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{{Image|Block diagram asymptotic gain.PNG|right|250px|Block diagram for asymptotic gain model<ref name=Gray-Meyer/>.}}
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{{Image|Signal-flow graph for asymptotic gain model.PNG|right|250px|Possible signal-flow graph for the asymptotic gain model.}}
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The '''asymptotic gain model'''<ref name=Middlebrook/> (also known as the '''Rosenstark method'''<ref name=Rosenstark/><ref name=Palumbo/>) is a representation of the gain of [[negative feedback amplifier]]s given by the asymptotic gain relation:
[[User:John R. Brews/Sample/Talk|Link to Talk page]]
:<math>G = G_{\infty} \left( \frac{T}{T + 1} \right) + G_0 \left( \frac{1}{T + 1} \right) \ ,</math>
where ''T'' is the [[return ratio]] with the input source disabled (equal to the negative of the [[loop gain]] in the case of a single-loop system composed of [[Electronic amplifier#Unilateral or bilateral|unilateral]] blocks), ''G<sub>∞</sub>'' is the asymptotic gain and ''G<sub>0</sub>'' is the direct transmission term. This form for the gain can provide intuitive insight into the circuit and often is easier to derive than a direct attack on the gain.


A block diagram that leads to the asymptotic gain expression is shown in the upper figure at right.  The asymptotic gain relation also can be expressed as a [[Signal-flow_graph|signal-flow graph]]. See lower of two figures.  The asymptotic gain model is a special case of the [[extra element theorem]].
In [[physics]] and [[chemistry]], <b>charge</b> is fundamentally related to [[field]]s and [[force]]s, and is a property of pieces of matter that leads to forces between spatially separate pieces of matter that likewise manifest that particular property. There are a wide variety of such charges. In the [[Standard Model]], there are three types of charge: ''color'', ''weak isospin'' and ''weak hypercharge''.<ref name=Donoghue/> These include the [[electric charge]] underlying [[electric current]] that enters [[Maxwell's equations]] for the [[Electromagnetism|electromagnetic field]]. In addition, there is [[mass]] that enters [[gravitation]].<ref name=Burgess/>


==Definition of terms==
These charges are ''conserved'' quantities and are related to currents describing their ''flux'' or motion. The ''conservation law'' relating the charge to its current is of the form:
As follows directly from limiting cases of the gain expression, the asymptotic gain ''G<sub>∞</sub>'' is simply the gain of the system when the return ratio approaches infinity:
:<math>G_{\infty} = G\  \Big |_{T \rightarrow \infty}\ , </math>


while the direct transmission term ''G<sub>0</sub>'' is the gain of the system when the return ratio is zero:
:<math>\text{div} \mathbf J + \frac{\partial}{\partial t} \rho =0 \ , </math>
:<math>G_{0} = G\ \Big |_{T \rightarrow 0}\ .</math>


==Advantages==
where ''div'' is the [[vector]] [[Divergence|divergence operator]], '''''J''''' is the vector current density, and ''&rho;'' is the charge density. For a volume enclosed by a surface, this equation can be expressed  by the statement that any change in the charge contained inside the closed surface is due to a current of said charge either entering or exiting through that surface.  
*This model is useful because it completely characterizes feedback amplifiers, including loading effects and the [[Electronic amplifier#Unilateral or bilateral|bilateral]] properties of amplifiers and feedback networks.
*Often feedback amplifiers are designed such that the return ratio ''T'' is much greater than unity. In this case, and assuming the direct transmission term ''G<sub>0</sub>'' is small (as it often is), the gain ''G'' of the system is approximately equal to the asymptotic gain ''G<sub>∞</sub>''.
*The asymptotic gain is (usually) only a function of passive elements in a circuit, and can often be found by inspection.
*The feedback topology (series-series, series-shunt, etc.) need not be identified beforehand as the analysis is the same in all cases.


==Implementation==
Such conservation laws are examples of [[Noether's theorem]], which states that every symmetry of a physical theory is related to a conservation law of this kind. This theorem is closely related to [[Curie's principle]]:
Direct application of the model involves these steps:
::The symmetry of an isolated system cannot decrease as the system evolves with time.<ref name=phase/>  
# Select a dependent source in the circuit.
# Find the [[return ratio]] for that source.
# Find the gain ''G<sub>∞</sub>'' directly from the circuit by replacing the circuit with one corresponding to ''T'' = ∞.
# Find the gain '' G<sub>0</sub>'' directly from the circuit by replacing the circuit with one corresponding to ''T'' = 0.
# Substitute the values for ''T, G<sub>∞</sub>'' and '' G<sub>0</sub>'' into the asymptotic gain formula.


These steps can be implemented directly in [[SPICE]] using the small-signal circuit of hand analysis. In this approach  the dependent sources of the devices are readily accessed. In contrast, for experimental measurements using real devices or SPICE simulations using numerically generated device models with inaccessible dependent sources, evaluating the return ratio requires [[Return_ratio#Other_Methods|special methods]].
The best known of these conservation laws are the conservation of momentum (the current is momentum density, the charge is mass density), related to translational symmetry of the laws of mechanics, conservation of angular momentum, related to the rotational symmetry of the laws of mechanics, and conservation of energy, related to the independence of the laws of mechanics from time translations. Such symmetries are intuitive for point particle mechanics, but for the physics of general fields some symmetries are quite non-intuitive.


==Connection with classical feedback theory==
A formal description of Noether's theorem as related to charge is that a current ''j<sup>a</sup>'' = (''j<sup>a</sup><sub>0</sub>'', '''''j<sup>a</sup>''''') satisfying:
Classical [[Negative_feedback_amplifier#Classical_model|feedback theory]] neglects feedforward (''G<sub>0</sub>''). If feedforward is dropped, the gain from the asymptotic gain model becomes
:<math>\partial^\mu j_u^a = \partial_t j_0^a + \nabla \cdot \mathbf{j}^a = 0 \ , </math>
which implies the conservation of the charge ''Q'' defined by:
:<math>Q = \int \ d^3x j_0^a \ , </math>
is a natural consequence of the ''j<sup>a</sup>'' being generators of a Lie group that is a symmetry group of the physical system.<ref name=Byers/>


::<math>G = G_{\infin} \frac {T} {1+T}  </math>
==Charge and exchange forces==
:::<math>=\frac {G_{\infin}T}{1+\frac{1} {G_{\infin}} G_{\infin} T} \ . </math>
Forces between particles are mediated by exchange of shared properties. For example, two nucleons in the same state of motion can exchange electric charge, producing an ''exchange force''. The Yukawa theory of nuclear force posited that nucleons (protons ''p'' and neutrons ''n'') could exchange electric charge by trading pions according to the reactions:<ref name= Arnikar/>


while in classical feedback theory, in terms of the open loop gain ''A'', the gain with feedback (closed loop gain) is:
:<math>n \Leftrightarrow p + \pi^-; \  p \Leftrightarrow n + \pi^+\ , </math>


::<math>A_{FB} = \frac {A} {1 + { \beta}_{FB} A} \ , </math>
and forces between like particles could be introduced by exchange of zero-charge pions:


Comparison of the two expressions indicates the feedback factor β<sub>FB</sub> is:
:<math> p \Leftrightarrow p + \pi^0; \ n \Leftrightarrow n + \pi^0 \ . </math>  


::<math> \beta_{FB} = \frac {1} {G_{\infin}} \ , </math>
These reactions do not conserve mass or energy, they are ''virtual reactions''. One common (although not universally accepted) "explanation" why violation is permissible is that such reactions occur very rapidly, and for very short times the energy uncertainty relation allows violation of these conservation rules.


while the open-loop gain is:
Besides electric charge, other properties can be exchanged, such as spin (Bartlett exchange), or position (Majorana exchange).


::<math> A  = G_{\infin} \ T \ . </math>
The swapping of shared properties is a symmetry operation, the exchange of identical particles, and as such is related to conserved quantities ''via'' Noether's theorem. For example, the nucleon can be thought of as a two-state particle with an ''isospin'' that is +1/2 for a neutron and −1/2 for a proton, so the change of one to the other is an isospin exchange, and symmetry of a theory under isospin exchange indicates the theory conserves isospin.<ref name=Neuenschwander/> In a quantized version of such a theory, isospin exchange could be moderated by the pion reactions above.


