid blue curve represents the analytic

value of mutual inductance and coupling coefficient and the as-

terisk and circle lines are the extracted results from the simula-

tions. The analytic and simulation results of the mutual induc-

tance agree very well within the communication range of 1 to

10 km. The extracted coupling coefficients from simulationsis around 2.5 times of the analytic values due to the discrepancy

between the simulated self-inductance and those derived analyt-

ically. Furthermore, the simulation results show that the varia-

tion of coupling coefficient between the two coils in free space

is almost independent of their operating frequency and it does

roll off at the rate of inverse cube of communication distance .

The coupling coefficient drops from 3.1e10 to 2.5e13 when

increases from 1 to 10 km.

To generate the capacity versus distance curves, the available

transmitter power is assumed to be 60 W and the noise tem-

perature of the system is K. The received power is

calculated with (15) and the noise power is the superposition ofthe thermal noise power and natural interference power. There-

fore, the capacity versus the transmitter/receiver loaded quality

factor is computed for both kHz and kHz and

plotted in Fig. 5(a) and (b) for several choices of distances. Fig. 5

shows there exists an optimal quality factors for each distance

that maximizes the link capacity as the transferred power and

the bandwidth of the link must be traded off. It is observed that

the optimal data rate of the system operating at kHz is ap-

proximately 2.5 times that of the system operating at kHz.

For example, at km, the highest data rate at kHz

is 1900 bps achieved at transmitter and receiver of 5, while,

at kHz it is only 800 bps achieved at the transmitter and

receiver of 4. Fig. 6 shows the effect of the available power

from source on the capacity performance of the same link with

a separation distance of km. The available transmitter

power varies from 125 to 375 W. The maximum data rates

for different available power level are 720 bps at 375 W, 600 bps

at 250 W, and 440 bps at 125 W. The corresponding transmitter

and receiver Q are at 12, 14, and 16, respectively. A lower avail-

able power from the source leads to a lower optimal capacity of

the system and a higher loaded transmitter Q requirement.

VI. CONCLUSIONS

The near field power transfer equation for inductively cou-

pled resonant loops are derived and validated by experimental

results. It has been demonstrated in the strong coupling case, for

each distance there is an optimum impedance matching condi-

tion that maximizes the power transfer efficiency over this dis-

tance. In the weak coupling case the received power in the near

field system goes down inversely with the sixth power of dis-

tance and a conjugate match to the loss of the coils in both the

transmitter and the receiver maximizes the power transferred

in this case. This paper also presents the theoretical analysis of

the capacity performance of an inductively coupled near fieldcommunication system based on the derived nearfield power

transfer equation and the information theory. It is concluded

that the capacity is limited respectively by thermal noise for

low-Q receiver and natural interference for high-Q receiver.

The capacity performance of an inductively coupled NFC link

operating at VLF is evaluated. It is observed that higher oper-

ating frequency provide greater optimal capacity than that in the

lower frequency in the air but requiring higher transmitter and

receiver Q.

APPENDIX

Derivation for Self and Mutual Inductance of Coils: Whenthe radius of a coil is much smaller than the wavelength and

its length, one can assume the magnetic field inside the coil is

uniformly distributed (solenoid approximation). The self-induc-

tance of a coil is defined as the ratio of magnetic flux linkage to

the current through the coil [19]

(A-1)

where is the magneticflux that arises in an area enclosed

by current in the conductor and is the mag-

netic flux density at the center of the coil itself. Magnetic flux

density at a point on the axis of the coil that carries the currentis given by [19]

(A-2)

At the center of the coil

(A-3)

Substitute (A-3) and into (A-1), the inductance is

given by

(A-4)


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The mutual inductance is defined as the ratio of flux

linkage of the circuit 2 to the current in [19]

(A-5)

where is the magneticflux that arises in an area enclosed

by due to current in the conductor . ,which can be obtained from (A-2), is the magnetic flux density

at the center of the . It should be noted that the magnetic

flux density through the area is assumed to be uniform and

identical to the one at the center in (A-5). This condition may

only be true when either the is much smaller than or

the distance is far greater than the radius of the . Sub-

stitute (A-2) and into (A-5), the mutual inductance

under the assumption of infinitesimal thickness of the coil

and the homogeneity of the magnetic field in the area of is

given by

(A-6)

Notice (A-6) does not satisfy symmetry in its expression due

to the smaller assumption made when (A-5) is derived.

