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1. What is its reverse saturation current, I0?

2. What is the load resistance that allows maximum power transfer?

3. What is the efficiency of this cell with the load above?

14.12 The power density of monochromatic laser light (586 nm) is to be monitored by a 1 x 1 mm silicon photodiode. The quantity observed is the short-circuit current generated by the silicon. Treat the diode as a perfect ideal device.

1. What current do you expect if the light level is 230 W/m2?

2. How does the temperature of the semiconductor affect this current? Of course, the temperature has to be lower than that which will destroy the device (some 150C, for silicon).

3. Instead of being shorted out, the diode is now connected to a load compatible with maximum electric output. Estimate the load voltage.

14.13 A silicon photocell being tested measures 4 by 4 cm. Throughout the tests, the temperature of the device is kept at 300 K. Assume the cell has no significant series resistance. Assume 100% quantum efficiency. The band-gap energy of silicon is 1.1 eV.

Initially, the cell is kept in the dark. When a current of 100 ¡iA is forced through it in the direction of good conduction, the voltage across the diode is 0.466 V.

Estimate the open-circuit voltage developed by the cell when exposed to bichromatic infrared radiation of 412 nm and 1300 nm wavelength. The power density at the shorter wavelength is 87W/m2, while at the longer wavelength, it is 93 W/m2.

14.14 What is the ideal efficiency of a photocell made from a semiconducting material with a band-gap energy, Wg = 2eV, when illuminated by radiation with the normalized spectral power distribution given as follows.

f dP/df

In this table the frequency, f, is in THz. Repeat for a semiconductor with Wg = 1 eV.

14.15 What is the theoretical efficiency of a photocell with a 2.5-V bandgap when exposed to 100 W/m2 solar radiation through a filter having the following transmittance characteristics:

Pass without attenuation all wavelengths between 600 and 1000 nm. Reject all else.

14.16 A photodiode is exposed to radiation of uniform spectral power density (p = constant) covering the range from 300 to 500 THz. Outside this range there is no radiation. The total power density is 2000 W/m2.

1. Assuming 100% quantum efficiency, what is the short-circuit pho-tocurrent of a diode having an active area of 1 by 1 cm?

2. When exposed to the radiation in Part 1 of this problem, the open-circuit voltage delivered by the diode is 0.498 V. A 1.0-V

voltage is applied to the diode (which is now in darkness) in the reverse conduction direction (i.e., in the direction in which it acts almost as an open circuit). What current circulates through the device? The temperature of the diode is 300 K.

14.17 The sun radiates (roughly) like a 6000-K black body. When the power density of such radiation is 1000 W/m2—"one sun"—the total photon flux is 4.46 x 1021 photons per m2 per second. Almost exactly half of these photons have energy equal or larger than 1.1 eV (the band-gap energy, Wg, of silicon).

Consider a small silicon photodiode with a 10 by 10 cm area. When 2 V of reversed bias is applied, the resulting current is 30 nA. This is, of course, the reverse saturation current, I0.

When the photodiode is short-circuited and exposed to black body radiation with a power density of 1000 W/m2, a short-circuit current, Iv, circulates.

1. Assuming 100% quantum efficiency (each photon creates one electron-hole pair and all pairs are separated by the p-n junction of the diode), what is the value of this current?

2. What is the open-circuit voltage of the photodiode at 300 K under the above illumination?

3. Observe that the V-I characteristics of a photodiode are very steep at the high current end. In other words, the best operating current is only slightly less than that of the short-circuit current. This knowledge will facilitate answering this question:

Under an illumination of 1000 W/m2, at 300K, what is the maximum power the photodiode can deliver to a load? What is the efficiency? Do this by trial and error and be satisfied with three significant figures in your answer. Consider an ideal diode with no internal resistance.

4. What is the load resistance that leads to maximum efficiency?

5. Now repeat the power, efficiency, and load resistance calculations for an illumination of 10,000 W/m2.

6. What happens to the efficiency and the optimal load resistance when the power density of the illumination on a photodiode increases?

14.18 Everything else being the same, the efficiency of a photodiode rises

1. when the operating temperature rises.

2. when the operating temperature falls.

3. when the light power density rises.

### 4. when the light power density falls.

14.19 A photodiode with a band-gap energy of Wg = 1.4 eV is exposed to monochromatic radiation (500 THz) with a power density P = 500 w/m2. The active area of the device is 10 by 10 cm.

Treat it as an ideal device in series with an internal resistance of 2 ml.

All measurements were made at 298 K. The open-circuit voltage is 0.555 V.

1. Estimate the short-circuit current.

2. How much power does the diode deliver to 200 ml load?

3. What is the efficiency of the device when feeding the 200 milliohm

14.20 Suggestion: To solve this problem, use a spreadsheet and tabulate all the pertinent values for different hour angles (from sunrise to sunset). 5° intervals in a will be adequate.