If the accuracy is adequate (usually it is), these formulas suggest an alternative evaluation of ''T'': evaluate the open-loop gain and ''G<sub>∞</sub>'' and use these expressions to find ''T''. Often these two evaluations are easier than evaluation of ''T'' directly.
Only if isospin symmetry in the theory can be produced by a continuous transformation (one depending upon some continuously variable parameter), does it lead to an isospin current conservation law.


==Examples==
==Electrodynamics==
The steps in deriving the gain using the asymptotic gain formula are outlined below for two negative feedback amplifiers. The single transistor example shows how the method works in principle for a transconductance amplifier, while the second two-transistor example shows the approach to more complex cases using a current amplifier.
In electrodynamics, two types of charge are known, ''magnetic'' and ''electric''. The distinguishing property of [[electric charge]] is that electric charges can be isolated, while while an isolated magnetic charge or [[magnetic monopole]] never has been observed.<ref name=Giancoli/> Electric charges interact with magnetic charges only when in relative motion one to the other.  


===Single-stage transistor amplifier===
The conservation of electric charge follows directly from Maxwell's equations. It also can be derived from Noether's theorem as a result of a ''gauge invariance'' of Maxwell's theory when that theory is expressed in terms of a [[Vacuum_(quantum_electrodynamic)#Quantization_of_the_fields|vector potential]]. Although this approach has continuity with much of modern field theory, it is somewhat unintuitive, as the "symmetry" of the recast Maxwell equations is simply due to introduction of a mathematical device that adds an unnecessary degree of freedom into the formulation thereby introducing this symmetry artificially.<ref name=Stamatescu/> Below is a digression on this topic.
{{Image|MOSFET Transresistance amplifier.PNG|right|200px| MOSFET transresistance feedback amplifier.}}
:The basic electric field '''''E''''' and magnetic field '''''B''''' of Maxwell's equations can be replaced by introduction of a ''scalar potential'' ''&phi;'' and a ''vector potential'' '''''A''''' using the relations:
{{Image|Small-signal transresistance amplifier.PNG|right|250px| Small-signal circuit for transresistance amplifier; the feedback resistor ''R<sub>f</sub>'' is placed below the amplifier to resemble the standard topology.}}
:::<math>\boldsymbol E = -\nabla \phi -\frac{\partial }{\partial t} \boldsymbol A , </math>
Consider the simple [[MOSFET]] feedback amplifier at right. The aim is to find the low-frequency, open-circuit, transresistance gain of this circuit ''G'' = {{nowrap|''v<sub>out</sub>'' / ''i <sub>in</sub>''}} using the asymptotic gain model.
:::<math>\boldsymbol B = -\nabla \times \boldsymbol A . </math>
:Although the potentials uniquely determine the fields, the reverse is not true. Different potentials produce the same fields; in particular the potentials denoted by primes below produce the same fields:
:::<math>\phi' = \phi +\frac{\partial }{\partial t} \Gamma , </math>
:::<math>\boldsymbol A' = \boldsymbol A  -\nabla \Gamma  . </math>
:Here ''&Gamma; = &Gamma;('''r''', t)'' is any continuous function of the space-time coordinates '''''r''''', ''t''. Consequently, a theory based upon potentials instead of fields has the additional symmetry that it is unchanged by substitution of primed potentials instead of the original potentials. This change of potentials from unprimed to primed is called a ''gauge transformation'' and this new symmetry leads directly to the continuity equation for electric charge:
:::<math>\nabla \boldsymbol  \cdot J + \frac{\partial }{\partial t} \rho = 0 \ .</math>
:This equation is a direct consequence of the Maxwell equations defining charge and current densities (in Heaviside-Lorentz units):
:::<math>\nabla \cdot \boldsymbol E = \rho \ ,  </math>
:::<math>\nabla \times \boldsymbol B -\frac{\partial }{\partial t}\boldsymbol E = \boldsymbol J \ . </math>
:However, using the potential formulation, the continuity equation is required if the theory is to be gauge invariant,<ref name=Cottingham/> and this requirement is consistent with Noether's theorem.


Under the circuit with the transistor is the [[small-signal]] equivalent circuit, where the transistor is replaced by its [[hybrid-pi model]].
In a [[Quantization of the electromagnetic field|quantized theory]] based upon the potential formulation of Maxwell's equations, the electrical force between charged particles is an exchange force mediated by trading charge-neutral [[photon]]s. The electromagnetic potentials exist as vibrations with certain allowed amplitudes determined by the number of photons employed, and field amplitudes are increased or decreased by adding or subtracting photons. Thus, the force exerted upon a charged particle as determined by the field it experiences, depends upon the number of photons in the corresponding potentials.


====Return ratio====
==Weak forces==
{{Image|Return ratio.PNG|right|300px|Small-signal circuit with return path broken and test current ''i<sub>t</sub>'' driving amplifier at the break.}}
Weak forces are mediated by the electric charged ''W<sup>+</sup>'' and ''W<sup>−</sup>'' particles and the electric charge neutral ''Z<sup>0</sup>'' particle, all with spin 1. The weak interaction is of short range, being effective over a distance of approximately 10<sup>−3</sup> fm. Analysis of the weak force parallels that of the electromagnetic force, apart from the huge mass of the exchanged particles compared to the photon. The "weak force" charge introduced that couples to this force is called ''flavor''.<ref name=Tipler/> It is customaryu to refer to lepton flavor, rather than lepton charge, and individual lepton flavors are attributed to each family: electron flavor ''L<sub>e</sub>'' for electrons; muon flavor ''L<sub>&mu;</sub>'' for muons; tau flavor ''L<sub>&tau;</sub>'' for taus.<ref name=Boyarkin/> p. 38
It is most straightforward to begin by finding the return ratio ''T'', because ''G<sub>0</sub>'' and ''G<sub>∞</sub>'' are defined as limiting forms of the gain as ''T'' tends to either zero or infinity. To take these limits, it is necessary to know what  parameters ''T'' depends upon. There is only one dependent source in this circuit, so as a starting point the return ratio related to this source is determined as outlined in the article on [[return ratio]].
However, the terminology is somewhat confused. Some authors do refer to both "weak charge" and to "lepton flavor".<ref name=Nagashima/> The issue may be that quarks and leptons behave differently under the weak force?


The [[return ratio]] is found using the figure. In the figure, the input current source is set to zero, By cutting the dependent source out of the output side of the circuit, and short-circuiting its terminals, the output side of the circuit is isolated from the input and the feedback loop is broken. A test current ''i<sub>t</sub>'' replaces the dependent source. Then the return current generated in the dependent source by the test current is found. The return ratio is then ''T'' ={{nowrap| −''i<sub>r</sub> / i<sub>t</sub>''.}} Using this method, and noticing that ''R<sub>D</sub>'' is in parallel with ''r<sub>O</sub>'', ''T'' is determined as:
==Nuclear forces==
:<math>T = g_m \left( R_D\ ||r_O \right) \approx g_m R_D \ , </math>
In 1935 [http://www.nobelprize.org/nobel_prizes/physics/laureates/1949/yukawa-bio.html Yukawa] invented the [[meson]] theory for explaining the forces holding atomic nucleii together, an assemblage of neutrons and protons. The theory led to the experimental observation of the [[pion]] or &pi;-meson and the [[muon]] or &mu;-meson. The behavior of nuclear forces was explained as an exchange of mesons. Today, mesons are considered to be quark-antiquark pairs, and a more refined theory of nuclear interactions is based upon [[quantum chromodynamics]]. Nuclear forces are not considered fundamental today, but are a consequence of the underlying ''strong forces'' between quarks, also called chromodynamic forces or color forces. On that basis, nuclear forces are an exchange force fundamentally based upon color, and only approximated by the Yukawa theory.
where the approximation is accurate in the common case where ''r<sub>O</sub>'' >> ''R<sub>D</sub>''. With this relationship it is clear that the limits ''T'' → 0, or ∞ are realized if we let [[transconductance]] ''g<sub>m</sub>'' → 0, or ∞.<ref name=note1/>


