Editing Electronic oscillator (section)

===Frequency of oscillation - the Barkhausen criterion===
{{main|Barkhausen stability criterion}}
{{multiple image
| align = right
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| header   =
| image1   = Oscillator diagram1.svg 
| image2   = Oscillator diagram2.svg
| width    = 150
| footer   = To determine the [[loop gain]], the [[feedback loop]] of the oscillator ''(left)'' is considered to be broken at some point ''(right)''.
}}

To determine the frequency(s) <math>\omega_0\;=\;2\pi f_0</math> at which a feedback oscillator circuit will oscillate, the [[feedback loop]] is thought of as broken at some point (see diagrams) to give an input and output port (for accuracy, the output port must be terminated with an impedance equal to the input port).    A sine wave is applied to the input <math>v_i(t) = V_ie^{j\omega t}</math> and the amplitude and phase of the sine wave after going through the loop <math>v_o = V_o e^{j(\omega t + \phi)}</math> is calculated<ref name="Sobot">{{cite book
 | last1  = Sobot
 | first1 = Robert 
 | title  = Wireless Communication Electronics: Introduction to RF Circuits and Design Techniques
 | publisher = Springer Science and Business Media
 | date   = 2012
 | location = 
 | pages  = 221–222
 | language = 
 | url    = https://books.google.com/books?id=SdGaiV6iup0C&dq=oscillator+gain+phase+barkhausen&pg=PA221
 | doi    = 
 | id     = 
 | isbn   = 978-1461411161
 }}</ref><ref name="Carr">{{cite book
 | last1  = Carr
 | first1 = Joe 
 | title  = RF Components and Circuits
 | publisher = Newnes
 | date   = 2002
 | location = 
 | pages  = 125–126
 | language = 
 | url    = https://books.google.com/books?id=V9gBTNvt3zIC&dq=barkhausen+inverting*180+%22phase+shift%22&pg=PA126
 | doi    = 
 | id     = 
 | isbn   = 0080498078
 }}</ref>
:<math>v_o = A v_f\,</math> &nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; <math>v_f = \beta(j\omega) v_i \,</math> &nbsp;&nbsp;&nbsp;&nbsp; so &nbsp;&nbsp;&nbsp;&nbsp; <math>v_o = A\beta(j\omega) v_i\,</math>
Since in the complete circuit <math>v_o</math> is connected to <math>v_i</math>, for oscillations to exist
:<math>v_o(t) = v_i(t)</math>
The ratio of output to input of the loop, <math>{v_o \over v_i} = A\beta(j\omega)</math>, is called the [[loop gain]]. So the condition for oscillation is that the loop gain must be one<ref name="Gonzalez" >{{cite book
 | last1  = Gonzalez
 | first1 = Guillermo 
 | title  = Foundations of Oscillator Circuit Design
 | publisher = Artech House
 | date   = 2006
 | location = 
 | url    = http://www.artechhouse.com/uploads/public/documents/chapters/Gonzalez-162_CH01.pdf
 | doi    = 
 | id     = 
 | isbn   = 9781596931633
 }}</ref>{{rp|p.3–5}}<ref name="Carr" /><ref name="Maas2">{{cite book
 | last1  = Maas
 | first1 = Stephen A.
 | title  = Nonlinear Microwave and RF Circuits
 | publisher = Artech House
 | date   = 2003
 | location = 
 | pages  = 537–540
 | language = 
 | url    = https://books.google.com/books?id=SSw6gWLG-d4C&dq=gain+phase&pg=PA537
 | doi    = 
 | id     = 
 | isbn   = 1580536115
 }}</ref><ref name="Lesurf">{{cite web
  | last = Lesurf
  | first = Jim
  | title = Feedback Oscillators
  | work = The Scots Guide to Electronics
  | publisher = School of Physics and Astronomy, Univ. of St. Andrewes, Scotland
  | date = 2006
  | url = https://www.st-andrews.ac.uk/~www_pa/Scots_Guide/RadCom/part4/page1.html
  | doi = 
  | access-date = 28 September 2015}}</ref>
:<math>A\beta(j\omega_0) = 1\,</math>
Since <math>A\beta(j\omega) </math> is a [[complex number]] with two parts, a [[Magnitude of Complex Number|magnitude]] and an angle, the above equation actually consists of two conditions:<ref name="Razavi">{{cite book
  | last = Razavi
  | first = Behzad
  | title = Design of Analog CMOS Integrated Circuits
  | publisher = The McGraw-Hill Companies
  | date = 2001
  | location =
  | pages = 482–484
  | url = https://books.google.com/books?id=hl6JZ8DKlFwC&dq=Barkhausen&pg=PA483
  | doi =
  | id =
  | isbn = 7302108862}}</ref><ref name="Lesurf" /><ref name="Carr" />
*The magnitude of the [[gain (electromagnetics)|gain]] ([[amplifier|amplification]]) around the loop at ω<sub>0</sub> must be unity
::<math>|A||\beta(j\omega_0)| = 1\, \qquad\qquad\qquad\qquad\qquad\qquad  \text{(1)} </math>
:so that after a trip around the loop the sine wave is the same [[amplitude]]. It can be seen intuitively that if the [[loop gain]] were greater than one, the amplitude of the sinusoidal signal would increase as it travels around the loop, resulting in a sine wave that [[exponential growth|grows exponentially]] with time, without bound.<ref name="Schubert">{{cite book
 | last1  = Schubert
 | first1 = Thomas F. Jr.
 | last2  = Kim
 | first2 = Ernest M.
 | title  = Fundamentals of Electronics. Book 4: Oscillators and Advanced Electronics Topics
 | publisher = Morgan and Claypool
 | date   = 2016
 | location = 
 | pages  = 926–928
 | language = 
 | url    = https://books.google.com/books?id=uNQlDAAAQBAJ&dq=nonlinearity&pg=PA927
 | doi    = 
 | id     = 
 | isbn   = 978-1627055697
 }}</ref>  If the loop gain were less than one, the signal would decrease around the loop, resulting in an exponentially decaying sine wave that decreases to zero.
*The sine wave at the end of the loop must be [[in phase]] with the wave at the beginning of the loop.<ref name="Carr" />  Since the sine wave is [[periodic function|periodic]] and repeats every 2π radians, this means that the [[phase shift]] around the loop at the oscillation frequency ω<sub>0</sub>  must be zero or a multiple of 2π [[radian]]s (360°)
::<math>\angle A + \angle \beta = 2 \pi n \qquad n \in 0, 1, 2...  \, \qquad\qquad \text{(2)}</math> 
Equations (1) and (2) are called the ''[[Barkhausen stability criterion]]''.<ref name="Lesurf" /><ref name="Gonzalez" />{{rp|p.3–5}}  It is a necessary but not a sufficient criterion for oscillation, so there are some circuits which satisfy these equations that will not oscillate.  An equivalent condition often used instead of the Barkhausen condition is that the circuit's [[closed loop transfer function]] (the circuit's complex [[electrical impedance|impedance]] at its output) have a pair of [[pole (complex analysis)|pole]]s on the [[imaginary axis]].

