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<report xmlns="http://www.bath.ac.uk/%7Epy8ieh/media/structure/report"
        xmlns:html="http://www.w3.org/TR/REC-html40">

   <header>
      <title> Forward and reverse bias of diodes and the effect of temperature on transistors</title>
      <author> Ian Hickson </author>
      <credit title="Tutor"> Dr Laughton </credit>
      <address> Department of Physics <br/> University of Bath <br/> Bath BA2 7AY </address>
      <date> November 1999 </date>

      <abstract> The forward and reverse biases of a germanium diode
      <component>OA91</component>, a silicon diode
      <component>IN4007</component>, and a silicon Schottky diode
      <component>BAT48</component> are analysed and it is found that
      the <component>OA91</component> is the least ideal, the
      <component>IN4007</component> has the smallest reverse
      saturation current (7.6nA), and the <component>BAT48</component>
      is the best wave rectifier. The effect of temperature on a
      <component>2N3055</component> transistor is then analysed and a
      positive exponential relationship is fitted.</abstract>

   </header>

   <body>

      <introduction>
         <p>This report details the results of the electronics
         laboratory experiment entitled "8: Diodes" (Bath, 1999). </p>
         <p> The theory behind the physics involved will be discussed
         in any good semiconductor or electronics book, such as
         "Microelectronics" (Millman, 1987).</p>
      </introduction>

      <section title="Forward Bias">
         
         <p>For each diode, the variation of current with applied
         forward voltage was measured using the following circuit.</p>

         <figure>
            <html:img src="forward-bias-circuit.gif" alt=""/>
            <caption num="1">Circuit for the analysis of the forward bias characteristic of diodes</caption>
         </figure>

         <p>The data was plotted (see next page, graphs 1-3) and the
         following results were found.</p>

         <table>
            <row>
               <header>Diode</header>
               <header><var>&eta;</var></header>
               <header><var>I</var><sub>0</sub></header>
               <header><var>T</var></header>
            </row>
            <row>
               <cell><component>OA91</component></cell>
               <cell>1</cell>
               <cell>1.0&micro;A</cell>
               <cell>404K</cell>
            </row>
            <row>
               <cell><component>IN4007</component></cell>
               <cell>2</cell>
               <cell>7.6nA</cell>
               <cell>276K</cell>
            </row>
            <row>
               <cell><component>BAT48</component></cell>
               <cell>1</cell>
               <cell>0.22&micro;A</cell>
               <cell>310K</cell>
            </row>
         </table>

         <p>See Appendix A for details on how the data was collected
         from the graphs. The value of <var>&eta;</var> was provided
         on the brief. </p>
         
         <p>The <component>OA91</component> breaks down very rapidly
         &mdash; above 0.2V (0.1mA) the relationship between the
         current and applied voltage no longer follows the theory used
         in Appendix A. However, both the
         <component>IN4007</component> and the
         <component>BAT48</component> follow the theory very closely.
         (On the graphs, the theoretical least squares fit is drawn as
         a solid line, and the data points are plotted as discrete
         dots.)</p>

         <p>The values of <var>I</var><sub>0</sub> for the
         <component>OA91</component> and the
         <component>BAT48</component> are within a factor of 10 of
         each other. The value of <var>I</var><sub>0</sub> for the
         <component>IN4007</component>, on the other hand, is
         considerably smaller. This should indicate that there will be
         very little (in fact, unmeasurable) current for the reverse
         bias of the <component>IN4007</component> diode. When this is
         tried, this is indeed what is found (see the next section).</p>

         <p>We would expect the values of <var>T</var> (temperature of
         the diode) to be very close to room temperature (aprox.
         295K). The more ideal the diode, the more likely the result
         is to be close to the room temperature (since to calculate
         the temperature ideality is assumed). The
         <component>OA91</component> diode is around 109 K away from
         the room temperature, while the <component>IN4007</component>
         and <component>BAT48</component> diodes are only 19K and 15K
         away respectively. This indicates that they follow the ideal
         relationship considerably more than the
         <component>OA91</component>. </p>