====Asymptotic gain====
==Chromodynamics==
Finding the asymptotic gain ''G<sub>∞</sub>'' provides insight, and usually can be done by inspection.  To find ''G<sub>∞</sub>'' we let ''g<sub>m</sub>'' → ∞ and find the resulting gain. The drain current, ''i<sub>D</sub>'' = ''g<sub>m</sub>'' ''v<sub>GS</sub>'', must be finite. Hence, as ''g<sub>m</sub>'' approaches infinity, ''v<sub>GS</sub>'' also must approach zero. As the source is grounded, ''v<sub>GS</sub>'' = 0 implies ''v<sub>G</sub>'' = 0 as well.<ref name=note2/> With ''v<sub>G</sub>'' = 0 and the fact that all the input current flows through ''R<sub>f</sub>'' (as the FET has an infinite input impedance), the output voltage is simply −''i<sub>in</sub> R<sub>f</sub>''. Hence
In the [[Standard Model]] of particle physics, [[quantum chromodynamics]] describes the ''strong force'', also called the ''color force'' or ''chromo force'', and relates it to the '''color charge''' as a property of [[quark]]s and [[gluon]]s.<ref name=Webb/> Similar to magnetic charge, ''color'' is not seen directly, as all observable particles have no overall color.<ref name=Watson/> As with electric and magnetic charge, color charge can be multiple valued, conventionally called ''red, green'' or ''blue''. Color charge is not assigned a numerical value; however, a superposition in equal amounts of all three colors leads to a "neutral" color charge, a somewhat stretched analogy with the superposition of red, green and blue light to produce white light.<ref name=Han/> Thus, protons and neutrons, which consist of three quarks with all three colors are color-charge neutral. Quark combinations are held together by exchange of combinations of eight different [[gluon]]s that also are color charged.<ref name=Rosen/><ref name=Gothard/><ref name=Greenberger/><ref name=Greenberg/>


:<math>G_{\infty} = \frac{v_{out}}{i_{in}} = -R_f\ .</math>
The color charges of ''anti''quarks are ''anti''colors. The combination of a quark and an antiquark to form a [[meson]], such as a [[pion]], [[kaon]] and so forth, leads to a neutral color charge.


Alternatively ''G<sub>∞</sub>'' is the gain found by replacing the transistor by an ideal amplifier with infinite gain - a [[nullor]].<ref name=Verhoeven/>
==Other charges==
The charges above are related to fields and forces and to a ''local'' (coordinate dependent) Noether's theorem. Other charges are known, however, that are connected to ''global'' or ''discrete'' symmetries (no continuous parametric dependence, such as a coordinate dependence) and so to a global Noether's theorem, and have no relation to forces or fields.


====Direct feedthrough====
One such charge in elementary particle theory is the '''baryonic charge''', ''B'', also referred to as a ''number'', with value +1 for all baryons (notably, neutrons and protons, but also others like the &Lambda; and &Sigma; particles) and −1 for all ''anti''baryons and zero for non-baryons. Quarks are an exception, and have a baryon number of 1/3. Unlike electric charge, which serves as a source for the [[Maxwell's equations|electromagnetic field]], baryon charge is not related to an associated "baryonic" field.<ref name=Boyarkin/>
To find the direct feedthrough ''G<sub>0</sub>'' we simply let ''g<sub>m</sub>'' → 0 and compute the resulting gain. The currents through ''R<sub>f</sub>'' and the parallel combination of ''R<sub>D</sub>'' || ''r<sub>O</sub>'' must therefore be the same and equal to ''i<sub>in</sub>''. The output voltage is therefore ''i<sub>in</sub> (R<sub>D</sub> || r<sub>O</sub>)''.


Hence
Finally, we mention the '''leptonic charge''' (also called lepton ''number'') carried by [[lepton]]s: electrons, muons, taus, and their associated [[neutrino]]s.<ref name=Han/> Lepton charge depends upon the ''flavor'' of the lepton<ref name=Tipler/> ''L<sub>e</sub>, L<sub>&mu;</sub>, L<sub>&tau;</sub>'' with values +1 for the electron, muon and tau meson, and −1 for their antiparticles.<ref name=Boyarkin/> The ''total'' lepton number ''L'' of a complex is:
:<math>G_0 = \frac{v_{out}}{i_{in}} = R_D\|r_O \approx R_D \ ,</math>
:<math>L=\sum_{j=e,\mu,\tau} L_j \ . </math>


where the approximation is accurate in the common case where ''r<sub>O</sub>'' >> ''R<sub>D</sub>''.
Non-leptons have a total lepton number ''L'' of zero. Within the Standard Model, lepton number is conserved for strong and electromagnetic interactions; however, it is not necessarily conserved in weak particle reactions.<ref name=Boyarkin/><ref name=Quinn/>


====Overall gain====
==References==
The overall [[Electronic_amplifier#Input_and_output_variables|transresistance gain]] of this amplifier is therefore:
{{reflist|refs=
<ref name= Arnikar>
{{cite book |title=Essentials of nuclear chemistry |author=Hari Jeevan Arnikar |url=http://books.google.com/books?id=C88GMy-p0AwC&pg=PA16 |pages=p. 16 ''ff'' |isbn=8122407129 |year=1995 |publisher=New Age International |edition=4th ed}}


:<math>G = \frac{v_{out}}{i_{in}} = -R_f \frac {g_m R_D}{1+g_m R_D} + R_D \frac{1}{1+g_m R_D} \ .</math>
</ref>


Examining this equation, it appears to be advantageous to make ''R<sub>D</sub>'' large in order make the overall gain approach the asymptotic gain, which makes the gain insensitive to amplifier parameters (''g<sub>m</sub>'' and ''R<sub>D</sub>''). In addition, a large first term reduces the importance of the direct feedthrough factor, which degrades the amplifier. One way to increase ''R<sub>D</sub>'' is to replace this resistor by an [[active load]], for example, a [[current mirror]].
<ref name=Boyarkin>
{{cite book |title=Introduction to Physics of Elementary Particles |author=O. M. Boyarkin|editor=O. M. Boyarkin, Alfred L. Heinzerton, eds |url=http://books.google.com/books?id=WFDs_SJgILQC&pg=PA37 |chapter=Chapter 3: Leptons and hadrons |pages=pp. 37-40 |isbn=160021200X |year=2007 |publisher=Nova Publishers}}


===Two-stage transistor amplifier===
</ref>
{{Image|Shunt-series feedback amplifier.PNG|right|200px|Two-transistor feedback amplifier; any source impedance ''R<sub>S</sub>'' is lumped in with the base resistor ''R<sub>B</sub>''.}}
{{Image|Using return ratio.PNG|right|350px|Three small-signal schematics used to discuss the asymptotic gain model; parameter α <nowiki>=</nowiki> β / ( β+1 ); resistor R<sub>C</sub> <nowiki>=</nowiki> R<sub>C1</sub>.}}
A figure at left shows a two-transistor amplifier with a feedback resistor ''R<sub>f</sub>''. This amplifier is often referred to as a ''shunt-series feedback'' amplifier, and analyzed on the basis that resistor ''R<sub>2</sub>'' is in series with the output and samples output current, while ''R<sub>f</sub>'' is in shunt (parallel) with the input and subtracts from the input current. See the article on [[Negative_feedback_amplifier#Two-port_analysis_of_feedback|negative feedback amplifier]] and references by Meyer or Sedra.<ref name=Gray-Meyer2/><ref name=Sedra1/> That is, the amplifier uses current feedback. It frequently is ambiguous just what type of feedback is involved in an amplifier, and the asymptotic gain approach has the advantage/disadvantage that it works whether or not you understand the circuit.


