WIRELESS ENERGY TRANSFER TO A PASSIVE RERADIATOR

I. N. Zhukov

(Presented by Professor, Doctor of Physico-Mathematical Sciences V. N. Kessenikh)

Application of an Elliptical Reflector

Passive reradiators are often used as antennas because of their practical convenience.

In communication lines a passive reradiator must be located at a great height. To create conditions of direct visibility, at a distance of 50 km, the height of the reradiator must be 70 meters. Consequently, it is necessary to transmit ultrahigh-frequency energy over a distance of 70 meters. When transmitting over this distance by cable, a large part of the energy is lost; waveguides are bulky and expensive; a single-wire line is unstable. At the same time, on centimeter waves it is possible to create sharply directional radiation and transmit energy to a passive reradiator with smaller losses than by other methods. The article considers the possibilities of such energy transfer in the range of 3–20 centimeters.

When radiating energy for broadcasting, communications, or radar, the distance at which the receiver will be located is unknown, and if it is known, it is very large. Therefore the radiation pattern of the antenna is calculated in the far zone, and in this zone it does not change depending on distance. With a passive reradiator, energy is transmitted to a definite, fixed distance. This advantage can also be used to construct a radiating system in such a way that the greatest directivity of radiation is at a distance equal to the distance to the reradiator. Such a radiating system can be created by using an elliptical surface for reflection. According to geometrical optics, energy radiated in one of the foci of an ellipse, after reflection from its surface, will be collected in the other focus (Fig. 1). The same result will be obtained if in-phase

Fig. 1

sources are placed on the surface of an ellipse.

radiators over the surface of a sphere. In other words, the ellipsoidal reflector creates an in-phase spherical surface with its center at the second focus. Thus, the second focus of the ellipsoid becomes a new source of radiation. By specifying the distances between the foci of the ellipsoid, one can calculate the surface of the ellipsoid so that energy is transmitted over any distance. When such a system is used for communication on the earth’s surface, an antenna with an ellipsoidal reflector must be installed on the surface of the earth, and above it a parabolic reflector with a focus coinciding with the second focus of the ellipsoid installed on the ground, and with an axis coinciding with the specified direction of radiation (Fig. 2).

Fig. 2.

Radiation pattern with an ellipsoidal reflector

Geometrical optics, being a zeroth approximation, does not give correct results in the case of large gradients of the energy flux. The true picture of the field near the second focus can be obtained by calculation according to the laws of electrodynamics. The radiation pattern is calculated from the distribution of the field in the aperture of the antenna. The field strength in the aperture can be determined using geometrical optics. We assume an equal amplitude of the field strength over the aperture, and determine the phase from the condition that the reflected rays converge at the second focus and that their lengths are equal (Fig. 3).

Fig. 3

Let us denote:

\(O_1\) — the first focus of the ellipse, \(O_2\) — the second focus of the ellipse,
\(f = O_1O_2\) — the distance between the foci,
\(\rho_1 = O_1V\) — the distance from the center of the aperture to the given point of the aperture,
\(q\) — the phase lead at the given point relative to the center of the aperture,
\(a\) — the radius of the aperture, \(\alpha = \dfrac{2\pi}{\lambda}\) — the wave number,

\[ q = \alpha\left(\sqrt{f^2+\rho_1^2}-f\right). \]

The distance between the foci \(f\) is equal to the distance over which energy must be transmitted.

Let us take the distance of energy transmission to be somewhat greater than the height of the antenna, \(f=80\) meters. We assume that the aperture radius \(a\) is not more than 3 meters,

\[ f \gg a > \rho_1 . \]

\[ q=\alpha\left(f+\frac{\rho_1^2}{2f}-\frac{\rho_1^4}{8f^3}+\frac{\rho_1^6}{16f^5}-\cdots-f\right). \]

In the series expansion, the third and subsequent terms may be neglected, since their effect on the phase is negligible,

\[ q=\alpha \frac{\rho_1^2}{2f}. \]

The phase in the aperture leads in proportion to the square of the distance from the center.

