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| {{Essay|August 2009|date=June 2011}}
| | Friends contact him Royal Seyler. Arizona is her beginning place and she will never move. His working day occupation is a cashier and his salary has been really fulfilling. Her friends say it's not good for her but what she enjoys doing is flower arranging and she is attempting to make it a occupation.<br><br>Here is my blog: [http://Livejasminbook.com/index.php?do=/profile-59346/info/ Livejasminbook.com] |
| In [[electronics]], '''Mason's invariant''', named after [[Samuel Jefferson Mason]], is a measure of the quality of [[transistor]]s.
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| "When trying to solve a seemingly difficult problem, Sam said to concentrate on the easier ones first; the rest, including the hardest ones, will follow," recalled [[Andrew Viterbi]], co-founder and former vice-president of [[Qualcomm]]. He had been a thesis advisee under Samuel Mason at [[MIT]], and this was one lesson he especially remembered from his professor.<ref name='news_office'>{{cite news | first= | last= | coauthors= | title=Entrepreneur endows chair; Rivest is holder | date=2000-08-09 | publisher=MIT | url =http://web.mit.edu/newsoffice/2000/rivest-0809.html | work =MIT News | pages = | accessdate = 2007-05-08 | language = }}</ref> A few years earlier, Mason had heeded his own advice when he defined a unilateral power gain for a linear two-port device, or U. After concentrating on easier problems with power gain in feedback [[amplifiers]], a [[figure of merit]] for all three-terminal devices followed that is still used today as Mason's Invariant.<ref name='Gupta'>{{cite journal|title=Power Gain in Feedback Amplifiers, a Classic Revisited|journal=IEEE Transactions on Microwave Theory and Techniques|date=May 1992|first=Madhu|last=Gupta|coauthors=|volume=40|issue=5|pages=864–879|id= |url=http://rfic.eecs.berkeley.edu/~niknejad/ee217sp05/mason.pdf|format=PDF|accessdate=2007-05-08|doi=10.1109/22.137392 |archiveurl = http://web.archive.org/web/20070418064920/http://rfic.eecs.berkeley.edu/~niknejad/ee217sp05/mason.pdf <!-- Bot retrieved archive --> |archivedate = 2007-04-18}}</ref>
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| == Origin ==
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| In 1953, [[transistors]] were only five years old, and they were the only successful, three-terminal [[passivity (engineering)|active device]]. They were beginning to be used for [[Radio frequency|RF]] applications, and they were limited to [[VHF]] frequencies and below. Mason wanted to find a figure of merit to compare [[transistors]], and this led him to discover that the unilateral power [[gain]] of a linear two-port device was an invariant figure of merit.<ref name="Gupta"/>
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| In his paper ''Power [[Gain]] in Feedback [[Amplifiers]]'' published in 1953, Mason stated in his introduction, <blockquote>
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| "A [[vacuum tube]], very often represented as a simple [[transconductance]] driving a passive [[Electrical impedance|impedance]], may lead to relatively simple [[amplifier]] designs in which the input impedance (and hence the power [[gain]]) is effectively infinite, the [[voltage]] [[gain]] is the quantity of interest, and the input circuit is isolated from the [[Electrical load|load]]. The [[transistor]], however, usually cannot be characterized so easily."<ref name='Mason'>{{cite journal|title=Power Gain in Feedback Amplifiers|journal=IRE Transactions on Circuit Theory|date=June 1954|first=Samuel|last=Mason|coauthors=|volume=1|issue=2|pages=20–25|id= |url=http://ieeexplore.ieee.org/Xplore/login.jsp?url=/iel5/8148/23422/01083579.pdf?arnumber=1083579|format=PDF|accessdate=2007-05-08|doi=10.1109/TCT.1954.1083579}}</ref>
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| </blockquote> He wanted to find a metric to characterize and measure the quality of transistors since up until then, no such measure existed. Little did Mason know, however, that he would discover an equation that is still used more than 50 years later and does much more than measure the quality of a transistor.
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| == Derivation of U ==
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| Mason first defined the device being studied with the three constraints listed below.<ref name="Gupta"/>
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| # The device has only two ports (at which power can be transferred between it and outside devices).
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| # The device is linear (in its relationships of currents and voltages at the two ports).
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| # The device is used in a specified manner (connected as an amplifier between a linear one-port source and a linear one-port [[Electrical load|load]]).
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| Then, according to Madhu Gupta in ''Power Gain in Feedback Amplifiers, a Classic Revisited'', Mason defined the problem as "being the search for device properties that are invariant with respect to transformations as represented by an embedding network" that satisfy the four constraints listed below.<ref name="Gupta"/>
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| # The embedding network is a four-port.
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| # The embedding network is linear.
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| # The embedding network is lossless.
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| # The embedding network is reciprocal.