Although the figure indicates the output node, it does not indicate the choice of output variable. In what follows, the output variable is selected as the short-circuit current of the amplifier, that is, the collector current of the output transistor. Other choices for output are discussed later.
<ref name=Byers>
{{cite web |title=The Life and Times of Emmy Noether: Contributions of Emmy Noether to Particle Physics |url=http://www.physics.ucla.edu/~cwp/articles/9411110.pdf |author=Nina Byers |year=1994 |publisher=UCLA/94/TEP/42; hep-th/9411110}} Presented at the International Conference on THE HISTORY OF ORIGINAL IDEAS AND
BASIC DISCOVERIES IN PARTICLE PHYSICS, Erice, Italy, 29 July - 4 August 1994.
</ref>


To implement the asymptotic gain model, the dependent source associated with either transistor can be used. Here the first transistor is chosen.
<ref name=Burgess>
{{cite book |title=Classical covariant fields |author=Mark Burgess |url=http://books.google.com/books?id=8k3NY53iMgsC&pg=PA325 |pages=pp. 325 ''ff'' |chapter=Chapter 12: Charge and current |isbn= 0521813638 |year=2004 |publisher=Cambridge University Press}}
</ref>


====Return ratio====
<ref name=Cottingham>
The circuit to determine the return ratio is shown in the top panel of the arrangement of three small-signal circuits. Labels show the currents in the various branches as found using a combination of [[Ohm's law]] and [[Kirchhoff's laws]]. Resistor ''R<sub>1</sub> = R<sub>B</sub> // r<sub>π1</sub>'' and ''R<sub>3</sub> = R<sub>C2</sub> // R<sub>L</sub>''. KVL from the ground of ''R<sub>1</sub>'' to the ground of ''R<sub>2</sub>'' provides:
{{cite book |title=An introduction to the Standard Model of particle physics |author=WN Cottingham, DA Greenwood |isbn=978-0-521-85249-4 |year=2007 |edition=2nd ed |publisher=Cambridge University Press |chapter=Gauge transformations |pages=p. 41 |url=http://books.google.com/books?id=Dm36BYq9iu0C&pg=PA41}}
</ref>


:<math> i_B = -v_{ \pi} \frac {1+R_2/R_1+R_f/R_1} {(\beta +1) R_2} \ . </math>
<ref name=Donoghue>
{{cite book |title=Dynamics of the standard model |author=John F. Donoghue, Eugene Golowich, Barry R. Holstein |url=http://books.google.com/books?id=hFasRlkBbpYC&pg=PA24 |pages=p. 24 |isbn= 0521476526 |year=1994 |publisher=Cambridge University Press}}
</ref>  


KVL provides the collector voltage at the top of ''R<sub>C</sub>'' as
<ref name=Giancoli>


:<math>v_C = v_{ \pi} \left(1+ \frac {R_f} {R_1} \right ) -i_B r_{ \pi 2} \ . </math>
{{cite book |title=Physics for scientists and engineers with modern physics |author=Douglas C. Giancoli |url=http://books.google.com/books?id=xz-UEdtRmzkC&pg=PA708 |pages=p. 708 |isbn=0132273594 |publisher=Pearson Education |edition=4rth ed}}
</ref>


Finally, KCL at this collector provides
<ref name=Gothard>


:<math> i_T = i_B - \frac {v_C} {R_{C}} \ . </math>
{{cite book |title=Encyclopedia of Physical Science, Volume 1 |author=Joe Rosen, Lisa Quinn Gothard |url=http://books.google.com/books?id=avyQ64LIJa0C&pg=PA278 |pages=p. 278 |isbn=0816070113 |year=2009 |publisher=Infobase Publishing}}


Substituting the first equation into the second and the second into the third, the return ratio is found as
</ref>


:<math>T = - \frac {i_R} {i_T} = -g_m \frac {v_{ \pi} }{i_T} </math>
<ref name=Greenberg>
:::<math> = \frac {g_m R_C} { \left( 1 + \frac {R_f} {R_1} \right) \left( 1+ \frac {R_C+r_{ \pi 2}}{( \beta +1)R_2} \right) +\frac {R_C+r_{ \pi 2}}{(\beta +1)R_1} } \ . </math>
{{cite journal |title=The color charge degree of freedom in particle physics |author=OW Greenberg |url=http://arxiv.org/abs/0805.0289v2  |year=2008 }} Chapter in ''Greenberger et al.'' below.


====Gain ''G<sub>0</sub>'' with ''T'' = 0 ====
</ref>
The circuit to determine ''G<sub>0</sub>'' is shown in the center panel of the small-circuit array. In the center panel, the output variable is  the output current β''i<sub>B</sub>'' (the short-circuit load current), which leads to the short-circuit current gain of the amplifier, namely β''i<sub>B</sub>'' / ''i''<sub>S</sub>:


::<math> G_0 = \frac { \beta i_B} {i_S} \ . </math>
<ref name=Greenberger>


Using [[Ohm's law]], the voltage at the top of ''R<sub>1</sub>'' is found as
{{cite book |title=Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy |chapter=Quantum chromodynamics (QCD)  |editor=Daniel M. Greenberger, Klaus Hentschel, Friedel Weinert |url=http://books.google.com/books?id=ekyAV8VtfuYC&pg=PA525 |pages=pp. 524 ''ff'' |isbn=3540706224 |year=2009 |publisher=Springer}}


::<math> ( i_S - i_R ) R_1 = i_R R_f +v_E \ \ ,</math>
</ref>


or, rearranging terms,
<ref name=Han>


::<math> i_S = i_R \left( 1 + \frac {R_f}{R_1} \right) +\frac {v_E} {R_1} \ . </math>
{{cite book |title=Quarks and gluons: a century of particle charges |author=M. Y. Han |url=http://books.google.com/books?id=LBb3z_-qPFoC&pg=PA116 |pages=p. 116 |isbn=9810237456 |publisher=World Scientific |year=1999}}


Using KCL at the top of ''R<sub>2</sub>'':
</ref>


::<math> i_R = \frac {v_E} {R_2} + ( \beta +1 ) i_B \ . </math>
<ref name=Nagashima>
{{cite book |author=Yorikiyo Nagashima, Yoichiro Nambu |url=http://books.google.com/books?id=J0l8s3pdOksC&pg=PA554 |pages=p. 554 |title=Elementary Particle Physics: Volume 1: Quantum Field Theory and Particles, Volume 1 |publisher=Wiley-VCH |year=2010 |isbn=3527409629}}  
</ref>


Emitter voltage ''v<sub>E</sub>'' already is known in terms of ''i<sub>B</sub>'' from the diagram in the center panel. Substituting the second equation in the first, ''i<sub>B</sub>'' is determined in terms of ''i<sub>S</sub>'' alone, and ''G<sub>0</sub>'' becomes:
<ref name=Neuenschwander>
{{cite book |title=Emmy Noether's Wonderful Theorem |url=http://books.google.com/books?id=QeZaVQWRnuQC&pg=PA192 |pages=p. 192 ''ff'' |chapter=§9.1 Conservation of properties and unitary transformations |author=Dwight E. Neuenschwander |isbn=0801896940 |year=2010 |publisher=The Johns Hopkins University Press}}
</ref>


::<math>G_0 = \frac { \beta  } {
<ref name=phase>
  ( \beta +1) \left( 1 + \frac{R_f}{R_1} \right ) +(r_{ \pi 2} +R_C ) \left[ \frac {1} {R_1} + \frac {1} {R_2} \left( 1 + \frac {R_f} {R_1} \right ) \right]
Some care is needed in looking at this principle because of the phenomenon of [[spontaneous symmetry breaking]]. For example, as a cubic ferroelectric material like BaTiO<sub>3</sub> is cooled below its Curie point, its cubic symmetry is replaced by a tetragonal ferroelectric symmetry as the frequency corresponding to a tetragonal elastic distortion tends to zero ([[Goldstone's theorem]]). The overall cubic symmetry of the crystal is retained because the crystal breaks into finite domains, each with a differently oriented tetragonal axis, so that ''statistically'' the symmetry of an infinite crystal still is cubic. For a general discussion, see {{cite book |title=Symmetry breaking |author=F. Strocchi |url=http://books.google.com/books?id=tPOcCWjvUK8C&printsec=frontcover |isbn=3540735925 |year=2008 |edition=2nd ed |publisher=Springer}}
} </math>
</ref>


Gain ''G<sub>0</sub>'' represents feedforward through the feedback network, and commonly is negligible.
<ref name=Quinn>
 
{{cite book |title=The Mystery of the Missing Antimatter |author=Helen R. Quinn, Yossi Nir |url=http://books.google.com/books?id=W_E2rAui1l8C&pg=PA130 |pages=p. 130 |chapter=Chapter 12: Baryon and Lepton number conservation? |publisher=Princeton University Press |year=2010 |isbn=1400835712}}
====Gain ''G<sub>&infin;</sub>'' with ''T'' &rarr; &infin;====
 
The circuit to determine ''G<sub>∞</sub>'' is shown in the bottom panel of the small-signal circuits. The introduction of the ideal op amp (a [[nullor]]) in this circuit is explained as follows. When ''T ''→  ∞, the gain of the amplifier goes to infinity as well, and in such a case the differential voltage driving the amplifier (the voltage across the input transistor ''r<sub>π1</sub>'') is driven to zero and (according to Ohm's law when there is no voltage) it draws no input current. On the other hand the output current and output voltage are whatever the circuit demands. This behavior is like a nullor, so a nullor can be introduced to represent the infinite gain transistor.
</ref>
 
