1 / 42

Stability of Nonlinear Circuits

ITSS'2007 – Pforzheim, July 7th-14th. Stability of Nonlinear Circuits. Giorgio Leuzzi University of L'Aquila - Italy. ITSS'2007 – Pforzheim, July 7th-14th. Motivation. Definition of stability criteria and design rules

boyd
Download Presentation

Stability of Nonlinear Circuits

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ITSS'2007 – Pforzheim, July 7th-14th Stability of Nonlinear Circuits Giorgio Leuzzi University of L'Aquila - Italy

  2. ITSS'2007 – Pforzheim, July 7th-14th Motivation Definition of stability criteria and design rules for the design of stable or intentionally unstablenonlinear circuits under large-signal operations (power amplifiers)(frequency dividers) Standard criteria are valid only under small-signal operations

  3. ITSS'2007 – Pforzheim, July 7th-14th Outline • Linear stability – a reminder: • Linearisation of a nonlinear (active) device • Stability criterion for N-port networks • Nonlinear stability – an introduction: • Dynamic linearisation of a nonlinear (active) device • The conversion matrix • Extension of the Stability criterion • Examples and perspectives • Frequency dividers • Chaos

  4. ITSS'2007 – Pforzheim, July 7th-14th Linear stability A nonlinear device can be linearised around a static bias point Example: a diode

  5. stable potentially unstable (negative resistance) (passive) ITSS'2007 – Pforzheim, July 7th-14th Linear stability The stability of the small-signal circuit is easily assessed Oscillation condition:

  6. Oscillation condition: ITSS'2007 – Pforzheim, July 7th-14th Linear stability Oscillation condition

  7. I Oscillation possible V ITSS'2007 – Pforzheim, July 7th-14th Linear stability Example: tunnel diode

  8. ITSS'2007 – Pforzheim, July 7th-14th Linear stability Example: tunnel diode oscillator Oscillation condition:

  9. ITSS'2007 – Pforzheim, July 7th-14th Linear stability Stability of a two-port network: transistor amplifier

  10. Oscillation condition: stable potentially unstable (negative resistance) (passive) ITSS'2007 – Pforzheim, July 7th-14th Linear stability Stability of a two-port network

  11. K - stability factor: ITSS'2007 – Pforzheim, July 7th-14th Linear stability Stability of a two-port network Stability condition

  12. ITSS'2007 – Pforzheim, July 7th-14th Linear stability Stability of a two-port network (stability circle) potentially unstable stable

  13. oscillation condition ITSS'2007 – Pforzheim, July 7th-14th Linear stability Intentional instability: oscillator

  14. ITSS'2007 – Pforzheim, July 7th-14th Linear stability Stability of an N-port network No stability factor available!

  15. ITSS'2007 – Pforzheim, July 7th-14th Outline • Linear stability – a reminder: • Linearisation of a nonlinear (active) device • Stability criterion for N-port networks • Nonlinear stability – an introduction: • Dynamic linearisation of a nonlinear (active) device • The conversion matrix • Extension of the Stability criterion • Examples and perspectives • Frequency dividers • Chaos

  16. ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability A nonlinear device can be linearised around a dynamic bias point Example: a diode driven by a large signal

  17. Small-signal linear time- dependent (periodic) circuit ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability The large signal is usually periodic (example: Local Oscillator) The time-varying conductance is also periodic

  18. ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Example: switched-diode mixer The diode is switched periodically on and off by the large-signal Local Oscillator

  19. Red lines: large-signal (Local Oscillator) circuit Blue lines: small-signal linear time-dependent circuit fLS 2fLS fs fLS-fs fLS+fs DC 2fLS-fs 2fLS+fs ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Spectrum of the signals in a mixer

  20. Input signal Frequency-converting element (diode) Passive loads at converted frequencies ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Linear representation of a time-dependent linear network (mixer) Conversion matrix

  21. Conversion matrix ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Stability of the N-port linear time-dependent frequency-converting network (linearised mixer) …can be treated as any linear N-port network!

  22. Conversion matrix Instability at fs frequency ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability One-port stability - the input reflection coefficient can be: stable potentially unstable (negative resistance)

  23. The stability depends onthe large-signal amplitude (power) ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Important remark: the conversion phenomenon, and therefore the Conversion matrix, depend on the Large-Signal amplitude Conversion matrix

  24. A large signal is applied |in| >1 A spurious signal appears at a small-signal frequency and all converted frequencies fLS 2fLS fs fLS-fs fLS+fs DC 2fLS-fs 2fLS+fs ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Instability at small-signal and converted frequencies

  25. Pout Pout(fLS) |in| <  Pout(fs) PI Pin(fLS) The amplifier is stable in linear conditions Nonlinear stability Instability in a power amplifier Bifurcation diagram mathematical real |in| > 

