View source for Colpitts oscillator

A '''Colpitts oscillator''', invented in 1918 by American engineer [[Edwin H. Colpitts]],<ref>{{cite patent |country-code=US |patent-number=1624537 |inventor-last=Colpitts |inventor-first=Edwin H. |title=Oscillation generator |publication-date=1 February 1918 |issue-date=12 April 1927 }}</ref> is one of a number of designs for [[LC oscillator]]s, [[electronic oscillator]]s that use a combination of [[inductor]]s (L) and [[capacitor]]s (C) to produce an oscillation at a certain frequency.  The distinguishing feature of the Colpitts oscillator is that the [[feedback]] for the active device is taken from a [[voltage divider]] made of two capacitors in series across the inductor.<ref name="Gottlieb">{{cite book   
  | last = Gottlieb
  | first = Irving Gottlieb
  | title = Practical Oscillator Handbook
  | publisher = Elsevier
  | year = 1997
  | location = US
  | pages = 151
  | url = https://books.google.com/books?id=e_oZ69GAuxAC&pg=PA151&dq=Colpitts+oscillator+%22capacitor+voltage+divider
  | doi = 
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  | isbn = 0750631023}}</ref><ref name="Carr">{{cite book   
  | last = Carr
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  | title = RF Components and Circuits 
  | publisher = Newnes
  | year = 2002
  | location = US
  | pages = 127
  | url = https://books.google.com/books?id=V9gBTNvt3zIC&pg=PA127&dq=Colpitts+oscillator+%22capacitive+voltage+divider
  | doi = 
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  | isbn = 0750648449}}</ref><ref name="Basak">{{cite book   
  | last = Basak
  | first = A. 
  | title = Analogue Electronic Circuits and Systems
  | publisher = Cambridge University Press
  | year = 1991
  | location = UK
  | pages = 153
  | url = https://books.google.com/books?id=vwnMPZx7wfkC&pg=PA153&dq=Colpitts+oscillator+%22capacitive+voltage+divider%
  | doi = 
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  | isbn = 0521360463}}</ref><ref name="Rohde">{{cite book   
  | last = Rohde
  | first = Ulrich L.
  | authorlink = 
  |author2=Matthias Rudolph
  | title = RF / Microwave Circuit Design for Wireless Applications, 2nd Ed.
  | publisher = John Wiley & Sons
  | year = 2012
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  | pages = 745–746
  | url = https://books.google.com/books?id=Y5ZiwX2Ap5cC&pg=PA745&dq=Colpitts+oscillator+%22capacitive+voltage+divider
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  | isbn = 978-1118431405}}</ref>

==Overview==

{| align="right"
| valign="top" | [[Image:Cb colp.svg|thumb|130px|Figure 1: Simple [[common-base]] Colpitts oscillator (with simplified [[biasing]])]]
| valign="top" | [[Image:Cc colp2.svg|thumb|130px|Figure 2: Simple [[common-collector]] Colpitts oscillator (with simplified [[biasing]])]] 
|}

The Colpitts circuit, like other LC oscillators, consists of a gain device (such as a bipolar junction [[transistor]], field-effect transistor, operational amplifier, or [[vacuum tube]]) with its output connected to its input in a [[feedback loop]] containing a parallel [[LC circuit]] ([[tuned circuit]]), which functions as a [[bandpass filter]] to set the frequency of oscillation.

A Colpitts oscillator is the electrical dual of a [[Hartley oscillator]], where the feedback signal is taken from an "inductive" voltage divider consisting of two coils in series (or a tapped coil).  Fig.&nbsp;1 shows the common-base Colpitts circuit.  ''L'' and the series combination of ''C''<sub>1</sub> and ''C''<sub>2</sub> form the parallel resonant [[tank circuit]], which determines the frequency of the oscillator.  The voltage across ''C''<sub>2</sub> is applied to the base-emitter junction of the transistor, as feedback to create oscillations.  Fig.&nbsp;2 shows the common-collector version.  Here the voltage across ''C''<sub>1</sub> provides feedback.  The frequency of oscillation is approximately the resonant frequency of the LC circuit, which is the series combination of the two capacitors in parallel with the inductor:
:<math>f_0 = \frac{1}{2\pi \sqrt{L \frac{C_1 C_2}{C_1 + C_2}}}.</math>
The actual frequency of oscillation will be slightly lower due to junction capacitances and resistive loading of the transistor.

As with any oscillator, the amplification of the active component should be marginally larger than the attenuation of the capacitive voltage divider, to obtain stable operation. Thus, a Colpitts oscillator used as a [[variable-frequency oscillator]] (VFO) performs best when a variable inductance is used for tuning, as opposed to tuning one of the two capacitors. If tuning by variable capacitor is needed, it should be done with a third capacitor connected in parallel to the inductor (or in series as in the [[Clapp oscillator]]).

