Editing Electronic oscillator

{{Short description|Type of electronic circuit}}

[[File:Animated schmitt-trigger-oscillator.gif|thumb|Simple [[relaxation oscillator]] made by [[Feedback|feeding back]] an inverting [[Schmitt trigger]]'s output voltage through a [[RC network]] to its input.]]

An '''electronic oscillator''' is an [[electronic circuit]] that produces a periodic, [[oscillation|oscillating]] or [[alternating current]] (AC) signal, usually a [[sine wave]], [[Square wave (waveform)|square wave]] or a [[triangle wave]],<ref name="Snelgrove">{{cite web
 | last = Snelgrove
 | first = Martin
 | title = Oscillator
 | website = McGraw-Hill Encyclopedia of Science and Technology, 10th Ed., Science Access online service
 | publisher = McGraw-Hill
 | year = 2011
 | url = http://accessscience.com/abstract.aspx?id=477900&referURL=http%3a%2f%2faccessscience.com%2fcontent.aspx%3fid%3d477900
 | access-date = March 1, 2012
 | archive-url = https://web.archive.org/web/20130719125711/http://accessscience.com/abstract.aspx?id=477900&referURL=http%3A%2F%2Faccessscience.com%2Fcontent.aspx%3Fid%3D477900
 | archive-date = July 19, 2013
 | url-status = dead
 }}</ref><ref name="Chattopadhyay">{{cite book
 | last = Chattopadhyay
 | first = D. 
 | title = Electronics (fundamentals And Applications)
 | publisher = New Age International
 | year = 2006
 | pages = 224–225
 | url = https://books.google.com/books?id=n0rf9_2ckeYC&q=%22negative+resistance%22&pg=PA224
 | isbn = 978-81-224-1780-7}}</ref><ref name=":0">{{Cite book|title=The Art of Electronics|last1=Horowitz|first1=Paul|last2=Hill|first2=Winfield|year=2015|isbn=978-0-521-80926-9|location=USA|pages=425}}</ref> powered by a [[direct current]] (DC) source. Oscillators are found in many electronic devices, such as [[radio receiver]]s, [[television set]]s, radio and television [[radio transmitter|broadcast transmitters]], [[computer]]s, [[computer peripheral]]s, [[mobile phone|cellphones]], [[radar]], and many other devices.<ref name="Snelgrove" />

Oscillators are often characterized by the [[frequency]] of their output signal:
*A [[Low-frequency oscillation|low-frequency oscillator]] (LFO) is an oscillator that generates a frequency below approximately 20&nbsp;Hz. This term is typically used in the field of audio [[synthesizer]]s, to distinguish it from an audio frequency oscillator.
*An audio oscillator<!--Please don't link - circular reference--> produces frequencies in the [[audio frequency|audio]] range, 20&nbsp;Hz to 20&nbsp;kHz.<ref name="Chattopadhyay" />
*A [[radio frequency]] (RF) oscillator produces signals above the audio range, more generally in the range of 100&nbsp;kHz to 100&nbsp;GHz.<ref name="Chattopadhyay" />

[[File:Integrierter Quarzoszillator (smial).jpg|thumb|[[Crystal oscillator]]|118x118px]]

There are two general types of electronic oscillators: the '''linear''' or '''harmonic oscillator''', and the '''nonlinear''' or '''[[relaxation oscillator]]'''.<ref name="Chattopadhyay" /><ref name="Garg">{{cite book
 | last = Garg
 | first = Rakesh Kumar
 | author2=Ashish Dixit |author3=Pavan Yadav
 | title = Basic Electronics
 | publisher = Firewall Media
 | year = 2008
 | pages = 280
 | url = https://books.google.com/books?id=9SOdnsHA2IYC&pg=PA280
 | isbn = 978-8131803028}}</ref>  The two types are fundamentally different in how oscillation is produced, as well as in the characteristic type of output signal that is generated.

The most-common linear oscillator in use is the [[crystal oscillator]], in which the output frequency is controlled by a [[piezo-electric]] [[resonator]] consisting of a vibrating [[quartz crystal]].  Crystal oscillators are ubiquitous in modern electronics, being the source for the [[clock signal]] in computers and digital watches, as well as a source for the signals generated in radio transmitters and receivers.  As a crystal oscillator's “native” output waveform is [[sinusoidal]], a signal-conditioning circuit may be used to convert the output to other waveform types, such as the [[Square wave (waveform)|square wave]] typically utilized in computer clock circuits.

==Harmonic oscillators==
[[File:Oscillator diagram1.svg|thumb|Block diagram of a feedback linear oscillator; an amplifier ''A'' with its output ''v<sub>o</sub>'' fed back into its input ''v<sub>f</sub>'' through a [[electronic filter|filter]], ''β(jω)''.]]

{{anchor|linear oscillator}}

[[Linear circuit|'''Linear''']] or '''harmonic oscillators''' generate a [[sinusoidal]] (or nearly-sinusoidal) signal. There are two types:

===Feedback oscillator===
The most common form of linear oscillator is an [[electronic amplifier]] such as a [[transistor]] or [[operational amplifier]] connected in a [[feedback loop]] with its output fed back into its input through a frequency selective [[electronic filter]] to provide [[positive feedback]].  When the power supply to the amplifier is switched on initially, [[electronic noise]] in the circuit provides a non-zero signal to get oscillations started.{{sfn|Gottlieb|1997|p=113–114}} The noise travels around the loop and is amplified and [[Filter (signal processing)|filtered]] until very quickly it converges on a [[sine wave]] at a single frequency.

