Editing Electronic oscillator (section)

==Theory of feedback oscillators==
A feedback oscillator circuit consists of two parts connected in a [[feedback loop]]; an [[amplifier]] <math>A</math> and an [[electronic filter]] <math>\beta(j\omega)</math>.<ref name="Rhea" />{{rp|p.1}}  The filter's purpose is to limit the frequencies that can pass through the loop so the circuit only oscillates at the desired frequency.<ref name="Schubert" />  Since the filter and wires in the circuit have [[electrical resistance|resistance]] they consume energy and the amplitude of the signal drops as it passes through the filter.  The amplifier is needed to increase the amplitude of the signal to compensate for the energy lost in the other parts of the circuit, so the loop will oscillate, as well as supply energy to the load attached to the output.

===Frequency of oscillation - the Barkhausen criterion===
{{main|Barkhausen stability criterion}}
{{multiple image
| align = right
| direction = horizontal
| 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.

===Frequency stability===
Temperature changes, other environmental changes, aging, and manufacturing tolerances will cause component values to "drift" away from their designed values.<ref name="Stephan1">{{cite book
 | last1  = Stephan
 | first1 = Karl 
 | title  = Analog and Mixed-Signal Electronics
 | publisher = John Wiley and Sons
 | date   = 2015
 | location = 
 | pages  = 192–193
 | language = 
 | url    = https://books.google.com/books?id=cDAABwAAQBAJ&pg=PA192
 | doi    = 
 | id     = 
 | isbn   = 978-1119051800
 }}</ref><ref name="Vidkjaer">{{cite web
  | last = Vidkjaer
  | first = Jens
  | title = Ch. 6: Oscillators
  | work = Class Notes: 31415 RF Communications Circuits
  | publisher = Technical Univ. of Denmark
  | date = 
  | url = http://rftoolbox.dtu.dk/book/Ch6.pdf
  | doi =
  | access-date = October 8, 2015}} p. 8-9</ref> Changes in ''frequency determining'' components such as the [[tank circuit]] in LC oscillators will cause the oscillation frequency to change, so for a constant frequency these components must have stable values. How stable the oscillator's frequency is to other changes in the circuit, such as changes in values of other components, gain of the amplifier, the load impedance, or the supply voltage, is mainly dependent on the [[Q factor]] ("quality factor") of the feedback filter.<ref name="Stephan1" /> Since the ''amplitude'' of the output is constant due to the nonlinearity of the amplifier (see Startup section below), changes in component values cause changes in the phase shift <math>\phi\;=\;\angle A\beta(j\omega)</math> of the feedback loop. Since oscillation can only occur at frequencies where the phase shift is a multiple of 360°, <math>\phi\;=\;360n^\circ</math>, shifts in component values cause the oscillation frequency <math>\omega_0</math> to change to bring the loop phase back to 360n°. The amount of frequency change <math>\Delta \omega</math> caused by a given phase change <math>\Delta \phi</math> depends on the slope of the loop phase curve at <math>\omega_0</math>, which is determined by the <math>Q </math><ref name="Stephan1" /><ref name="Vidkjaer" /><ref name="Huijsing">{{cite book
 | last1  = Huijsing
 | first1 = Johan
 | last2  = van de Plassche
 | first2 = Rudy J.
 | last3  = Sansen
 | first3 = Willy
 | title  = Analog Circuit Design
 | publisher = Springer Scientific and Business Media
 | date   = 2013
 | location = 
 | pages  = 77
 | language = 
 | url    = https://books.google.com/books?id=B8fSBwAAQBAJ&pg=PA77
 | doi    = 
 | id     = 
 | isbn   = 978-1475724622
 }}</ref>
<ref name="Kazimierczuk">{{cite book
 | last1  = Kazimierczuk
 | first1 = Marian K. 
 | title  = RF Power Amplifiers, 2nd Ed.
 | publisher = John Wiley and Sons
 | date   = 2014
 | location = 
 | pages  = 586–587
 | language = 
 | url    = https://books.google.com/books?id=-U7YBAAAQBAJ&pg=PA587
 | doi    = 
 | id     = 
 | isbn   = 978-1118844335
 }}</ref>
:<math>{d\phi \over d\omega}\Bigg|_{\omega_0} = -{2Q \over \omega_0}\,</math> &nbsp;&nbsp;&nbsp;&nbsp; so &nbsp;&nbsp;&nbsp;&nbsp;  <math>\Delta \omega = -{\omega_0 \over 2Q}\Delta \phi \,</math>
RC oscillators have the equivalent of a very low <math>Q</math>, so the phase changes very slowly with frequency, therefore a given phase change will cause a large change in the frequency. In contrast, LC oscillators have [[tank circuit]]s with high <math>Q</math> (~10<sup>2</sup>).  This means the phase shift of the feedback network increases rapidly with frequency near the [[resonant frequency]] of the tank circuit.<ref name="Stephan1" /> So a large change in phase causes only a small change in frequency.  Therefore, the circuit's oscillation frequency is very close to the natural resonant frequency of the [[tuned circuit]], and doesn't depend much on other components in the circuit.  The quartz crystal resonators used in [[crystal oscillator]]s have even higher <math>Q</math> (10<sup>4</sup> to 10<sup>6</sup>)<ref name="Kazimierczuk" /> and their frequency is very stable and independent of other circuit components.