The field strength at any point of the aperture is

\[ E_p=E_0 e^{jq}=E_0 e^{j\alpha \frac{\rho_1^2}{2f}}, \]

where \(E_0\) is the field strength at the center of the aperture.

Let us determine the field strength in the focal plane, i.e., in the plane passing through the second focus perpendicular to the axis of the ellipsoid (Fig. 4).

Fig. 4

\[ E=\int_F \frac{-E_p}{\lambda r}\,e^{-j\alpha r}\,dF; \]

\(r=VW\) is the distance from the radiation point \(V\) to the point \(W\) at which the field strength is determined,

\(\rho_2=O_2W\) is the distance from the second focus \(O_2\) to the point \(W\) at which the field strength is determined,

\(F\) is the aperture area.

\[ r=\sqrt{f^2+\rho_1^2+\rho_2^2-2\rho_1\rho_2\cos\varphi}; \]

\[ f\gg \rho_1 \quad \text{and} \quad f\gg \rho_2, \]

\[ r=f+\frac{\rho_1^2+\rho_2^2-2\rho_1\rho_2\cos\varphi}{2f}, \]

\[ E=\int_{0}^{a}\int_{-\frac{\pi}{2}}^{\frac{3}{2}\pi} -\frac{E_0}{\lambda r}\, e^{j\alpha\frac{\rho_1^2}{2f}} e^{-j\alpha\left(f+\frac{\rho_1^2+\rho_2^2-2\rho_1\rho_2\cos\varphi}{2f}\right)} \rho_1\,d\rho_1\,d\varphi . \]

In the denominator one may assume that \(r=f\),

\[ E=\frac{E_0}{\lambda f}\, e^{-j\alpha\left(f+\frac{\rho_2^2}{2f}\right)} \int_{0}^{a}\int_{-\frac{\pi}{2}}^{\frac{3}{2}\pi} e^{j\alpha\frac{\rho_1\rho_2}{f}\cos\varphi}\, \rho_1\,d\rho_1\,d\varphi . \]

Introduce a new variable \(\Theta=-\frac{\pi}{2}-\varphi\),

\[ E=\frac{E_0}{\lambda f}\, e^{-j\alpha\left(f+\frac{\rho_2^2}{2f}\right)} \int_{0}^{a}\int_{-\pi}^{\pi} e^{j\alpha\frac{\rho_1\rho_2}{f}\sin\Theta}\, \rho_1\,d\rho_1\,d\Theta . \]

The second integral gives the Bessel function of zero order \((I_0)\).

\[ E=\frac{E_0}{\lambda f}\, e^{-j\alpha\left(f+\frac{\rho_2^2}{2f}\right)} \,2\pi\int_{0}^{a} I_0\left(\frac{\alpha\rho_1\rho_2}{f}\right) \rho_1\,d\rho_1 . \]

Integration gives the Bessel function of first order \((I_1)\),

\[ E=E_0\,\frac{a}{\rho_2}\, I_1\left(\frac{\alpha a\rho_2}{f}\right) e^{-j\alpha\left(f+\frac{\rho_2^2}{2f}\right)} = \]

\[ =E_0\,\frac{2F}{\lambda f}\cdot \frac{ I_1\left(\frac{\alpha a\rho_2}{f}\right) }{ \frac{\alpha a\rho_2}{f} } e^{-j\alpha\left(f+\frac{\rho_2^2}{2f}\right)} . \]

Let us determine the field intensity on a spherical surface of radius \(f\), passing through the second focus, with its center at the first focus (Fig. 5).

