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 for the design of stable or intentionally unstable nonlinear circuits (power amplifiers) (frequency dividers) under large-signal operations Standard criteria are valid only under small-signal operations 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 ITSS'2007 – Pforzheim, July 7th-14th Linear stability A nonlinear device can be linearised around a static bias point Example: a diode Vo I I ( t ) I 0 g V t V 0 I0 Rs v t v (t) g i( t ) Vth 1 Vo g Rs V V0 ITSS'2007 – Pforzheim, July 7th-14th Linear stability The stability of the small-signal circuit is easily assessed Rs s d s d Vs v (t) g Vth Vo Oscillation condition: s d 1 d 1 stable d 1 potentially unstable (negative resistance) s 1 (passive) ITSS'2007 – Pforzheim, July 7th-14th Linear stability Oscillation condition s d a a d s d a Oscillation condition: s d a a s d 1 ITSS'2007 – Pforzheim, July 7th-14th Linear stability Example: tunnel diode C I L 1 2 Rs Vo V s C 1 Rs d L 2 -g Vth Vo Oscillation possible d 1 ITSS'2007 – Pforzheim, July 7th-14th Linear stability Example: tunnel diode oscillator C s d L 1 2 Rs -g Vth Vo Oscillation condition: s s d 1 f0 1 2 LC 1 d Rs 1 g ITSS'2007 – Pforzheim, July 7th-14th Linear stability 2 L Stability of a two-port network: transistor amplifier 2 L Vo 0 1 1 Rs v (t) RL 0 s Vs in out L ITSS'2007 – Pforzheim, July 7th-14th Linear stability Stability of a two-port network s in out L Vs Oscillation condition: s in 1 in 1 stable in 1 potentially unstable (negative resistance) s 1 (passive) ITSS'2007 – Pforzheim, July 7th-14th Linear stability Stability of a two-port network s in out L Vs in 1 Stability condition in S 11 S 12 S 21 L 1 S 22 L K - stability factor: k 1 S 11 2 S 22 2 2 S 12 S 21 S 11 S 22 S 12 S 21 2 ITSS'2007 – Pforzheim, July 7th-14th Linear stability Stability of a two-port network s in out L Vs in f L k 1 k 1 (stability circle) L L in 1 stable in 1 in 1 potentially unstable ITSS'2007 – Pforzheim, July 7th-14th Linear stability k 1 Intentional instability: oscillator s in out in in 1 L s 1 in oscillation condition L ITSS'2007 – Pforzheim, July 7th-14th Linear stability Stability of an N-port network in f L 1 , L 2 , L 3 , L 4 , L 5 1 in s L1 L2 L3 L4 L5 No stability factor available! 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 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 V _ L S (t ) I I ( t ) I LS t g t V t V LS t I LS (t) Rs V i( t ) v t Rs V LS (t) v (t ) g (t ) 1 g t Vth V _ L S (t ) ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability The large signal is usually periodic (example: Local Oscillator) The time-varying conductance is also periodic g(t) I g t n Gn e jn LS t Rs v (t ) g (t ) Vth I LS (t) V _ L S (t ) V t V LS (t) Small-signal linear timedependent (periodic) circuit ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Example: switched-diode mixer V _ L S (t ) The diode is switched periodically on and off by the large-signal Local Oscillator Rs v (t) RL g (t) g (t) Rs v (t) RL t ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Spectrum of the signals in a mixer Red lines: large-signal (Local Oscillator) circuit Blue lines: small-signal linear time-dependent circuit fLS Rs 2fLS v (t) fs DC fLS-fs fLS+fs V _ L S (t ) 2fLS-fs 2fLS+fs ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Linear representation of a time-dependent linear network (mixer) Input signal Frequency-converting element (diode) fs fLS+ fs 2 fLS+ fs fLS-fs Conversion matrix 2 fLS-fs 3 fLS-fs Passive loads at converted frequencies ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Stability of the N-port linear time-dependent frequency-converting network (linearised mixer) fs fLS+ fs 2 fLS+ fs fLS-fs Conversion matrix 2 fLS-fs 3 fLS-fs …can be treated as any linear N-port network! ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability One-port stability - the input reflection coefficient can be: in 1 stable in 1 potentially unstable (negative resistance) in fs fLS+ fs 2 fLS+ fs fLS-fs Conversion matrix Instability at fs frequency 2 fLS-fs s 3 fLS-fs in ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Important remark: the conversion phenomenon, and therefore the Conversion matrix, depend on the Large-Signal amplitude g(t) I The stability depends on the large-signal amplitude (power) I LS (t) V t fs V LS (t) fLS+ fs 2 fLS+ fs fLS-fs Conversion matrix 2 fLS-fs 3 fLS-fs ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Instability at small-signal and converted frequencies A large signal is applied fLS 2fLS fs DC fLS-fs |in| >1 A spurious signal appears at a small-signal frequency and all converted frequencies fLS+fs 2fLS-fs 2fLS+fs Nonlinear stability Instability in a power amplifier Bifurcation diagram mathematical Pout Pout(fLS) real |in| < |in| > Pout(fs) The amplifier is stable in linear conditions PI Pin(fLS ) Nonlinear stability Design procedure – one port (1) First step: Harmonic Balance analysis at n•f0 S(nf0) f0 L(nf0) Pin(f0) 2f0 DC Z0 Second step: Conversion matrix at fs and converted frequencies f0 fs f0+fs f0-fs 2f0+fs 2f0-fs 3f0+fs 2f0 DC fs Nonlinear stability Design procedure – one port (2) Third step: Conversion matrix reduction to a one-port fs f0+fs f0-fs 2f0+fs 2f0-fs 3f0+fs S(fs) in(fs) fs stable in 1 potentially unstable 2f0 design choice Fourth step: verification of the stability at fs in 1 f0 s 1 in Oscillation condition yes/no Nonlinear stability Design procedure – two port (1): same as for one port First step: Harmonic Balance analysis at n•f0 S(nf0) f0 L(nf0) Pin(f0) 2f0 DC Z0 Second step: Conversion matrix at fs and converted frequencies f0 fs f0+fs f0-fs 2f0+fs 2f0-fs 3f0+fs 2f0 DC fs ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Design procedure – two port (2) f0 fs 2f0 Third step: Conversion matrix reduction to a two-port f0+fs fs f0+fs f0-fs 2f0+fs 2f0-fs 3f0+fs S(fs) L(f0+fs) Fourth step: verification of the stability of the two-port k 1 k 1 stable potentially unstable L yes/no s 1 in Oscillation condition ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Design procedure – N port (1): same as for one and two port First step: Harmonic Balance analysis at n•f0 S(nf0) f0 L(nf0) Pin(f0) 2f0 DC Z0 Second step: Conversion matrix at fs and converted frequencies f0 fs f0+fs f0-fs 2f0+fs 2f0-fs 3f0+fs 2f0 DC fs ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Design procedure – N port (2) Third step: Conversion matrix reduction to a one-port fs f0+fs f0-fs 2f0+fs 2f0-fs 3f0+fs S(fs) in(fs) …and simultaneous optimisation of all the loads at converted frequencies until: in 1 stable in 1 s 1 in intentionally unstable (maybe) 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! fLS 2fLS fs DC fLS-fs …otherwise the