A flat array of silicon photodiodes is set up at 32° N. The array faces south and is mounted at an elevation angle that maximizes the year-long energy collection, assuming perfectly transparent air.

1. What is the elevation angle of the array?

2. On April 15, 2002, how does the insolation on the array vary throughout the day? Plot the insolation, P, versus the time of day, t, in hours.

### 3. What is the average insolation on the collector.

4. Assuming ideal silicon photodiodes with a reverse saturation current density of 10nA/m2, what is the average power delivered during the day (from sunrise to sunset) if a perfect load follower is used, that is, if the load is perfectly matched at all the different instantaneous values of insolation? What is the average overall efficiency?

5. Estimate the average power collected if the array is connected to a load whose resistance maximizes the efficiency at noon. In other words, the average power when no load-follower is used.

14.21 To simplify mathematical manipulation, we will postulate a very simple (and unrealistic) spectral power distribution:

1. If A = 10~12 Wm~2 Hz-1, what is the power density of the radiation?

2. Assuming 100% quantum efficiency, what is the short-circuit current density, Jv, in an ideal photodiode having a band-gap energy smaller than the energy of 300 THz photon?

3. At 300 K, assuming a reverse saturation current density of J0 = 10-7 A/m2, what is the open-circuit voltage of the photocell?

4. At what load voltage, Vm, does this photocell deliver its maximum power output?

5. What is the current density delivered by the photocell when max imum power is being transferred to the load?

load?

0, otherwise.

0, otherwise.

6. What is the efficiency of the photocell?

7. What is the load resistance under the above conditions?

8. Repeat all the above for a light power density of 2 W/m2.

9. What would be the efficiency at these low light levels if the load resistance had the optimum value for 200 W/m2?

14.22 It is hoped that high-efficiency cascaded photocells can be produced at a low cost. This consists of a sandwich of two cells of different band gaps. The bottom cell (the one with the smaller band gap) can be made using CuInxGai_xSe2, known as CIGS. This material has a band gap of about 1 eV and has been demonstrated as yielding cells with 15% efficiency.

The question here is what band gap of the top cell yields the largest efficiency for the combined cascaded cells. Assume radiation from a black body at 6000 K. Assume no losses; that is, consider only the theoretical efficiency.

14.23 The V-I characteristic of a photocell can be described by a rather complex mathematical formula, which can be handled with a computer but is too complicated for an in-class exam. To simplify handling, we are adopting, rather arbitrarily, a simplified characteristic consisting of two straight lines as shown in the figure above. The position of point C, of maximum output, varies with the Iv/I0 ratio. Empirically,

Now consider silicon photodiodes operating at 298 K. These diodes form a panel, 1 m2 in area, situated in Palo Alto (latitude 37.4° N, longitude 125° W). The panel faces true south and has an elevation of 35°. In practice, the panel would consist of many diodes in a series/parallel connection. In the problem here, assume that the panel has a single enormous photodiode.

Calculate the insolation on the surface at a 1130 PST and at 1600 PST on October 27. Assume clear meteorological conditions.

Assume that the true solar time is equal to the PST.

1. Calculate the insolation on the collector at the two moments mentioned.

2. What are the short-circuit currents (Iv) under the two illuminations? Consider the sun as a 6000-K black body.

3. When exposed to the higher of the two insolations, the open-circuit voltage of the photodiode is 0.44 V. What is the power and delivered to a load at 1130 and at 1600? The resistance of the load is, in each case, that which maximizes the power output for that case. What are the load resistances? What are the efficiencies?

4. Suppose that at 1600 the load resistance used was the same as that which optimized the 1130 output. What are the power in the load and the efficiency?

5. Let the load resistance be the same at both 1130 and 1600, but, unlike Question 4, not necessarily the resistance that optimizes the output at 1130. The idea is to operate the panel at slightly lower efficiency at 1130 and at somewhat higher efficiency than that of Question 4 at 1600 in the hope that the overall efficiency can be improved. What is the value of this common load resistance?

14.24 1. What is the ideal (theoretical) efficiency of a gallium phosphide photocell exposed to the radiation of a 6000-K black body? For your information: the corresponding efficiency for silicon is 43.8%.

2. What is the efficiency of an ideal silicon photocell when illuminated by monochromatic light with a frequency of 266 THz?

3. What is the efficiency of an ideal silicon photocell when illuminated by monochromatic light with a frequency of 541.6 THz?

4. A real silicon photocell measuring 10 by 10 cm is exposed to 6000-K black body radiation with a power density of 1000.0 W/m2. The temperature of the cell is 310 K. The measured open-circuit voltage is 0.493 V. When short-circuited, the measured current is 3.900 A. The power that the cell delivers to a load depends, of course, on the exact resistance of this load. By properly adjusting the load, the power can be maximized. What is this maximum power?

