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The '''return ratio''' of a dependent source in a linear electrical circuit is the ''negative'' of the ratio of the current (voltage) returned to the site of the dependent source to the current (voltage) of a replacement independent source. The terms ''loop gain'' and ''return ratio'' are often used interchangeably; however, they are necessarily equivalent only in the case of a single feedback loop system with unilateral blocks. <ref name=Spencer/>
The '''Standard Model''' of particle physics is the mathematical theory that describes the [[weak force|weak]], [[Maxwell's equations|electromagnetic]] and [[Strong force|strong]] interactions between [[lepton]]s and [[quark]]s, the basic particles of particle physics. This model is very strongly supported by experimental observations, and is considered to be a major achievement (perhaps the most outstanding achievement) of theoretical physics. It does not, however, treat the [[Gravitation|gravitational force]], inclusion of which remains an elusive goal of the ultimate "theory of everything". The Standard Model is accordingly not consistent with [[general relativity]]. The theory is consistent with [[special relativity]].
 
==Calculating the return ratio==
The steps for calculating the return ratio of a source are as follows:<ref name=Gray-Meyer/>
#  Set all independent sources to zero.
#  Select the dependent source for which the return ratio is sought.
#  Place an independent source of the same type (voltage or current) and polarity in parallel with the selected dependent source.
#  Move the dependent source to the side of the inserted source and cut the two leads joining the dependent source to the independent source.
#  For a '''voltage source''' the return ratio is minus the ratio of the voltage across the dependent source divided by the voltage of the independent replacement source.
#  For a '''current source''', short-circuit the broken leads of the dependent source. The return ratio is minus the ratio of the resulting short-circuit current to the current of the independent replacement source.
 
=== Other Methods ===
The above steps can be implemented in [[SPICE]] simulations by replacing non-linear devices by their small-signal model equivalents. These steps are not feasible where dependent sources inside devices are not directly accessible, for example, when SPICE itself is used to generate the small-signal circuit numerically, or when measuring the return ratio experimentally. When small-signal models cannot be used, an added problem is finding how to break a a loop without affecting the [[bias point]] and altering the results. Middlebrook<ref name=Middlebrook/> and Rosenstark<ref name=Rosenstark/> have proposed several methods for experimental evaluation of return ratio (loosely referred to by these authors as simply ''loop gain''), and similar methods have been adapted for use in [[SPICE]] by Hurst.<ref name=Hurst/><ref name=Spectrum/> or Roberts,<ref name=Roberts/> or Sedra,<ref name=Sedra/> and especially Tuinenga.<ref name=Tuinenga/>
 
==Example: Collector-to-base biased bipolar amplifier==
{{Image|Bipolar transresistance feedback amplifier.PNG|right|150px| Collector-to-base biased bipolar amplifier.}}
{{Image|Inserting source for return ratio.PNG|right|700px| ''Left'' - small-signal circuit corresponding to bipolar amplifier; ''Center'' - inserting independent source and marking leads to be cut; ''Right''  - cutting the dependent source free and short-circuiting broken leads.}}
 
The figure at right shows a bipolar amplifier with feedback bias resistor ''R<sub>f</sub>'' driven by a [[Norton's theorem|Norton signal source]]. In the tableaux of small-signal circuits, the left panel shows the corresponding small-signal circuit obtained by replacing the transistor with its [[hybrid-pi model]]. The objective is to find the return ratio of the dependent current source in this amplifier.<ref name=Spencer2/> To reach the objective, the steps outlined above are followed. The center panel shows the application of these steps up to Step 4, with the dependent source moved to the left of the inserted source of value ''i<sub>t</sub>'', and the leads targeted for cutting marked with an <font color=red>''X''</font>. The right panel shows the circuit set up for calculation of the return ratio ''T'', which is
 
::<math> T = - \frac {i_r} {i_t} \ . </math>
 
The return current is  
 
::<math> i_r = g_m v_{\pi} \ . </math>
 
The feedback current in ''R<sub>f</sub>'' is found by [[current division]] to be:
::<math>i_f = \frac {R_D\mathit{\parallel}r_O} {R_D\mathit{\parallel}r_O +R_F +r_{\pi}\mathit{\parallel} R_S} \  i_t \ . </math>
 
The base-emitter voltage ''v<sub>π</sub>'' is then, from [[Ohm's law]]:
 
::<math> v_{\pi} = -i_f \ ( r_{\pi}\mathit{\parallel} R_S ) \ . </math>
 
Consequently,
::<math> T = g_m  (r_{\pi}\mathit{\parallel} R_S ) \  \frac {R_D\mathit{\parallel}r_O} {R_D\mathit{\parallel}r_O +R_F +r_{\pi}\mathit{\parallel} R_S}\ .  </math>
 
=== Application in asymptotic gain model ===
The overall transresistance gain of this amplifier can be shown to be:
 
::<math> G = \frac {v_{out}} {i_{in}} =  \frac {(1-g_m R_F)R_1 R_2} {R_F+R_1+R_2+g_m R_1R_2} \ , </math>
 
with ''R<sub>1</sub> = R<sub>S</sub> || r<sub>π</sub>'' and ''R<sub>2</sub> = R<sub>D</sub> || r<sub>O</sub>''.
 
