Editing Electronic oscillator (section)

===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.