\[ r=\sqrt{f^2+\rho_1^2-2f\rho_1\sin\psi\cos\varphi} \approx f+\frac{\rho_1^2}{2f}-\rho_1\sin\psi\cos\varphi . \]

\[ E= \int\limits_{0}^{a} \int\limits_{-\frac{\pi}{2}}^{\frac{3}{2}\pi} \frac{E_{0}}{\lambda r} e^{j\alpha\frac{\rho_{1}^{2}}{2f}} e^{-ikr}\,\rho_{1}\,d\rho_{1}\,d\varphi = \]

\[ = -\frac{E_{0}}{\lambda f}\,e^{-j\alpha f} \int\limits_{0}^{a} \int\limits_{-\frac{\pi}{2}}^{\frac{3}{2}\pi} e^{j\alpha\rho_{1}\sin\psi\cos\varphi}\, \rho_{1}\,d\rho_{1}\,d\varphi . \]

Fig. 5

After integration we obtain

\[ E=E_{0}\frac{2F}{f\lambda}\, \frac{I_{1}(\alpha a\sin\psi)}{\alpha a\sin\psi}\, e^{-j\alpha f}. \]

The constancy of phase over the spherical surface and the lag proportional to the square of the distance from the axis show that we have a spherical wave diverging from the center of the aperture. The amplitude of the field strength on the sphere varies in proportion to a Bessel function of the first order divided by its argument.

The same result is obtained for the radiation pattern of an in-phase surface at a very large distance. With an elliptical reflecting surface, the radiation pattern on a sphere passing through the second focus is the same as with a parabolic reflecting surface in the far zone. Consequently, an elliptical reflecting surface brings the region of formation of the radiation pattern closer. The concentration of energy per unit solid angle in the near zone, that is, in the region of an unformed radiation pattern, is less than in the far zone when the radiation pattern has formed. Therefore an elliptical reflecting surface gives an increase in the concentration of energy near the second focus as compared with a parabolic reflecting surface. An elliptical reflecting surface gives the maximum possible concentration of energy for the given distance, wavelength, and aperture surface.

Gain produced by an elliptical reflecting surface

Let us compare the field strength on the axis produced by parabolic and elliptical reflecting surfaces. In this case the radius of the sphere \(R \ne f\).

On the axis

\[ \psi=0, \qquad \sin \psi=0. \]

Therefore

\[ r=R+\frac{\rho_1^2}{2R}. \]

For a parabolic reflector \(E_p=E_0\),

\[ E=-\frac{E_0}{\lambda R}\int_0^a \int_{-\frac{\pi}{2}}^{\frac{3}{2}\pi} e^{-j\alpha\left(R+\frac{\rho_1^2}{2R}\right)}\rho_1\,d\rho_1\,d\varphi, \]

\[ E=2E_0\sin \frac{\alpha a^2}{4R}e^{-j\alpha\left(R+\frac{a^2}{4R}\right)}. \]

The amplitude is

\[ E_m=2E_0\sin \frac{F}{2\lambda R}. \]

For an elliptical reflector

\[ E_p=E_0 e^{j\alpha \frac{\rho_1^2}{2f}} \]

\[ E=-\frac{E_0}{\lambda R}\int_0^a \int_{-\frac{\pi}{2}}^{\frac{3}{2}\pi} e^{-j\alpha\left(R+\frac{\rho_1^2}{2R}\right)} e^{j\alpha\frac{\rho_1^2}{2f}}\rho_1\,d\rho_1\,d\varphi = \]

\[ =E_0\frac{2f}{R-f}\sin\left(\alpha a^2\frac{R-f}{4Rf}\right) e^{-j\alpha\left(R-\frac{R-f}{4Rf}a^2\right)}. \]

The amplitude is

\[ E_m=E_0\frac{2f}{R-f}\sin\left(\frac{F}{2\lambda R}\cdot\frac{R-f}{f}\right). \]

The field strength at the point of the second focus \(R=f\), without focusing, is

\[ E_m=2E_0\sin \frac{F}{2\lambda f}, \]

with focusing,

\[ E_m=E_0\frac{F}{\lambda f}. \]

The focusing gain is equal to the ratio of the field strength with focusing to the field strength without focusing. We determine the gain at the point of the second focus:

\[ K=\frac{E_{\text{with foc.}}}{E_{\text{without foc.}}}; \qquad K=\frac{\frac{F}{2\lambda f}}{\sin \frac{F}{2\lambda f}}. \]

The focusing gain is equal to the ratio of the argument to its sine. This ratio is greater, the larger the argument. Consequently, focusing gives a gain that is greater the larger the area of the radiating surface and the smaller the distance to the second focus and the shorter the wavelength.

