Electrodynamic Model of Microstrip Rectenna D.V. Semenikhina1 A.I. Semenikhin2

1 1 Taganrog technological institute of the South federal university, Russia, GSP-17, Taganrog, Nekrasovskiy, 44 e-mail: airpu@tsure.ru, tel. (the fax) +78634 371733 2 Taganrog technological institute of the South federal university, Russia, GSP-17, Taganrog, Nekrasovskiy, 44 e-mail: airpu@tsure.ru, tel. (the fax) +78634 371733

Abstract The microstrip rectenna model as an infinite periodic array of rectangular microstrip elements with nonlinear loads included between strips is considered. The dependences of rectified voltage value on parameters of nonlinear loads and incident plane electromagnetic wave are investigated.

1 INTRODUCTION

Introduction of energy transmission systems by a microwave beam (ETRMB) was brake by high cost of these projects. However such ETRMR aspects as energy feed of flying machines [1] and an exchange of energy between space objects [2] already found its application. The research laboratory of communication of Japan has made experiments under program ETHER on transfer of the microwave-energy on the greater height for control of airplane [1] in which it was possible to reach high conversion efficiency of the microwave electromagnetic energy to direct current. Rectenna is the device directly taking energy from a microwave bunch and reforming it to energy of direct current. Quality of all ETRMR depends on its characteristics; it defines transformation efficiency and weight and size parameters of the whole system. Besides, the electromagnetic field scattered by rectenna on fundamental frequency and on harmonics’ frequencies can aggravate electromagnetic situations on electronic platforms of flying machines and also it essentially influences on ecological atmosphere. Rectenna usually structurally represents the antenna array, which consists of great number of receiving-rectifying elements (RRE) which basic component is the antenna with nonlinear load (NL) [1, 2]. The rectenna theoretical analysis is known as a rule for RRE in the variety of the elementary oscillators loaded with the diode. Perfection of the rectenna goes on the way of increase of the efficiency and reduction of weight and size characteristics of the RRE, decrease of the levels of the field harmonics scattered on RRE. It is possible on the basis of expansion of classes of used RRE and nonlinear loads. Perspective on all above-mentioned

parameters are rectennas executed on patch technology. To increase their characteristics RRE of complex topology are applied, for example, in the form of log-periodic antenna type. However, methods of the theoretical analysis microstrip rectennas were not almost developed, that attracts significant time and material inputs on experimental search of the best parameters while designing such rectennas.

2 THEORY

In the given work in the basis of rectenna development there was the decision of a scattering problem of plane monochromatic wave by an infinite periodic microstrip array of any form having NL included between strips and also in breaks of additional rods between a patch and the screen.

2.1 Formulation of the problem

Let voltage-current (VI) characteristics on nonlinear loads (of microstrip array) be set in the form of [3]:

,/1

Pdtdubuai

where P is a degree of a polynom; i , u are current through load and voltage in input terminal of load;

ba , are coefficients defined by electrophysical properties of load. We consider that the structure is located in the homogeneous isotropic linear environment with 11,~

aa parameters. We enter Cartesian coordinate system (CCS) as shown in fig.1. The period of an arrangement of strip elements on axis х is equal to d1, on y is d2. The CCS axis z passes through the middle of one of the periods. The dielectric substrate thickness d has 22 ,~

aa parameters. We shall define the field of scattered periodic nonlinear loaded structure while inciting on it the plane wave on frequency ω under angles i i.

978-1-61284-978-2/11/$26.00 ©2011 IEEE

175

a2 a2d2

d1z

y

x

Nonlinearload

i

i

d V2

Figure 1: Model of Microstrip Rectenna.

On surface z = 0 on all sites which have been not occupied by nonlinear elements and conductors the condition of continuity of field tangential components is satisfied. We shall designate area z > 0 as 1V . On surface lS of nonlinear elements located in 2V region (limited by plane z = 0, short surface shS and nonlinear elements ldS included in them) and on surface of nonlinear elements which are on the plane z = 0, nonlinear boundary conditions (NBC) which are certain on the method stated in [4] should be satisfied.

