ANALYSIS OF ANALOG CIRCUITS USING VOLTERRA SERIEStimor/DCourse/Chp1_2.pdf1. Phenomena caused by...

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(C) 1999- Timo Rahkonen, University of Oulu, Oulu, Finland

ANALYSIS OF ANALOG CIRCUITS USING V OLTERRA SERIES

BY:

TIMO RAHKONEN

ELECTRONICS LABORATORYDEPARTMENT OF ELECTRCAL ENGINEERING AND INFOTECH OULU

UNIVERSITY OF OULU

90571 OULUFINLAND

email [email protected]

2


CONTENTS

1. Phenomena caused by nonlinearities2. Ways to model nonlinear effects3. Volterra analysis4. Nonlinearities in IC devices5. Example circuits6. Linearization methods7. Noise in nonlinear circuits

3


1. EFFECTS OF NONLINEARITIES

Phenomena caused by nonlinearities

• Finite escape time (goes to unstability quickly)• Multiple operating points• Limit cycle oscillations• Subharmonic, harmonic, or almost-periodic oscillations• Staircase functions• Chaos• Multiple modes of operation or existence (e.g. freezing of water)

• Signal-dependant gain (Variations in loop gain in control systems, gain compression in telecomms)• Harmonic and IMD frequency components• Frequency translation of signals and noise• Jump resonance (in nonlinear filters)• Interference rectification to baseband

4


TIME DOMAIN EFFECTS

In this simple example, some time domain effects areshown. The system is described by a nonlinear diffe-rential equation:

Two important effects are seen:

• As the nonlinearity changes the effective value ofg(u), thetime constant in the nonlinear system isdifferent from that in a linear system anddependson signal amplitude

• The present amplitude affects the height of the stepresponse. Thus,superposition does not apply.

tddu 1

C---- iin t( ) g u( ) u t( )⋅–( )⋅=

g u( ) g0 g1 u t( )2⋅+=

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Different time constant

Amplitude compression due toprevious history

LINEARNONLIN

TIME s

Cg(u)

u(t)

i in

5


MEASUREMENTS OF NONLINEAR SYSTEMS

Non-forced system

• Stability analysis

Single-tone tests

• Harmonic distortion (HD)• Gain compression/expansion (P-1dB, AM-AM and AM-PM)• Large-signal impedances (large signal s-params, oscillator analysis)• Root-locus analysis vs. varying gain

2-tone tests

• Intermodulation distortion (IMD)• Cross-modulation distortion• Desensitation

Wideband tests

• Adjacent channel power leakage (ACP)

6


A CLASSIFIFICA TION OF SYSTEMS

Linear memoryless systems

• analysed by solving one matrix equation

Linear systems with memory

• phasor calculations or linear ODE• sinusoidal response obtained from a matrix equation with complex coefficients

Nonlinear memoryless systems

• immediate nonlinear time response• usually no closed form solution• possible to model with series expansion (if converges)• equations solved iterativelly (e.g. Newton-Raphson)

Nonlinear systems with memory

• described by nonlinear differential equations (initial or boundary value problems)• generally no closed form solutions• sinusoidal responses can be calculated using Volterra analysis (if converges)

7


CIRCUITS WITH MEMOR Y

In nonlinear circuits, even order nonlinearity rectifiesdc component out of input signal. Thus, the dc biaspoint varies with signal level. The settling of dc biaspoint may be slow because of:

DC decoupling in wideband amplifiers

The DC point is set by feedback, but when dc point isvaried, also the voltage over decoupling cap needs tochange. This time constant is generally much slowerthan slowest signal component.

Active biasing

The active bias sensing used commonly BJT poweramplifiers employs slow pnps and may create longtime constants

Thermal time constants

Thermal time constants on silicon are in the range of 1- 100 us, package and PCB may cause time constantsof several seconds.

