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

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

3. 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 ITSS'2007 – Pforzheim, July 7th-14th

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

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

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

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

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

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

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

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

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

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

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

15. 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 ITSS'2007 – Pforzheim, July 7th-14th

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

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

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

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

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

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

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

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

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

25. Nonlinear stability Instability in a power amplifier Bifurcation diagram real mathematical P out (f s ) P in (f LS ) P out P out (f LS ) PIPIPIPI | in | >  | in | <  The amplifier is stable in linear conditions

26. Nonlinear stability Design procedure – one port (1) Second step: Conversion matrix at f s and converted frequencies fsfsfsfs 2f 0 -f s f 0 +f s 2f 0 +f s f 0 -f s 3f 0 +f s fsfsfsfs 2f 0 f0f0f0f0 DC First step: Harmonic Balance analysis at nf 0 Z0Z0Z0Z0  L (nf 0 ) P in (f 0 )  S (nf 0 ) 2f 0 f0f0f0f0 DC

27. Nonlinear stability Design procedure – one port (2) Third step: Conversion matrix reduction to a one-port  S (f s )  in (f s ) fsfsfsfs 2f 0 -f s f 0 +f s 2f 0 +f s f 0 -f s 3f 0 +f s fsfsfsfs 2f 0 f0f0f0f0 Oscillation condition Fourth step: verification of the stability at f s stable potentially unstable yes/no design choice

28. Nonlinear stability Design procedure – two port (1): same as for one port Second step: Conversion matrix at f s and converted frequencies fsfsfsfs 2f 0 -f s f 0 +f s 2f 0 +f s f 0 -f s 3f 0 +f s fsfsfsfs 2f 0 f0f0f0f0 DC First step: Harmonic Balance analysis at nf 0 Z0Z0Z0Z0  L (nf 0 ) P in (f 0 )  S (nf 0 ) 2f 0 f0f0f0f0 DC

29. Nonlinear stability Design procedure – two port (2) Third step: Conversion matrix reduction to a two-port fsfsfsfs 2f 0 -f s f 0 +f s 2f 0 +f s f 0 -f s 3f 0 +f s fsfsfsfs 2f 0 f0f0f0f0 f 0 +f s  S (f s )  L (f 0 +f s ) Oscillation condition Fourth step: verification of the stability of the two-port stable potentially unstable yes/no ITSS'2007 – Pforzheim, July 7th-14th

30. Nonlinear stability Design procedure – N port (1): same as for one and two port Second step: Conversion matrix at f s and converted frequencies fsfsfsfs 2f 0 -f s f 0 +f s 2f 0 +f s f 0 -f s 3f 0 +f s fsfsfsfs 2f 0 f0f0f0f0 DC First step: Harmonic Balance analysis at nf 0 Z0Z0Z0Z0  L (nf 0 ) P in (f 0 )  S (nf 0 ) 2f 0 f0f0f0f0 DC ITSS'2007 – Pforzheim, July 7th-14th

31. Nonlinear stability Design procedure – N port (2) Third step: Conversion matrix reduction to a one-port fsfsfsfs 2f 0 -f s f 0 +f s 2f 0 +f s f 0 -f s 3f 0 +f s  S (f s )  in (f s ) …and simultaneous optimisation of all the loads at converted frequencies until: stable intentionally unstable (maybe) ITSS'2007 – Pforzheim, July 7th-14th

32. Nonlinear stability Design procedure – important remark f LS 2f LS DC fsfsfsfs 2f LS -f s f LS -f s f LS +f s 2f LS +f s 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 ITSS'2007 – Pforzheim, July 7th-14th

33. 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 ITSS'2007 – Pforzheim, July 7th-14th

34. 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 ITSS'2007 – Pforzheim, July 7th-14th

35. 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! ITSS'2007 – Pforzheim, July 7th-14th

36. 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 ITSS'2007 – Pforzheim, July 7th-14th

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

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

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

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

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

42. 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 ITSS'2007 – Pforzheim, July 7th-14th