Conversion matrix changes as well. fLS+fs 2fLS-fs 2fLS+fs This is not easy from a network-synthesis point of view ITSS'2007 – Pforzheim, July 7th-14th Nonlinear stability Design problem: commercial software Currently, no commercial CAD software allows easy implementation of the design scheme SUBCKT ID=S1 NET="S_conv_matrix_real_imag_1_dBm" A relatively straightforward procedure has been set up in Microwave Office (AWR) 0.8 S(2 ,2 ) i n p ut_n e two rk _2 f0 1.0 Smith_loads S(1 ,1 ) o utpu t_ n etwork PORTF P=13 Z=50 Ohm Freq=fs GHz Pwr=Ps dBm Swp Max 1.728GHz 6 0. S(2 ,2 ) i n p ut_n e two rk _3 f0 PORT P=1 Z=50 Ohm 2. 0 S(1 ,1 ) o utpu t_ n etwork _ 2 f0 S(1 ,1 ) o utpu t_ n etwork _ 3 f0 S(2 ,2 ) d es i g n ed _ i n p ut_n e twork 0. 4 S(2 ,2 ) d es i g n ed _ i n p ut_n e twork _ 2 f0 PORT P=14 Z=50 Ohm PORT P=15 Z=50 Ohm PORT P=18 Z=50 Ohm PORT P=16 Z=50 Ohm PORT P=17 Z=50 Ohm PORT P=10 Z=50 Ohm PORT P=11 Z=50 Ohm PORT P=12 Z=50 Ohm S(2 ,2 ) i n p ut_n e two rk 0 3. S(2 ,2 ) d es i g n ed _ i n p ut_n e twork _ 3 f0 0 4. S(1 ,1 ) d es i g n ed _ o utpu t_ n etwo rk 12 11 10 1 9 2 8 3 7 PORT P=9 Z=50 Ohm 5 .0 S(1 ,1 ) d es i g n ed _ o utpu t_ n etwo rk _ 2f0 0 .2 S(1 ,1 ) d es i g n ed _ o utpu t_ n etwo rk _ 3f0 PORT P=3 Z=50 Ohm PORT P=2 Z=50 Ohm 2 PORT P=5 Z=50 Ohm 2 f0<f<2*f0 0<f<f0 2*f0<f<3*f0 PORT P=4 Z=50 Ohm 2 f0<f<2*f0 2 2 0<f<f0 4 4 SUBCKT ID=S3 NET="output_couplers" 3 5 1 PORT1 P=19 Z=50 Ohm Pwr=Pin dBm 2 1 P_METER3 ID=P1 2 W 1 2 SUBCKT ID=S4 NET="input_network" 1 2 SUBCKT ID=S2 NET="biased_transistor" 1 2 SUBCKT ID=S5 NET="output_network" I 2 1 V 3 RES ID=R1 R=50 Ohm PORT P=3 Z=50 Ohm .0 -2 -1.0 -0.8 -0 .6 .4 -0 SUBCKT ID=S1 5 NET="input_couplers" 3 10.0 4.0 5.0 3.0 2.0 1.0 0.8 S(2 ,2 ) l i ne a r_ de s i g n ed _ am p l i fi e r 0.6 DPWRSMP ID=U6 R=50 PORT P=2 Z=50 Ohm PORT P=8 Z=50 Ohm 4 - 0. 2*f0<f<3*f0 10.0 S(1 ,1 ) l i ne a r_ de s i g n ed _ am p l i fi e r S(2 ,2 ) l i ne a r_ am p l i fi er DPWRSMP ID=U5 R=50 4 PORT P=6 Z=50 Ohm S(1 ,1 ) l i ne a r_ am p l i fi er 1 0<f<f0 3 DPWRSMP ID=U4 R=50 4 PORT P=1 Z=50 Ohm PORT P=10 Z=50 Ohm 0.4 3 0.2 PORT 1 P=11 f0<f<2*f0 Z=50 Ohm 3 DPWRSMP ID=U1 R=50 4 2 PORT 1 P=12 2*f0<f<3*f0 Z=50 Ohm 0 DPWRSMP ID=U2 R=50 4 2 0<f<f0 1 3 DPWRSMP ID=U3 R=50 4 PORT P=7 Z=50 Ohm 4 5 6 PORT P=7 Z=50 Ohm -3 .0 3 -4 .0 - 5. 0 PORT 1 P=8 f0<f<2*f0 Z=50 Ohm 3 -10. 0 PORT 1 P=9 2*f0<f<3*f0 Z=50 Ohm Swp Min 0.576GHz PORT P=4 Z=50 Ohm PORT P=5 Z=50 Ohm PORT P=6 Z=50 Ohm It is advisable that commercial Companies make the Conversion matrix and multifrequency design available to the user 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 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! 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 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 ITSS'2007 – Pforzheim, July 7th-14th Examples Frequency divider-by-two at 100-MHz 1 Rout @ 50 MHz 2 2 F RE Q = 10 0 M H z 1 0 Rout < 0 ITSS'2007 – Pforzheim, July 7th-14th Examples Frequency divider-by-two at 100-MHz ITSS'2007 – Pforzheim, July 7th-14th Examples Chaotic behaviour Bifurcation diagram id For increasing amplitude of the input signal, many different frequencies appear Vs ITSS'2007 – Pforzheim, July 7th-14th Examples Chaotic behaviour The spectrum becomes dense with spurious frequencies, and the waveform becomes 'chaotic' 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