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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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motivation

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

outline

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
linear stability

ITSS'2007 – Pforzheim, July 7th-14th

Linear stability

A nonlinear device can be linearised around a static bias point

Example: a diode

linear stability1

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:

linear stability2

Oscillation condition:

ITSS'2007 – Pforzheim, July 7th-14th

Linear stability

Oscillation condition

linear stability3

I

Oscillation possible

V

ITSS'2007 – Pforzheim, July 7th-14th

Linear stability

Example: tunnel diode

linear stability4

ITSS'2007 – Pforzheim, July 7th-14th

Linear stability

Example: tunnel diode oscillator

Oscillation condition:

linear stability5

ITSS'2007 – Pforzheim, July 7th-14th

Linear stability

Stability of a two-port network:

transistor amplifier

linear stability6

Oscillation condition:

stable

potentially unstable (negative resistance)

(passive)

ITSS'2007 – Pforzheim, July 7th-14th

Linear stability

Stability of a two-port network

linear stability7

K - stability factor:

ITSS'2007 – Pforzheim, July 7th-14th

Linear stability

Stability of a two-port network

Stability condition

linear stability8

ITSS'2007 – Pforzheim, July 7th-14th

Linear stability

Stability of a two-port network

(stability circle)

potentially

unstable

stable

linear stability9

oscillation condition

ITSS'2007 – Pforzheim, July 7th-14th

Linear stability

Intentional instability: oscillator

linear stability10

ITSS'2007 – Pforzheim, July 7th-14th

Linear stability

Stability of an N-port network

No stability

factor

available!

outline1

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
nonlinear stability

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

nonlinear stability1

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

nonlinear stability2

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

nonlinear stability3

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

nonlinear stability4

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

nonlinear stability5

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!

nonlinear stability6

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)

nonlinear stability7

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

nonlinear stability8

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

nonlinear stability9

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| > 

nonlinear stability10

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)

nonlinear stability11

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)

nonlinear stability12

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

nonlinear stability13

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

nonlinear stability14

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

nonlinear stability15

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

nonlinear stability16

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

nonlinear stability17

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

outline2

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
examples

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!

examples1

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

examples2

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

examples3

Rout @ 50 MHz

ITSS'2007 – Pforzheim, July 7th-14th

Examples

Frequency divider-by-two at 100-MHz

Rout < 0

examples4

ITSS'2007 – Pforzheim, July 7th-14th

Examples

Frequency divider-by-two at 100-MHz

examples5

Bifurcation diagram

id

Vs

ITSS'2007 – Pforzheim, July 7th-14th

Examples

Chaotic behaviour

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

examples6

ITSS'2007 – Pforzheim, July 7th-14th

Examples

Chaotic behaviour

The spectrum becomes dense with spurious frequencies, and the waveform becomes 'chaotic'

conclusions

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