8


REMINDER OF IMPOR TANT TRANSFORM PAIRS

In linear systems, time domain convolution can be seen in frequency domain as multiplication of spectrums.However, in nonlinear systems, the spectral effects of time domain multiplication can be found by convolvingthetwo-sidedspectrums of the inputs.

y t( ) h t( )* x t( )=

h τ( )x t τ–( ) τdo

t

∫=

Linear system

Y s( ) H s( ) X s( )⋅=

ω

Y s( ) H s( )* X s( )=y t( ) x t( ) x t( )⋅=

Nonlinear system

ω ω

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OUTPUT SPECTRUM OF A 2-TONE TEST

Supposing a 2-tone test, the output spectrum of Nthorder nonlinearity can simply be obtained by convol-ving a 2-sided spectrum N times with itself. Note thatthe phase of the negative frequency components isopposite to positive frequency components.

In Matlab, it looks this (this is handy in analysing e.g.image rejecting mixers)

%Convolve spectrumsspek = 0.5*[ 0 0 0 0 0 1 1 0]; % SSB spect.spek1 = [fliplr(spek) 0 spek]; % make 2-sided spec.

spek2 = conv(spek1,spek1);spek3 = conv(spek2,spek1);spek4 = conv(spek3,spek1);spek5 = conv(spek4,spek1);

-40 -30 -20 -10 0 10 20 30 400

0.5

1

1.5

-40 -30 -20 -10 0 10 20 30 400

0.5

1

1.5

-40 -30 -20 -10 0 10 20 30 400

0.5

1

1.5

-40 -30 -20 -10 0 10 20 30 400

1

2

3

4

-40 -30 -20 -10 0 10 20 30 400

1

2

3

4

Order

1

2

3

4

5

A1A1*

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... WIDEBAND SIGNAL

Here, a flat wideband spectrum is convolved with itselfN times.

spek = 1*[ 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 ];spek1 = [fliplr(spek) 0 spek];

spek2 = 0.25*conv(spek1,spek1);spek3 = 0.25/3*conv(spek2,spek1);spek4 = 0.125*conv(spek3,spek1);spek5 = 0.125*conv(spek4,spek1);

(coeffs are arbitratry)

-60 -40 -20 0 20 40 600

0.5

1

1.5

-60 -40 -20 0 20 40 600

0.5

1

1.5

-60 -40 -20 0 20 40 600

0.5

1

1.5

-60 -40 -20 0 20 40 600

0.5

1

1.5

-60 -40 -20 0 20 40 600

0.5

1

1.5

Order

1

2

3

4

5

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CHANGES IN OPERATING MODE

Consider an example, where a controller for coolingand freezing soil samples is required. Thermal energythat needs to be transferred from soil varies nonlinea-rily with temperature, as freezing of water releaseshugh amounts of temperature. Also the thermal capaci-tance of water differs from that of ice. Thus, the designof the controller is not trivial.

In electronics, e.g. the gate capacitance of powerMOSFETs shows similar type of behaviour.

Thermalenergy J

Thermalcapacitance

Temp

Temp

12


2. WAYS TO ANALYZE NONLINEARITIES

2.1 Taylor series expansions

2.2 Describing function method ( fundamental tone analysis)

• Nonlinear control theory (also sigma-delta converter gain analysis)• AM-AM and AM-PM• oscillator large signal gain analysis

2.3 Nonlinear control theory (for stability analysis)

• Visualisation by state equations and phase portrait trajectories• Lyupunov stability analysis

2.4 Volterra analysis (in detail in Chp. 3)

• Extension of small-signal sinusoidal analysis into nonlinear circuits• Contains both amplitude and phase information

2.5 Numerical simulation (see Kundert99)

• Transient analysis• Harmonic balance• Envelope harmonic balance• Shooting methods

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2.1 TAYLOR SERIES ANALYSIS OF AN N-TONE TEST

Suppose a Taylor expansion of a nonlinear function y=f(x)

where

and x(t) contains the ac component only, i.e. the effect of DC bias x0 is included in K0. For small-signal ana-lysis, the series must be expanded around the operating point. In a N-tone test,

or, which is not so tedious to calculate

y t( ) f x t( )( ) K0 K1 x t( ) x0–( )⋅ K2 x t( ) x0–( )2⋅ K3 x t( ) x0–( )3⋅ ...+ + + += =