2.2 Integral Equations

The conclusion of integrated relations for fields is based on application of Lorentz lemma for scattered fields written down for each spectral component (n number) and application of boundary conditions of continuity of field tangential components on z = 0 boundary on the sites which have not been occupied by nonlinear loads. Let's designate the surfaces limiting areas 1V as 1S ; 0S

is z = 0 surface; 2,12,1 , mn

mn HE

are auxiliary fields of an

unit magnetic dipole which current changes with n frequency orientated as 2,1b

(indexes 1 and 2 mean an

accessory to 2,1V ), satisfying a boundary condition on

ideal conductor surface at z = 0; 22 , mn

mn HE

are similar

auxiliary fields satisfying the same condition at z = 0, z = -d; 2,12,1 , nn HE

are unknown complex amplitudes

of fields in areas 2,1V on n frequencies. The equation system should be completely determined by the equations in points on short surface and the loads included in them. We shall choose as auxiliary dipole the unit electric dipole orientated along axis z (we shall designate its fields as 22 , e

nen HE

), and also we

shall set an auxiliary source in the form of an elementary magnetic-current circular loop in local cylindrical system of coordinates in point р(a,za) located on nonlinear load surface included in short break (a is a radius of the load included in short equal

to short radius). Let lmn

lmn HE 22 ,

are auxiliary fields

excited by magnetic loop. At a conclusion of the integrated equations for magnetic current harmonics on surface nonlinear elements, it is necessary to consider NBC [3] combining electric 1e

nJ and magnetic 1mnJ surface

current harmonics. We shall set nonlinear elements in a strip plane as concentrated loads included in a gap between strips so that the current in them can flow only along one of coordinates (x or y). Assuming in Lorentz's lemma that xib

1 or yib

1 in

view of NBC we shall receive two more equations. Totally we have a system of nonlinear integrated equations (SNIE):

,

,111

,11)(

11)(

311

)(

21

1

1

хlym

nS

mn

iny

ymnx

iny

S

ymny

inx

mysq

s

msy

q

myqn

nxs

msy

mysnnx

mnynx

SpdSHJHdSHE

HEJJJ

yxСJJ

yxВJ

yxA

, ,,11,1

,11)(

11)(

311

)(

21

1

1

ylxm

nS

mn

inx

xmnx

iny

S

xmny

inx

mxsq

s

msx

q

mxqn

nys

msx

mxsnny

mnxny

SpdSHJHdSHE

HEJJJ

xyСJJ

xyВJ

xyA

,,

0

22

2222

1

shze

nzSS

enz

zen

S

mn

zen

S

mn

SpdSEJ

dSHJdSHJ

ldsh

ld

,,2222

22222

32222

1

ldSS

lmnz

enz

S

lмn

мn

S

lmn

mn

s

msq

ms

q

mqn

ns

msn

msn

mnn

SpdSEJdSHJ

dSHJJJJ

zСJJzBzJA

ldshld

)1(,

,)

()(

1

,22,22,2

,11,1,1

0 1

ylxl

S

bmn

mn

SS

bmnz

enz

bmn

S

bmn

S

mn

in

bmnx

iny

bmny

inx

SSSp

dSHJdSЕJdSH

HJHbdSHEHE

ldldsh

where 1

,m

ynxJ , 22 , mn

enz JJ are intensities of electric and

magnetic surface currents on loads between strips, on shorts and loads included in it, accordingly;

332211 ,, binaCbinaBbinaA nnn ;

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ylxl SS , are the surfaces occupied by nonlinear elements, an electric current through which flows along

yx, , accordingly [4]. Infinite SNIE (1) concerning complex amplitudes of current harmonics by means of Floquet theorem and Puisson summation formulas [4] can result in equation system concerning currents on array zero-order period. The received equation system allows to define intensities of magnetic surface current harmonics being sources of the secondary (scattered) electromagnetic field. The system is solved by moment method; roof-top basic functions are taken for surface currents (triangular basic function is on the coordinate conterminous with a direction of current flow and piece-constant function is on the second coordinate). The considered electrodynamic model of a microstrip array with NL can serve also as rectenna model (“models” as the decision of an electrodynamic problem is received for an infinite array), and theoretically with any strip topology. The cell of such array represents RRE having the nonlinear load intended for detecting the incident wave.

