Energy Convers. Mgmt Vol. 29, No. 2, pp. 137-140, 1989 0196-8904/89 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1989 Maxwell Pergamon Macmillan pie

E N E R G Y C O N V E R S I O N T H R O U G H P A R A M E T R I C

A M P L I F I C A T I O N - - I T S A P P L I C A B I L I T Y O N P H O T O N

E X C I T A T I O N I N S E M I C O N D U C T O R S

A. J. VARKEYt National Centre for Energy Research and Development, University of Nigeria, Nsukka, Nigeria

(Received 20 May 1987; received for publication 15 February 1989)

Abstract---Conversion of light energy into electricity through parametric amplification in an electrical circuit has been recently proposed [I] [H. Wetzel and H. Tributsch, Sol. Energy 37, 65 (1986)]. A theoretical expression for the periodicity of the external agency that stimulates variation in the energy defining parameter of the circuit is derived. Practical applicability of systems that involve photon induced amplification is also discussed, in view of their conversion efficiency and production cost. A mechano- electrical system is suggested for future work in this area.

Parametric amplification Energy conversion

I N T R O D U C T I O N

Direct conversion of solar energy into electricity through photovoltaic and thermoelectric mechanisms has been a major area of research in the past few decades. Because of limitations on the properties of materials available for fabrication of solar cells and thermoelectric generators, the efficiency of such devices has been limited. Even though the technological advancement in this field has gone through a period of successive achievements, the target of $1/peak watt for residential application of these devices is still far-fetched. This balance of system cost restricts the use of solar cells. The a-Si technology and the tandem structure of single crystal Si cells should still be regarded as successful only from the experimental point of view in terms of their economic viability.

As regards its practical applicability in rural and agro-based industrial areas, the existing solar energy conversion devices need further cost reduction. While attempts may be made on one side to achieve this, a parallel search for alternate methods in electricity generation appears a challenge to workers in this field. One such device is based on the mechanism of parametric amplification [2]. In this, a periodic variation in one of the energy defining parameters of an electrical circuit results in the amplification of the stored energy in each cycle, which may be conveniently converted into electricity in another circuit through proper coupling. Such a device, in which oscillations are produced in the electrical circuit by photosensitive capacitance of inductance, has already been proposed [1]. In the following, a theoretical analysis of the device is given, and its practical applicability is discussed.

T H E O R E T I C A L A N A L Y S I S

Consider an L - C circuit in which the capacitor, of capacitance Co, is initially charged to q0. Neglecting resistance, the differential equation is

d2q + q

dt ---i ~00 = 0

or

q2

Equation (2) represents an ellipse. On the t~-q

tAddress for correspondence: Department of Physics and

(i)

q2 + ~ = constant. (2)

plane, it can be represented as in Fig. 1. Point "a"

Astronomy, University of Nigeria, Nsukka, Nigeria.

137

138 VARKEY:

d C

'1

PARAMETRIC AMPLIFICATION

h ,_ q

g

Fig. 1. Representation of charging and discharging of the capacitor in the q-q plane.

corresponds to charge q0, where the energy is purely electrostatic. A solution to equation (1) is

t q ( t ) = qo cos ~ (3)

~ t .'. I ( t ) = sin./~---~. (4)

Equation (3) is represented by the curve a--c in the figure. Clearly, during time

where T is the period of oscillation of the circuit, the charge q0 discharges completely. Energy is now purely electromagnetic at point "c". If the capacitance is decreased by an external agency, say by AC0, by either increasing the distance between its plates or by decreasing the permittivity of the dielectric at point "a", the electrostatic energy stored in the capacitor increases by

qo 2 qo 2

2(c0- ac0) 2c0" For AC0<< Co, this can be approximated to ½AC0 V 2. This increment is represented by a-b in the q-axis. On subsequent discharge, the q-q transformation follows the path b-d. Alternatively, if AC0 decreases slowly and periodically with a frequency equal to half the resonant frequency of the circuit, the curve a-d represents the transformation. Obviously, such a variation should be of the form

t AC(t) = A Co sin L~/~0" (5)

Correspondingly, the instantaneous value of the capacitance

t C ( t ) = Co - AC0 sin L~/~0" (6)

In both cases, the extra energy supplied to the circuit by the external agency, by decreasing the capacitance (this is the same as increasing the p.d.), is converted into electromagnetic energy within the system (c--d in the q axis). During charging, the electromagnetic energy becomes electrostatic (d-e in the figure) in time t = T/4 . From equation (5), it follows that the capacitance increases by AC0 during this period. This results in more charges of opposite sign being produced in the capacitor plates. Again, during discharge in time T/4 , a decrease in C introduces the same energy increment into the circuit, and the energy becomes electromagnetic again (point g in the figure). During the next charging, the point moves to "h" in the q-axis. Thus, in one cycle, the net increase

VARKEY: PARAMETRIC AMPLIFICATION 139

in energy is given by a-h. This energy can be extracted from the device during the cycle (the point moves back to "a") as electrical energy by coupling it to another circuit containing a load, and the process continues.