K1dfdx------

x x0== K2

12--- d2 f

dx2---------

x x0=

⋅= K316--- d3 f

dx3---------

x x0=

⋅= Kn1n!----- dn f

dxn---------

x x0=

⋅=

x t( ) Ai ωit θi+( )cos⋅i 1=

N

∑=

x t( )Ai

2----- e

j ωit θi+( )e

j– ωit θi+( )+( )⋅

i 1=

N

∑Ai˜ e

jωit⋅ Ai˜ ∗ e

jωit–⋅+

2------------------------------------------------------------

i 1=

N

∑= =

14


Supposing that the input signal to a nth order nonlinearity is a sum of Q sinusoids

where Ai may be complex, containing the phase information, and A-k = Ak* . Then, akx(t)k will be k-times sum

which does k-times convolution of the line spectrums. In a 2-tone test the output will be seen at frequencies

where

x t( ) 12--- Ai e

jωit⋅i Q–=

Q

∑⋅= y ak x t( )k⋅k 1=

n

∑=

Kk x t( )k⋅Kk

2k

------ ...

i2 Q–=

Q

∑i1 Q–=

Q

∑ Ai1ik Q–=

Q

∑ ... Aik j ωi1 ... ωik+ +( )t( )exp⋅ ⋅ ⋅ ⋅ ⋅=

ωout M ω1⋅ L± ω2⋅±=

M L+ order of nonlinearity=

15


A1 A2A2* A1* x 1/2

x 1/4

A222A1A2

A12

x 1/8

A23A13

A22A1*+(2A2A1*)A2 = 3A22A1*

2A2A1*

16


Here, the amplitudes of various components caused by a two-tone test in a 5th order nonlinearity are shown.These correspond to single-sided spectrum, i.e. sine or cosine amplitudes. If a 2-sided spectrum is required, allamplitude values except the DC component must be divided by 2.

E.g. the envelope term is (1-sided)

or (as 2-sided spectrum)

Table 1: Frequency components in a 2-tone test: DC and envelope terms

f K1 K2 / 2 K3 / 4 K4 / 8 K5 / 16

DC A1A1*+

A2A2*

12A1A1*A2A2

*

+3A12A1

*2

+3A22A2

*2

2A2A1* 12A1A1

2*A2

+12A1*A2

2A2*

6A1*2A2

2

K2A2A1* 3K4

2---------- A1A1

*2A2 A1

*A2

2A2

*+( )⋅+

j ω2 ω1–( )t( )exp⋅

A2A1*

K2 1.5K4 A1A1*

A2A2*

+( )⋅+( )⋅2

----------------------------------------------------------------------------------------------- j ω2 ω1–( )t( )exp⋅A2

*A1 K2 1.5K4 A1A1

*A2A2

*+( )⋅+( )⋅

2----------------------------------------------------------------------------------------------- j– ω2 ω1–( )t( )exp⋅+

ω2 ω1–

2 ω2 ω1–( )

17


Table 1: Frequency components in a 2-tone test: fundamental and IM components

f K1 K2 / 2 K3 / 4 K4 / 8 K5 / 16

10A13A2

*2

3A12A2

* 30A12A2A2

*2

+20A13A1

*A2*

A1 6A1A2*A2

+3A1A1A1*

60A12A1

*A2A2*

+30A1A22A2

*2

+10A13A1

*2

A2 6A1A1*A2

+3A2A2A2*

60A1A1*A2

2A2*

+30A12A1

*2A2

+10A23A2

*2

3A1*A2

2 30A1A1*2A2

2

+20A1*A2

3A2*

10A1*2A2

3

3ω1 2ω2–

2ω1 ω2–

ω1

ω2

2ω2 ω1–

3ω2 2ω1–

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Table 1: Frequency components in a 2-tone test: 2nd harmonic

f K1 K2 / 2 K3 / 4 K4 / 8 K5 / 16

4A13A2

*

A12 12A1

2A2A2*

+ 4A13A1

*

2A1A2 12A12A1

*A2

+12A1A22A2

*

A22 12A1A1

*A22

+ 4A23A2

*

4A1*A2

3

3ω1 ω2–

2ω1

ω1 ω2+

2ω2

3ω2 ω1–

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Table 1: Frequency components in a 2-tone test: 3rd harmonic