3 ANALYSIS OF NUMERICAL RESULTS

Numerical research of model microstrip rectenna characteristics is curried out for the infinite periodic microstrip array having NL with square-law V-I characteristic (a2≠0) without rods between strip and ground. We investigate dependences of a direct component of the voltage on NL input terminal (induced by incident wave) and reflection coefficients of the array from parameters of loads and amplitude H0 of incident magnetic field on frequency f = 10 GHz. The value of the voltage 0U rectified on load is defined through value of zero harmonic of magnetic surface current on NL mJ0 (found as a result of the numerical decision of

SNIE) as |||| 00mJlU , where l is distance between

load input terminals. The array reflection coefficients [4] show the attitude of amplitudes of Floquet harmonics of the scattered field on frequency n to amplitude of the incident field on the fundamental frequency. On fig.2-3 dependences of the rectified voltage and modules the reflection coefficients of zero-order Floquet modes on the fundamental frequency, the second and third frequency harmonics from value of coefficients а2 for square-law member V-I characteristic are shown. The array sizes are set so that on these frequencies they belong to single mode domain of periodicity (i.e. the space spectrum of the reflected field on frequencies , 2, 3 contains only zero-order Floquet harmonic, we shall designate its index “00”). Calculations are carried out both for passive arrays with positive linear conductivity of loads

а1 (fig.2, 3), and for active arrays with negative linear conductivity а1 (fig.2, 4).

10-3

10-2

10-1

a20 0.025 0.05 0.075

U0

a1=-0.05

a1=-0.02

in V,

a1=0.01a1=-0.007

Figure 2: Rectified Voltage.

Rn00

10-3

10-210-1

0 0.05 0.0 57 a2

n=1n=2

n=3

0.0 52

Figure 3: Reflection Coefficients for Passive Load.

n=3

n=2

n=1Rn00

a2 0 0.01 0.02

1

1.5

0.5

Figure 4: Reflection Coefficients for Active Load.

The voltage value rectified on NL with growth а2 first increases on square law and then almost linearly. For some value of coefficient for square-law member of the V-I characteristic а2=а2max, the voltage value

|| 0U rectified on load reaches the maximum. Value

а2max is defined by linear conductivity of load, its absolute and relative geometrical sizes (in comparison with the sizes of the cell). In case of passive loads the reflection coefficients of a zero-order Floquet mode on the second frequency harmonic || 00

2R also reaches the maximum for а2=а2max (fig.2, 3); in the same point the graphic of reflection coefficient on the fundamental frequency

|)(| 2001 aR has the characteristic minimum. It means,

for this value а2 the greatest swapping of incident wave energy on harmonic frequencies is carried out, the transformation efficiency of the microwave field energy to the rectified voltage will be maximal. For the increase in values а2>а2max the module of the reflection

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coefficients |)(| 2001 aR increases to 1, and values

|)(| 200

3,2 aR and |)(| 20 aU decrease. It is possible to explain by change of V-I characteristic type of load for а2>а2max with close to the diode detector characteristic (for а2а2max) on almost parabolic (for а2>>а2max). We shall note that there exists а2=а2min for which the graphic of reflection coefficient of the third frequency harmonic |)(| 2

003 aR has the minimum and

|)(||)(| min2003min2

002 aRaR (difference of modules

of these reflection coefficients makes approximately 100 dB). In some vicinity of a point а2min a correlation between reflection coefficients is |)0(||)(| 2