Feedback oscillator circuits can be classified according to the type of frequency selective filter they use in the feedback loop:<ref name="Chattopadhyay" /><ref name="Garg" />

*In an ''[[RC oscillator]]'' circuit, the filter is a network of [[resistor]]s and [[capacitor]]s.<ref name="Chattopadhyay" /><ref name="Garg" /> RC oscillators are mostly used to generate lower frequencies, for example in the audio range.  Common types of RC oscillator circuits are the [[phase shift oscillator]] and the [[Wien bridge oscillator]]. LR oscillators, using [[inductor]] and resistor filters also exist, however they are much less common due to the required size of an inductor to achieve a value appropriate for use at lower frequencies.

{{anchor|LC oscillator}}

[[File:Oscillator comparison.svg|thumb|right|400px|Two common LC oscillator circuits, the Hartley and Colpitts oscillators]]

*In an ''[[LC circuit| LC oscillator]]'' circuit, the filter is a [[tuned circuit]] (often called a ''tank circuit'') consisting of an [[inductor]] (L) and [[capacitor]] (C) connected together, which acts as a [[resonator]].<ref name="Chattopadhyay" /><ref name="Garg" />  Charge flows back and forth between the capacitor's plates through the inductor, so the tuned circuit can store electrical energy oscillating at its [[resonant frequency]]. The amplifier adds power to compensate for resistive energy losses in the circuit and supplies the power for the output signal. LC oscillators are often used at [[radio frequency|radio frequencies]],<ref name="Chattopadhyay" /> when a tunable frequency source is necessary, such as in [[signal generator]]s, tunable radio [[transmitter]]s and the [[local oscillator]]s in [[radio receiver]]s.  Typical LC oscillator circuits are the [[Hartley oscillator|Hartley]], [[Colpitts oscillator|Colpitts]]<ref name="Chattopadhyay" /> and [[Clapp oscillator|Clapp]] circuits.
*In a [[Crystal oscillator#Crystal oscillator circuits|''crystal oscillator'' circuit]] the filter is a [[piezoelectric]] crystal (commonly a [[quartz crystal]]).<ref name="Chattopadhyay" /><ref name="Garg" />  The crystal mechanically vibrates as a [[resonator]], and its frequency of vibration determines the oscillation frequency. Since the [[resonant frequency]] of the crystal is determined by its dimensions, crystal oscillators are fixed frequency oscillators, their frequency can only be adjusted over a tiny range of less than one percent.<ref name="Terman" /><ref name="Misra" /><ref name="Scroggie" />{{sfn|Gottlieb|1997|p=39-40}}  Crystals have a very high [[Q-factor]] and also better temperature stability than tuned circuits, so crystal oscillators have much better frequency stability than LC or RC oscillators. Crystal oscillators are the most common type of linear oscillator, used to stabilize the frequency of most [[radio transmitter]]s, and to generate the [[clock signal]] in computers and [[quartz clock]]s. Crystal oscillators often use the same circuits as LC oscillators, with the crystal replacing the [[tuned circuit]];<ref name="Chattopadhyay" /> the [[Pierce oscillator]] circuit is also commonly used. Quartz crystals are generally limited to frequencies of 30&nbsp;MHz or below.<ref name="Chattopadhyay" />  Other types of resonators, [[dielectric resonator]]s and [[surface acoustic wave]] (SAW) devices, are used to control higher frequency oscillators, up into the [[microwave]] range.  For example, SAW oscillators are used to generate the radio signal in [[cell phone]]s.<ref>{{Cite web|last=APITech|title=SAW Technology|url=https://info.apitech.com/saw-technology-va|access-date=2021-05-12|website=info.apitech.com|language=en}}</ref>
{{clear}}

===Negative-resistance oscillator===
{{multiple image
| align = right
| direction = horizontal
| header   =
| image1   = Negative resistance oscillator.svg
| width1   = 190
| image2   = Ganna gjenerators M31102-1.jpg
| width2   = 162
| footer   = ''(left)'' Typical block diagram of a negative resistance oscillator. In some types the negative resistance device is connected in parallel with the resonant circuit.  ''(right)'' A negative-resistance microwave oscillator consisting of a [[Gunn diode]] in a [[cavity resonator]].  The negative resistance of the diode excites microwave oscillations in the cavity, which radiate out the aperture into a [[waveguide]].
}}

In addition to the feedback oscillators described above, which use [[two-port network|two-port]] amplifying active elements such as transistors and operational amplifiers, linear oscillators can also be built using [[port (circuit theory)|one-port]] (two terminal) devices with [[negative resistance]],<ref name="Chattopadhyay" /><ref name="Garg" /> such as [[magnetron]] tubes, [[tunnel diode]]s, [[IMPATT diode]]s and [[Gunn diode]]s.<ref name=" Raisanen" /><ref name="Solymar" />{{rp|p.197–198}}{{sfn|Gottlieb|1997|p=103}}  Negative-resistance oscillators are usually used at high frequencies in the [[microwave]] range and above, since at these frequencies feedback oscillators perform poorly due to excessive phase shift in the feedback path.