The above discussion neglected the resistance of the circuit. Including the resistance R, the differential equation becomes,

d2q . R dg q ~-i + ~ --~ + ~ 0 = 0 . (7)

The form of the solution to this depends on the value of R. If R > 0, the amplitude decreases in every cycle. For R2<<4L/C, we get underdamped oscillations, and the solution is

q(t) = qo exp ~ cos (t/~L~o) (8)

where the exponential term is due to R. This means that, to obtain sustained oscillations, energy must be added to the circuit to compensate for the energy dissipation in R. Obviously, an additional increment in q,

or an equivalent charge in C

is required. Combining equations (5) and (10), the total charge in Co becomes

Ar~0 = AC0 sm = + C0 1 - exp . (1 l) -~t°' x/LC ° " - ~

The first term on the r.h.s, is the effective change in C, while the second term takes care of energy dissipation. For t = T/4, this term becomes

- n Co n R N / ~ '

for small values of Co. Therefore, the periodic variation in AC(t) takes the form,

c0- R sin

to compensate the heat loss in the circuit. Thus, equation (11) becomes

t r C R ~ s i n t = ( A C 0 + n V / ~ ) t AC°( t ) t ° t=ACos in~ +C°4 LCo Co-~R sin Lx//~0. (12)

The maximum amplitude

ACmax=ACo+Co4RN/~. (13)

Obviously, the only constraint for parametric amplification is that AC0 should be positive. That is

AC~x > C0~ ~/L" (14)

That means the external agency should be able to produce a change in C more than the r.h.s, term. The greater the change, the higher the amplification. For various values of R, L and C of practical interest, calculation shows that the dissipation term is negligible. The variation in C with time from equation (12) is

C(t) = Co -- AC0(t)tot. (15)

140 VARKEY: PARAMETRIC AMPLIFICATION

C

C O -

Co -ACo (t) E ~ /I" 7- tot

"-~Co- AC max

Fig. 2. Schematic representation of variation in C with time. a and b correspond to discharging and charging, respectively.

This is shown schematically in Fig. 2.

PRACTICAL APPLICABILITY

The above discussion was based on the variation of capacitance in the circuit. A similar argument will be valid for the inductance also. A practical application of this mechanism may be envisaged in an electrical circuit having a semiconductor p - n junction capacitance that can be varied upon illumination. However, it has the following limitations:

(a) The depletion layer capacitance of the junction is very small, of the order of 103-105 pF/cm 2 (abrupt/linearly graded)[3]. Therefore, the change in capacitance and, hence, the energy increment (which is the output of the device) is negligibly small.

(b) The visible range of the solar spectrum has a frequency ~ 1015 Hz. For an electrical circuit with a p - n junction as capacitor, to have a resonant frequency of this order, an impractical- ibly low inductance is required.

(c) A high frequency circuit radiates most of its energy as electromagnetic radiation, which minimizes the efficiency of the device.

(d) The variation in the capacitance of a semiconductor junction involves a change in the depletion layer thickness by photo-excitation of the charge carriers and subsequent creation of photovoltage. This being the major process in a photovoltaic system, no comparable gain can be expected of a photo-parametric device, either in the technological aspect or from the point of view of production cost.

It is, therefore, concluded that use of the semiconductor junction capacitance for photo-paramet- ric amplification has to be ruled out. However, a mechano-electrical system is suggested, whereby the inductance of an electrical circuit can be varied by a periodically varying mechanical agency with a frequency low enough to minimize energy loss by radiation. In view of its importance in energy conversion, such mechanical agencies could be activated by wind energy or solar thermal energy.

SUMMARY

Parametric amplification is an alternate means of energy conversion into electricity, other than the already existing semiconductor devices. The behaviour of an electrical circuit in which amplification is effected by periodic variation in the capacitance has been discussed. Photon- induced periodic variation in the depletion layer capacitance of a semiconductor junction for such devices is found to be of no significant practical interest. A mechano-electrical system in which wind energy or solar thermal energy that activates a mechanical agency to produce a variation in the inductance of an electrical circuit is suggested for future work.

R E F E R E N C E S

1. H. Wetzel and H. Tributsch, Sol. Energy 37, 65 (1986). 2. R. J. Illiot and A. F. Gibson, Introduction to Solid State Physics and its Applications, p. 384. Macmillan,

New York (1978). 3. S. M. Eze, Physics of Semiconductor Devices, p. 89. Wiley, New York (1969).