Parameter Characterizing Energy Concentration

At short distances it is possible to concentrate energy and create a field strength greater than that in the aperture of the antenna. At greater distances this cannot be done. Energy can be concentrated the farther, the shorter the wavelength and the larger the dimensions of the antenna. Thus, the possibility of concentrating energy, and consequently also the efficiency of transmission, depend on three quantities: distance, wavelength, and antenna size.

It is desirable to have a single parameter characterizing the possibility of concentrating energy at a distance and, consequently, the possibility of transmitting energy. The wave parameter $p$, known in physics, may serve as such a parameter.

\[ p=\frac{\sqrt{\lambda R}}{D}, \]

where $\lambda$ is the wavelength,
$R$ is the distance of energy transmission,
$D$ is the diameter of the transmitting antenna.

The smaller the wave parameter, the better the conditions for energy transmission.

Let us introduce the concentration coefficient $M$.

The larger $M$, the better the possibilities for energy transmission:

\[ M=\frac{\pi}{4p^2}=\frac{F}{\lambda R}. \]

Let us express the relations obtained earlier in terms of the concentration coefficient and the wave parameter. The field strength on the axis with focusing is

\[ E_m=E_0\frac{F}{\lambda f}=ME_0=E_0\frac{\pi D^2}{4\lambda f}=E_0\frac{\pi}{4p^2}. \]

For $M=1$, i.e. $p=0.886$, the field strength at the second focus is equal to the field strength in the antenna aperture. If $M>1$, i.e. $p<0.886$, we have $E_m>E_0$. A field strength is produced that is greater than the field strength in the antenna. It is precisely in this case that a significant concentration of energy occurs, the action of focusing is fully manifested, and we have a significant increase in field strength due to focusing.

The field strength on the axis without focusing is

\[ E_m=2E_0\sin\frac{F}{2\lambda f} =2E_0\sin\frac{M}{2} =2E_0\sin\frac{\pi D^2}{2\lambda f^4} =2E_0\sin\frac{\pi}{8p^2}. \]

The focusing gain is

\[ K=\frac{\dfrac{M}{2}}{\sin\dfrac{M}{2}} = \frac{\dfrac{\pi}{8p^2}}{\sin\dfrac{\pi}{8p^2}}. \]

The strengthening of the focusing depends only on the concentration coefficient or on the wave parameter (Fig. 6). This dependence gives an increase in the field strength. The power gain will be twice as large.

The field strength in the focal plane also depends on the distance \(\rho_2\) from the axis. If this dependence is excluded, then the field strength can be expressed in terms of the energy concentration coefficient or the wave parameter. Let \(\rho_2=a\), that is, let us determine the field strength at a distance equal to the antenna radius. This is practically the most interesting case.

Fig. 6

It is undesirable to make the reflector larger than the antenna at the top, since it is under more severe meteorological conditions. If the reflector is made smaller than the antenna, then the secondary radiation pattern will broaden. In this case the conditions are

\[ E=E_0 I_1(2M)=E_0 I_1\left(\frac{\pi}{2p^2}\right). \]

The field strength at the edge of a reflector equal to the antenna depends only on the concentration coefficient or on the wave parameter.

The greatest efficiency of transmission will occur when the reflector uses the entire main lobe of the antenna radiation pattern. In this case, at the edge of the reflector there must be a zero of the field strength:

\[ E=0,\ \text{i.e.}\ I_1(2M)=I_1\left(\frac{\pi}{2p^2}\right)=0. \]

This will be for

\[ 2M=\frac{\pi}{2p^2}=3.83; \]

\[ M=1.915,\qquad p=0.64. \]

For these values of \(M\) and \(p\) we obtain the greatest transmission efficiency (about \(87\%\)).

