# Colpitts振盪器

## 概述

 图1：简单的共基極Colpitts振荡器（简化偏置） 图2：简单的共集极Colpitts振荡器（简化偏置）

${\displaystyle f_{0}={1 \over 2\pi {\sqrt {L\left({C_{1}C_{2} \over C_{1}+C_{2}}\right)}}}}$

## 理论

${\displaystyle Z_{in}={\frac {v_{1}}{i_{1}}}}$

${\displaystyle v_{1}}$ 為輸入電壓，${\displaystyle i_{1}}$ 為輸入電流，電壓 ${\displaystyle v_{2}}$ 的值是根據下式：

${\displaystyle v_{2}=i_{2}Z_{2}}$

${\displaystyle Z_{2}}$ 的值為 ${\displaystyle C_{2}}$ 的阻抗。流入 ${\displaystyle C_{2}}$ 的電流值為 ${\displaystyle i_{2}}$，這個值是另外兩個電流值的和：

${\displaystyle i_{2}=i_{1}+i_{s}}$

${\displaystyle i_{s}=g_{m}\left(v_{1}-v_{2}\right)}$

${\displaystyle g_{m}}$ 是BJT的跨導（transconductance）。另外一個電流值 ${\displaystyle i_{1}}$ 的表示式為：

${\displaystyle i_{1}={\frac {v_{1}-v_{2}}{Z_{1}}}}$

${\displaystyle Z_{in}=Z_{1}+Z_{2}+g_{m}Z_{1}Z_{2}}$

${\displaystyle R_{in}=g_{m}\cdot Z_{1}\cdot Z_{2}}$

${\displaystyle Z_{1}}$${\displaystyle Z_{2}}$ 為同號複數，${\displaystyle R_{in}}$ 便會是負阻抗（negative resistance）。若 ${\displaystyle Z_{1}}$${\displaystyle Z_{2}}$${\displaystyle j{\omega }C_{1}}$${\displaystyle j{\omega }C_{2}}$ 代入 ${\displaystyle R_{in}}$

${\displaystyle R_{in}={\frac {-g_{m}}{\omega ^{2}C_{1}C_{2}}}}$

${\displaystyle R_{in}=-30\ \Omega }$

${\displaystyle A_{v}=g_{m}\cdot R_{p}\geq 4}$
Hartley和Colpitts振荡器对比

${\displaystyle R_{in}=-g_{m}\omega ^{2}L_{1}L_{2}}$

### 振荡幅度

${\displaystyle V_{C}=2I_{C}R_{L}{\frac {C_{2}}{C_{1}+C_{2}}}}$

## 参考文献

1. ^ US 1624537,Colpitts, Edwin H.,「Oscillation generator」,发表于1 February 1918,发行于12 April 1927
2. ^ Gottlieb, Irving Gottlieb. Practical Oscillator Handbook. US: Elsevier. 1997: 151. ISBN 0750631023.
3. ^ Carr, Joe. RF Components and Circuits. US: Newnes. 2002: 127. ISBN 0750648449.
4. ^ Basak, A. Analogue Electronic Circuits and Systems. UK: Cambridge University Press. 1991: 153. ISBN 0521360463.
5. ^ Rohde, Ulrich L.; Matthias Rudolph. RF / Microwave Circuit Design for Wireless Applications, 2nd Ed.. John Wiley & Sons. 2012: 745–746. ISBN 1118431405.
6. ^
7. ^ S. Sarkar, S. Sarkar, B. C. Sarkar. "Nonlinear Dynamics of a BJT Based Colpitts Oscillator with Tunable Bias Current" 互联网档案馆存檔，存档日期2014-08-14.. IJEAT ISSN: 2249–8958, Volume-2, Issue-5, June 2013. p. 1.
8. ^ Razavi, B. Design of Analog CMOS Integrated Circuits. McGraw-Hill. 2001.
9. ^ Theron Jones. "Design a Crystal Oscillator to Match Your Application"[失效連結]. Maxim tutorial 5265 Sep 18, 2012, Maxim Integrated Products, Inc
10. ^ Trade-Offs in Analog Circuit Design: The Designer's Companion, Part 1 By Chris Toumazou, George S. Moschytz, Barrie Gilbert [1]

## 延伸阅读

• Lee, T. The Design of CMOS Radio-Frequency Integrated Circuits. Cambridge University Press. 2004.
• Ulrich L. Rohde, Ajay K. Poddar, Georg Böck "The Design of Modern Microwave Oscillators for Wireless Applications ", John Wiley & Sons, New York, NY, May, 2005, ISBN 978-0-471-72342-4.
• George Vendelin, Anthony M. Pavio, Ulrich L. Rohde " Microwave Circuit Design Using Linear and Nonlinear Techniques ", John Wiley & Sons, New York, NY, May, 2005, ISBN 978-0-471-41479-7.