===Practical example===

[[Image:NPN Colpitts oscillator collector coil.svg|thumb|300px|Figure 3: Practical{{dubious|Figure 3|date=March 2016}} common-base Colpitts oscillator with an oscillation frequency of ~50&nbsp;MHz]]

Fig.&nbsp;3 shows a working example with component values. Instead of [[bipolar junction transistor]]s, other active components such as [[field-effect transistors]] or [[vacuum tube]]s, capable of producing gain at the desired frequency, could be used.

The capacitor at the base provides an AC path to ground for parasitic inductances that could lead to unwanted resonance at undesired frequencies.<ref>[http://www.ece.ucsb.edu/Faculty/rodwell/Classes/ece218b/notes/Oscillators2.pdf University of California Santa Barbara Untitled Publication], p. 3.</ref> Selection of the base's biasing resistors is not trivial. Periodic oscillation starts for a critical bias current and with the variation of the bias current to a higher value chaotic oscillations are observed.<ref>S. Sarkar, S. Sarkar, B. C. Sarkar. [http://www.ijeat.org/attachments/File/v2i5/E1662062513.pdf "Nonlinear Dynamics of a BJT Based Colpitts Oscillator with Tunable Bias Current"] {{webarchive|url=https://web.archive.org/web/20140814113015/http://www.ijeat.org/attachments/File/v2i5/E1662062513.pdf |date=2014-08-14 }}. ''IJEAT'' {{ISSN|2249-8958}}, Volume-2, Issue-5, June 2013. p. 1.</ref>

==Theory==
[[Image:Colpitts ideal model.svg|thumb|right|250px|Ideal Colpitts oscillator model (common-collector configuration)]]

One method of oscillator analysis is to determine the input impedance of an input port neglecting any reactive components. If the impedance yields a [[negative resistance]] term, oscillation is possible. This method will be used here to determine conditions of oscillation and the frequency of oscillation.

An ideal model is shown to the right. This configuration models the common collector circuit in the section above. For initial analysis, parasitic elements and device non-linearities will be ignored. These terms can be included later in a more rigorous analysis. Even with these approximations, acceptable comparison with experimental results is possible.

Ignoring the inductor, the input [[Electrical impedance|impedance]] at the base can be written as

:<math>Z_\text{in} = \frac{v_1}{i_1},</math>

where <math>v_1</math> is the input voltage, and <math>i_1</math> is the input current. The voltage <math>v_2</math> is given by

:<math>v_2 = i_2 Z_2,</math>

where <math>Z_2</math> is the impedance of <math>C_2</math>. The current flowing into <math>C_2</math> is <math>i_2</math>, which is the sum of two currents:

:<math>i_2 = i_1 + i_s,</math>

where <math>i_s</math> is the current supplied by the transistor. <math>i_s</math> is a dependent current source given by

:<math>i_s = g_m (v_1 - v_2),</math>

where <math>g_m</math> is the [[transconductance]] of the transistor. The input current <math>i_1</math> is given by

:<math>i_1 = \frac{v_1 - v_2}{Z_1},</math>

where <math>Z_1</math> is the impedance of <math>C_1</math>. Solving for <math>v_2</math> and substituting above yields

:<math>Z_\text{in} = Z_1 + Z_2 + g_m Z_1 Z_2.</math>

The input impedance appears as the two capacitors in series with the term <math>R_\text{in}</math>, which is proportional to the product of the two impedances:

:<math>R_\text{in} = g_m Z_1 Z_2.</math>

If <math>Z_1</math> and <math>Z_2</math> are complex and of the same sign, then <math>R_\text{in}</math> will be a [[negative resistance]]. If the impedances for <math>Z_1</math> and <math>Z_2</math> are substituted, <math>R_\text{in}</math> is

:<math>R_\text{in} = \frac{-g_m}{\omega^2 C_1 C_2}.</math>

If an inductor is connected to the input, then the circuit will oscillate if the magnitude of the negative resistance is greater than the resistance of the inductor and any stray elements. The frequency of oscillation is as given in the previous section.