The current gain is read directly off the schematic:
 
::<math> G_{ \infty } = \frac { \beta i_B } {i_S} =  \left( \frac {\beta} {\beta +1} \right)  \left( 1 + \frac {R_f} {R_2} \right) \ . </math>
 
====Comparison with classical feedback theory====
Using the classical model, the feed-forward is neglected and the feedback factor β<sub>FB</sub> is (assuming transistor β >> 1):


::<math> \beta_{FB} = \frac {1} {G_{\infin}} \approx  \frac {1} {(1+ \frac {R_f}{R_2} )} = \frac {R_2} {(R_f + R_2)} \ , </math>
<ref name=Rosen>


and the open-loop gain ''A'' is:
{{cite book |title=Encyclopedia of physics |author=Joe Rosen |url=http://books.google.com/books?id=HQWNJyRV6kMC&pg=PA85 |pages=p. 85 |isbn=0816049742 |publisher=Infobase Publishing |year=2004}}


::<math>A = G_{\infin}T \approx  \frac {\left( 1+\frac {R_f}{R_2} \right) g_m R_C} { \left( 1 + \frac {R_f} {R_1} \right) \left( 1+ \frac {R_C+r_{ \pi 2}}{( \beta +1)R_2} \right) +\frac {R_C+r_{ \pi 2}}{(\beta +1)R_1} }  \ . </math>
====Overall gain====
The above expressions  can be substituted into the asymptotic gain model equation to find the overall gain ''G''. The resulting gain is the ''current'' gain of the amplifier with a short-circuit load.
=====Gain using alternative output variables=====
In the figure showing the amplifier with transistors, ''R<sub>L</sub>'' and ''R<sub>C2</sub>'' are in parallel.
To obtain the transresistance gain, say ''A''<sub>ρ</sub>, that is, the gain using voltage as output variable, the short-circuit current gain ''G'' is multiplied by ''R<sub>C2</sub> // R<sub>L</sub>'' in accordance with [[Ohm's law]]:
::<math> A_{ \rho} = G \left( R_{C2} // R_{L} \right) \ . </math>
The ''open-circuit'' voltage gain is found from ''A''<sub>ρ</sub> by setting ''R''<sub>L</sub> → ∞.
To obtain the current gain when load current ''i<sub>L</sub>'' in load resistor ''R''<sub>L</sub> is the output variable, say ''A''<sub>i</sub>, the formula for [[current division]] is used: ''i<sub>L</sub> = i<sub>out</sub> × R<sub>C2</sub> / ( R<sub>C2</sub> + R<sub>L</sub> )'' and the short-circuit current gain ''G'' is multiplied by this [[Voltage_divider#Loading_effect|loading factor]]:
::<math> A_i = G \left( \frac {R_{C2}} {R_{C2}+ R_{L}} \right) \ . </math>
Of course, the short-circuit current gain is recovered by setting ''R''<sub>L</sub> = 0 Ω.
==References and notes==
{{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
|year= 2001
|edition=Fourth Edition
|publisher=Wiley
|location=New York
|isbn=0-471-32168-0
|url=http://worldcat.org/isbn/0-471-32168-0
|nopp=true
|pages=Figure 8.42 p. 604}}
</ref>
</ref>


<ref name=Gray-Meyer2>
<ref name=Stamatescu>
{{cite book
{{cite book |title=Approaches to fundamental physics: an assessment of current theoretical ideas |author=K.-H. Rehren, E Seiler |editor=Ion-Olimpiu Stamatescu, Erhard Seiler, eds |quote=Gauge symmetry was originally observed within Maxwell's theory of classical electrodynamics as an ambiguity related to the artificial introduction of unobservable potentials in order to solve two of Maxwell's four equations. |url=http://books.google.com/books?id=2Vpa6PxOs9IC&pg=PA401 |pages=p. 401 |isbn=3540711155 |year=2007 |publisher=Springer}}
|author=P R Gray, P J Hurst, S H Lewis, and R G Meyer
|title=Analysis and Design of Analog Integrated Circuits
|year= 2001
|edition=Fourth Edition
|publisher=Wiley
|location=New York
|isbn=0-471-32168-0
|url=http://worldcat.org/isbn/0471321680
|pages=586–587}}
</ref>
</ref>


<ref name=Middlebrook>
<ref name=Tipler>
 
{{cite journal |author=RD Middlebrook |title= Design-oriented analysis of feedback amplifiers|journal=Proc. of National Electronics Conference|volume= Vol. XX |date=Oct. 1964 |pages= pp. 1-4}}


</ref>
{{cite book |title=Physics for scientists and engineers: Elementary modern physics, Volume 3 |author=Paul Allen Tipler |chapter=Summary table |pages=p. 1409 |url=http://books.google.com/books?id=i58oVr_Mbs4C&pg=PA1409 |isbn=1429201347 |edition=6th ed |publisher=Macmillan |year=2007}}
 


<ref name=note1>
Although changing ''R<sub>D</sub> // r<sub>O</sub>'' also could force the return ratio limits, these resistor values affect other aspects of the circuit as well. It is the ''control parameter'' of the dependent source that must be varied because it affects ''only'' the dependent source.
</ref>
</ref>


<ref name=note2>
<ref name=Watson>
Because the input voltage ''v<sub>GS</sub>'' approaches zero as the return ratio gets larger, the amplifier input impedance also tends to zero, which means in turn (because of [[current division]]) that the amplifier works best if the input signal is a current. If a Norton source is used, rather than an ideal current source, the formal equations derived for ''T'' will be the same as for a Thévenin voltage source.  Note that in the case of input current, ''G<sub>∞</sub>'' is a [[Electronic amplifier#Input and output variables|transresistance]] gain.
</ref>


<ref name=Palumbo>
{{cite book |title=The quantum quark |author=Andrew Watson |url=http://books.google.com/books?id=ip50x8IOfnEC&pg=PA170 |pages=pp. 170 ''ff'' |isbn=0521829070 |year=2004 |publisher=Cambridge University Press}}
{{cite book
|author=Palumbo, Gaetano & Salvatore Pennisi
|title=Feedback amplifiers: theory and design
|year= 2002
|publisher=Kluwer Academic
|location=Boston/Dordrecht/London
|isbn=0792376439
|url=http://worldcat.org/isbn/0792376439
|pages=§3.3 pp. 69–72}}
</ref>


<ref name=Rosenstark>
{{cite book
|author=Rosenstark, Sol
|title=Feedback amplifier principles
|page=15
|year= 1986
|publisher=Collier Macmillan
|location=NY
|isbn=0029478103
|url=http://worldcat.org/isbn/0029478103}}
</ref>
</ref>


<ref name=Sedra1>
<ref name=Webb>
{{cite book
|author=A. S. Sedra and K.C. Smith
|title=Microelectronic Circuits
|year= 2004
|edition=Fifth Edition
|pages=Example 8.4, pp. 825–829 and PSpice simulation pp. 855–859
|publisher=Oxford
|location=New York
|isbn=0-19-514251-9
|url=http://worldcat.org/isbn/0-19-514251-9
|nopp=true}}
</ref>


<ref name=Verhoeven>
{{cite book |title=Out of this world: colliding universes, branes, strings, and other wild ideas of modern physics |url=http://books.google.com/books?id=3AJdTYu3m5sC&pg=PA190 |pages=p. 190 |isbn=0387029303 |author=Stephen Webb |publisher=Springer |year=2004}}
{{cite book
|author=Verhoeven C J M van Staveren A Monna G L E Kouwenhoven M H L & Yildiz E
|title=Structured electronic design: negative feedback amplifiers
|year= 2003
|publisher=Kluwer Academic
|location=Boston/Dordrecht/London
|isbn=1402075901
|pages=§2.3 - §2.5 pp. 34–40
|url=http://books.google.com/books?id=p8wDptzCMrUC&pg=PA24&dq=isbn=1402075901&sig=cxJIK6hgY7wKfWc7cV6ZVHT-iDc#PPA35,M1}}
</ref>
</ref>
}}