f K1 K2 / 2 K3 / 4 K4 / 8 K5 / 16

5A14A2

*

A13 20A1

3A2A2*

+5A14A1

*

3A12A2 30A1

2A22A2

*

+20A13A1

*A2

3A1A22 30A1

2A1*A2

2

+20A1A23A2

*

A23 20A1A1

*A23

+5A24A2

*

5A1*A2

4

4ω1 ω2–

3ω1

2ω1 ω2+

ω1 2ω2+

3ω2

4ω2 ω1–

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Table 1: Frequency components in a 2-tone test: 4th and 5th harmonics

f K1 K2 / 2 K3 / 4 K4 / 8 K5 / 16

A14

4A13A2

6A12A2

2

4A1A23

A24

A15

5A14A2

10A13A2

2

10A12A2

3

5A1A24

A25

4ω1

3ω1 ω2+

2ω1 2ω2+

ω1 3ω2+

4ω2

5ω1

4ω1 ω2+

3ω1 2ω2+

2ω1 3ω2+

ω1 4ω2+

5ω2

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0 ω2−

ω1

2ω1−

ω2

ω1

ω2

2ω2−

ω1

2ω1

ω1+

ω2

2ω2

3ω1

2ω1+

ω2

3ω2

ω1+

2ω2

Κ2Α

2

0.5Κ

2Α2

0.5Κ

2Α2

Κ2Α

2

Κ2Α

2

(3/4

)Κ3Α

3

(3/4

)Κ3Α

3

(3/4

)Κ3Α

3

(3/4

)Κ3Α

3

(1/4

)Κ3Α

3

(1/4

)Κ3Α

3

Κ1Α

+(9/

4)Κ 3

Α3

Κ1Α

+(9/

4)Κ 3

Α3

Κ2Α2

(3/4)Κ3Α3

1:2

1:3

IM3 IM3 IM3 IM3HD3 HD3IM2 HD2HD2IM2DC

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2ND ORDER PRODUCTS

In general, the multiplication halves the amplitude:

but the 2nd order nonlinearity involves a multiply-by-2operation:

Thus, mixing products are created without attenuation

x( )cos y( )cos⋅ x y+( )cos x y–( )cos+2

---------------------------------------------------------=

v1 v2+( ) v12 2v1v2 v2

2+ +=

1st ordervoltages

2nd ordervoltages

2nd order nonlinarity

v1 v2

2v1v2

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Al AhAh* Al*

x 1/2AhA2hAlA2l

A2hAsum

A2lAenv

Adc+

K2

A2Asum+AlA2hAhA2l+AlAsum

A2hAl+AhAenv

AdcAh+AenvAl+AsumAl*+A2hAh*

x 1/2

x 1/2

24


SOME COMMON APPROXIMA TIONS USING TAYLOR EXPANSIONS

Supposing, that components have a single, memoryless nonlinearity, some figures of merit (as amplitudes oramplitude ratios) can be calculated using a Taylor series of the nonlinearity

by adding a 1-tone or 2-tone excitation in x(t) and comparing the amplitudes of fundamental and harmonic orIM components

Table 1: Amplitudes for various figures of merit

HD2/F0 IM2/F0

HD3/F0 IM3/F0

IIP2(harm) IIP2(IM)

IIP3(harm) IIP3(IM)

1dBcompr.

y K0 K1 x⋅ K2 x2⋅ K3 x

3⋅ ...+ + + +=

A K2 K1⁄ 2⁄⋅ A K2 K1⁄⋅

A2

K3 K1⁄ 4⁄⋅ A2

3 K3 K1⁄ 4⁄⋅

2 K1 K2⁄⋅ K1 K2⁄

2 K1 K3⁄⋅ 2 K1 3K3⁄⋅

0.38K1

K3---------⋅

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COMMON MEASURES

IIP3 (input intercept point), dB scale

IIP3, linear scale

IIP3 Pin

P1 PIM3–

2-------------------------+=

iip3 vin

v1vIM3------------⋅=

log(Vin)

log(

Vout

)

IIP3

IIP3

log(Vin)

+10dB/dek

+30dB/dek

+50dB/dek

IM3IM5

~A3 ~A5

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MORE GENERAL EXPRESSIONS

Suppose a M-tone test signal, nth order nonlinearity,and response at frequency fΣ, where

is valid. Then nth order output yn(t) is as shown left,where Volterra kernel Hn() is evaluated at a propercombination of tones f1,f2,..fM and the gain coeffi-cient of the tone fΣ is Bn(m).

