Editing RC oscillator (section)

==Description==
RC oscillators are a type of [[feedback]] oscillator; they consist of an amplifying device, a [[transistor]], [[vacuum tube]], or [[op-amp]], with some of its output energy fed back into its input through a network of [[resistor]]s and [[capacitor]]s, an [[RC network]], to achieve [[positive feedback]], causing it to generate an oscillating sinusoidal voltage.<ref name="Mancini">{{cite web
  | last1 =  Mancini
  | first1 = Ron
  | first2 = Richard
  | last2 = Palmer
  | title = Application Report SLOA060: Sine-Wave Oscillator
  | publisher = Texas Instruments Inc.
  | date = March 2001
  | url = http://www.ti.com/lit/an/sloa060/sloa060.pdf
  | access-date = August 12, 2015}}</ref><ref name="Gottlieb">{{cite book
 | last1  = Gottlieb
 | first1 = Irving
 | title  = Practical Oscillator Handbook
 | publisher = Elsevier
 | date   = 1997
 | pages  = 49–53
 | url    = https://books.google.com/books?id=e_oZ69GAuxAC
 | isbn   = 0080539386
 }}</ref><ref name="Coates1">{{cite web
  | last = Coates
  | first = Eric
  | title = Oscillators Module 1 - Oscillator Basics
  | website = Learn About Electronics
  | publisher = Eric Coates
  | date = 2015
  | url = http://www.learnabout-electronics.org/Oscillators/osc10.php
  | access-date = August 7, 2015}}</ref>  They are used to produce lower [[frequency|frequencies]], mostly [[audio frequency|audio frequencies]], in such applications as audio [[signal generator]]s and electronic musical instruments.<ref name="Coates2">{{cite web
  | last = Coates
  | first = Eric
  | title = Oscillators Module 3 - AF Sine Wave Oscillators
  | website = Learn About Electronics
  | publisher = Eric Coates
  | date = 2015
  | url = http://www.learnabout-electronics.org/Downloads/Oscillators-module-03.pdf
  | access-date = August 7, 2015}}</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 = 81-224-1780-9}}</ref>  At [[radio frequency|radio frequencies]], another type of feedback oscillator, the LC oscillator is used, but at frequencies below 100&nbsp;kHz the size of the [[inductor]]s and [[capacitor]]s needed for the LC oscillator become cumbersome, and RC oscillators are used instead.<ref name="DAEnotes">{{cite web
  | title = RC Feedback Oscillators
  | work = Electronics tutorial
  | publisher = DAEnotes
  | date = 2013
  | url = http://www.daenotes.com/electronics/digital-electronics/rc-feedback-oscillators
  | access-date = August 9, 2015}}</ref>  Their lack of bulky inductors also makes them easier to integrate into microelectronic devices.  Since the oscillator's frequency is determined by the value of resistors and capacitors, which vary with temperature, RC oscillators do not have as good frequency stability as [[crystal oscillator]]s.

The frequency of oscillation is determined by the [[Barkhausen stability criterion|Barkhausen criterion]], which says that the circuit will only oscillate at frequencies for which the [[phase shift]] around the [[feedback loop]] is equal to 360° (2π radians) or a multiple of 360°, and the [[loop gain]] (the amplification around the feedback loop) is equal to one.<ref name="Rao">{{cite book
 | last1  = Rao
 | first1 = B.
 | last2  = Rajeswari
 | first2 = K.
 | last3  = Pantulu
 | first3 = P.
 | title  = Electronic Circuit Analysis
 | publisher = Pearson Education India
 | date   = 2012
 | location = India
 | pages  = 8.2–8.6, 8.11
 | url    = https://books.google.com/books?id=yooVw9u8GMwC&pg=SA8-PA11
 | isbn   = 978-8131754283
 }}</ref><ref name="Mancini" />   The purpose of the feedback RC network is to provide the correct phase shift at the desired oscillating frequency  so the loop has 360° phase shift, so the [[sine wave]], after passing through the loop will be in phase with the sine wave at the beginning and reinforce it,  resulting in positive feedback.<ref name="DAEnotes" />   The amplifier provides [[gain (electronics)|gain]] to compensate for the energy lost as the signal passes through the feedback network, to create sustained oscillations.  As long as the gain of the amplifier is high enough that the total gain around the loop is unity or higher, the circuit will generally oscillate.