  26. f0 2f0 First step: Harmonic Balance analysis at n•f0 L(nf0) S(nf0) DC Pin(f0) Z0 Second step: Conversion matrix at fs and converted frequencies fs f0+fs f0 f0-fs 2f0+fs 2f0 3f0+fs 2f0-fs fs DC Nonlinear stability Design procedure – one port (1)

  27. Third step: Conversion matrix reduction to a one-port f0 in(fs) S(fs) fs 2f0 Fourth step: verification of the stability at fs design choice fs f0+fs f0-fs 2f0+fs stable 3f0+fs 2f0-fs potentially unstable yes/no Oscillation condition Nonlinear stability Design procedure – one port (2)

  28. f0 2f0 First step: Harmonic Balance analysis at n•f0 L(nf0) S(nf0) DC Pin(f0) Z0 Second step: Conversion matrix at fs and converted frequencies fs f0+fs f0 f0-fs 2f0+fs 2f0 3f0+fs 2f0-fs fs DC Nonlinear stability Design procedure – two port (1): same as for one port

  29. L(f0+fs) S(fs) f0 fs 2f0 f0+fs Fourth step: verification of the stability of the two-port fs f0+fs yes/no f0-fs 2f0+fs stable 3f0+fs 2f0-fs potentially unstable Oscillation condition ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Design procedure – two port (2) Third step: Conversion matrix reduction to a two-port

  30. f0 2f0 First step: Harmonic Balance analysis at n•f0 L(nf0) S(nf0) DC Pin(f0) Z0 Second step: Conversion matrix at fs and converted frequencies fs f0+fs f0 f0-fs 2f0+fs 2f0 3f0+fs 2f0-fs fs DC ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Design procedure – N port (1): same as for one and two port

  31. in(fs) S(fs) fs f0+fs …and simultaneous optimisation of all the loads at converted frequencies until: f0-fs 2f0+fs stable 3f0+fs 2f0-fs intentionally unstable (maybe) ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Design procedure – N port (2) Third step: Conversion matrix reduction to a one-port

  32. fLS 2fLS fs fLS-fs fLS+fs DC 2fLS-fs 2fLS+fs ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Design procedure – important remark Loads at small-signal and converted frequencies are designed for stability/intentional instability Loads at fundamental frequency and harmonics must not be changed! …otherwise the Conversion matrix changes as well. This is not easy from a network-synthesis point of view

  33. ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Design problem: commercial software Currently, no commercial CAD software allows easy implementation of the design scheme A relatively straightforward procedure has been set up in Microwave Office (AWR) It is advisable that commercial Companies make the Conversion matrix and multi-frequency design available to the user

  34. ITSS'2007 – Pforzheim, July 7th-14th Outline • Linear stability – a reminder: • Linearisation of a nonlinear (active) device • Stability criterion for N-port networks • Nonlinear stability – an introduction: • Dynamic linearisation of a nonlinear (active) device • The conversion matrix • Extension of the Stability criterion • Examples and perspectives • Frequency dividers • Chaos

  35. ITSS'2007 – Pforzheim, July 7th-14th Examples Frequency divider-by-three based on a 3 GHz FET amplifier Harmonic Balance analysis of a 3-GHz stable amplifier Remark: a Harmonic Balance analysis will not detect an instability at a spurious frequency, not a priori included in the signal spectrum!

  36. ITSS'2007 – Pforzheim, July 7th-14th Examples Frequency divider-by-three based on a 3 GHz FET amplifier Spectra for increasing input power of the stable 3-GHz amplifier Spectra from time-domain analysis

  37. ITSS'2007 – Pforzheim, July 7th-14th Examples Frequency divider-by-three based on a 3 GHz FET amplifier Spectra for increasing input power of the modified amplifier

  38. Rout @ 50 MHz ITSS'2007 – Pforzheim, July 7th-14th Examples Frequency divider-by-two at 100-MHz Rout < 0

  39. ITSS'2007 – Pforzheim, July 7th-14th Examples Frequency divider-by-two at 100-MHz

  40. Bifurcation diagram id Vs ITSS'2007 – Pforzheim, July 7th-14th Examples Chaotic behaviour For increasing amplitude of the input signal, many different frequencies appear

  41. ITSS'2007 – Pforzheim, July 7th-14th Examples Chaotic behaviour The spectrum becomes dense with spurious frequencies, and the waveform becomes 'chaotic'

  42. ITSS'2007 – Pforzheim, July 7th-14th Conclusions • Nonlinear stability: • The approach based on the dynamic linearisation of a nonlinear (active) device is a natural extension of the linear stability approach • Can be studied by means of the well-known Conversion matrix • Design criteria are available, even though not yet implemented in commercial software • Stability criterion for N-port networks still missing

More Related