0032

003 aRaR . For

such values of coefficient а2 of the load’s V-I characteristic rectenna can be distinguished by a nonlinear radar as an object with “electronic nonlinearity” as it is considered that in a spectrum of rereflected fields from a objects with artificial NL prevails the second instead of the third harmonic of the scattered field (as for objects with natural nonlinear contacts) . For а2>а2max values || 00

2R and || 003R are

one of though always || 002R > || 00

3R . For an active array (а1 <0) (fig.3, 4) with the increase of а2 values || 00

3,2R smoothly grow up to a point

а2=а2max (the graphic |)(| 2001 aR also in a point а2max

has the minimum, see fig.5). On harmonics value of the reflection coefficients for active loads it is more than for passive ones for identical parameters а2. Thus the value of the voltage rectified on NL and the array reflection coefficients on frequencies of harmonics are defined by V-I characteristic coefficients а2 and а1of he load. Growth of the rectified voltage with change of load V-I characteristic coefficients at work of a array as rectenna always is accompanied by increase in amplitude of a plane wave reradiated by the array on the second frequency harmonic under the angle equal to angle of incidence. The dependence character of the rectified voltage from amplitude of an incident wave |)(| 00 HU (fig.5) also is defined by the correlation V-I characteristic coefficients of loads (i.e. V-I characteristic type). In case of loads which V-I characteristic close to reference point looks like the characteristic of the diode detector (a curve 1* on fig.5), for some 0H the value of the rectified voltage reaches the maximum, and then in some range of increase 0H decreases (see graphic 1 on fig.5). For other V-I characteristic type of loads ("cubic" and "parabolic") (fig.5, curves 2*, 3*), graphics |)(| 00 HU have sites of square-law growth for H0< 0.2 A/m and slower growth for H0> 0.2 A/m (graphics 2, 3 on fig.5).

10-4

10-6

10-2

0 0.5

21

3

H0,A/m

U0 in V

2*

3*

1*

,

Figure 5: Rectified Voltage.

From fig.5 it is visible that at work of a microstrip array as rectenna it is more favorable to use NL having V-I characteristic type 1* if the amplitude incident waves on rectenna is small (H0< 0.5 A/m) and loads with V-I characteristic type 2* or 3* when the greater amplitude incident wave is provided. However, it is necessary to consider that values of reflection coefficients on the second and third frequency harmonics nonlinearly depend on amplitude H0 as well.

4. CONCLUSION

The values of NL V-I characteristic coefficients are found for which the rectified voltage is maximal. It is shown that if the amplitude of incident wave is small rectenna efficiencies higher if active NL are applied, otherwise if passive NL are applied. In case of active loads the amplitude of parasitic fields is higher on the maximum frequency harmonics rereflected by rectenna.

References

[1] Fujino Yoshiyuki, Fujita Masaharu. “Development of a High-Efficiency Rectenna for Wireless Power Transmission - Application to Microwave- Powered Airship Experiment”. J. Commun. Res. Lab., V.43. №3. PP. 367-374. 1996.

[2] J.0. McSpadden, Nobuyuki Kaya, A.M. Brown, Kai Chang. “A Reciving Rectifying Antenna for The International Space Year-Microwave Energy Transmission in Space (ISY-METS) Roket Experiment”. 29th Intersoc. Energy Convers. Conf., Monterey, Calif., Aug. 7-11, 1994: Collect. Jechn. Pap. Pt.2.-Washington (D.C.). PP. 723-727. 1994.

[3] B.M. Petrov, D.V. Semenikhina, A.I.Panihev. “A New Analysis Method of Nonlinear Scattering for Solution EMC Problems”. 11 Internat. Wroclaw Symp. on Electromag. Compat. 1992.

[4] D.V. Semenikhina, I.E. Dekalo. “The Analysis of the Frequent-Space Field Characteristics of the Annenna Array With Nonlinear Inclusions”. 14 Internat. Wroclaw Symp. on Electromag. Compat. 1998.

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