}}
==miscellaneous==
*[http://www.amazon.com/Quantum-Mathematicians-Encyclopedia-Mathematics-Applications/dp/052163265X#reader_052163265X Noether's theorem]
*[http://books.google.com/books?id=htJbAf7xA_oC&pg=PA278&dq=weak+isospin&hl=en&ei=lc5kTrC_BungiAKu9YioCg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDIQ6AEwAQ#v=onepage&q=weak%20isospin&f=false essay on charge]
*[http://books.google.com/books?id=AwhkM6hVj-wC&pg=PA6&dq=this+invariance+%22Quantum+electrodynamics+possesses%22&hl=en&ei=8vRbTrCrNKnRiAKg86TvDg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCoQ6AEwAA#v=onepage&q=this%20invariance%20%22Quantum%20electrodynamics%20possesses%22&f=false gauge theory]
*[http://books.google.com/books?id=cDJw3dWjw_UC&pg=PA208&dq=photon+exchange+force+Noether&hl=en&ei=C_JbTvuWIOfSiAKd27TtDg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDIQ6AEwATgK#v=onepage&q&f=false symmetry breaking]
*[http://books.google.com/books?id=YPz2KsNBrz4C&pg=PA39&dq=photon+exchange+force+Noether&hl=en&ei=vu9bTsKDDarQiAK4kaDbDg&sa=X&oi=book_result&ct=result&resnum=10&ved=0CF8Q6AEwCQ#v=onepage&q&f=false gauge theory]
*[http://www.physics.wustl.edu/~alford/p551/noether.pdf derivation of Noether from change in lagrangian]
*[http://books.google.com/books?id=0AroLuyXAX0C&pg=PA11&dq=photon+exchange+force+Noether&hl=en&ei=vu9bTsKDDarQiAK4kaDbDg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CFoQ6AEwCA#v=onepage&q=photon%20exchange%20force%20Noether&f=false Lagrangian is a function of the symmetry current]
*[http://www.amazon.com/Theoretical-Nuclear-Subnuclear-Physi-Walecka/dp/9812388982#reader_9812388982 Lagrangian of QCD and Noether's theorem]
*[http://books.google.com/books?id=ip50x8IOfnEC&pg=PA58&dq=photon+exchange+force&hl=en&ei=--lbTpzVAaXQiAKI2InEDg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEsQ6AEwBg#v=onepage&q=photon%20exchange%20force&f=false exchange forces]
*[http://books.google.com/books?id=0QyQC9cvhtMC&pg=PA66&dq=photon+exchange+force&hl=en&ei=--lbTpzVAaXQiAKI2InEDg&sa=X&oi=book_result&ct=result&resnum=8&ved=0CFAQ6AEwBw#v=onepage&q=photon%20exchange%20force&f=false derive Coulomb's law as an exchange]
*[http://books.google.com/books?id=6-7TE5N0vbIC&pg=PA23&dq=lepton+flavor+charge&hl=en&ei=En1ZTtroO_PSiAK6rtjDCQ&sa=X&oi=book_result&ct=result&resnum=6&ved=0CEgQ6AEwBTgK#v=onepage&q=lepton%20flavor%20charge&f=false flavor not conserved]
*[http://books.google.com/books?id=6-7TE5N0vbIC&pg=PA23&dq=lepton+flavor+charge&hl=en&ei=En1ZTtroO_PSiAK6rtjDCQ&sa=X&oi=book_result&ct=result&resnum=6&ved=0CEgQ6AEwBTgK#v=onepage&q=lepton%20flavor%20charge&f=false three coupling constants: &alpha;, &alpha;<sub>s</sub>, &alpha;<sub>W</sub>]
*[http://books.google.com/books?id=tpU18JqcSNkC&pg=PA656&dq=lepton+flavor+charge&hl=en&ei=En1ZTtroO_PSiAK6rtjDCQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CD0Q6AEwAzgK#v=onepage&q=lepton%20flavor%20charge&f=false weak charge also called flavor charge]
*[http://books.google.com/books?id=wJUDIBstnMQC&pg=PA37&dq=lepton+number&hl=en&ei=Bm9ZTqjZKNHXiAL9s72uCQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDMQ6AEwAQ#v=onepage&q=lepton%20number&f=false lepton and baryon number]
*[http://books.google.com/books?id=w9Dz56myXm8C&pg=PA353&dq=introduces+the+gauge+theories+that+describe&hl=en&ei=H1FZTvfWBMbSiAKv2anJCQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA#v=onepage&q=introduces%20the%20gauge%20theories%20that%20describe&f=false Griffiths: gauge theories]
*[http://books.google.com/books?id=HNcQ_EiuTxcC&pg=PA165&dq=the+phase+is+just+the+gauge+function&hl=en&ei=9U9ZTobpJKjYiALJ0qnPCQ&sa=X&oi=book_result&ct=result&resnum=7&ved=0CEoQ6AEwBg#v=onepage&q=the%20phase%20is%20just%20the%20gauge%20function&f=false the phase is just the gauge function]
*[http://books.google.com/books?id=jPEkmI2TLnkC&pg=PA6&dq=introduces+the+gauge+theories+that+describe&hl=en&ei=0lFZToe2AebViAKfucWkCQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDAQ6AEwATgK#v=onepage&q&f=false Gauge field interactions]
*[http://books.google.com/books?id=IcL31owjU3gC&pg=PA126&dq=conserved+generator+current&hl=en&ei=SB5VTrfMKIXPiALbrN3wDA&sa=X&oi=book_result&ct=result&resnum=9&ved=0CFQQ6AEwCDge#v=onepage&q=conserved%20generator%20current&f=false generators of group induced by a current are charges]
*[http://books.google.com/books?id=IcL31owjU3gC&pg=PA129&dq=conserved+generator+current&hl=en&ei=SB5VTrfMKIXPiALbrN3wDA&sa=X&oi=book_result&ct=result&resnum=9&ved=0CFQQ6AEwCDge#v=onepage&q=conserved generator current&f=false does a conserved current imply a symmetry?] p. 129
*[http://books.google.com/books?id=n8Mmbjtco78C&pg=PA77  Noether current; basics]
*[http://books.google.com/books?id=J0l8s3pdOksC&pg=PA731&dq=conserved+generator+current&hl=en&ei=ER1VTtfZJKHjiAKdpOjFDA&sa=X&oi=book_result&ct=result&resnum=10&ved=0CFgQ6AEwCTgU#v=onepage&q=conserved%20generator%20current&f=false basics again]
*[http://books.google.com/books?id=J0l8s3pdOksC&pg=PA733&dq=conserved+generator+current&hl=en&ei=ER1VTtfZJKHjiAKdpOjFDA&sa=X&oi=book_result&ct=result&resnum=10&ved=0CFgQ6AEwCTgU#v=onepage&q=conserved%20generator%20current&f=false conserved quantities and forces]
*[http://books.google.com/books?id=oQn5ybiQKAoC&pg=PA676&dq=conserved+generator+current&hl=en&ei=SB5VTrfMKIXPiALbrN3wDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwADge#v=onepage&q=conserved%20generator%20current&f=false conservation of current]
*[http://books.google.com/books?id=n8Mmbjtco78C&pg=PA79&dq=conserved+generator+current&hl=en&ei=ZBlVTvG2KcXjiAKchKTyDA&sa=X&oi=book_result&ct=result&resnum=10&ved=0CFkQ6AEwCQ#v=onepage&q=conserved%20generator%20current&f=false Zee]
*[http://books.google.com/books?id=LCInXoalY2wC&pg=PA316&dq=conserved+generator+current&hl=en&ei=ZBlVTvG2KcXjiAKchKTyDA&sa=X&oi=book_result&ct=result&resnum=8&ved=0CE8Q6AEwBw#v=onepage&q=conserved%20generator%20current&f=false Noether current]
*[http://books.google.com/books?id=ZtthVxxc3SkC&pg=PA59&dq=conserved+generator+current&hl=en&ei=ER1VTtfZJKHjiAKdpOjFDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwADgU#v=onepage&q=conserved%20generator%20current&f=false formal math statement]
*[http://books.google.com/books?id=YgkfZgFdui8C&pg=PA188&dq=vector+current+coupling&hl=en&ei=ABhVToDxBuPjiAL_3JXrDA&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDQQ6AEwAg#v=onepage&q=vector%20current%20coupling&f=false conserved vector current]
*[http://books.google.com/books?id=Lxr2S-zjOgYC&pg=PA358&dq=weak+isospin+hypercharge+electric&hl=en&ei=5xFVTunTJ4PYiALz57X3DA&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDwQ6AEwAw#v=onepage&q=weak%20isospin%20hypercharge%20electric&f=false coupling factors e, gI, gY]
*[http://books.google.com/books?id=hFasRlkBbpYC&pg=PA53&dq=weak+isospin+hypercharge+electric&hl=en&ei=5xFVTunTJ4PYiALz57X3DA&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDUQ6AEwAg#v=onepage&q=weak%20isospin%20hypercharge%20electric&f=false Donoghue on charge]
*[http://books.google.com/books?id=wez0SGmemagC&pg=PA99&dq=weak+isospin+hypercharge+electric&hl=en&ei=5xFVTunTJ4PYiALz57X3DA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCoQ6AEwAA#v=onepage&q=weak%20isospin%20hypercharge%20electric&f=false Dodd on charge]
*[http://books.google.com/books?id=w9Dz56myXm8C&pg=PA344&dq=weak+isospin+hypercharge+electric&hl=en&ei=5xFVTunTJ4PYiALz57X3DA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDAQ6AEwAQ#v=onepage&q=weak%20isospin%20hypercharge%20electric&f=false Griffiths on charge]
*[http://books.google.com/books?id=ZdaE2agLxY8C&pg=PA1&dq=electric+charge+electroweak&hl=en&ei=uw1VTrG3IbHSiAKahtXcDA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDAQ6AEwAQ#v=onepage&q=electric%20charge%20electroweak&f=false charges]
*[http://books.google.com/books?id=WFDs_SJgILQC&pg=PA38&dq=lepton+charge&hl=en&ei=vDtJTqDsOePKiALM6vDsAQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CD8Q6AEwAw#v=onepage&q=lepton%20charge&f=false lepton flavor, not lepton charge]
*[http://books.google.com/books?id=cOnjDfQQX0UC&pg=PA332&dq=lepton+charge&hl=en&ei=vDtJTqDsOePKiALM6vDsAQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC4Q6AEwAA#v=onepage&q=lepton%20charge&f=false Rowlands]
*[http://books.google.com/books?id=c60mCxGRMR8C&pg=PA892&dq=lepton+charge&hl=en&ei=vDtJTqDsOePKiALM6vDsAQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDMQ6AEwAQ#v=onepage&q=lepton%20charge&f=false Harris]
*[http://books.google.com/books?id=P_T0xxhDcsIC&pg=PA127&dq=lepton+charge&hl=en&ei=vDtJTqDsOePKiALM6vDsAQ&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDoQ6AEwAg#v=onepage&q=lepton%20charge&f=false Schutz]
*[http://books.google.com/books?id=krTli3-XL4AC&pg=PA495&dq=color+chromodynamic+force&hl=en&ei=hh9NTp2VDpPKiALc5JiFAQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDkQ6AEwAw#v=onepage&q=color%20chromodynamic%20force&f=false Bord]
*[http://books.google.com/books?id=w9Dz56myXm8C&pg=PA67&dq=color+chromodynamic+force&hl=en&ei=hh9NTp2VDpPKiALc5JiFAQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CD8Q6AEwBA#v=onepage&q=color%20chromodynamic%20force&f=false Griffiths]
*[http://arxiv.org/PS_cache/arxiv/pdf/0901/0901.1903v1.pdf QCD]
*[http://books.google.com/books?id=8TxnB4uGUxkC&pg=SA36-PA26&dq=pion+pion+color+dipole+interaction&hl=en&ei=mBZhTrLOKY_YiAK0wN24Dg&sa=X&oi=book_result&ct=result&resnum=6&ved=0CEIQ6AEwBTgK#v=onepage&q&f=false virtual]