         <p>The overwhelming source of error was the fluctuations in
         the power supply caused by the number of appliances being
         used in the lab at the time. Other errors, such as the
         limitations of the equipment, were swamped by the
         fluctuations, and so have not been taken into account when
         drawing the errors bars.</p>

      </section>

      <section title="Reverse Bias">

         <p>To measure the reverse bias characteristics of the diodes,
         the following circuit was used:</p>
         
         <figure>
            <html:img src="reverse-bias-circuit.gif" alt=""/>
            <caption num="2">Circuit for the analysis of the reverse bias characteristic of diodes</caption>
         </figure>

         <p>Again the data was plotted (see next page, graphs 4 and 5)
         and the following results were found.</p>

         <table>
            <row>
               <header>Diode</header>
               <header><var>&eta;</var></header>
               <header><var>I</var><sub>0</sub></header>
            </row>
            <row>
               <cell>OA91</cell>
               <cell>1</cell>
               <cell>0.9&micro;A</cell>
            </row>
            <row>
               <cell>BAT48</cell>
               <cell>1</cell>
               <cell>1.0&micro;A</cell>
            </row>
         </table>

         <p>As predicted in the previous section, the
         <component>IN4007</component> has a very high reverse bias
         resistance and so no direct readings could be taken of
         <var>I</var><sub>0</sub>.</p>

         <p>For the <component>OA91</component>, it can be seen from
         the graph that the relationship between the reverse current
         and applied voltage may be linear rather than exponential
         (although an exponential relationship <em>could</em> be
         fitted within the error bars), in which case the data should
         fit:</p>

         <table xmlns="http://www.w3.org/TR/REC-HTML40">
            <tr>
               <td>|<var>I</var>|</td>
               <td>=</td>
               <td><var>I</var><sub>0</sub></td>
               <td>+</td>
               <td><var>&alpha;</var>|<var>V</var>|</td>
            </tr>
         </table>

         <p>...where <var>&alpha;</var> is some constant of
         proportionality (the gradient of the line), in this case
         0.1S.</p>

         <p>For the <component>BAT48</component>, the relationship
         appears to be exponential (although it should be noted that
         here a linear relationship could just be fitted within the
         error bars). The form of the relationship is:</p>

         <table xmlns="http://www.w3.org/TR/REC-HTML40">
            <tr>
               <td>|<var>I</var>|</td>
               <td>=</td>
               <td><var>I</var><sub>0</sub></td>
               <td>exp</td>
               <td>(</td>
               <td><var>&beta;</var>|<var>V</var>|</td>
               <td>)</td>
            </tr>
         </table>

         <p>...where <var>&beta;</var> is a constant, in this case
         0.548V<sup>-1</sup>.</p>

         <p>In both of these cases, the theory (see Appendix A) would
         suggest that the relationship should be independent of V, if
         the diodes were ideal. (The ideal relationship is simply
         <var>I</var> = <var>I</var><sub>0</sub>.) That there is some
         dependence indicates that the diodes are <em>not</em>
         ideal.</p>

      </section>

      <section title="Half wave rectifier">

         <p>The following circuit was used to examine the performance
         of the <component>IN4007</component> and
         <component>BAT48</component> diodes as half wave
         rectifiers.</p>
         
         <figure>
            <html:img src="rectifier.gif" alt=""/>
            <caption num="3">Half wave rectifier circuit for the analysis of diodes</caption>
         </figure>

         <p>An oscilloscope was placed across the 1k&ohm; resistor,
         and the frequency was varied from 1kHz to 1MHz for the two
         diodes. Sketches of the resulting waves are included in
         Appendix C, at the back. </p>

         <p>Qualitatively, the <component>BAT48</component> diode is
         much `better' at rectifying the sinusoidal wave than the
         <component>IN4007</component> &mdash; for example at 100kHz,
         the IN4007 lets through almost the entire amplitude of the
         wave (0.42V out of 0.45V), while the
         <component>BAT48</component> only has a small imperfection
         (less than 5% of the total amplitude of the wave). </p>

         <p>However, the efficiency of both reduces as the frequency
         increases: by 1MHz, the <component>IN4007</component> is
         having almost no effect on the wave, and the
         <component>BAT48</component> has imperfections in the order
         of 20% of the total peak-to-peak amplitude.</p>
         