This explains why mixed terms have higher amplitudethan e.g. pure harmonics.

mii 1=

M

∑ n=

f Σ m1 f 1 m2 f 2 ... mM f M+ + +=

yn t( ) 12--- Bn m( ) Hn( ) j2πf Σt( )exp⋅ ⋅∑⋅=

Hn( ) H f 1 .. f 1 f 2 .. f 2 .. f M .. f M, , , , , , , , ,( )=

m1 m2 mM times

Bn m( )n! A1

m1 A2m2 ... AM

mM⋅ ⋅ ⋅ ⋅

2n 1– m1! m2! ... mM!⋅ ⋅ ⋅ ⋅-----------------------------------------------------------------=

27


OTHER SIGNAL TYPES

2-tone test is easy for characterization but otherwisenot very practical. Using wideband test signals it ismore difficult to separate distortion and signal, or dis-tortion and noise floor. Couple of measures are used:

• In modulated RF signals, distortion products onneighbouring channel is measured (ACP = adjacentchannel power) and distortion on top of the desiredsignal is simply ignored.

• In audio and baseband amplifiers, the distortion liesmostly on top of the desired signal band. E.g., 0.1%THD for wideband input signal may be spesified, butit is difficult to measure. H. Sjöland (Design andAnalysis of Highly Linear Integrated WidebandAmplifiers, PhD thesis, Lund Univ, Sweden, 1997)has derived suitable amplitudes and frequencies sothat a single-tone sine test can be used to predictwideband performance.

ACP

signal

distortion

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EFFECT OF DC BIAS TO COEFFICIENTS

In distortion analysis, one is interested in nonlinearity coefficients at the operating point. If the Taylor series isexpanded at some other point, one needs to use X0 + x, where X0 is the operating point, instead of plain x.When expanded, this changes the numerical values of all lower order coefficients. New coefficients can beobtained using Pascal’s triangle rule.

Note: Only linear and exponential functions have equal amount of distortion at all DC points

uo ci UB us+( )i⋅i 0=

5

∑=

c0 c1 UB⋅ c2 UB2⋅ c3 UB

3⋅ c4 UB4⋅ c5 UB

5⋅+ + + + +( )=

c1 2c2 UB⋅ 3c3 UB2⋅ 4c4 UB

3⋅ 5c5 UB4⋅+ + + +( ) us⋅+

c2 3c3 UB⋅ 6c4 UB2⋅ 10c5 UB

3⋅+ + +( ) us2⋅+

c3 4c4 UB⋅ 10c5 UB2⋅+ +( ) us

3⋅+

c4 5c5 UB⋅+( ) us4⋅+

c5 us5⋅+

29


Example: Coefficients of a quadratic MOS transistor

Suppose that a MOSFET behaves like

Now substitute , where VGS0 is the DC operating point, and you obtain

Thus, the original function that has only the 2nd order term shows now all lower order terms, and the ratio oflinear and second order term depends (in this type of nonlinearity) on the DC bias:

Clearly, the DC bias affects the amount of nonlinearity.

iD β vGS VT–( )2⋅=

vGS VGS0 vgs+=

iD β VGS0 vgs VT–+( )2⋅ β VGS0 VT–( )2⋅ 2β VGS0 VT–( ) vgs⋅ ⋅ β vgs2⋅+ += =

K2′K2K1------- β

2β VGS0 VT–( )⋅------------------------------------------- 1

2 VGS0 VT–( )-----------------------------------= = =

30


Example: Offset in differential stages

Fully differential behaviour: u1 = ui, u2 = -ui, uo = f(u1)-f(u2)

This cancels all even order terms in the output.Now some dc offset is added in the input signals: u1 = ui + uoff, u2 = -ui, uo = f(u1)-f(u2)