14.25 Consider a solar cell made of semiconducting nanocrystals with a band-gap energy of Wg = 0.67 eV. What is the theoretical efficiency when the solar cell is exposed to the radiation of a 6000-K black body? Assume that photons with less than 3.3 Wg create each one single-electron/hole pair and that those with more than 3.3 Wg create 2 electron/hole pairs each owing to impact ionization.

14.26 The Solar Power Satellites proposed by NASA would operate at 2.45 GHz. The power density of the beam at ionospheric heights (400 km) was to be 230W/m2. The collector on the ground was designed to use dipole antennas with individual rectifiers of the Schottky barrier type. These dipoles were dubbed rectennas.

The satellites would have been geostationary (they would be on a 24-hr equatorial orbit with zero inclination and zero eccentricity).

### 1. Calculate the orbital radius of the satellites.

2. Calculate the microwave power density on the ground at a point directly below the satellite (the subsatellite point). Assume no absorption of the radiation by the atmosphere.

3. The total power delivered to the load is 5 GW. The rectenna system has 70% efficiency. Assume uniform power density over the illuminated area. What is the area that the ground antenna farm must cover?

4. A dipole antenna abstracts energy from an area given in the text. How many dipoles must the antenna farm use?

5. Assuming (very unrealistically) that the only part of each rectenna that has any mass is the dipole itself, and assuming that the half-wave dipole is made of extremely thin aluminum wire, only 0.1 mm in diameter, what is the total mass of aluminum used in the antenna farm?

6. How many watts must each dipole deliver to the load?

7. If the impedance of the rectenna is 70 ohms, how many volts does each dipole deliver?

14.27 The Solar Power Satellite radiates 6 GW at 2.45 GHz. The transmitting antenna is mounted 10 km from the center of gravity of the satellite. What is the torque produced by the radiation?

14.28 Compare the amount of energy required to launch a mass, m, from the surface of the Earth to the energy necessary to launch the same mass from the surface of the moon. "Launch" here means placing the mass in question an infinite distance from the point of origin. Consult the Handbook of Chemistry and Physics (CRC) for the pertinent astronomical data.

14.29 Consider a simple spectral distribution:

The total power density is P = 1000 W/m2.

a. A silicon diode having 100% quantum efficiency is exposed to this radiation. What is the short-circuit current density delivered by the diode?

b. At 300K, the reverse saturation current density, J0, of the diode is 40 yA/m2. What is the open-circuit voltage generated by the diode?

c. If the photodiode has an effective area of 10 by 10 cm, what load resistance will result in the largest possible output power?

14.30 1. Five identical photodiodes are connected in series and feed a single-cell water electrolyzer.

The whole system operates at 300 K, and the photodiode is exposed to a light power density that causes a photon flux of 2.5 x 1021 photons per second per square meter to interact with the device. Quantum efficiency is 100%.

For each photodiode, the reverse saturation current, I0, is 0.4 yA, and the series resistance is 10 mfi. The active area of the diode is 10 by 10 cm.

The electrolyzer can be represented by a 1.6-V voltage source, in series with a 100-mQ resistance.

What is the hydrogen production rate in grams/day? 2. What photon flux is sufficient to just start the electrolysis?

14.31 What is the frequency at which a human body radiates the most heat energy per 1-Hz bandwidth?

14.32 The ideal photocells can exceed the black body spectrum efficiency if used in a configuration called a cascade. Thus the ideal efficiency of silicon cells exposed to a 6000-K black body radiation is 43.8%. If in cascade with an ideal cell with a 1.8 eV band gap, the efficiency is 56%. Clearly, using more cells the efficiency goes up further.

What is the efficiency of a cascade arrangement consisting of an infinite number of cells, the first having a band gap of 0 eV and each succeeding one having a band gap infinitesimally higher than that of the preceding cell? The cell with the highest band gap is on top (nearest the light source).

14.33 A high-precision photometer (300 K) equipped with a very narrow band-pass filter made the following measurements:

Light power density in a 1-MHz-wide band centered around 200 THz: 2.0 x 10"11 W/m2

Light power density in a 1-MHz-wide band centered around 300 THz: 2.7 x 10"11 W/m2

Assuming the radiation came from a black body, what is the temperature of the black body?

14.34 A silicon diode, operating at 300 K, is exposed to 6000-K black body radiation with a power density of 1000 W/m2. Its efficiency is 20% when a load that maximizes power output is used. Estimate the open-circuit voltage delivered by the diode.

14.35 A GaAs photodiode, operating at 39 C, is exposed to 5500 K black body radiation with a power density of 675 W/m2. The open-circuit voltage of the device is 0.46 V. What is the efficiency of the photodiode when delivering energy to a 2-milliohm m2 load?

## Solar Stirling Engine Basics Explained

The solar Stirling engine is progressively becoming a viable alternative to solar panels for its higher efficiency. Stirling engines might be the best way to harvest the power provided by the sun. This is an easy-to-understand explanation of how Stirling engines work, the different types, and why they are more efficient than steam engines.

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