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<document xmlns="http://cnx.rice.edu/cnxml" xmlns:md="http://cnx.rice.edu/mdml/0.4" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:bib="http://bibtexml.sf.net/" id="new36">
  <name>Common Emitter Models</name>
  <metadata>
  <md:version>2.10</md:version>
  <md:created>2000/08/04</md:created>
  <md:revised>2007/08/14 10:02:11.933 GMT-5</md:revised>
  <md:authorlist>
      <md:author id="wlw">
      <md:firstname>Bill</md:firstname>
      
      <md:surname>Wilson</md:surname>
      <md:email>wlw@madriver.net</md:email>
    </md:author>
  </md:authorlist>

  <md:maintainerlist>
    <md:maintainer id="wlw">
      <md:firstname>Bill</md:firstname>
      
      <md:surname>Wilson</md:surname>
      <md:email>wlw@madriver.net</md:email>
    </md:maintainer>
    <md:maintainer id="lizzardg">
      <md:firstname>Elizabeth</md:firstname>
      
      <md:surname>Gregory</md:surname>
      <md:email>elizabeth.gregory@gmail.com</md:email>
    </md:maintainer>
    <md:maintainer id="jsilv">
      <md:firstname>Jeffrey</md:firstname>
      <md:othername>M</md:othername>
      <md:surname>Silverman</md:surname>
      <md:email>JSilverman@astro.berkeley.edu</md:email>
    </md:maintainer>
    <md:maintainer id="gerardw">
      <md:firstname>Gerard</md:firstname>
      
      <md:surname>Wysocki</md:surname>
      <md:email>gerardw@rice.edu</md:email>
    </md:maintainer>
  </md:maintainerlist>
  
  <md:keywordlist>
    <md:keyword>emitter</md:keyword>
    <md:keyword>models</md:keyword>
  </md:keywordlist>

  <md:abstract>Common Emitter Models</md:abstract>
</metadata>

  <content>
    <para id="para1">
      Let's go ahead and draw a model for the transistor in the <cnxn target="fig10">common emitter configuration</cnxn>. We again
      have a diode connected between the base and the emitter, and a
      new current controlled current source between collector and
      emitter. There is one small caveat which we need to keep in mind
      however when drawing the common emitter circuit.  The diode we
      see in the base circuit is <emphasis>not</emphasis> the same one
      as we had in the common base model. In the common base model, it
      was true that

    <equation id="eqn12">
      <m:math>
        <m:apply><m:eq/>
          <m:ci><m:msub>
            <m:mi>I</m:mi>
            <m:mi>E</m:mi>
          </m:msub></m:ci>
          <m:apply><m:times/>
            <m:ci><m:msub>
              <m:mi>I</m:mi>
              <m:mi>sat</m:mi>
            </m:msub></m:ci>
            <m:apply><m:minus/>
              <m:apply>
               <m:exp/>
                <m:apply><m:divide/>
                  <m:apply><m:times/>
                    <m:ci>q</m:ci>
                    <m:ci><m:msub>
                      <m:mi>V</m:mi>
                      <m:mi>BE</m:mi>
                    </m:msub></m:ci>
                  </m:apply>
                  <m:apply><m:times/>
                    <m:ci>k</m:ci>
                    <m:ci>T</m:ci>
                  </m:apply>
                </m:apply>
              </m:apply>
              <m:cn>1</m:cn>
            </m:apply>
          </m:apply>
        </m:apply>
      </m:math>
    </equation>

  </para>

  <figure id="fig10">
    <media type="image/png" src="3_10.png"/>
    <caption>Discrete model for the common emitter configuration
    </caption>
  </figure>

  <para id="para3">
    For the base however, only a small fraction of the
    current that goes through this "diode" actually goes in
    through the base, the rest is coming in through the collector.
    Thus we have to make a couple of changes

    <equation id="eqn13">
      <m:math>
        <m:apply><m:eq/>
          <m:ci><m:msub>
            <m:mi>I</m:mi>
            <m:mi>C</m:mi>
          </m:msub></m:ci>