===Tunability===
The frequency of RC and LC oscillators can be tuned over a wide range by using variable components in the filter.  A [[microwave cavity]] can be tuned mechanically by moving one of the walls. In contrast, a [[crystal oscillator|quartz crystal]] is a mechanical [[resonator]] whose [[resonant frequency]] is mainly determined by its dimensions, so a crystal oscillator's frequency is only adjustable over a very narrow range, a tiny fraction of one percent.{{sfn|Gottlieb|1997|p=39-40}}<ref name="Froehlich">{{cite book
 | last1  = Froehlich
 | first1 = Fritz E.
 | last2  =  Kent
 | first2 = Allen
 | title  = The Froehlich/Kent Encyclopedia of Telecommunications, Volume 3
 | publisher = CRC Press
 | date   = 1991
 | location = 
 | pages  = 448
 | language = 
 | url    = https://books.google.com/books?id=QQcfD_iWlPYC&dq=%22crystal+oscillator%22+tuning+stiffness&pg=PA448
 | doi    = 
 | id     = 
 | isbn   = 0824729021
 }}</ref><ref name="Misra">{{cite book
 | last1  = Misra
 | first1 = Devendra 
 | title  = Radio-Frequency and Microwave Communication Circuits: Analysis and Design
 | publisher = John Wiley
 | date   = 2004
 | location = 
 | pages  = 494
 | language = 
 | url    = https://books.google.com/books?id=7nWN_pGQKnMC&dq=%22crystal+oscillator%22+pulling+%22tuning+range%22&pg=PA494
 | doi    = 
 | id     = 
 | isbn   = 0471478733
 }}</ref><ref name="Terman">{{cite book
 | last1  = Terman
 | first1 = Frederick E.
 | title  = Radio Engineer's Handbook
 | publisher = McGraw-Hill
 | date   = 1943
 | location = 
 | pages  = 497
 | language = 
 | url    = http://www.itermoionici.it/letteratura_files/Radio-Engineers-Handbook.pdf
 | doi    = 
 | id     = 
 | isbn   =
 }}</ref><ref name="FrequencyManagement">{{cite web
  | title = Oscillator Application Notes
  | work = Support
  | publisher = Frequency Management International, CA
  | date = 
  | url = http://www.frequencymanagement.com/web_pdfs/cat_pdfs_applicationNotes/45-49.PDF
  | format = 
  | doi = 
  | access-date = October 1, 2015}}</ref>
<ref name="Scroggie">{{cite book
 | last1  = Scroggie
 | first1 = M. G.
 | last2  = Amos
 | first2 = S. W.
 | title  = Foundations of Wireless and Electronics
 | publisher = Elsevier
 | date   = 2013
 | location = 
 | pages  = 241–242
 | language = 
 | url    = https://books.google.com/books?id=ihABBQAAQBAJ&dq=%22fixed+frequency%22+%22hardly+anything+about+it+that+can+vary%22&pg=PA242
 | doi    = 
 | id     = 
 | isbn   = 978-1483105574
 }}</ref>
<ref name="Vig">Vig, John R. and Ballato, Arthur "Frequency Control Devices" in {{cite book
 | last1  = Thurston
 | first1 = R. N. 
 | last2  = Pierce
 | first2 = Allan D.
 | last3  = Papadakis
 | first3 = Emmanuel P.
 | title  = Reference for Modern Instrumentation, Techniques, and Technology: Ultrasonic Instruments and Devices II
 | publisher = Elsevier 
 | date   = 1998
 | location = 
 | pages  = 227
 | language = 
 | url    = https://books.google.com/books?id=0lvCkB6y4dwC&dq=%22crystal+oscillator%22+tuning+stiffness&pg=PA227
 | doi    = 
 | id     = 
 | isbn   = 0080538916
 }}</ref>   Its frequency can be changed slightly by using a [[trimmer capacitor]] in series or parallel with the crystal.{{sfn|Gottlieb|1997|p=39-40}}