In RC oscillator circuits which use a single inverting amplifying device, such as a transistor, tube, or an op amp with the feedback applied to the inverting input, the amplifier provides 180° of the phase shift, so the RC network must provide the other 180°.<ref name="DAEnotes" />   Since each capacitor can provide a maximum of 90° of phase shift, RC oscillators require at least two frequency-determining capacitors in the circuit (two [[pole (complex analysis)|pole]]s), and most have three or more,<ref name="Mancini" /> with a comparable number of resistors.

This makes tuning the circuit to different frequencies more difficult than in other types such as the LC oscillator, in which the frequency is determined by a single LC circuit so only one element must be varied. Although the frequency can be varied over a small range by adjusting a single circuit element, to tune an RC oscillator over a wide range two or more resistors or capacitors must be varied in unison, requiring them to be ''ganged'' together mechanically on the same shaft.<ref name="Gottlieb" /><ref name="Coates3">[http://www.learnabout-electronics.org/Downloads/Oscillators-module-03.pdf Eric Coates, 2015, AF Sine Wave Oscillators, p. 10]</ref>    The oscillation frequency is proportional to the inverse of the capacitance or resistance, whereas in an LC oscillator the frequency is proportional to inverse square root of the capacitance or inductance.<ref name="Groszkowski">{{cite book
 | last1  = Groszkowski
 | first1 = Janusz 
 | title  = Frequency of Self-Oscillations
 | publisher = Elsevier
 | date   = 2013
 | pages  = 397–398
 | url    = https://books.google.com/books?id=H_ZFBQAAQBAJ&pg=PA
 | isbn   = 978-1483280301
 }}</ref> So a much wider frequency range can be covered by a given variable capacitor in an RC oscillator.  For example, a variable capacitor that could be varied over a 9:1 capacitance range will give an RC oscillator a 9:1 frequency range, but in an LC oscillator it will give only a 3:1 range.

Some examples of common RC oscillator circuits are listed below:

[[Image:RC phase shift oscillator.svg|thumb|225px|A phase-shift oscillator]]

===Phase-shift oscillator===
{{main|Phase-shift oscillator}}
In the [[phase-shift oscillator]] the feedback network is three identical cascaded RC sections.<ref>{{citation |last=Department of the Army |title=Basic Theory and Application of Transistors |series=Technical Manuals |orig-year=1959 |year=1962 |publisher=Dover |id=TM 11-690 |pages=178–179}}</ref>  In the simplest design the capacitors and resistors in each section have the same value <math>\scriptstyle R\;=\;R1\;=\;R2\;=\;R3</math> and <math>\scriptstyle C\;=\;C1\;=\;C2\;=\;C3</math>.  Then at the oscillation frequency each RC section contributes 60° phase shift for a total of 180°.  The oscillation frequency is
:<math>f = \frac{\sqrt{6}}{2\pi RC}</math>
The feedback network has an attenuation of 1/29, so the op-amp must have a gain of 29 to give a loop gain of one for the circuit to oscillate 
:<math>R_\mathrm{fb} = 29\cdot R</math>

{{Breakafterimages}}
[[Image:Twin T oscillator.svg|thumb|225px|A twin-T oscillator]]

===Twin-T oscillator===
Another common design is the "Twin-T" oscillator as it uses two "T" RC circuits operated in parallel.<!-- Amos should be a ref. --> One circuit is an R-C-R "T" which acts as a [[low-pass filter]]. The second circuit is a C-R-C "T" which operates as a [[high-pass filter]]. Together, these circuits form a bridge which is tuned at the desired frequency of oscillation. The signal in the C-R-C branch of the Twin-T filter is advanced, in the R-C-R - delayed, so they may cancel one another for frequency <math>f=\frac{1}{2\pi RC}</math> if <math>x=2</math>; if it is connected as a [[negative feedback]] to an amplifier, and x>2, the amplifier becomes an oscillator.  (Note: <math>x = C2/C1 = R1/R2</math>.)

===Quadrature oscillator===
The quadrature oscillator uses two cascaded [[op-amp]] integrators in a feedback loop, one with the signal applied to the inverting input or two integrators and an invertor.  The advantage of this circuit is that the sinusoidal outputs of the two op-amps are 90° [[out of phase]] (in quadrature).  This is useful in some communication circuits.

It is possible to stabilize a quadrature oscillator by squaring the sine and cosine outputs, adding them together, ([[Pythagorean trigonometric identity]]) subtracting a constant, and applying the difference to a multiplier that adjusts the loop gain around an inverter.  Such circuits have a near-instant amplitude response to the constant input and extremely low distortion.