      </section>

      <section title="Effect of temperature">

         <p>The following circuit was used to determine the
         temperature-current relationship for the
         <component>2N3055</component> transistor:</p>
         
         <figure>
            <html:img src="temperature-circuit.gif" alt=""/>
            <caption num="4">Circuit for the analysis of temperature dependence of current for the <component>2N3055</component> transistor</caption>
         </figure>

         <p>The data collected is plotted on the last graph (see the
         next page, graph 6).</p>

         <p>According to the brief, the temperature dependence of
         reverse current can be described by</p>

         <table xmlns="http://www.w3.org/TR/REC-HTML40">
            <tr>
               <td><var>I</var><sub>0</sub></td>
               <td>&prop;</td>
               <td>exp</td>
               <td class="tall">(</td>
               <td>-</td>
               <td><table>
                     <tr>
                        <td><var>E</var
                           ><sub>G</sub></td>
                     </tr>
                     <tr>
                        <td><var>k</var
                           ><var>T</var></td>
                     </tr>
                  </table></td>
               <td class="tall">)</td>
            </tr>
         </table>

         <p>This can also be written as:</p>

         <table xmlns="http://www.w3.org/TR/REC-HTML40">
            <tr>
               <td><var>I</var><sub>0</sub></td>
               <td>=</td>
               <td><var>A</var></td>
               <td>exp</td>
               <td class="tall">(</td>
               <td>-</td>
               <td><table>
                     <tr>
                        <td><var>E</var
                           ><sub>G</sub></td>
                     </tr>
                     <tr>
                        <td><var>k</var
                           ><var>T</var></td>
                     </tr>
                  </table></td>
               <td class="tall">)</td>
            </tr>
         </table>

         <p>...where <var>A</var> is the constant of
         proportionality.</p>

         <p>Unfortunately, the data collected did not appear to match
         this at all, indicating either that the equation provided was
         incorrect, or that an error was made during the collection of
         the data.</p>

         <p>A least squares fit analysis of the data shows that it
         fits a positive exponential relation:</p>

         <table xmlns="http://www.w3.org/TR/REC-HTML40">
            <tr>
               <td><var>I</var></td>
               <td>=</td>
               <td><var>I</var><sub>0</sub></td>
               <td>exp</td>
               <td>(</td>
               <td>&gamma;T</td>
               <td>)</td>
            </tr>
         </table>

         <p>...where <var>I</var><sub>0</sub> = 27.2nA and
         <var>&gamma;</var> = 50.7 mK<sup>-1</sup>. Since there is no
         theory behind these numbers -- they are purely empirical --
         no value of <var>E</var><sub>G</sub> can be derived or
         estimated.</p>

         <p>Note that both sets of data have been used to find the
         fit. This is because during both the heating and cooling, the
         temperature of the transistor was lagged with respect to the
         casing, which is what the probe was in contact with. During
         the heating, the transistor was slightly cooler than the
         casing, and during the cooling, the transistor was slightly
         warmer than the casing. The net effect should be that they
         cancel out.</p>

      </section>

      <appendix title="Appendix A: Methods used for data analysis of the forward bias">
         <p>The brief (Bath, 1999) stated the following equation as a
         good approximation of diode behaviour:</p>

         <table xmlns="http://www.w3.org/TR/REC-HTML40">
            <tr>
               <td><var>I</var></td>
               <td>=</td>
               <td><var>I</var><sub>0</sub></td>
               <td class="tall">(</td>
               <td>exp</td>
               <td class="tall">(</td>
               <td><table>
                     <tr>
                        <td><var>q</var
                           ><var>V</var></td>
                     </tr>
                     <tr>
                        <td><var>&eta;</var
                           ><var>k</var
                           ><var>T</var></td>
                     </tr>
                  </table></td>
               <td class="tall">)</td>
               <td>-</td>
               <td>1</td>
               <td class="tall">)</td>
            </tr>
         </table>