===Startup and amplitude of oscillation===
The [[Barkhausen stability criterion|Barkhausen criterion]] above, eqs. (1) and (2), merely gives the frequencies at which steady-state oscillation is possible, but says nothing about the amplitude of the oscillation, whether the amplitude is stable, or whether the circuit will start oscillating when the power is turned on.<ref name="Stephan2">{{cite book
 | last1  = Stephan
 | first1 = Karl 
 | title  = Analog and Mixed-Signal Electronics
 | publisher = John Wiley and Sons
 | date   = 2015
 | location = 
 | pages  = 187–188
 | language = 
 | url    = https://books.google.com/books?id=cDAABwAAQBAJ&pg=PA188
 | doi    = 
 | id     = 
 | isbn   = 978-1119051800
 }}</ref><ref name="Gonzalez" />{{rp|p.5}}<ref name="ECE3434">{{cite web
  | title = Sinusoidal Oscillators
  | work = Course notes: ECE3434 Advanced Electronic Circuits
  | publisher = Electrical and Computer Engineering Dept., Mississippi State University
  | date = Summer 2015
  | url = http://courses.ece.msstate.edu/ece3434/notes/oscillators/Oscillator.doc
  | format = DOC
  | doi = 
  | access-date = September 28, 2015}}, p. 4-7</ref>  For a practical oscillator two additional requirements are necessary:
*In order for oscillations to start up in the circuit from zero, the circuit must have "excess gain"; the loop gain for small signals must be greater than one at its oscillation frequency<ref name="Lesurf" /><ref name="Schubert" /><ref name="Razavi" /><ref name="Gonzalez" />{{rp|p.3–5}}<ref name="ECE3434" /> 
::<math>|A\beta(j\omega_0)| > 1\,</math>
*For stable operation, the feedback loop must include a [[linear circuit|nonlinear]] component which reduces the gain back to unity as the amplitude increases to its operating value.<ref name="Lesurf" /><ref name="Schubert" />
A typical rule of thumb is to make the small signal loop gain at the oscillation frequency 2 or 3.<ref name="Rhea">{{cite book
 | last1  = Rhea
 | first1 = Randall W.
 | title  = Discrete Oscillator Design: Linear, Nonlinear, Transient, and Noise Domains
 | publisher = Artech House 
 | date   = 2014
 | location = 
 | language = 
 | url    = https://books.google.com/books?id=4Op56QdHFPUC&pg=PA11
 | doi    = 
 | id     = 
 | isbn   = 978-1608070480
 }}</ref>{{rp|p=11}}<ref name="Razavi" />  When the power is turned on, oscillation is started by the power turn-on transient or random [[electronic noise]] present in the circuit.<ref name="Gonzalez" />{{rp|p.5}}{{sfn|Gottlieb|1997|p=113–114}} Noise guarantees that the circuit will not remain "balanced" precisely at its unstable DC equilibrium point ([[Q point]]) indefinitely. Due to the narrow passband of the filter, the response of the circuit to a noise pulse will be sinusoidal, it will excite a small sine wave of voltage in the loop. Since for small signals the loop gain is greater than one, the amplitude of the sine wave increases exponentially.<ref name="Lesurf" /><ref name="Schubert" />

During startup, while the amplitude of the oscillation is small, the circuit is approximately [[Linear circuit|linear]], so the analysis used in the Barkhausen criterion is applicable.<ref name="Rhea" />{{rp|p=144,146}}  When the amplitude becomes large enough that the amplifier becomes [[Linear circuit|nonlinear]], generating harmonic distortion, technically the [[frequency domain]] analysis used in normal amplifier circuits is no longer applicable, so the "gain" of the circuit is undefined.  However the filter attenuates the harmonic components produced by the nonlinearity of the amplifier, so the fundamental frequency component <math>\sin \omega_0 t</math> mainly determines the loop gain<ref name="Toumazou">{{cite book
 | last1  = Toumazou
 | first1 = Chris
 | last2  = Moschytz
 | first2 = George S.
 | last3  = Gilbert
 | first3 = Barrie
 | title  = Trade-Offs in Analog Circuit Design: The Designer's Companion, Part 1
 | publisher = Springer Science and Business Media
 | date   = 2004
 | location = 
 | pages  = 565–566
 | language = 
 | url    = https://books.google.com/books?id=VoBIOvirkiMC&dq=nonlinear&pg=PA565
 | doi    = 
 | id     = 
 | isbn   = 9781402080463
 }}</ref> (this is the "[[harmonic balance]]" analysis technique for nonlinear circuits).