In negative-resistance oscillators, a resonant circuit, such as an [[LC circuit]], [[crystal oscillator|crystal]], or [[Resonator#Cavity resonators|cavity resonator]], is connected across a device with [[negative differential resistance]], and a DC bias voltage is applied to supply energy.  A resonant circuit by itself is "almost" an oscillator; it can store energy in the form of electronic oscillations if excited, but because it has electrical resistance and other losses the oscillations are [[Harmonic oscillator#Damped harmonic oscillator|damped]] and decay to zero.<ref name="Edson">{{cite book
  | last = Edson
  | first = William A.
  | title = Vacuum Tube Oscillators
  | publisher = John Wiley and Sons
  | date = 1953
  | location = 
  | pages = 7–8
  | language = 
  | url = https://archive.org/details/dli.ernet.504339/page/7/mode/2up
  | archive-url= 
  | archive-date=
  | doi = 
  | id = 
  | isbn = 
  | mr = 
  | zbl = 
  | jfm =}}</ref><ref name="Solymar">{{cite book
  | last1  = Solymar
  | first1 = Laszlo
  | last2  = Walsh
  | first2 = Donald
  | title = Electrical Properties of Materials
  | publisher = Oxford University Press
  | date = 2009
  | location = 
  | pages = 181–182
  | language = 
  | url = https://books.google.com/books?id=AiWyp0NQW6UC&q=%22negative+resistance%22
  | archive-url= 
  | archive-date=
  | doi = 
  | id = 
  | isbn = 9780191574351
  | mr = 
  | zbl = 
  | jfm =}}</ref>  The negative resistance of the active device cancels the (positive) internal loss resistance in the resonator, in effect creating a resonator circuit with no damping, which generates spontaneous continuous oscillations at its [[resonant frequency]].

The negative-resistance oscillator model is not limited to one-port devices like diodes; feedback oscillator circuits with [[two-port network|two-port]] amplifying devices such as transistors and [[vacuum tube|tubes]] also have negative resistance.{{sfn|Gottlieb|1997|p=104}}<ref name="Kung">{{cite web
 | last = Kung
 | first = Fabian Wai Lee
 | title = Lesson 9: Oscillator Design
 | website = RF/Microwave Circuit Design
 | publisher = Prof. Kung's website, Multimedia University
 | year = 2009
 | url = http://pesona.mmu.edu.my/~wlkung/ADS/rf/lesson9.pdf
 | access-date = October 17, 2012
 | archive-url = https://web.archive.org/web/20150722165131/http://pesona.mmu.edu.my/~wlkung/ADS/rf/lesson9.pdf
 | archive-date = July 22, 2015
 | url-status = dead
 }}, Sec. 3 Negative Resistance Oscillators, pp. 9–10, 14</ref><ref name=" Raisanen" /><ref name="Ellinger">{{cite book
 | last = Ellinger
 | first = Frank
 | title = Radio Frequency Integrated Circuits and Technologies, 2nd Ed
 | publisher = Springer
 | year = 2008
 | location = USA
 | pages = 391–394
 | url = https://books.google.com/books?id=0pl9xYD0QNMC&pg=PA391
 | isbn = 978-3540693246}}</ref>  At high frequencies, three terminal devices such as transistors and FETs are also used in negative resistance oscillators. At high frequencies these devices do not need a feedback loop, but with certain loads applied to one port can become unstable at the other port and show negative resistance due to internal feedback.  The negative resistance port is connected to a tuned circuit or resonant cavity, causing them to oscillate.<ref name="Kung" /><ref name=" Raisanen">{{cite book
 | last =  Räisänen
 | first = Antti V.
 | author2=Arto Lehto
 | title = Radio Engineering for Wireless Communication and Sensor Applications
 | publisher = Artech House
 | year = 2003
 | location = USA
 | pages = 180–182
 | url = https://books.google.com/books?id=m8Dgkvf84xoC&pg=PA181
 | isbn = 978-1580535427}}</ref><ref name="Maas" /> High-frequency oscillators in general are designed using negative-resistance techniques.<ref name="Kung" /><ref name=" Raisanen" /><ref name="Ellinger" />

=== List of harmonic oscillator circuits ===
Some of the many harmonic oscillator circuits are listed below:

{| class="wikitable" style="float:right;margin:0 0 1em 1em;"
|+ style="font-size: 80%"|Amplifying devices used in oscillators and approximate maximum frequencies<ref name=" Raisanen" />
|-
! Amplifying device
! Frequency
|-
| [[Triode]] vacuum tube
| ~1&nbsp;GHz
|-
| [[Bipolar junction transistor|Bipolar transistor]] (BJT)
| ~20&nbsp;GHz
|-
| [[Heterojunction bipolar transistor]] (HBT)
| ~50&nbsp;GHz
|-
| [[MESFET|Metal–semiconductor field-effect transistor]] (MESFET)
| ~100&nbsp;GHz
|-
| [[Gunn diode]], fundamental mode
| ~100&nbsp;GHz
|-
| [[Magnetron]] tube
| ~100&nbsp;GHz
|-
| [[High electron mobility transistor]] (HEMT)
| ~200&nbsp;GHz
|-
| [[Klystron]] tube
| ~200&nbsp;GHz
|-
| [[Gunn diode]], harmonic mode
| ~200&nbsp;GHz
|-
| [[IMPATT]] diode
| ~300&nbsp;GHz
|-
| [[Gyrotron]] tube
| ~600&nbsp;GHz
|-
|}

* [[Armstrong oscillator]], a.k.a. Meissner oscillator
* [[Hartley oscillator]]
* [[Colpitts oscillator]]
* [[Clapp oscillator]]
* [[Seiler oscillator]]
* [[Vackář oscillator]]
* [[Pierce oscillator]]
* [[Tri-tet oscillator]]
* [[Cathode follower oscillator]]
* [[Wien bridge oscillator]]
* [[Phase-shift oscillator]]
* Cross-coupled oscillator
* [[Dynatron oscillator]]
* [[Opto-electronic oscillator]]
* [[Robinson oscillator]]
{{clear}}

==Relaxation oscillator==
{{Main|Relaxation oscillator}}

[[File:OpAmpHystereticOscillator.svg|thumb|A popular [[op-amp]] [[Relaxation oscillator#Comparator–based electronic relaxation oscillator|relaxation oscillator]].]]