For the example oscillator above, the emitter current is roughly 1&nbsp;[[Ampere|mA]]. The transconductance is roughly 40&nbsp;[[Siemens (unit)|mS]]. Given all other values, the input resistance is roughly

:<math>R_\text{in} = -30\ \Omega.</math>

This value should be sufficient to overcome any positive resistance in the circuit. By inspection, oscillation is more likely for larger values of transconductance and smaller values of capacitance.  A more complicated analysis of the common-base oscillator reveals that a low-frequency amplifier voltage gain must be at least 4 to achieve oscillation.<ref>Razavi, B.  Design of Analog CMOS Integrated Circuits. McGraw-Hill. 2001.</ref>  The low-frequency gain is given by

:<math>A_v = g_m R_p \ge 4.</math>

[[File:Oscillator comparison.svg|thumb|Comparison of Hartley and Colpitts oscillators]]
If the two capacitors are replaced by inductors, and magnetic coupling is ignored, the circuit becomes a [[Hartley oscillator]]. In that case, the input impedance is the sum of the two inductors and a negative resistance given by

:<math>R_\text{in} = -g_m \omega^2 L_1 L_2.</math>

In the Hartley circuit, oscillation is more likely for larger values of transconductance and larger values of inductance.

The above analysis also describes the behavior of the [[Pierce oscillator]]. The Pierce oscillator, with two capacitors and one inductor, is equivalent to the Colpitts oscillator.<ref>Theron Jones. [http://www.maximintegrated.com/en/app-notes/index.mvp/id/5265 "Design a Crystal Oscillator to Match Your Application"] {{Webarchive|url=https://web.archive.org/web/20150122233938/http://www.maximintegrated.com/en/app-notes/index.mvp/id/5265 |date=2015-01-22 }}. Maxim tutorial 5265 Sep 18, 2012, Maxim Integrated Products, Inc.</ref> Equivalence can be shown by choosing the junction of the two capacitors as the ground point. An electrical dual of the standard Pierce oscillator using two inductors and one capacitor is equivalent to the [[Hartley oscillator]].

===Oscillation amplitude===
The amplitude of oscillation is generally difficult to predict, but it can often be accurately estimated using the [[describing function]] method.

For the common-base oscillator in Figure&nbsp;1, this approach applied to a simplified model predicts an output (collector) voltage amplitude given by<ref>Chris Toumazou, George S. Moschytz, Barrie Gilbert. [https://books.google.com/books?id=VoBIOvirkiMC&lpg=PA568&ots=MD4aYrSVjr&dq=the%20tank%20voltage%20amplitude%20is%20calculated%20to%20be&pg=PA568#v=onepage&q=the%20tank%20voltage%20amplitude%20is%20calculated%20to%20be&f=false Trade-Offs in Analog Circuit Design: The Designer's Companion, Part 1].</ref>

:<math>
V_C = 2 I_C R_L \frac{C_2}{C_1 + C_2},
</math>

where <math>I_C</math> is the bias current, and <math>R_L</math> is the load resistance at the collector.

This assumes that the transistor does not saturate, the collector current flows in narrow pulses, and that the output voltage is sinusoidal (low distortion).

This approximate result also applies to oscillators employing different active device, such as [[MOSFET]]s and [[vacuum tubes]].

==References==
{{reflist}}

==Further reading==
{{refbegin}}
* {{cite book |last=Lee |first=T. |author-link=Thomas H. Lee (engineering professor) |title=The Design of CMOS Radio-Frequency Integrated Circuits |publisher=Cambridge University Press |date=December 2003 |isbn=978-0521835398}}
* {{cite book |first1=Ulrich L. |last1=Rohde |first2=Ajay K. |last2=Poddar |first3=Georg |last3=Böck |title=The Design of Modern Microwave Oscillators for Wireless Applications |publisher=John Wiley & Sons |location=New York, NY |date=May 2005 |isbn=0-471-72342-8}}.
* {{cite book |first1=George |last1=Vendelin |first2=Anthony M. |last2=Pavio |first3=Ulrich L. |last3=Rohde |title=Microwave Circuit Design Using Linear and Nonlinear Techniques |publisher=John Wiley & Sons |location=New York, NY |date=May 2005 |isbn=0-471-41479-4}}.
* {{cite journal |first1=Ulrich L. |last1=Rohde |first2=Anisha M. |last2=Apte |title=Everything You Always Wanted to Know About Colpitts Oscillators |journal=IEEE Microwave Magazine |volume=17 |issue=6 |pages=59–76 |date=August 2016 |doi=10.1109/MMM.2016.2561498}}
* {{cite conference |first1=Anisha M. |last1=Apte |first2=Ajay K. |last2=Poddar |first3=Ulrich L. |last3=Rohde |first4=Enrico |last4=Rubiola |title=Colpitts oscillator: A new criterion of energy saving for high performance signal sources |date=2016 |conference=IEEE International Frequency Control Symposium |doi=10.1109/FCS.2016.7546729}}
{{refend}}

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