The sine wave cannot grow indefinitely; in all real oscillators some nonlinear process in the circuit limits its amplitude,<ref name="Lesurf" /><ref name="Roberge">{{cite book
 | last1  = Roberge
 | first1 = James K. 
 | title  = Operational Amplifiers: Theory and Practice
 | publisher = John Wiley and Sons
 | date   = 1975
 | location = 
 | pages  = 487–488
 | language = 
 | url    = http://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/textbook/MITRES_6-010S13_chap12.pdf
 | doi    = 
 | id     = 
 | isbn   =  0471725854
 }}</ref>{{sfn|Gottlieb|1997|p=120}} reducing the gain as the amplitude increases, resulting in stable operation at some constant amplitude.<ref name="Lesurf" />   In most oscillators this nonlinearity is simply the [[Limiting|saturation]]  (limiting or [[clipping (signal processing)|clipping]]) of the amplifying device, the [[transistor]], [[vacuum tube]] or [[op-amp]].<ref name="Tang">{{cite book
 | last1  = van der Tang
 | first1 = J.
 | last2  = Kasperkovitz
 | first2 = Dieter
 | last3  = van Roermund
 | first3 = Arthur
 | title  = High-Frequency Oscillator Design for Integrated Transceivers
 | publisher = Springer Science and Business Media
 | date   = 2006
 | location = 
 | pages  = 51
 | language = 
 | url    = https://books.google.com/books?id=0rniokw7bLkC&dq=%22amplitude+stabilization%22+self-limiting&pg=PT51
 | doi    = 
 | id     = 
 | isbn   = 0306487160
 }}</ref><ref name="Razavi2">[https://books.google.com/books?id=hl6JZ8DKlFwC&pg=PA487&dq=saturation+nonlinearity+%22amplifier+limiting%22   Razavi, Behzad (2001)  ''Design of Analog CMOS Integrated Circuits'', p. 487-489]</ref><ref name="Gonzalez" />{{rp|p.5}}  The maximum voltage swing of the amplifier's output is limited by the DC voltage provided by its power supply.  Another possibility is that the output may be limited by the amplifier [[slew rate]].

As the amplitude of the output nears the [[power supply]] voltage rails, the amplifier begins to saturate on the peaks (top and bottom) of the sine wave, flattening or "[[clipping (signal processing)|clipping]]" the peaks.<ref name="Carter" />  To achieve the maximum amplitude sine wave output from the circuit, the amplifier should be [[bias (electrical engineering)|bias]]ed midway between its clipping levels.  For example, an op amp should be biased midway between the two supply voltage rails.  A common-emitter transistor amplifier's collector voltage should be biased midway between cutoff and saturation levels.   

Since the output of the amplifier can no longer increase with increasing input, further increases in amplitude cause the equivalent gain of the amplifier and thus the loop gain to decrease.<ref name="ECE3434" />  The amplitude of the sine wave, and the resulting clipping, continues to grow until the loop gain is reduced to unity, <math>|A\beta(j\omega_0)|\;=\;1\,</math>, satisfying the Barkhausen criterion, at which point the amplitude levels off and [[steady state]] operation is achieved,<ref name="Lesurf" /> with the output a slightly distorted sine wave with peak amplitude determined by the supply voltage.  This is a stable equilibrium; if the amplitude of the sine wave increases for some reason, increased clipping of the output causes the loop gain <math>|A\beta(j\omega_0)|</math> to drop below one temporarily, reducing the sine wave's amplitude back to its unity-gain value.  Similarly if the amplitude of the wave decreases, the decreased clipping will cause the loop gain to increase above one, increasing the amplitude.

