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### Stability of Nonlinear Circuits

ITSS'2007 – Pforzheim, July 7th-14th

Giorgio Leuzzi

University of L'Aquila - Italy

ITSS'2007 – Pforzheim, July 7th-14th

MotivationDefinition 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

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 stabilityA nonlinear device can be linearised around a static bias point

Example: a diode

potentially unstable (negative resistance)

(passive)

ITSS'2007 – Pforzheim, July 7th-14th

Linear stabilityThe stability of the small-signal circuit is easily assessed

Oscillation condition:

ITSS'2007 – Pforzheim, July 7th-14th

Linear stabilityExample: tunnel diode oscillator

Oscillation condition:

ITSS'2007 – Pforzheim, July 7th-14th

Linear stabilityStability of a two-port network:

transistor amplifier

stable

potentially unstable (negative resistance)

(passive)

ITSS'2007 – Pforzheim, July 7th-14th

Linear stabilityStability of a two-port network

ITSS'2007 – Pforzheim, July 7th-14th

Linear stabilityStability of a two-port network

Stability condition

ITSS'2007 – Pforzheim, July 7th-14th

Linear stabilityStability of a two-port network

(stability circle)

potentially

unstable

stable

ITSS'2007 – Pforzheim, July 7th-14th

Linear stabilityStability of an N-port network

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 stabilityA nonlinear device can be linearised around a dynamic bias point

Example: a diode driven by a large signal

dependent (periodic) circuit

ITSS'2007 – Pforzheim, July 7th-14th

Nonlinear stabilityThe large signal is usually periodic (example: Local Oscillator)

The time-varying conductance is also periodic

ITSS'2007 – Pforzheim, July 7th-14th

Nonlinear stabilityExample: switched-diode mixer

The diode is switched periodically on and off by the large-signal Local Oscillator

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 stabilitySpectrum of the signals in a mixer

Frequency-converting element (diode)

Passive loads at converted frequencies

ITSS'2007 – Pforzheim, July 7th-14th

Nonlinear stabilityLinear representation of a time-dependent linear network (mixer)

Conversion

matrix

matrix

ITSS'2007 – Pforzheim, July 7th-14th

Nonlinear stabilityStability of the N-port linear time-dependent frequency-converting network (linearised mixer)

…can be treated as any linear N-port network!

matrix

Instability at fs frequency

ITSS'2007 – Pforzheim, July 7th-14th

Nonlinear stabilityOne-port stability - the input reflection coefficient can be:

stable

potentially unstable (negative resistance)

The stability depends onthe large-signal amplitude (power)

ITSS'2007 – Pforzheim, July 7th-14th

Nonlinear stabilityImportant remark: the conversion phenomenon, and therefore the Conversion matrix, depend on the Large-Signal amplitude

Conversion

matrix

|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 stabilityInstability at small-signal and converted frequencies

Pout(fLS)

|in| <

Pout(fs)

PI

Pin(fLS)

The amplifier is stable in linear conditions

Nonlinear stabilityInstability in a power amplifier

Bifurcation diagram

mathematical

real

|in| >

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 stabilityDesign procedure – one port (1)

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 stabilityDesign procedure – one port (2)

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 stabilityDesign procedure – two port (1): same as for one port

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 stabilityDesign procedure – two port (2)

Third step: Conversion matrix reduction to a two-port

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 stabilityDesign procedure – N port (1): same as for one and two port

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 stabilityDesign procedure – N port (2)

Third step: Conversion matrix reduction to a one-port

2fLS

fs

fLS-fs

fLS+fs

DC

2fLS-fs

2fLS+fs

ITSS'2007 – Pforzheim, July 7th-14th

Nonlinear stabilityDesign 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

ITSS'2007 – Pforzheim, July 7th-14th

Nonlinear stabilityDesign 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

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

ExamplesFrequency 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

ExamplesFrequency 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

ExamplesFrequency divider-by-three based on a 3 GHz FET amplifier

Spectra for increasing input power of the modified amplifier

id

Vs

ITSS'2007 – Pforzheim, July 7th-14th

ExamplesChaotic behaviour

For increasing amplitude of the input signal, many different frequencies appear

ITSS'2007 – Pforzheim, July 7th-14th

ExamplesChaotic 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

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