It is seen that dc and other even order terms arise again, due to the offset.

uo1 a0 a+1

ui⋅ a2 ui2⋅ a3 ui

3⋅ a4 ui4⋅ a5 ui

5⋅+ + + +=

uo2 a0 a–1

ui⋅ a2 ui2⋅ a3 ui

3⋅– a4 ui4⋅ a5 ui

5⋅–+ +=

uo1 uo2– 2 a1 ui⋅ a3 ui3⋅ a5 ui

5⋅+ +( )⋅=

uo1 uo2– 2 a1 ui⋅ a3 ui3⋅ a5 ui

5⋅+ +( )⋅≈

uoff a1 2a2ui 3a3ui2

4a4ui3

5a5ui4

+ + + +( )⋅+

31


2.2 DESCRIBING FUNCTION ANALYSIS

Describing function (DF) analysis can be used under following assumptions:

• Input is a single-tone sinusoid and only the fundamental output frequency is of importance• The systems contains only one nonlinearity• The nonlinearity is memoryless

DF is performed by calculating the gain and phase of the fundamental at the output of the nonlinearity byexpanding the output into a Fourier series (This is excatly what is done when measuring AM-AM and AM-PMusing a vector analyzer). Typical applications are:

• Find gain variations for root locus analysis of nonlinear control systems• Lately used for stability analysis of high order sigma-delta converters• AM-AM and AM-PM curves

32


EXAMPLE: A SINEW AVE IN AN IDEALLIMITER

Let us drive a sinusoid in an ideal limiter clipping atinput voltages +/-1. Amplitudes and and amplitude-dependent gains of 5 first harmonics are measuredusing Fourier analysis (FFT in Matlab) and shown inthe next figure. In describing function analysis, onlythegain of the fundamental frequency is usually interes-ting.

Similar models have been used in other nonlinearitiesthat appear often in control systems:

• saturation• comparison (relay)• hysteresis

Vin

Vout

33


00.

20.

40.

60.

81

1.2

1.4

1.6

1.8

2-1

-0.50

0.51

VOUT

TIM

E

00.

51

1.5

22.

53

3.5

40

0.2

0.4

0.6

0.81

1.2

1.4

VIN

VOUT

00.

51

1.5

22.

53

3.5

40

0.2

0.4

0.6

0.81

1.2

1.4

VIN

GAIN

FU

ND

.

HD

3

HD

5

FU

ND

HD

3

4A/π

1:3

1:5

34


OSCILLA TOR GAIN AN ALYSIS

Intr oduction

In oscillator circuits, compressing nonlinearity isessential in maintaining the oscillation amplitudefinite. It also reduces noise, as due to compression, vol-tage (amplitude) gain is reduced.

Minimum required gain for oscillation can be solvedwith small signal analysis. However, oscillation amp-litude vs. bias current is difficult to estimate because itis essentially a nonlinear phenomena. Oscillation isalso autonomous phenomena, the frequency of whichis not exactly known a priori.

Describing function analysis can be used to find DCand fundamental currents. E.g. when ideal (exponen-tial) BJT is driven by sinusoidal voltage, output con-sists of Bessel functions (Vittoz IEEE j. SSC June1988).

35


AMPLITUDE VS. DC CURRENT INCOLPITTS OSCILLA TOR

Huang (Power Consumption vs. LO Amplitude forCMOS Colpitts Oscillators, CICC97) has analyzed theoscillator shown left. The MOS operates usually inclass C (duty cycle < 50%), but average DC and funda-mental signal currents components of ID can still becalculated using square law behaviour of the MOS, asshown in Huang97.From the fundamental Id(1), ‘large-signal gain’ GM

can be calculated as GM = Id(1)/UA. This can be rela-ted to to DC current consumption and oscillation requi-rements.

VGS

ID

36


EXAMPLE OF LARGE SIGN ALNEGATIVE IMPED ANCE ANALYSIS

General principle (Kurokawa)

Simulate oscillator impedance and phase as a functionof signal amplitude. This can be done by sinusoidaltest signal which is chosen according to followingrules:

• In a series resonance (as here), current can flow onlyat the resonance frequency. Thus, for impedancesimulation, the resonator can be modeled as purelysinusoidal current source whose amplitude is swept.