A '''nonlinear''' or '''[[relaxation oscillator]]''' produces a non-sinusoidal output, such as a [[Square wave (waveform)|square]], [[sawtooth wave|sawtooth]] or [[triangle wave]].<ref name="Garg" /><ref name="Graf">{{cite book   
  | last = Graf
  | first = Rudolf F. 
  | title = Modern Dictionary of Electronics
  | publisher = Newnes
  | date = 1999
  | pages = 638
  | url = https://books.google.com/books?id=uah1PkxWeKYC&pg=PA638
  | isbn = 0750698667}}</ref><ref name=" Morris">{{cite book   
  | last =  Morris
  | first = Christopher G. Morris
  | title = Academic Press Dictionary of Science and Technology
  | publisher = Gulf Professional Publishing
  | date = 1992
  | pages = 1829
  | url = https://books.google.com/books?id=nauWlPTBcjIC&pg=PA1829
  | isbn = 0122004000 }}</ref><ref name="Du">{{cite book   
  | last = Du
  | first = Ke-Lin
  |author2=M. N. S. Swamy 
  | title = Wireless Communication Systems: From RF Subsystems to 4G Enabling Technologies
  | publisher = Cambridge Univ. Press
  | date = 2010
  | pages = 443
  | url = https://books.google.com/books?id=5dGjKLawsTkC&q=%22relaxation+oscillator&pg=PA443
  | isbn = 978-1139485760}}</ref> It consists of an energy-storing element (a [[capacitor]] or, more rarely, an [[inductor]]) and a nonlinear switching device (a [[Latch (electronics)|latch]], [[Schmitt trigger]], or [[negative resistance]] element) connected in a [[feedback loop]].{{sfn|Gottlieb|1997|p=69-73}}<ref name="HyperPhysics">{{cite web
  | last = Nave
  | first = Carl R.
  | title = Relaxation Oscillator Concept
  | work = HyperPhysics
  | publisher = Dept. of Physics and Astronomy, Georgia State Univ.
  | date = 2014
  | url = http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/relaxo.html
  | accessdate = February 22, 2014}}</ref> The switching device periodically charges the storage element with energy and when its voltage or current reaches a threshold discharges it again, thus causing abrupt changes in the output waveform.<ref name="van der Tang">{{cite book
  | last1  = van der Tang
  | first1 = J.     
  | last2  = Kasperkovitz
  | first2 = Dieter
  | last3  = van Roermund
  | first3 = Arthur H.M.
  | title = High-Frequency Oscillator Design for Integrated Transceivers
  | publisher = Springer Science and Business Media 
  | date = 2006
  | location = 
  | pages = 20
  | language = 
  | url = https://books.google.com/books?id=0rniokw7bLkC&dq=%22ring+oscillator%22+oscillator+relaxation&pg=PA20
  | archive-url= 
  | archive-date=
  | doi = 
  | id = 
  | isbn = 0306487160
  | mr = 
  | zbl = 
  | jfm =}}</ref>{{rp|p.20}}  Although in the past negative resistance devices like the [[unijunction transistor]], [[thyratron]] tube or [[neon lamp]] were used,{{sfn|Gottlieb|1997|p=69-73}} today relaxation oscillators are mainly built with [[integrated circuits]] like the [[555 timer IC]].

Square-wave relaxation oscillators are used to provide the [[clock signal]] for [[sequential logic]] circuits such as timers and [[digital counter|counters]], although [[crystal oscillator]]s are often preferred for their greater stability.<ref name="Patrick">{{cite book
  | last1  = Patrick
  | first1 = Dale R.
  | last2  = Fardo
  | first2 = Stephen W.
  | last3  = Richardson
  | first3 = Ray E.
  | title = Electronic Devices and Circuit Fundamentals
  | publisher = CRC Press
  | date = 2023
  | language = 
  | url = https://books.google.com/books?id=tZK4EAAAQBAJ&pg=PT767
  | doi = 
  | id = 
  | isbn = 9781000879773
  | mr = 
  | zbl = 
  | jfm =}}</ref>{{rp|p.20.1}} Triangle-wave or sawtooth oscillators are used in the timebase circuits that generate the horizontal deflection signals for [[cathode-ray tube]]s in analogue [[oscilloscope]]s and [[television]] sets.<ref name="Patrick" />{{rp|p.20.1}} They are also used in [[voltage-controlled oscillator]]s (VCOs), [[inverter]]s and [[switching power supply|switching power supplies]], [[dual-slope ADC|dual-slope analog to digital converter]]s (ADCs), and in [[function generator]]s to generate square and triangle waves for testing equipment. In general, relaxation oscillators are used at lower frequencies and have poorer frequency stability than linear oscillators.