Efficiency of wireless energy transfer

Let us define the efficiency as the ratio of the reflected energy \(P_o\) to the energy radiated by the antenna \(P_\Sigma\).

\[ \eta = \frac{P_o}{P_\Sigma} = \frac{\displaystyle \int_F \frac{E^2}{120\pi}\, dF} {\displaystyle \frac{E_o^2}{120\pi}\, F}. \]

We assume that all the energy incident on the reflector is reflected, and, moreover, that all the assumptions made in the preceding calculations are valid.

\[ \eta = \frac{2a}{\lambda f} \int_0^a \int_{-\frac{\pi}{2}}^{+\frac{\pi}{2}} \frac{ I_1^2\!\left(\dfrac{a\alpha\rho_2}{f}\right) }{ \dfrac{a\alpha\rho_2}{f} } \, d\rho_2\, dz = \frac{4\pi a}{\lambda f} \int_0^a \frac{ I_1^2\!\left(\dfrac{a\alpha\rho_2}{f}\right) }{ \dfrac{a\alpha\rho_2}{f} } \, d\rho_2 . \]

Introduce the new variable \(x=\dfrac{a\alpha\rho_2}{f}\); then

\[ \eta = 2 \int_0^{2M} \frac{I_1^2(x)}{x}\, dx = 2 \int_0^{\frac{\pi}{2p^2}} \frac{I_1^2(x)}{x}\, dx . \]

Carrying out graphical integration, we determine the efficiency as a function of the concentration coefficient or of the wave parameter. Under the assumptions made, the concentration coefficient, or the wave parameter, uniquely determines the efficiency (Fig. 7).

Fig. 7

From the graph we see that for \(p=1\), i.e. \(M=\dfrac{\pi}{4}=0.785\),

\[ \eta = 40\%. \]

For \(p > 1\) the efficiency becomes rather low. From this one may conclude that the wave parameter should be kept between 0.64 and 1, and the concentration coefficient between 0.785 and 1.915:

\[ 0.64 < p < 1, \qquad 0.785 < M < 1.915. \]

The wave parameter depends on three quantities, of which the distance is specified, \(f = 80\) meters. Therefore, the efficiency depends on two quantities: the size of the antenna and the wavelength. Consequently, for the range \(3 \div 20\) cm the efficiency graph can be expanded and represented as the dependence of the efficiency on the antenna diameter with the wavelength as a parameter (Fig. 8). It is evident that the working region of effective use of the system lies in the upper part of the graph, at efficiencies above 50%.

Fig. 8

The possibility of using centimeter waves for wireless energy transmission depends strongly on the technical feasibility of constructing large antennas and reflectors. If it is possible to build an antenna 2 meters in diameter, then at a wavelength of 3 cm one can obtain an almost maximum transmission efficiency of about 80%.

With a diameter of 3 meters we obtain a good result at a wavelength of 7 cm and a satisfactory one at a wavelength of 10 cm. With a decrease in the communication distance and, consequently, in the height at which the reradiator is located, the dimensions of the antenna decrease substantially.

Conclusions

In wireless energy transmission it is advisable to use an elliptical reflector instead of a parabolic one. Changing the shape of the reflector increases the transmitted power by 20%. The transmission efficiency increases by the same amount.

The possibilities for energy transfer are determined by the concentration coefficient or by the wave parameter. It is desirable to have the concentration coefficient as large as possible, within the range \(0.785 \div 1.915\), and the wave parameter as small as possible, within the range \(0.64 \div 1\).

The maximum transfer efficiency of \(87\%\) is obtained at a concentration coefficient of \(1.915\).

The most convenient wavelength for communication is \(7\) cm. At this wavelength and with an antenna diameter of 3 meters, energy can be transmitted over a distance of 80 meters with a transfer efficiency of \(80\%\).