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| He next showed that all transformations that satisfy the above constraints can be accomplished with just three simple transformations performed sequentially. Similarly, this is the same as representing an embedding network by a set of three embedding networks nested within one another. The three mathematical expressions can be seen below.<ref name="Gupta"/>
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| 1. Reactance padding:
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| <math>
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| \begin{bmatrix}
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| Z'_{11} & Z'_{12} \\
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| Z'_{21} & Z'_{22}
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| Z_{11}+jx_{11} & Z_{12}+jx_{12} \\
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| Z_{21}+jx_{21} & Z_{22}+jx_{22}
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| \end{bmatrix}
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| </math>
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| 2. Real Transformations:
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| <math>
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| \begin{bmatrix}
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| Z'_{11} & Z'_{12} \\
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| Z'_{21} & Z'_{22}
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| n_{11} & n_{12} \\
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| n_{21} & n_{22}
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| \end{bmatrix}
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| \begin{bmatrix}
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| Z_{11} & Z_{12} \\
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| Z_{21} & Z_{22}
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| \end{bmatrix}
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| \begin{bmatrix}
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| n_{11} & n_{12} \\
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| n_{21} & n_{22}
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| \end{bmatrix}
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| </math>
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| 3. Inversion:
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| <math>
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| \begin{bmatrix}
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| Z'_{11} & Z'_{12} \\
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| Z'_{21} & Z'_{22}
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| Z_{11} & Z_{12} \\
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| Z_{21} & Z_{22}
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| \end{bmatrix}^{-1}
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| </math>
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| Mason then considered which quantities remained invariant under each of these three transformations. His conclusions, listed respectively to the transformations above, are shown below. Each transformation left the values below unchanged.<ref name="Gupta"/>
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| 1. Reactance padding:
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| <math>
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| \left [ Z-Z_{t} \right ]
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| </math>
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| and
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| <math>
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| \left [ Z+Z^{*} \right ]
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| </math>
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| 2. Real transformations:
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| <math>
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| \left [ Z-Z_{t} \right ]
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| \left [ Z+Z^{*} \right ]
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| </math>
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| and
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| <math>
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| \dfrac{\det{\left [ Z-Z_{t} \right ]}}{\det{\left [ Z+Z^{*} \right ]}}
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| </math>
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| 3. Inversion:
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| The magnitudes of the two determinants and the sign of the denominator in the above fraction remain unchanged in the inversion transformation. Consequently, the quantity invariant under all three conditions is:<ref name="Gupta"/>
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| : <math>
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| \begin{align}
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| U & =\dfrac{|\det{\left [ Z-Z_t \right ]}|}{\det{\left [ Z+Z^{*} \right ]}} \\
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| & =
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| \dfrac{|Z_{12}-Z_{21}|^{2}}{4 (\operatorname{Re}[Z_{11}] Re[Z_{22}]-\operatorname{Re}[Z_{12}] \operatorname{Re}[Z_{21}])} \\
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| & =
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| \dfrac{|Y_{21}-Y_{12}|^{2}}{4 (\operatorname{Re}[Y_{11}] \operatorname{Re}[Y_{22}]-\operatorname{Re}[Y_{12}] \operatorname{Re}[Y_{21}])}
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| \end{align}
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| </math>
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| == Importance ==
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| Mason's Invariant, or U, is the only device characteristic that is invariant under lossless, reciprocal embeddings. In other words, U can be used as a figure of merit to compare any three-terminal, active device. For example, a factory producing [[BJT]]s can calculate U of the [[transistors]] it is producing and compare their quality to the other [[BJT]]s on the market. Furthermore, U can be used as an indicator of activity. If U is greater than one, the two-port device is active; otherwise, that device is passive. This is especially useful in the [[microwave]] engineering community. Though originally published in a circuit theory journal, Mason's paper becomes especially relevant to [[microwave]] engineers since U is usually slightly greater to or equal to one in the [[microwave]] frequency range. When U is smaller than or considerably larger than one, it becomes relatively useless.<ref name="Gupta"/>
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| While Mason's Invariant can be used as a figure of merit across all operating frequencies, its value at '''''ƒ''<sub>max</sub>''' is especially useful. ''F''<sub>max</sub> is the maximum [[oscillation]] frequency of a device, and it is discovered when <math>U(f_\max) = 1</math>. This frequency is also the frequency at which the maximum stable [[gain]] G<sub>ms</sub> and the maximum available [[gain]] G<sub>ma</sub> of the device become one. Consequently, ''ƒ''<sub>max</sub> is a characteristic of the device, and it has the significance that it is the maximum frequency of [[oscillation]] in a circuit where only one active device is present, the device is embedded in a passive network, and only single [[sinusoidal]] signals are of interest.<ref name="Gupta"/>
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| == Conclusion ==
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| In his revisit of Mason's paper, Gupta states, "Perhaps the most convincing evidence of the utility of the concept of a unilateral power [[gain]] as a device figure of merit is the fact that for the last three decades, practically every new, active, two-port device developed for high frequency use has been carefully scrutinized for the achievable value of U..."<ref name="Gupta"/> This assumption is appropriate because "U<sub>max</sub>" or "maximum unilateral [[gain]]" is still listed on [[transistor]] specification sheets, and Mason's Invariant is still taught in some undergraduate electrical engineering curricula—ones that Mason, no doubt, had a hand in innovating. Though now it has been over five decades, Mason's finding of an invariant device characteristic still plays a significant role in [[transistor]] design.
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| ==See also==
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| *[[Power gain]]
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| *[[Scattering parameters]]
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| *[[Two-port network]]
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| == References ==
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| {{Reflist}}
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| [[Category:Electronic engineering]]
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| [[Category:Two-port networks]]
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Friends contact him Royal Seyler. Arizona is her beginning place and she will never move. His working day occupation is a cashier and his salary has been really fulfilling. Her friends say it's not good for her but what she enjoys doing is flower arranging and she is attempting to make it a occupation.
Here is my blog: Livejasminbook.com