Limited-memory BFGS: Difference between revisions

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The matrix stored by BFGS approximates the inverse Hessian, not the Hessian. This is noted earlier in this sentence, so this seems to just be a typo.
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A '''bias tee''' is a three port network used for setting the DC [[biasing|bias]] point of some electronic components without disturbing other components.  The bias tee is a [[diplexer]]. The low frequency port is used to set the bias; the high frequency port passes the radio frequency signals but blocks the biasing levels; the combined port connects to the device, which sees both the bias and RF. It is called a ''tee'' because the 3 ports are often arranged in the shape of a '''T'''.
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==Design==
[[Image:Bias t.png|right|thumb|250px|Equivalent circuit of a bias tee.]]
Conceptually, the bias tee can be viewed as an ideal [[capacitor]] that allows [[alternating current|AC]] through but blocks the [[direct current|DC]] bias and an ideal [[inductor]] that blocks AC but allows DC.  Although some bias tees can be made with a simple inductor and capacitor, wideband bias tees are considerably more complicated because practical components have parasitic elements.
 
Bias tees are designed for transmission line environments.  Typically, the characteristic impedance ''Z<sub>0</sub>'' will be 50&nbsp;ohms or 75&nbsp;ohms.  The [[Electrical impedance|impedance]] of the capacitor (X<sub>C</sub>) is chosen to be much less than ''Z<sub>0</sub>'', and the impedance of the inductor (X<sub>L</sub>) is chosen to be much greater than ''Z<sub>0</sub>''.
 
:<math>X_C = {1 \over j \omega C} = {1 \over j 2 \pi f C} \ ; \ X_L = j \omega L = j 2 \pi f L</math>
 
Where ''&omega;'' is the frequency in radians per second and ''f'' is the frequency in hertz.
 
Bias tees are designed to operate over a range of signal frequencies.  The reactances are chosen to have minimal impact at the lowest frequency.
 
For wide range bias tees, the inductor must be large at the lowest frequency.  A large inductor will have a stray capacitance (which creates its self-resonant frequency). At a high enough frequency, the stray capacitance presents a low impedance shunt path for the signal, and the bias tee becomes ineffective.  Practical wide band bias tees must use circuit topologies that avoid the shunt path.  For example, a Picosecond Pulse Labs 5580 works from 10&nbsp;kHz to 15&nbsp;GHz.  {{Harv|Andrews|2000|p=3}}  Consequently, the simple design would need an inductance of at least 800&nbsp;μH (''X<sub>L</sub>'' about ''j''500&nbsp;ohms at 10&nbsp;kHz), and that inductor must still look like an inductor at 15&nbsp;GHz.  However, a commercial 820&nbsp;μH inductor has a self-resonant frequency of only 1.8&nbsp;MHz—four orders of magnitude too low.<ref>[http://search.digikey.com/scripts/DkSearch/dksus.dll?Detail&name=T1812-821J-ND Tamura surface mount inductor at DigiKey]</ref>
 
== Application ==
A bias tee is used to insert [[Direct Current|DC]] power into an [[Alternating Current|AC]] [[Signal (electrical engineering)|signal]] to power remote [[Antenna (radio)|antenna]] [[amplifier]]s or other devices.  It is usually positioned at the receiving end of the coaxial cable to pass DC power from an external source to the coaxial cable running to powered device.  A bias “T” consists of a feed [[inductor]] to deliver DC to a [[electrical connector|connector]] on the device side and a blocking [[capacitor]] to keep DC from passing through to the [[Receiver (radio)|receiver]]. The [[Radio frequency|RF]] signal is connected directly from one connector to the other with only the blocking capacitor in series.  The internal blocking [[diode]] prevents damage to the bias “T” if reverse supply [[voltage]] is applied.
 
Bias tees are used in a variety of applications, but are generally used to provide an RF signal and (DC) power to a remote device where running two separate cables would not be advantageous.<ref name="HamtvbiasT">{{Citation | title = Bias T for Antenna Mount Preamps | year = 2007 | url = http://www.hamtv.com/pdffiles/biasT.pdf | accessdate = 2008-08-08 }}</ref>  Biasing is often used with [[photodiodes]] (vacuum and solid state), [[Microchannel plate detector]]s, [[transistor]]s, and [[triode]]s, so that high frequencies from the signal do not leak into a common power supply rail. Conversely, noise from the power supply does not appear on the signal line. Other examples include:  [[Power over Ethernet]],<ref name="Intel">{{Citation | title = Power Over Ethernet Bias T Power Module | url = http://download.intel.com/support/wireless/wlan/pro2011b/accesspoint/poe_diagram.pdf | accessdate = 2008-08-08 }}</ref><ref>PoE is a dubious bias tee. End of span power insertion is done with common mode injection between two signal pairs. Midspan insertion is done on unused pairs rather than the signal line.</ref> active antennas, low-noise amplifiers, and down converters. <ref name="WiNRADiO">{{Citation | title = WR-BT-650 HF/VHF Power Injector (Bias 'T') | url = http://www.winradio.co.uk/home/bt-650.htm | accessdate = 2008-08-08 | archiveurl= http://web.archive.org/web/20080705210313/http://www.winradio.co.uk/home/bt-650.htm| archivedate= 5 July 2008 <!--DASHBot-->| deadurl= no}}</ref>
 