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In physics and chemistry, charge is fundamentally related to fields and forces, and is a property of pieces of matter that leads to forces between spatially separate pieces of matter that likewise manifest that particular property. There are a wide variety of such charges. In the Standard Model, there are three types of charge: color, weak isospin and weak hypercharge.[1] These include the electric charge underlying electric current that enters Maxwell's equations for the electromagnetic field. In addition, there is mass that enters gravitation.[2]

These charges are conserved quantities and are related to currents describing their flux or motion. The conservation law relating the charge to its current is of the form:

where div is the vector divergence operator, J is the vector current density, and ρ is the charge density. For a volume enclosed by a surface, this equation can be expressed by the statement that any change in the charge contained inside the closed surface is due to a current of said charge either entering or exiting through that surface.

Such conservation laws are examples of Noether's theorem, which states that every symmetry of a physical theory is related to a conservation law of this kind. This theorem is closely related to Curie's principle:

The symmetry of an isolated system cannot decrease as the system evolves with time.[3]

The best known of these conservation laws are the conservation of momentum (the current is momentum density, the charge is mass density), related to translational symmetry of the laws of mechanics, conservation of angular momentum, related to the rotational symmetry of the laws of mechanics, and conservation of energy, related to the independence of the laws of mechanics from time translations. Such symmetries are intuitive for point particle mechanics, but for the physics of general fields some symmetries are quite non-intuitive.

A formal description of Noether's theorem as related to charge is that a current ja = (ja0, ja) satisfying:

which implies the conservation of the charge Q defined by:

is a natural consequence of the ja being generators of a Lie group that is a symmetry group of the physical system.[4]

Charge and exchange forces

Forces between particles are mediated by exchange of shared properties. For example, two nucleons in the same state of motion can exchange electric charge, producing an exchange force. The Yukawa theory of nuclear force posited that nucleons (protons p and neutrons n) could exchange electric charge by trading pions according to the reactions:[5]

and forces between like particles could be introduced by exchange of zero-charge pions:

These reactions do not conserve mass or energy, they are virtual reactions. One common (although not universally accepted) "explanation" why violation is permissible is that such reactions occur very rapidly, and for very short times the energy uncertainty relation allows violation of these conservation rules.

Besides electric charge, other properties can be exchanged, such as spin (Bartlett exchange), or position (Majorana exchange).

The swapping of shared properties is a symmetry operation, the exchange of identical particles, and as such is related to conserved quantities via Noether's theorem. For example, the nucleon can be thought of as a two-state particle with an isospin that is +1/2 for a neutron and −1/2 for a proton, so the change of one to the other is an isospin exchange, and symmetry of a theory under isospin exchange indicates the theory conserves isospin.[6] In a quantized version of such a theory, isospin exchange could be moderated by the pion reactions above.

Only if isospin symmetry in the theory can be produced by a continuous transformation (one depending upon some continuously variable parameter), does it lead to an isospin current conservation law.

Electrodynamics

In electrodynamics, two types of charge are known, magnetic and electric. The distinguishing property of electric charge is that electric charges can be isolated, while while an isolated magnetic charge or magnetic monopole never has been observed.[7] Electric charges interact with magnetic charges only when in relative motion one to the other.

The conservation of electric charge follows directly from Maxwell's equations. It also can be derived from Noether's theorem as a result of a gauge invariance of Maxwell's theory when that theory is expressed in terms of a vector potential. Although this approach has continuity with much of modern field theory, it is somewhat unintuitive, as the "symmetry" of the recast Maxwell equations is simply due to introduction of a mathematical device that adds an unnecessary degree of freedom into the formulation thereby introducing this symmetry artificially.[8] Below is a digression on this topic.

The basic electric field E and magnetic field B of Maxwell's equations can be replaced by introduction of a scalar potential φ and a vector potential A using the relations:
Although the potentials uniquely determine the fields, the reverse is not true. Different potentials produce the same fields; in particular the potentials denoted by primes below produce the same fields:
Here Γ = Γ(r, t) is any continuous function of the space-time coordinates r, t. Consequently, a theory based upon potentials instead of fields has the additional symmetry that it is unchanged by substitution of primed potentials instead of the original potentials. This change of potentials from unprimed to primed is called a gauge transformation and this new symmetry leads directly to the continuity equation for electric charge:
This equation is a direct consequence of the Maxwell equations defining charge and current densities (in Heaviside-Lorentz units):
However, using the potential formulation, the continuity equation is required if the theory is to be gauge invariant,[9] and this requirement is consistent with Noether's theorem.

In a quantized theory based upon the potential formulation of Maxwell's equations, the electrical force between charged particles is an exchange force mediated by trading charge-neutral photons. The electromagnetic potentials exist as vibrations with certain allowed amplitudes determined by the number of photons employed, and field amplitudes are increased or decreased by adding or subtracting photons. Thus, the force exerted upon a charged particle as determined by the field it experiences, depends upon the number of photons in the corresponding potentials.