In general, the phase shift of the feedback network increases with increasing frequency so there are only a few discrete frequencies (often only one) which satisfy the second equation.<ref name="Lesurf" /><ref name="Schubert" /> If the amplifier gain <math>A</math>  is high enough that the loop gain is unity (or greater, see Startup section) at one of these frequencies, the circuit will oscillate at that frequency. Many amplifiers such as common-emitter [[transistor]] circuits are "inverting", meaning that their output voltage decreases when their input increases.<ref name="Razavi" /><ref name="Carr" />  In these the amplifier provides 180° [[phase shift]], so the circuit will oscillate at the frequency at which the feedback network provides the other 180° phase shift.<ref name="Gonzalez" />{{rp|p.3–5}}<ref name="Carr" />

At frequencies well below the [[pole (complex analysis)|pole]]s of the amplifying device, the amplifier will act as a pure gain <math>A</math>, but if the oscillation frequency <math>\omega_0</math> is near the amplifier's [[cutoff frequency]] <math>\omega_C</math>, within <math>0.1\omega_C</math>, the active device can no longer be considered a 'pure gain', and it will contribute some [[phase shift]] to the loop.<ref name="Gonzalez" />{{rp|p.3–5}}<ref name="Carter">{{cite book
 | last1  = Carter
 | first1 = Bruce
 | last2  = Mancini
 | first2 = Ron
 | title  = Op Amps for Everyone, 3rd Ed.
 | publisher = Elsevier
 | date   = 2009
 | location = 
 | pages  = 
 | language = 
 | url    = https://books.google.com/books?id=nnCNsjpicJIC&pg=PA346
 | doi    = 
 | id     = 
 | isbn   = 9781856175050
 }}</ref>{{rp|p.345–347}}

An alternate mathematical stability test sometimes used instead of the Barkhausen criterion is the [[Nyquist stability criterion]].<ref name="Gonzalez" />{{rp|p.6–7}}  This has a wider applicability than the Barkhausen, so it can identify some of the circuits which pass the Barkhausen criterion but do not oscillate.