          <m:apply><m:times/>
            <m:ci>α</m:ci>
            <m:ci><m:msub>
              <m:mi>I</m:mi>
              <m:mi>E</m:mi>
            </m:msub></m:ci>
          </m:apply>

          <m:apply><m:times/>
            <m:ci>α</m:ci>
            <m:ci><m:msub>
              <m:mi>I</m:mi>
              <m:mi>sat</m:mi>
            </m:msub></m:ci>
             <m:apply><m:minus/>
              <m:apply>
                <m:exp/>
                <m:apply><m:divide/>
                  <m:apply><m:times/>
                    <m:ci>q</m:ci>
                    <m:ci><m:msub>
                      <m:mi>V</m:mi>
                      <m:mi>BE</m:mi>
                    </m:msub></m:ci>
                  </m:apply>
                  <m:apply><m:times/>
                    <m:ci>k</m:ci>
                    <m:ci>T</m:ci>
                  </m:apply>
                </m:apply>
              </m:apply>
              <m:cn>1</m:cn>
            </m:apply>
          </m:apply>
        </m:apply>
      </m:math>
    </equation>

    <equation id="eqn14">
      <m:math>
        <m:apply><m:eq/>

          <m:ci><m:msub>
            <m:mi>I</m:mi>
            <m:mi>B</m:mi>
          </m:msub></m:ci>

          <m:apply><m:divide/>
            <m:ci><m:msub>
              <m:mi>I</m:mi>
              <m:mi>C</m:mi>
            </m:msub></m:ci>
            <m:ci>β</m:ci>
          </m:apply>

          <m:apply><m:times/>
            <m:apply><m:divide/>
              <m:apply><m:times/>
                <m:ci>α</m:ci>
                <m:ci><m:msub>
                  <m:mi>I</m:mi>
                  <m:mi>sat</m:mi>
                </m:msub></m:ci>
              </m:apply>
              <m:ci>β</m:ci>
            </m:apply>
            <m:apply><m:minus/>
              <m:apply>
               <m:exp/>
                <m:apply><m:divide/>
                  <m:apply><m:times/>
                    <m:ci>q</m:ci>
                    <m:ci><m:msub>
                      <m:mi>V</m:mi>
                      <m:mi>BE</m:mi>
                    </m:msub></m:ci>
                  </m:apply>
                  <m:apply><m:times/>
                    <m:ci>k</m:ci>
                    <m:ci>T</m:ci>
                  </m:apply>
                </m:apply>
              </m:apply>
              <m:cn>1</m:cn>
            </m:apply>
          </m:apply>
        </m:apply>
      </m:math>
    </equation>

    So the operational equation for the diode in the base circuit
    still is the usual exponential function of

      <m:math>
        <m:ci><m:msub>
          <m:mi>V</m:mi>
          <m:mi>BE</m:mi>
        </m:msub>
       </m:ci>
      </m:math>,
    except that it now has a saturation current of

      <m:math>
        <m:apply><m:divide/>
          <m:apply><m:times/>
            <m:ci>α</m:ci>
            <m:ci><m:msub>
              <m:mi>I</m:mi>
              <m:mi>sat</m:mi>
            </m:msub>
           </m:ci>
          </m:apply>
          <m:ci>β</m:ci>
        </m:apply>
      </m:math>

    instead of just

      <m:math>
        <m:ci><m:msub>
          <m:mi>I</m:mi>
          <m:mi>sat</m:mi>
        </m:msub></m:ci>
      </m:math>.
  </para>

  <para id="para7">
    In principle you could put this model into a circuit, and
    analyze it to find all of the necessary voltages and
    currents. However, this would not be very convenient. The
    base-emitter junction is connected by a diode, which as we know, has a very
    non-linear I-V relationship. It would be nice if we could come up
    with a <emphasis>linear</emphasis> model which, at least over some
    limited range of inputs, we could use with confidence.
  </para>
  </content>
  
</document>
