Entrainment Phenomena in Nonlinear Oscillations

  • Hans Georg Brachtendorf
  • Rainer Laur
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 17)


Entrainment or injection locking is the underlying effect of synchronization. It can therefore be observed in a variety of fields including physics, biology and electronic engineering. In recent years various circuit designs have been developed using injection locking for the design of i.e. quadrature oscillators, frequency dividers and circuits exhibiting low phase noise. On the other hand, unwanted temporary entrainment known as pulling can be a severe cause of performance degradation for zero-IF or low-IF transceivers. Therefore entrainment effects have been studied since decades (i.e. Andronov and Witt, Adler, Kurokawa). A general theory is still missing.

In this paper, we give a theory of injection phenomena based on a perturbation technique employing Floquet’s theory. The theory is valid as long as the injected signal power is sufficiently small.


Phase Noise Nonlinear Oscillation Impulse Train Arnold Tongue Colpitts Oscillator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler, R.: A study of locking phenomena in oscillators. Proc. IRE 34, 351–357 (1946)CrossRefGoogle Scholar
  2. 2.
    Andronov, A., Witt, A.: Zur Theorie des Mitnehmens von van der Pol. Archiv für Elektrotechnik 24, 99–110 (1930)CrossRefGoogle Scholar
  3. 3.
    Brachtendorf, H.G.: Theorie und Analyse von autonomen und quasiperiodisch angeregten elektrischen Netzwerken. Eine algorithmisch orientierte Betrachtung. Universität Bremen, Bremen (2001). HabilitationsschriftGoogle Scholar
  4. 4.
    Demir, A.: Floquet Theory and Nonlinear Perturbation Analysis for Oscillators with Differential-Algebraic Equations. Tech. Rep. ITD-98-33478N, Bell-Laboratories (1998)Google Scholar
  5. 5.
    Demir, A., Roychowdhury, J.: A reliable and efficient procedure for oscillator ppv computation, with phase noise macromodeling applications. IEEE Trans. Comp. Aided Des. Integrated Circ. Syst. 22(2), 188–197 (2003). doi:10.1109/TCAD.2002.806599CrossRefGoogle Scholar
  6. 6.
    Harutyunyan, D., Rommes, J., ter Maten, J., Schilders, W.: Simulation of mutually coupled oscillators using nonlinear phase macromodels. IEEE Trans. Comp. Aided Des. Integrated Circ. Syst. 28(10), 1456–1466 (2009). doi:10.1109/TCAD.2009.2026359CrossRefGoogle Scholar
  7. 7.
    Kaertner, F.X.: Analysis of white and f  − α noise in oscillators. Int. J. Circ. Theor. Appl. 18, 485–519 (1990)Google Scholar
  8. 8.
    Kinget, P., Melville, R., Long, D., Gopinathan, V.: An injection-locking scheme for precision quadrature generation. IEEE J. Solid-State Circ. 37(7), 845 –851 (2002). doi:10.1109/JSSC.2002.1015681CrossRefGoogle Scholar
  9. 9.
    Kurokawa, K.: Injection locking of microwave solid-state oscillators. Proc. IEEE 61, 1386–1410 (1973)CrossRefGoogle Scholar
  10. 10.
    Laur, R., Brachtendorf, H.G.: Computerized method for determination and optimization of the synchronization region of a circuit or system (2000). Patent DE10062414Google Scholar
  11. 11.
    Tiebout, M.: A cmos direct injection-locked oscillator topology as high-frequency low-power frequency divider. IEEE J. Solid-State Circ. 39(7), 1170–1174 (2004). doi:10.1109/JSSC.2004.829937CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.University of Upper AustriaHagenbergAustria
  2. 2.Institute for Electromagnetic Theory and MicroelectronicsUniversity of BremenBremenGermany

Personalised recommendations