[[Ring oscillator]]s are built of a ring of active delay stages, such as [[inverter (logic gate)|inverter]]s.<ref name="van der Tang" />{{rp|p.20}} Generally the ring has an odd number of inverting stages, so that there is no single stable state for the internal ring voltages. Instead, a single transition propagates endlessly around the ring.

Some of the more common relaxation oscillator circuits are listed below:

*[[Multivibrator]]
*[[Pearson–Anson effect|Pearson–Anson oscillator]]
*[[Ring oscillator]]
*[[Delay-line oscillator]]
*[[Royer oscillator]]

==Voltage-controlled oscillator (VCO)==
{{Main|Voltage-controlled oscillator}}
An oscillator can be designed so that the oscillation frequency can be varied over some range by an input voltage or current. These [[voltage controlled oscillator]]s are widely used in [[phase-locked loop]]s, in which the oscillator's frequency can be locked to the frequency of another oscillator. These are ubiquitous in modern communications circuits, used in [[electronic filter|filters]], [[modulator]]s, [[demodulator]]s, and forming the basis of [[frequency synthesizer]] circuits which are used to tune radios and televisions.

Radio frequency VCOs are usually made by adding a [[varactor]] diode to the [[tuned circuit]] or resonator in an oscillator circuit. Changing the DC voltage across the varactor changes its [[capacitance]], which changes the [[resonant frequency]] of the tuned circuit. Voltage controlled relaxation oscillators can be constructed by charging and discharging the energy storage capacitor with a voltage controlled [[current source]]. Increasing the input voltage increases the rate of charging the capacitor, decreasing the time between switching events.

==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.

==History==
The first practical oscillators were based on [[electric arc]]s, which were used for lighting in the 19th century. The current through an [[arc lamp|arc light]] is unstable due to its [[negative resistance]], and often breaks into spontaneous oscillations, causing the arc to make hissing, humming or howling sounds{{sfn|Hong|2001|p=161-165}} which had been noticed by [[Humphry Davy]] in 1821, [[Benjamin Silliman]] in 1822,<ref>{{cite book | url=https://archive.org/details/firstprinciples05sillgoog | page=[https://archive.org/details/firstprinciples05sillgoog/page/n653 629] |quote=Davy Silliman Hissing. |title=First Principles of Physics: Or Natural Philosophy, Designed for the Use of Schools and Colleges |publisher=H.C. Peck & T. Bliss |last1=Silliman |first1=Benjamin |year=1859}}</ref> [[Auguste Arthur de la Rive]] in 1846,<ref>{{cite web |url=https://archive.org/details/wirelesstelephon00ruhmrich|title=Wireless telephony, in theory and practice |year=1908 |publisher=N.Y. Van Nostrand}}</ref> and [[David Edward Hughes]] in 1878.<ref>{{cite journal |doi=10.1007/BF00611436 |title= The humming telephone as an acoustic maser |journal=[[Optical and Quantum Electronics]]|volume = 23 |issue=8 |pages=995–1010 |year=1991 |author1-link=Lee Wendel Casperson |last1=Casperson |first1=L. W |bibcode= 1991OQEle..23..995C |s2cid=119956732}}</ref> [[Ernst Lecher]] in 1888 showed that the current through an electric arc could be oscillatory.<ref name="Anders">{{cite book |last1=Anders |first1=André |title=Cathodic Arcs: From Fractal Spots to Energetic Condensation |publisher=Springer Science and Business Media |date=2009 |pages=31–32 |url=https://books.google.com/books?id=rwIUhsbBHQYC&pg=PA31 |isbn=978-0387791081}}</ref><ref name="Cady">{{cite journal |last1=Cady |first1=W. G. |last2=Arnold |first2=H. D. |title=On the electric arc between metallic electrodes |journal=American Journal of Science |volume=24 |issue=143 |pages=406 |date=1907 |url=https://books.google.com/books?id=0KD0BuvSOokC&pg=PA406 |access-date=April 12, 2017}}</ref><ref name="Ayrton">{{cite journal |title=Notes |journal=The Electrical Review |volume=62 |issue=1578 |pages=812 |date=February 21, 1908 |url=https://books.google.com/books?id=it9QAAAAYAAJ&pg=PA312 |access-date=April 12, 2017}}</ref>

