Christophe LETELLIER
20/03/2015

**Edwin Colpitts**

**Edwin Henry Colpitts** (1872-1949) was the chief of the research branch for the Western Electric. He contributed to the development of the thermionic vacuum tube as an amplifying element in a telephone repeater for radio telephony. In 1915, Colpitts proposed an oscillator using capacitive coupling (Fig. 1) as an alternative to the inductive coupling used by the oscillator invented by his colleague Ralph Hartley. He patented this oscillator [1] and only published this circuit few years later with Edward Craft [2]. He published a review paper on carrier multiplex telephony and telegraphy [3]. He and his co-workers achieved significant advances in the development of an oscillator, now known as the Colpitts oscillator. The theoretical circuit and the circuit to use in practice the Colpitts oscillator are shown in Figs 1a and 1b, respectively (these circuits are reproduced from [4]).

**Fig. 1. The Colpitts circuit.**

A chaotic attractor produced by a Colpitts circuit was found in 1994 by Michael P. Kennedy [5]. He was motivated in doing this by a statement in Nhat Nguyen’s Ph.D. Thesis [6] according which multioscillations "*is a phenomenon in which
two or more oscillations exist simultaneously in the steady state.
In other words, there are parasitic or unwanted oscillations existing
together with a main oscillation. Due to the multiple oscillations,
the resultant steady-state signal is severely distorted and hence has
limited application in communication systems.*" The Colpitts circuit was found to be described by the adimensional model [7]

where

, *x* is the voltage at *C*_{1}, *y* at *C*_{2}, and *z* is the
current across the inductance *L*. Parameters are

where
is the resonant frequency of the unloaded *LC*-circuit and *g* is the loop gain of the oscillator.

A chaotic attractor is found for , and *Q* = 4.679 as shown in Fig. 2a. Initial conditions are *x*_{0}=1, *y*_{0}=*z*_{0}=0. A Poincaré section for this attractor is conveniently chosen as

A first-return map to this Poincaré section presents three monotonic branches, one increasing and two decreasing branches (Fig. 2b).

**Fig. 2. Chaotic attractor produced by the Colpitts oscillator.**

The smooth maximum between the increasing and the first decreasing branches implies that the route to chaos is a period-doubling cascade (Fig. 3).

**Fig. 3. Bifurcation diagram versus parameter Q.**

The Colpitts oscillator has no symmetry property. Nevertheless, it is related to the Chua circuit when its inversion symmetry is modded out, that is, when the nonlinearity of the Chua circuit is transformed into an asymmetric one : the vector field of the Colpitts oscillator is in this case "*linearly conjugated to that of the asymmetric Chua circuit*"
[8].

[1] **E. H. Colpitts**, *System for the transmission of intelligence*, US Patent 1137384 A, for the Western Electric Co, 1915.

[2] **E. B. Craft & E. H. Colpitts**, Radio telephony, *Proceedings of the American Institute of Electrical Engineers*, **38** (3), 337-375, 1919.

[3] **E. H. Colpitts & O. B. Blackwell**, Carrier current telephony and telegraphy, *Transactions of the American Institute of Electrical Engineers*, **40**, 205-300, 1921.

[4] **H. J. Van Der Bijl**, *The thermionic vacuum tube and its applications*, McGraw-Hill, 1920.

[5] **M. P. Kennedy**, Chaos in the Colpitts oscillator, *IEEE Transactions on Circuits and Systems*, **41** (11), 771-774, 1994.

[6] N. M. Nguyen, *Monolithic microwave oscillators and amplifiers*, EECS Department
University of California, Berkeley, Technical Report No. UCB/ERL M91/36,
1991.

[7] **G. M. Maggio, O. De Feo & M. P. Kennedy**, Nonlinear analysis of the Colpitts oscillator and applications to design, *IEEE Transactions on Circuits and Systems*, **46** (9), 1118-1130, 1999.

[8] **M. P. Kennedy**, On the relationship between the chaotic Colpitts oscillator and Chua’s oscillator, *IEEE Transactions on Circuits and Systems*, **42** (6), 376-379, 1995.