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[[File:Gain-bandwidth product.svg|thumb|right|300px|Adding [[negative feedback]] limits the amplification but improves frequency response of the amplifier.]]
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The '''gain–bandwidth product''' (designated as '''GBWP''', '''GBW''', '''GBP''' or '''GB''') for an [[amplifier]] is the product of the amplifier's [[Bandwidth (signal processing)|bandwidth]] and the [[gain]] at which the bandwidth is measured.
  <li>[http://www.heliygroup.com/index.php?option=com_kunena&func=view&catid=3&id=98386&Itemid=84&lang=en#98386 http://www.heliygroup.com/index.php?option=com_kunena&func=view&catid=3&id=98386&Itemid=84&lang=en#98386]</li>
 
 
For devices such as [[operational amplifier]]s that are designed to have a simple one-pole [[frequency]] response, the gain–bandwidth product is nearly independent of the gain at which it is measured; in such devices the gain–bandwidth product will also be equal to the unity-gain bandwidth of the amplifier (the bandwidth within which the amplifier gain is at least 1).<ref>
  <li>[http://passerelle.ethiopie.free.fr/spip.php?article81/ http://passerelle.ethiopie.free.fr/spip.php?article81/]</li>
{{cite book
 
| title = Analog And Digital Electronics
  <li>[http://www.haokepa.com/forum.php?mod=viewthread&tid=593370&fromuid=34862 http://www.haokepa.com/forum.php?mod=viewthread&tid=593370&fromuid=34862]</li>
| author = U. A. Bakshi and A. P. Godse
 
| publisher = Technical Publications
  <li>[http://colossuscorporation.net/appicker/index.php?option=com_kunena&func=view&catid=2&id=498768&Itemid=534#498768 http://colossuscorporation.net/appicker/index.php?option=com_kunena&func=view&catid=2&id=498768&Itemid=534#498768]</li>
| year = 2009
 
| isbn = 978-81-8431-708-4
</ul>
| pages = 2–5 <!-- this really is page 2-5, not pages 2–5 -->
| url = http://books.google.com/books?id=Nw-1RETEUFUC&pg=SA1-PA105
}}</ref> 
For an amplifier in which negative feedback reduces the gain to below the [[open-loop gain]], the gain–bandwidth product of the closed-loop amplifier will be approximately equal to that of the open-loop amplifier.
According to S. Srinivasan, "The parameter characterizing the frequency dependence of the operational amplifier gain is the finite gain–bandwidth product (GB)."<ref>Srinivasan, S. "A universal compensation scheme for active filters." International Journal of Electronics 42.2 (Feb. 1977): 141. Science & Technology Collection. EBSCO. Dallas Public Library <http://www.dplibrary.org>, Dallas, TX, USA. retrieved 31 July 2009 from <http://search.ebscohost.com/login.aspx?direct=true&db=syh&AN=5259750&site=ehost-live>.</ref>
 
== Relevance to design ==
 
This quantity is commonly specified for [[operational amplifier]]s, and allows [[circuit design]]ers to determine the maximum gain that can be extracted from the device for a given frequency (or bandwidth) and vice versa.
 
When adding [[LC circuit]]s to the input and output of an amplifier the gain rises and the bandwidth decreases, but the product is generally bounded by the gain–bandwidth product.
 
=== Examples ===
 
If the GBWP of an operational amplifier is 1&nbsp;MHz, it means that the gain of the device falls to unity at 1&nbsp;MHz. Hence, when the device is wired for unity gain, it will work up to 1&nbsp;MHz (GBWP&nbsp;=&nbsp;gain&nbsp;×&nbsp;bandwidth, therefore if BW&nbsp;=&nbsp;1&nbsp;MHz, then gain&nbsp;=&nbsp;1) without excessively distorting the signal. The same device when wired for a gain of 10 will work only up to 100&nbsp;kHz, in accordance with the GBW product formula. Further, if the maximum frequency of operation is 1&nbsp;Hz, then the maximum gain that can be extracted from the device is 1{{e|6}}.
 
We can also analytically show that for <math>\omega >> \omega_c</math> GBWP is constant.
 
Let <math>A_1(\omega)</math> be a first-order transfer function given by:
 
<math>A_1(\omega)= \frac{{{H_0}}}{{\sqrt {1 + {{\left( {\frac{\omega }{{{\omega_c}}}} \right)}^2}} }}</math>
 
We will show that:
 
<math>GBWP_{\omega  >  > {\omega_c}} = {A_1}(\omega )\cdot\omega  \approx const.</math>
 
Proof:
 
<math>GBWP = {A_1}(\omega )\cdot\omega  = \frac{{{H_0}}}{{\sqrt {1 + {{\left( {\frac{\omega }{{{\omega_c}}}} \right)}^2}} }}\cdot\omega \simeq \frac{{{H_0}}}{{\sqrt {{{\left( {\frac{\omega }{{{\omega_c}}}} \right)}^2}} }}\cdot\omega = {H_0}\cdot{\omega_c} = const.</math>
 
Example for <math>\omega = 5\cdot \omega_c</math>
 
<math>GBWP = \frac{{{H_0}}}{{\sqrt {\frac{{\omega_c^2 + 25{\omega _c}^2}}{{\omega_c^2}}} }}\cdot5{\omega_c} = \frac{5}{{\sqrt {26} }}{H_0}\cdot{\omega_c} = 0.98\cdot{H_0}\cdot{\omega_c}</math>
 
Note that the error in this case is only about 2%.
 
==Transistors==
 
For [[transistor]]s, the current-gain–bandwidth product is known as the {{math|''f''<sub>''T''</sub>}} or ''transition frequency''.<ref>
{{cite book
| title = Principles of transistor circuits: introduction to the design of amplifiers, receivers, and digital
| edition = 9th
| author = Stanley William Amos and Mike James
| publisher = Newnes
| year = 2000
| isbn = 978-0-7506-4427-3
| page = 169
| url = http://books.google.com/books?id=mfec2Zw_b7wC&pg=PA169
}}</ref><ref>
{{cite book
| title = Fundamentals of semiconductor devices
| author = M K Achuthan and K N Bhat
| publisher = Tata McGraw-Hill Education
| year = 2007
| isbn = 978-0-07-061220-4
| page = 408
| url = http://books.google.com/books?id=REQkwBF4cVoC&pg=PA408
}}</ref>
It is calculated from the low-frequency (a few [[kilohertz]]) current gain under specified test conditions, and the ''cutoff frequency'' at which the current gain drops by 3 decibels (70% amplitude); the product of these two values can be thought of as the frequency at which the current gain would drop to 1, and the transistor current gain between the cutoff and transition frequency can be estimated by dividing {{math|''f''<sub>''T''</sub>}} by the frequency. Usually, transistors must be applied at frequencies well below {{math|''f''<sub>''T''</sub>}} to be useful as amplifiers and oscillators.<ref>Martin Hartley Jones  ''A practical introduction to electronic circuits'', Cambridge University Press, 1995 ISBN 0-521-47879-0  page 148</ref> In a bipolar junction transistor, frequency response declines owing to the internal capacitance of the junctions. The transition frequency varies with collector current, reaching a maximum for some value and declining for greater or lesser collector current.
 
== References ==
 
<references />
 
== External links ==
* [http://masteringelectronicsdesign.com/an-op-amp-gain-bandwidth-product/ "Op-amp gain-bandwidth-product"] masteringelectronicsdesign.com
 
{{DEFAULTSORT:Gain-bandwidth product}}
[[Category:Electronic amplifiers]]

Latest revision as of 21:01, 31 May 2014

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