An oscillator was built by [[Elihu Thomson]] in 1892{{sfn|Morse|1925|p=23}}<ref>{{cite patent |inventor-first=Elihu |inventor-last=Thomson |title=Method of and Means for Producing Alternating Currents |country-code=US |patent-number=500630 |publication-date=18 July 1892 |issue-date=4 July 1893 |doi=}}</ref> by placing an [[LC circuit|LC tuned circuit]] in parallel with an electric arc and included a magnetic blowout. Independently, in the same year, [[George Francis FitzGerald]] realized that if the damping resistance in a resonant circuit could be made zero or negative, the circuit would produce oscillations, and, unsuccessfully, tried to build a negative resistance oscillator with a dynamo, what would now be called a [[parametric oscillator]].<ref name="Fitzgerald">G. Fitzgerald, ''On the Driving of Electromagnetic Vibrations by Electromagnetic and Electrostatic Engines'', read at the January 22, 1892, meeting of the Physical Society of London, in {{cite book |editor-last=Larmor |editor-first=Joseph |title=The Scientific Writings of the late George Francis Fitzgerald |publisher=Longmans, Green and Co. |date=1902 |location=London |pages=277–281 |url=https://books.google.com/books?id=G0bPAAAAMAAJ&pg=PA277}}</ref>{{sfn|Hong|2001|p=161-165}} The arc oscillator was rediscovered and popularized by [[William Duddell]] in 1900.{{sfn|Morse|1925|pp=80&ndash;81}}<ref>{{cite patent |inventor-first=William du Bois |inventor-last=Duddell |inventorlink=William Duddell |title=Improvements in and connected with Means for the Conversion of Electrical Energy, Derived from a Source of Direct Current, into Varying or Alternating Currents |country-code=GB |patent-number=190021629 |publication-date=29 Nov 1900 |issue-date=23 Nov 1901 |doi=}}</ref> Duddell, a student at London Technical College, was investigating the hissing arc effect.  He attached an [[LC circuit]] (tuned circuit) to the electrodes of an arc lamp, and the negative resistance of the arc excited oscillation in the tuned circuit.{{sfn|Hong|2001|p=161-165}} Some of the energy was radiated as sound waves by the arc, producing a musical tone. Duddell demonstrated his oscillator before the London [[Institute of Electrical Engineers]] by sequentially connecting different tuned circuits across the arc to play the national anthem "[[God Save the King|God Save the Queen]]".{{sfn|Hong|2001|p=161-165}} Duddell's "singing arc" did not generate frequencies above the audio range.<!-- no mag blowout --> In 1902 Danish physicists [[Valdemar Poulsen]] and P. O. Pederson were able to increase the frequency produced into the radio range by operating the arc in a hydrogen atmosphere with a magnetic field, inventing the [[Poulsen arc]] [[radio transmitter]], the first continuous wave radio transmitter, which was used through the 1920s.{{sfn|Morse|1925|p=31}}<ref>{{cite patent |inventor-first=Valdemar |inventor-last=Poulsen |inventor-link=Valdemar Poulsen |title=Improvements relating to the Production of Alternating Electric Currents |country-code=GB |patent-number=190315599 |publication-date= |issue-date=14 July 1904 |doi=}}</ref><ref>{{cite patent |inventor-first=Valdemar |inventor-last=Poulsen |inventor-link=Valdemar Poulsen |title=Method of Producing Alternating Currents with a High Number of Vibrations |country-code=US |patent-number=789449 |publication-date= |issue-date=9 May 1905 |doi=}}</ref>

[[File:Parallel rod push-pull 120MHz oscillator.jpg|thumbnail|A 120&nbsp;MHz oscillator from 1938 using a parallel rod [[transmission line]] resonator ([[Lecher line]]). Transmission lines are widely used for UHF oscillators.]]
The vacuum-tube feedback oscillator was invented around 1912, when it was discovered that feedback ("regeneration") in the recently invented [[audion|audion (triode)]] [[vacuum tube]] could produce oscillations.  At least six researchers independently made this discovery, although not all of them can be said to have a role in the invention of the oscillator.<ref name=" Hempstead">{{cite book |last=Hempstead |first=Colin |author2=William E. Worthington |title=Encyclopedia of 20th-Century Technology |volume=2 |publisher=Taylor & Francis |year=2005 |page=648 |url=https://books.google.com/books?id=0wkIlnNjDWcC&pg=PA648 |isbn=978-1579584641}}</ref>{{sfn|Hong|2001|p=156}}  In the summer of 1912, [[Edwin Armstrong]] observed oscillations in audion [[radio receiver]] circuits<ref name="Fleming">{{cite book |last=Fleming |first=John Ambrose |title=The Thermionic Valve and its Developments in Radiotelegraphy and Telephony |publisher=The Wireless Press |year=1919 |location=London |pages=148&ndash;155 |url=https://books.google.com/books?id=ZBtDAAAAIAAJ&pg=PA148}}</ref> and went on to use positive feedback in his invention of the [[regenerative receiver]].{{sfn|Hong|2003|p=9-10}}<ref name="Armstrong">{{cite journal |last=Armstrong |first=Edwin H. |title=Some recent developments in the Audion receiver |journal=Proc. IRE |volume=3 |issue=9 |pages=215&ndash;247 |date=September 1915 |url=http://www.ieee.org/documents/00573757.pdf |archive-url=https://web.archive.org/web/20130728164117/http://www.ieee.org/documents/00573757.pdf |url-status=dead |archive-date=July 28, 2013 |access-date=August 29, 2012 |doi= 10.1109/jrproc.1915.216677|s2cid=2116636}}</ref>  Austrian [[Alexander Meissner]] independently discovered positive feedback and invented oscillators in March 1913.<ref name="Fleming" />{{sfn|Hong|2003|p=13}} [[Irving Langmuir]] at General Electric observed feedback in 1913.{{sfn|Hong|2003|p=13}}  Fritz Lowenstein may have preceded the others with a crude oscillator in late 1911.{{sfn|Hong|2003|p=5}}  In Britain, H. J. Round patented amplifying and oscillating circuits in 1913.<ref name="Fleming" />  In August 1912, [[Lee De Forest]], the inventor of the audion, had also observed oscillations in his amplifiers, but he didn't understand the significance and tried to eliminate it{{sfn|Hong|2003|pp=6&ndash;7}}<ref name="Hijiya">{{cite book |last=Hijiya |first=James A. |title=Lee De Forest and the Fatherhood of Radio |publisher=Lehigh University Press |year=1992 |pages=89&ndash;90 |url=https://books.google.com/books?id=JYylHhmoNZ4C&pg=PA89 |isbn=978-0934223232}}</ref> until he read Armstrong's patents in 1914,{{sfn|Hong|2003|p=14}} which he promptly challenged.<ref name="Nahin">{{cite book
 | last = Nahin
 | first = Paul J.
 | title = The Science of Radio: With Matlab and Electronics Workbench Demonstration, 2nd Ed
 | publisher = Springer
 | year = 2001
 | pages = 280
 | url = https://books.google.com/books?id=V1GBW6UD4CcC&pg=PA82
 | isbn = 978-0387951508}}</ref>  Armstrong and De Forest fought a protracted legal battle over the rights to the "regenerative" oscillator circuit<ref name="Nahin" />{{sfn|Hong|2001|pp=181&ndash;189}} which has been called "the most complicated patent litigation in the history of radio".{{sfn|Hong|2003|p=2}} De Forest ultimately won before the Supreme Court in 1934 on technical grounds, but most sources regard Armstrong's claim as the stronger one.<ref name="Hijiya" /><ref name="Nahin" />