Taking consideration of the symmetry between the coils, a more

general form of the mutual inductance between two coils is

yielded as follows:

(A-7)

For comparisons, the exact expression for coefficient of mutual

inductance given in [15] is

(A-8)

where

(A-9)

and and are complete elliptic integrals offirst and second

kind.The mutual inductances between two coils having same ra-

dius (2 cm) and different radii (2 and 10 cm) aligned along the

same axis are evaluated for distance varying from 2 to 50 cm

using the approximate expression (A-7) and the exact expres-

sion (A-8). The results are plotted in Fig. 7, which show good

agreements until the coils are extremely close to each other.

Power Received Due to VLF/ELF Natural Noise: The

voltage received at the receiver due to ELF/VLF

natural interference at angular frequency can be found using

Maxwells equation

(A-10)

Fig. 7. Mutual Inductance between two coils calculated using the approximateexpression in the paper and the exact expression in [15].

where is the magnetic flux density due to ELF/VLF Noisefloor. is the number of turns and is the cross-section area

of the loop. The received power at a particular frequency is

(A-11)

Total received power due to ELF/VLF interference is obtained

by integrating the received power for all the frequencies within

the bandwidth

(A-12)

In [20] vertical electric and horizontal magneticflux density is

given in dB in relative to . In (A-12), needs to be

integrated over the bandwidth of the system. The value of

is approximately - for frequencies up to 0.1 MHz.

Hence the received natural noise power is given by

(A-13)

Substituting the area , inductance of the receiver coil using

(1), termination factor in

(A-13), the received power takes the form

(A-14)

where

-

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Umar Azad received the B.E. degree in electricalengineering from the College of Electrical andMechanical Engineering, National University ofSciences and Technology, Rawalpindi, Pakistan, in2007, the M.S. degree in electrical engineering fromthe University of California at Los Angeles in 2010,and is currently working toward the Ph.D. degree inelectromagnetics at the University of California atLos Angeles.

He has been working in the Digital MicrowaveLab, University of California at Los Angeles, since

January of 2009. His research interests include electrically small antennas,wireless power transfer, and near-field communication systems

Hengzhen Crystal Jing received the Ph.D. degreein electrical engineering from the University of Cal-ifornia at Los Angeles in 2008.

She is currently the Product Design Engineer atQuinStar Technology Inc., Torrance, CA, focusingon design and development of millimeter wave trans-ceivers, antennas. Herresearch interest in theUniver-sity of California at Los Angeles was switched reso-nant antennas for UWB pulse transmission and near

field communication system.

Yuanxun Ethan Wang (S96M99SM10) re-ceived the B.S. degree in electrical engineering fromthe University of Science and Technology of China(USTC), Hefei, China, in 1993, and the M.S. andPh.D. degrees in electrical engineering from theUniversity of Texas at Austin in 1996 and 1999,respectively.

From 1999 to 2002, he worked as a Research En-gineer and Lecturer in the Department of ElectricalEngineering, University of California at Los Angelesand became an Assistant Professor in November

2002. He is now an Associate Professor. He has worked on radar systems formore than 15 years and is a Technical Consultant of several local microwaveand radar companies in California. He has published more than 100 journal andconference papers and graduated 10 Ph.D.s. His research is in the general areaof microwave and radar systems with emphasis on antennas and phased arrays,high efficiency power amplifiers and transmitters and integrated RF front-ends.His researches blend thedigital processing technologies and concepts into RFand microwave system design, which often lead to new system architecturesand novel antenna and circuit configurations.

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