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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

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Stability of Nonlinear Circuits

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  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

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