The first and most widely used relaxation oscillator circuit, the [[astable multivibrator]], was invented in 1917 by French engineers Henri Abraham and Eugene Bloch.<ref name="Abraham">{{cite journal
 | last = Abraham
 | first = H.
 | author2=E. Bloch
 | title = Measurement of period of high frequency oscillations
 | journal = Comptes Rendus
 | volume = 168
 | pages = 1105
 | year = 1919
 }}</ref><ref name="Glazebrook">{{cite book
 | last = Glazebrook
 | first = Richard
 | title = A Dictionary of Applied Physics, Vol. 2: Electricity
 | publisher = Macmillan and Co. Ltd.
 | year = 1922
 | location = London
 | pages = 633–634
 | url = https://books.google.com/books?id=bavQAAAAMAAJ&pg=PA633
 }}</ref><ref name="Calvert">{{cite web
 | last = Calvert
 | first = James B.
 | title = The Eccles-Jordan Circuit and Multivibrators
 | publisher =  Dr. J. B. Calvert website, Univ. of Denver
 | year = 2002
 | url = http://mysite.du.edu/~etuttle/electron/elect36.htm
 | access-date = May 15, 2013}}</ref>  They called their cross-coupled, dual-vacuum-tube circuit a ''multivibrateur'', because the square-wave signal it produced was rich in [[harmonic]]s,<ref name="Glazebrook" /><ref name="Calvert" /> compared to the sinusoidal signal of other vacuum-tube oscillators.

Vacuum-tube feedback oscillators became the basis of radio transmission by 1920.  However, the [[triode]] vacuum tube oscillator performed poorly above 300&nbsp;MHz because of interelectrode capacitance.<ref>{{Cite web |last=Peto |first=David Charles |title=The vacuum tube triode at ultra high frequencies |url=https://core.ac.uk/download/pdf/36724402.pdf |access-date=July 8, 2024}}</ref>  To reach higher frequencies, new "transit time" (velocity modulation) vacuum tubes were developed, in which electrons traveled in "bunches" through the tube.  The first of these was the [[Barkhausen–Kurz tube|Barkhausen–Kurz oscillator]] (1920), the first tube to produce power in the [[Ultra high frequency|UHF]] range.  The most important and widely used were the [[klystron]] (R. and S. Varian, 1937) and the cavity [[magnetron]] (J. Randall and H. Boot, 1940).