The amount of [[harmonic distortion]] in the output is dependent on how much excess loop gain the circuit has:<ref name="ECE3434" /><ref name="Rhea" />{{rp|p=12}}<ref name="Carter" /><ref name="Schubert" />
*If the small signal loop gain is made close to one, just slightly greater, the output waveform will have minimum distortion, and the frequency will be most stable and independent of supply voltage and load impedance.  However, the oscillator may be slow starting up, and a small decrease in gain due to a variation in component values may prevent it from oscillating.
*If the small signal loop gain is made significantly greater than one, the oscillator starts up faster, but more severe clipping of the sine wave occurs, and thus the resulting distortion of the output waveform increases.  The oscillation frequency becomes more dependent on the supply voltage and current drawn by the load.<ref name="Carter" /> 
An exception to the above are high [[Q factor|Q]] oscillator circuits such as [[crystal oscillator]]s; the narrow bandwidth of the crystal removes the harmonics from the output, producing a 'pure' sinusoidal wave with almost no distortion even with large loop gains.

===Design procedure===
Since oscillators depend on nonlinearity for their operation, the usual linear [[frequency domain]] circuit analysis techniques used for amplifiers based on the [[Laplace transform]], such as [[root locus]] and gain and phase plots ([[Bode plot]]s), cannot capture their full behavior.<ref name="Stephan2" />  To determine startup and transient behavior and calculate the detailed shape of the output waveform, [[electronic circuit simulation]] computer programs like [[SPICE]] are used.<ref name="Stephan2" />  A typical design procedure for oscillator circuits is to use linear techniques such as the [[Barkhausen stability criterion]] or [[Nyquist stability criterion]] to design the circuit, use a rule of thumb to choose the loop gain, then simulate the circuit on computer to make sure it starts up reliably and to determine the nonlinear aspects of operation such as harmonic distortion.<ref name="Schubert" /><ref name="Stephan2" />  Component values are tweaked until the simulation results are satisfactory.  The distorted oscillations of real-world (nonlinear) oscillators are called [[limit cycle]]s and are studied in [[nonlinear control theory]].

===Amplitude-stabilized oscillators===
In applications where a 'pure' very low [[harmonic distortion|distortion]] sine wave is needed, such as precision [[signal generator]]s,  a nonlinear component is often used in the feedback loop that provides a 'slow' gain reduction with amplitude.  This stabilizes the loop gain at an amplitude below the saturation level of the amplifier, so it does not saturate and "clip" the sine wave.  Resistor-diode networks and [[field effect transistor|FETs]] are often used for the nonlinear element.  An older design uses a [[thermistor]] or an ordinary [[incandescent light bulb]]; both provide a resistance that increases with temperature as the current through them increases.

As the amplitude of the signal current through them increases during oscillator startup, the increasing resistance of these devices reduces the loop gain.  The essential characteristic of all these circuits is that the nonlinear gain-control circuit must have a long [[time constant]], much longer than a single [[frequency|period]] of the oscillation.  Therefore, over a single cycle they act as virtually linear elements, and so introduce very little distortion.  The operation of these circuits is somewhat analogous to an [[automatic gain control]] (AGC) circuit in a radio receiver.  The [[Wein bridge oscillator]] is a widely used circuit in which this type of gain stabilization is used.<ref name="Mancini">{{cite book
  | last = Mancini
  | first = Ron 
  | title = Op Amps for Everyone: Design Reference
  | publisher = Newnes
  | date = 2003
  | location = 
  | pages = 247–251
  | language = 
  | url = https://books.google.com/books?id=0zqU01lKPCEC&dq=wein+bridge+oscillator&pg=PA247
  | archive-url= 
  | archive-date=
  | doi = 
  | id = 
  | isbn = 9780750677011
  | mr = 
  | zbl = 
  | jfm =}}</ref>

===Frequency limitations===
At high frequencies it becomes difficult to physically implement feedback oscillators because of shortcomings of the components.  Since at high frequencies the tank circuit has very small capacitance and inductance, [[parasitic capacitance]] and [[parasitic inductance]] of component leads and PCB traces become significant.  These may create unwanted feedback paths between the output and input of the active device, creating instability and oscillations at unwanted frequencies ([[parasitic oscillation]]). Parasitic feedback paths inside the active device itself, such as the interelectrode capacitance between output and input, make the device unstable. The [[input impedance]] of the active device falls with frequency, so it may load the feedback network.  As a result, stable feedback oscillators are difficult to build for frequencies above 500&nbsp;MHz, and negative resistance oscillators are usually used for frequencies above this.