The [[telephone line]] for [[plain old telephone service]] and some early microphones use a bias tee circuit -- often with a [[gyrator]] replacing the inductor -- this enables a thin cable with only 2 conductors to send power from the system to the device, and send audio from the device back to the system.
Modern microphones often use 3 conductors in a [[phantom power]] circuit very similar to a bias tee circuit.
 
== Construction ==
 
There are several bias tee designs.
 
=== A particular construction ===
 
The construction of the horizontal bar of the T is based on the rigid [[coaxial cable]] with air as dielectric. The radius is chosen to be as large as possible without allowing higher modes.
The design of a bias "T" is based upon power going out to the remote device, but not being seen by the base station or receiver.  It does this by using a capacitor on the RF output terminal, effectively creating an open circuit for the DC current.<ref name="patent6229408">{{citation | inventor1-last = Jovanovich | inventor1-first = Alan | inventor2-last = Lam | inventor2-first = For Sander | title =  Zero loss bias "T" | issue-date = 2001 | patent-number = 6229408 | country-code = US}}</ref>  The incoming RF signal, or the one from the antenna, is the output for the DC power.  This front-end of a bias "T" typically consists of a [[bandpass filter]], a low noise [[amplifier]], and a mixer coupled to a local oscillator.<ref name="patent6229408" />
 
==== Capacitor ====
At one point a small slice is cut out of the center conductor, therefore a capacitor is formed and low frequencies are blocked. This kind of capacitor has the advantage that it is nearly invisible to higher frequencies. To pass frequencies down to 1&nbsp;MHz the capacitance has to be increased. A dielectric like NPO multiplies the capacitance by a factor of 65. The thickness of the capacitor has to be minimal without leading to electric breakdown in the dielectric, this means to avoid any peaks in the electric field and this means smooth electrodes with rounded edges and a dielectric protruding between the electrodes (doorknob design). A stack of capacitors can be used, but every capacitor needs access to the surface of the inner conductor, because if it's hidden behind another capacitor the high frequencies won't see it, because the electric field needs a lot of time to travel through a dielectric with a high dielectric constant
 
==== Coil ====
A small coil made of fine wire with an air core or MnFeZn-core connects the inner conductor of one of the sides of the capacitor with the a port in the outer conductor leading down the T.  Frequencies above 1&nbsp;GHz hit the coil from the side and apply an equal electric field to the whole coil. Therefore no higher modes are excited within the coil. Because of the inductiveness of the coil almost no current leaks from the center conductor to the port. Frequencies between 1&nbsp;MHz and 1&nbsp;GHz do leak into this port, so there is a second coil with a cone shaped core outside of the outer conductor, but inside of a housing to avoid interference with other components. This cone acts like a tapered transmission line transformer. It starts with a high impedance, so a lot of power will be reflected, but the rest will travel down the coil and there is some leakage into the low frequency port.
 
==== Oscillations ====
Any oscillations in the capacitor or the coil or the composed [[LC circuit]] are damped by the dielectric and the core. Also the small coil should have about 10 ohm resistance to further damp oscillations and avoid ripple on the transmitted spectrum.
 
==See also==
* [[Diplexer]]
 
== References ==
{{reflist|1}}
 
*{{Citation
|last=Andrews
|first=James R.
|title= Broadband Coaxial Bias Tees
|date= November 2000
|publisher= Picosecond Pulse Labs
|series= Application Note
|volume= AN-1e
|location= Boulder, CO
|url= http://www.picosecond.com/objects/AN-01e.pdf
|doi=}}
 
== External links ==
* [[S parameters]] for a bias T [http://qucs.sourceforge.net/tech/node47.html]
 
[[Category:Multiplexing]]
[[Category:Microwave technology]]

Revision as of 02:29, 28 February 2014

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