         <p>...where <var>I</var> is the current through the diode
         (positive for a forward bias and negative for reverse bias);
         <var>I</var><sub>0</sub> is the reverse saturation current;
         <var>q</var> is the electronic charge; <var>V</var> is the
         applied voltage (positive for a forward bias and negative for
         reverse bias); <var>&eta;</var> is the ideality factor;
         <var>k</var> is the Boltzmann constant and <var>T</var> is
         the junction temperature in Kelvin.</p>

         <p>For positive and large V (as in forward bias), this
         equation simplifies to:</p>

         <table xmlns="http://www.w3.org/TR/REC-HTML40">
            <tr>
               <td><var>I</var></td>
               <td>=</td>
               <td><var>I</var><sub>0</sub></td>
               <td>exp</td>
               <td class="tall">(</td>
               <td><table>
                     <tr>
                        <td><var>q</var
                           ><var>V</var></td>
                     </tr>
                     <tr>
                        <td><var>&eta;</var
                           ><var>k</var
                           ><var>T</var></td>
                     </tr>
                  </table></td>
               <td class="tall">)</td>
            </tr>
         </table>

         <p>...as the exponential term dominates the <span>-1</span> term.</p>

         <p>Using Excel, one can get a least-squares-fit for the equation </p>

         <table xmlns="http://www.w3.org/TR/REC-HTML40">
            <tr>
               <td><var>y</var></td>
               <td>=</td>
               <td><var>c</var></td>
               <td>exp</td>
               <td>(</td>
               <td><var>b</var><var>x</var></td>
               <td>)</td>
            </tr>
         </table>

         <p>...by plotting an "Exponential Trend Line" (Microsoft,
         1997). By equating <var>I</var>=<var>y</var>,
         <var>I</var><sub>0</sub>=<var>c</var>,
         <var>V</var>=<var>x</var> and </p>

         <table xmlns="http://www.w3.org/TR/REC-HTML40">
            <tr>
               <td><var>b</var></td>
               <td>=</td>
               <td><table>
                     <tr>
                        <td><var>q</var></td>
                     </tr>
                     <tr>
                        <td><var>&eta;</var
                           ><var>k</var
                           ><var>T</var></td>
                     </tr>
                  </table></td>
            </tr>
         </table>

         <p>...values for <var>I</var><sub>0</sub>=<var>c</var> and
         <var>T</var> can be derived.</p>

      </appendix>

      <appendix title="Appendix B: Device Characteristics">
         <p>The following data is given in the relevant databooks for the devices.</p>

         <table>
            <row>
               <header>Diode</header>
               <header>Maximum reverse voltage</header>
               <header>Maximum forward current</header>
               <header>Source</header>
            </row>
            <row>
               <cell>OA91</cell>
               <cell>90v</cell>
               <cell>150mA</cell>
               <cell>Data sheet</cell>
            </row>
            <row>
               <cell>IN4007</cell>
               <cell>50V</cell>
               <cell>10A</cell>
               <cell>Mullard, February 1985</cell>
            </row>
            <row>
               <cell>BAT48</cell>
               <cell>40V</cell>
               <cell>350mA</cell>
               <cell>Thomson Semiconductors Data Sheet</cell>
            </row>
         </table>

      </appendix>


      <appendix title="Appendix C: Sketches of results for the Half Wave Rectifier">
         <p>These hand-drawn sketches are included separately.</p>
      </appendix>

      <references>

         <ref id="diodes">
           <author>Bath University Physics Department</author> 
           <pubyear>1999</pubyear> 
           <chapter>Experiment 8: Diodes</chapter>
           <resource>Electronics Laboratory Experiments</resource>
           <publisher location="Bath, UK">Bath University Physics Department</publisher>
         </ref>
         
         <ref id="electronics">
           <author>Jacob Milman, Arvin Grabel</author> 
           <pubyear>1987</pubyear> 
           <resource>Microelectronics 2nd Edition</resource>
           <publisher>McGraw-Hill, Inc.</publisher>
           <page>49</page>
         </ref>
         
         <ref id="excelhelp">
           <author>Microsoft</author> 
           <pubyear>1997</pubyear> 
           <chapter>Equations, chart trendlines</chapter>
           <resource>Microsoft Excel Help</resource>
           <publisher location="Redmond, WA">Microsoft Corporation</publisher>
         </ref>

      </references>

   </body>

</report>