Weak forces

Weak forces are mediated by the electric charged W+ and W− particles and the electric charge neutral Z0 particle, all with spin 1. The weak interaction is of short range, being effective over a distance of approximately 10−3 fm. Analysis of the weak force parallels that of the electromagnetic force, apart from the huge mass of the exchanged particles compared to the photon. The "weak force" charge introduced that couples to this force is called flavor.[10] It is customaryu to refer to lepton flavor, rather than lepton charge, and individual lepton flavors are attributed to each family: electron flavor Le for electrons; muon flavor Lμ for muons; tau flavor Lτ for taus.[11] p. 38 However, the terminology is somewhat confused. Some authors do refer to both "weak charge" and to "lepton flavor".[12] The issue may be that quarks and leptons behave differently under the weak force?

Nuclear forces

In 1935 Yukawa invented the meson theory for explaining the forces holding atomic nucleii together, an assemblage of neutrons and protons. The theory led to the experimental observation of the pion or π-meson and the muon or μ-meson. The behavior of nuclear forces was explained as an exchange of mesons. Today, mesons are considered to be quark-antiquark pairs, and a more refined theory of nuclear interactions is based upon quantum chromodynamics. Nuclear forces are not considered fundamental today, but are a consequence of the underlying strong forces between quarks, also called chromodynamic forces or color forces. On that basis, nuclear forces are an exchange force fundamentally based upon color, and only approximated by the Yukawa theory.

Chromodynamics

In the Standard Model of particle physics, quantum chromodynamics describes the strong force, also called the color force or chromo force, and relates it to the color charge as a property of quarks and gluons.[13] Similar to magnetic charge, color is not seen directly, as all observable particles have no overall color.[14] As with electric and magnetic charge, color charge can be multiple valued, conventionally called red, green or blue. Color charge is not assigned a numerical value; however, a superposition in equal amounts of all three colors leads to a "neutral" color charge, a somewhat stretched analogy with the superposition of red, green and blue light to produce white light.[15] Thus, protons and neutrons, which consist of three quarks with all three colors are color-charge neutral. Quark combinations are held together by exchange of combinations of eight different gluons that also are color charged.[16][17][18][19]

The color charges of antiquarks are anticolors. The combination of a quark and an antiquark to form a meson, such as a pion, kaon and so forth, leads to a neutral color charge.

Other charges

The charges above are related to fields and forces and to a local (coordinate dependent) Noether's theorem. Other charges are known, however, that are connected to global or discrete symmetries (no continuous parametric dependence, such as a coordinate dependence) and so to a global Noether's theorem, and have no relation to forces or fields.

One such charge in elementary particle theory is the baryonic charge, B, also referred to as a number, with value +1 for all baryons (notably, neutrons and protons, but also others like the Λ and Σ particles) and −1 for all antibaryons and zero for non-baryons. Quarks are an exception, and have a baryon number of 1/3. Unlike electric charge, which serves as a source for the electromagnetic field, baryon charge is not related to an associated "baryonic" field.[11]

Finally, we mention the leptonic charge (also called lepton number) carried by leptons: electrons, muons, taus, and their associated neutrinos.[15] Lepton charge depends upon the flavor of the lepton[10] Le, Lμ, Lτ with values +1 for the electron, muon and tau meson, and −1 for their antiparticles.[11] The total lepton number L of a complex is:

Non-leptons have a total lepton number L of zero. Within the Standard Model, lepton number is conserved for strong and electromagnetic interactions; however, it is not necessarily conserved in weak particle reactions.[11][20]

References

  1. ↑ John F. Donoghue, Eugene Golowich, Barry R. Holstein (1994). Dynamics of the standard model. Cambridge University Press, p. 24. ISBN 0521476526. 
  2. ↑ Mark Burgess (2004). “Chapter 12: Charge and current”, Classical covariant fields. Cambridge University Press, pp. 325 ff. ISBN 0521813638. 
  3. ↑ Some care is needed in looking at this principle because of the phenomenon of spontaneous symmetry breaking. For example, as a cubic ferroelectric material like BaTiO3 is cooled below its Curie point, its cubic symmetry is replaced by a tetragonal ferroelectric symmetry as the frequency corresponding to a tetragonal elastic distortion tends to zero (Goldstone's theorem). The overall cubic symmetry of the crystal is retained because the crystal breaks into finite domains, each with a differently oriented tetragonal axis, so that statistically the symmetry of an infinite crystal still is cubic. For a general discussion, see F. Strocchi (2008). Symmetry breaking, 2nd ed. Springer. ISBN 3540735925. 
  4. ↑ Nina Byers (1994). The Life and Times of Emmy Noether: Contributions of Emmy Noether to Particle Physics. UCLA/94/TEP/42; hep-th/9411110. Presented at the International Conference on THE HISTORY OF ORIGINAL IDEAS AND BASIC DISCOVERIES IN PARTICLE PHYSICS, Erice, Italy, 29 July - 4 August 1994.
  5. ↑ Hari Jeevan Arnikar (1995). Essentials of nuclear chemistry, 4th ed. New Age International, p. 16 ff. ISBN 8122407129. 
  6. ↑ Dwight E. Neuenschwander (2010). “§9.1 Conservation of properties and unitary transformations”, Emmy Noether's Wonderful Theorem. The Johns Hopkins University Press, p. 192 ff. ISBN 0801896940. 
  7. ↑ Douglas C. Giancoli. Physics for scientists and engineers with modern physics, 4rth ed. Pearson Education, p. 708. ISBN 0132273594. 
  8. ↑ K.-H. Rehren, E Seiler (2007). Ion-Olimpiu Stamatescu, Erhard Seiler, eds: Approaches to fundamental physics: an assessment of current theoretical ideas. Springer, p. 401. ISBN 3540711155. “Gauge symmetry was originally observed within Maxwell's theory of classical electrodynamics as an ambiguity related to the artificial introduction of unobservable potentials in order to solve two of Maxwell's four equations.” 
  9. ↑ WN Cottingham, DA Greenwood (2007). “Gauge transformations”, An introduction to the Standard Model of particle physics, 2nd ed. Cambridge University Press, p. 41. ISBN 978-0-521-85249-4. 
  10. ↑ 10.0 10.1 Paul Allen Tipler (2007). “Summary table”, Physics for scientists and engineers: Elementary modern physics, Volume 3, 6th ed. Macmillan, p. 1409. ISBN 1429201347. 
  11. ↑ 11.0 11.1 11.2 11.3 O. M. Boyarkin (2007). “Chapter 3: Leptons and hadrons”, O. M. Boyarkin, Alfred L. Heinzerton, eds: Introduction to Physics of Elementary Particles. Nova Publishers, pp. 37-40. ISBN 160021200X. 
  12. ↑ Yorikiyo Nagashima, Yoichiro Nambu (2010). Elementary Particle Physics: Volume 1: Quantum Field Theory and Particles, Volume 1. Wiley-VCH, p. 554. ISBN 3527409629. 
  13. ↑ Stephen Webb (2004). Out of this world: colliding universes, branes, strings, and other wild ideas of modern physics. Springer, p. 190. ISBN 0387029303. 
  14. ↑ Andrew Watson (2004). The quantum quark. Cambridge University Press, pp. 170 ff. ISBN 0521829070. 
  15. ↑ 15.0 15.1 M. Y. Han (1999). Quarks and gluons: a century of particle charges. World Scientific, p. 116. ISBN 9810237456. 
  16. ↑ Joe Rosen (2004). Encyclopedia of physics. Infobase Publishing, p. 85. ISBN 0816049742. 
  17. ↑ Joe Rosen, Lisa Quinn Gothard (2009). Encyclopedia of Physical Science, Volume 1. Infobase Publishing, p. 278. ISBN 0816070113. 
  18. ↑ (2009) “Quantum chromodynamics (QCD)”, Daniel M. Greenberger, Klaus Hentschel, Friedel Weinert: Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy. Springer, pp. 524 ff. ISBN 3540706224. 
  19. ↑ OW Greenberg (2008). "The color charge degree of freedom in particle physics". Chapter in Greenberger et al. below.
  20. ↑ Helen R. Quinn, Yossi Nir (2010). “Chapter 12: Baryon and Lepton number conservation?”, The Mystery of the Missing Antimatter. Princeton University Press, p. 130. ISBN 1400835712. 

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