This expression can be rewritten in the form used by the [[asymptotic gain model]], which expresses the overall gain of a feedback amplifier in terms of several independent factors that often are more easily derived separately than is the gain itself, and that often provide insight into the circuit. This form is:
 
::<math> G = \ G_{ \infty } \frac {T} {1+T}  + G_0 \frac {1} { 1+T} \ \ , </math>
 
where the so-called '''asymptotic gain''' ''G<sub>&infin;</sub>'' is the gain at infinite ''g<sub>m</sub>'', namely:
 
::<math> G_{\infty} = - R_F \ , </math>
 
and the so-called '''feed forward''' or '''direct feedthrough''' ''G<sub>0</sub>'' is the gain for zero ''g<sub>m</sub>'', namely:
 
::<math> G_{0} = \frac { R_1 R_2 } {R_F +R_1 +R_2}\ . </math>
 
For additional applications of this method, see [[Asymptotic gain model#Return ratio|asymptotic gain model]].


==References==
==References==
{{Reflist|refs=
{{Reflist|refs=


<ref name=Gray-Meyer>
{{cite book
|author=Paul R. Gray, Hurst P J Lewis S H & Meyer RG
|title=Analysis and design of analog integrated circuits
|page=§8.8 pp. 599-613
|year= 2001
|edition=Fourth Edition
|publisher=Wiley
|location=New York
|isbn=0-471-32168-0
|url=http://worldcat.org/isbn/0-471-32168-0}}
</ref>
<ref name=Hurst>
{{cite journal |url=http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=99170 |author=Hurst, PJ |title=Exact simulation of feedback circuit parameters |journal=IEEE Trans. on Circuits and Systems |volume=38 |issue=11 |year=1991 |pages= pp.1382-1389}}</ref>
<ref name=Middlebrook>
{{cite journal |url=http://www.informaworld.com/smpp/content~content=a771365730~db=all |author=Middlebrook, RD |title=Loop gain in feedback systems 1|journal= Int. J. of Electronics |volume=38 |issue=4 |year=1975 |pages= pp. 485-512 }}
</ref>
<ref name=Roberts>
{{cite book
|author=Gordon W. Roberts & Sedra AS
|title=SPICE
|edition=Second Edition
|year= 1997
|pages=Chapter 8; pp. 256-262
|publisher=Oxford University Press
|location=New York
|isbn=0-19-510842-6
|url=http://worldcat.org/isbn/0-19-510842-6}}
</ref>
<ref name=Rosenstark>
{{cite journal |url=http://www.informaworld.com/smpp/content~content=a777774065~db=all |author=Rosenstark, Sol|title=Loop gain measurement in feedback amplifiers|journal=Int. J. of Electronics|volume=57|issue=3 |year=1984|pages= pp.415-421}}
</ref>
<ref name=Sedra>
{{cite book
|author=Adel S Sedra & Smith KC
|title=Microelectronic circuits
|edition=Fifth Edition
|year= 2004
|pages=Example 8.7; pp. 855--859
|publisher=Oxford University Press
|location=New York
|isbn=0-19-514251-9
|url=http://worldcat.org/isbn/0-19-514251-9}}
</ref>
<ref name=Spectrum>
{{cite web |url=http://www.spectrum-soft.com/news/spring97/loopgain.shtm |publisher=Spectrum Software |title=Simulating loop gain |work=Spectrum Newsletters | date=Spring, 1997|accessdate=2011-06-30}}
</ref>
<ref name=Spencer>
{{cite book
|author=Richard R Spencer & Ghausi MS
|title=Introduction to electronic circuit design
|page=p. 723
|year= 2003
|publisher=Prentice Hall/Pearson Education
|location=Upper Saddle River NJ
|isbn=0-201-36183-3
|url=http://worldcat.org/isbn/0-201-36183-3}}
</ref>
<ref name=Spencer2>
{{cite book
|author=Richard R Spencer & Ghausi MS
|title=Reference cited earlier
|chapter=Example 10.7 pp. 723-724
|isbn=0-201-36183-3
|url=http://worldcat.org/isbn/0-201-36183-3}}
</ref>


<ref name=Tuinenga>
{{cite book
|author=Paul W Tuinenga
|title=SPICE: a guide to circuit simulation and analysis using PSpice
|edition=Third Edition
|year= 1995
|pages=Chapter 8: ''Loop gain analysis''
|publisher=Prentice-Hall
|location=Englewood Cliffs NJ
|isbn=0134360494
|url=http://worldcat.org/isbn/0134360494}}
</ref>


}}
}}

Revision as of 09:48, 28 August 2011

The Standard Model of particle physics is the mathematical theory that describes the weak, electromagnetic and strong interactions between leptons and quarks, the basic particles of particle physics. This model is very strongly supported by experimental observations, and is considered to be a major achievement (perhaps the most outstanding achievement) of theoretical physics. It does not, however, treat the gravitational force, inclusion of which remains an elusive goal of the ultimate "theory of everything". The Standard Model is accordingly not consistent with general relativity. The theory is consistent with special relativity.

References

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