Mathematical conditions for feedback oscillations, now called the [[Barkhausen stability criterion|Barkhausen criterion]], were derived by [[Heinrich Georg Barkhausen]] in 1921. He also showed that all linear oscillators must have negative resistance.<ref name="Duncan">The requirements for negative resistance in oscillators were first set forth by [[Heinrich Barkhausen]] in 1907 in [https://archive.org/details/dasproblemdersc01barkgoog ''Das Problem Der Schwingungserzeugung''] according to {{cite journal
  | last = Duncan
  | first = R. D.
  | title = Stability conditions in vacuum tube circuits
  | journal = Physical Review
  | volume = 17
  | issue = 3
  | page = 304
  | date = March 1921
  | url = https://books.google.com/books?id=rCgKAAAAIAAJ&q=%22negative+resistance&pg=PA304
  | doi = 10.1103/physrev.17.302
  | access-date = July 17, 2013 | bibcode = 1921PhRv...17..302D }}:  "''For alternating current power to be available in a circuit which has externally applied only continuous voltages, the average power consumption during a cycle must be negative...which demands the introduction of negative resistance ''[which]'' requires that the phase difference between voltage and current lie between 90° and 270°...''[and for nonreactive circuits]'' the value 180° must hold... The volt-ampere characteristic of such a resistance will therefore be linear, with a negative slope...''"</ref>  The first analysis of a nonlinear electronic oscillator model, the [[Van der Pol oscillator]], was done by [[Balthasar van der Pol]] in 1927.<ref name="Van der Pol">{{cite journal
 | last = Van der Pol
 | first = Balthazar
 | title = On relaxation-oscillations
 | journal = The London, Edinburgh and Dublin Philosophical Magazine
 | volume = 2
 | issue = 7
 | pages = 978–992
 | year = 1927
 | url = http://audiophile.tam.cornell.edu/randdocs/classics/vanderpol.pdf
 | doi = 10.1080/14786442608564127
 }}</ref>  He originated the term "relaxation oscillation" and was first to distinguish between linear and relaxation oscillators.  He showed that the stability of the oscillations ([[limit cycle]]s) in actual oscillators was due to the [[linear circuit|nonlinearity]] of the amplifying device.   Further advances in mathematical analysis of oscillation were made by [[Hendrik Wade Bode]] and [[Harry Nyquist]]<ref name="Nyquist">{{cite journal
 | last = Nyquist
 | first = H.
 | title = Regeneration Theory
 | journal = Bell System Tech. J.
 | volume = 11
 | issue = 1
 | pages = 126–147
 | date = January 1932
 | url = http://www3.alcatel-lucent.com/bstj/vol11-1932/articles/bstj11-1-126.pdf
 | doi = 10.1002/j.1538-7305.1932.tb02344.x
 | s2cid = 115002788
 | access-date = December 5, 2012}} on [http://www.alcatel-lucent.com   Alcatel-Lucent website]</ref> in the 1930s.  In 1969 Kaneyuki Kurokawa derived necessary and sufficient conditions for oscillation in negative-resistance circuits,<ref name="Kurokawa">{{cite journal
 | last = Kurokawa
 | first = Kaneyuki
 | title = Some Basic Characteristics of Broadband Negative Resistance Oscillator Circuits
 | journal = Bell System Tech. J.
 | volume = 48
 | issue = 6
 | pages = 1937–1955
 | date = July 1969
 | url = http://www3.alcatel-lucent.com/bstj/vol48-1969/articles/bstj48-6-1937.pdf
 | doi = 10.1002/j.1538-7305.1969.tb01158.x
 | access-date = December 8, 2012}}  Eq. 10 is a necessary condition for oscillation; eq. 12 is a sufficient condition,</ref> which form the basis of modern microwave oscillator design.<ref name="Maas">{{cite book
 | last = Maas
 | first = Stephen A.
 | title = Nonlinear Microwave and RF Circuits, 2nd Ed
 | publisher = Artech House
 | year = 2003
 | pages = 542–544
 | url = https://books.google.com/books?id=SSw6gWLG-d4C&pg=PA542
 | isbn = 978-1580534840}}</ref>

==See also==
*[[Injection locked oscillator]]
*[[Numerically controlled oscillator]]
*[[Extended interaction oscillator]]
*[[Variable-frequency drive]]
*[[Thin-film bulk acoustic resonator]]

==References==
{{Reflist|30em}}
*{{cite book
  | last = Gottlieb
  | first = Irving M.
  | title = Practical Oscillator Handbook
  | publisher = Elsevier
  | date = 1997
  | location =
  | pages =
  | url = https://books.google.com/books?id=e_oZ69GAuxAC&dq=pull+pulling&pg=PA40
  | doi =
  | id =
  | isbn = 0080539386}} 
*{{cite book |last=Hong |first=Sungook |title=Wireless: From Marconi's Black-Box to the Audion |publisher=MIT Press |year=2001 |url=https://books.google.com/books?id=UjXGQSPXvIcC&pg=PA165 |isbn=978-0262082983}}
*{{cite journal |last=Hong |first=Sungook |title=A history of the regeneration circuit: From invention to patent litigation |year=2003 |publisher=Seoul National University, Seoul, Korea |url=http://www.ieeeghn.org/wiki/images/0/08/Hong.pdf |accessdate=August 29, 2012 |archive-date=February 1, 2014 |archive-url=https://web.archive.org/web/20140201200653/http://www.ieeeghn.org/wiki/images/0/08/Hong.pdf }}
*{{Citation |last=Morse |first=A. H. |year=1925 |title=Radio: Beam and Broadcast: Its story and patents |location=London |publisher=Ernest Benn |url=https://archive.org/details/radiobeamandbroa029214mbp}}.

==Further reading==
* {{cite book
  | last1  = Rohde
  | first1 = Ulrich
  | last2  = Poddar
  | first2 = Ajay
  | last3  = Bock
  | first3 = Georg
  | title = The Design of Modern Microwave Oscillators for Wireless Applications: Theory and Optimization
  | publisher = John Wiley and Sons
  | date = 2005
  | location = 
  | pages = 
  | language = 
  | url = https://books.google.com/books?id=GrvgJe8aujcC
  | archive-url= 
  | archive-date=
  | doi = 
  | id = 
  | isbn = 0-471-72342-8
  | mr = 
  | zbl = 
  | jfm =}}
* {{cite book
  | last = Rubiola
  | first = Enrico 
  | title = Phase Noise and Frequency Stability in Oscillators
  | publisher = Cambridge University Press
  | date = 2010
  | location = 
  | pages = 
  | language = 
  | url = https://books.google.com/books?id=ncv7QwAACAAJ
  | archive-url= 
  | archive-date=
  | doi = 
  | id = 
  | isbn = 9780521153287
  | mr = 
  | zbl = 
  | jfm =}}

==External links==
{{Commons category|Electronic oscillators}}
* [http://www.howstuffworks.com/oscillator.htm Howstuffworks: oscillator].
* [http://www.ieee.li/pdf/viewgraphs/oscillator_oddities.pdf Oscillator Oddities].
* [http://www.ieee.li/pdf/viewgraphs/precision_frequency_generation.pdf Tutorial on Precision Frequency Generation].

{{Electronic oscillators}}

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{{DEFAULTSORT:Electronic Oscillator}}
[[Category:Electronic oscillators| ]]