• In a parallel resonance, voltage can be developedonly at the resonance frequency. Thus, the resonatorcan be modeled as a sinusoidal voltage source.

Then, Zosc = Vin(fund.)/Iampl (or Vampl/Iin(fund))vs. output amplitude can be plotted together with -Zresvs. signal frequency. Their crossing point shows bothoscillation frequency from Zres(f) curve and amplitudefrom Zosc(ampl).

Cval

Cval

Zres ZampIB

Rs

Cs

Ls

-Zres(f)

Zosc(Vol)

Re(Z)

VoVin

+

-

37


STABILITY AN ALYSIS IN SIGMA-DEL TA A/D CONVERTERS

In sigma-delta converters, the gain of the quantizers affects stability as shown in signal and noise transfer fun-ctions below

The gain of the comparator varies Vout/Vin(max) to infinity. It can be estimated statistically by driving theconverter with a set of dc voltages, measuring dc and ac powers in the input of the comparator and their corre-lation with the output y(t):

From this it is apparent that when input dc voltage is increased, the effective comparator gain decreases, whichis bad for stability.

STF z λsig,( )λsig H z( )⋅

1 λsig H z( )⋅+------------------------------------= NTF z λnoi,( ) 1

1 λnoi H z( )⋅+------------------------------------=

λsig

E y t( ) udc⋅

udc2

---------------------------------= λnoi

E y t( ) uac t( )⋅

E uac2

t( )î

----------------------------------------=

H(z)-

nλnoi

H(z)-

λsig ysignal ynoise

u u

x 0

38


SIGMA-DEL TA CONVERTER TOPOLOGIES

It has been found (Karema94) that in many converter topologies,

where bmax is the largest coefficient in the transfer function.

Table 1: Coefficients for sigma-delta converters

Order b5 b4 b3 b2 b1

1 1

2 2 1

3 16 8 1

4 64 32 4 1

5 512 256 64 8 1

λmean λsig λnoi1.5

bmax------------≈= = λmin

12bmax---------------≈

IN IN IN

IN IN IN

b1 bNb2

bN bN-1 b1

clk

clk

39


ROOT LOCUS EXAMPLE

Below right is a root locus of a third order sigma-delta A/D converter shown left. Its is seen that noise transferfunction of the modulator is stable if comparator gain λ is larger than 0.016. Because bmax = 16, the expectedminimum lambda is 1/32 = 0.03. Thus, this modulator is stable when not overdriven. (Karema PhD thesis1994)

NTF z λ,( ) 1 z 1––( )3

1 16λ 3–( )z 1– 3 24λ–( )z 2– 9λ 1–( )z 3–+ + +------------------------------------------------------------------------------------------------------------------=

-1 -0.6 -0.2 0 0.2 0.6 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.016

0.016

0.001

1 168

clk

- - -z-1

1-z-1

z-1

1-z-1

z-1

1-z-1

λ

λ = 0.001 ... 0.15

u

40


STAIRCASE ZONES

Too low dc gain causes idle zones (called devil’ s stair-case) in the response of low order sigma delta conver-ters. The width of these zones is directly related to dc

gain: if lossy integrator is modelled as 1/(1+pz-1), thewidth of the center dead zone in 1st orderΣ∆ is

+/- (1-p)/(1+p)

Such a staircase function can be modelled as an integ-ral function of Cantor dust, and similar behaviour canbe found in many nonlinear systems, including

• input-output characteristics of low order sigma-deltaconverters

• proper frequency ratio for injection locking of oscil-lators

• slope of least square error (LSE) line fit when both xand y values are quantized

• Josephson junction circuits

(M. Kennedy, IEEE TCAS-II August 89)

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

in

out

integrate

41


NOISE MIXING IN Σ−∆-CONVERTERS

Deterministic limit cycle signals and 2nd order non-linearities in output D/A converters can mix the quanti-zation noise down to baseband, thus reducing SNR. Toreduce this, quantization noise must be randomized by

• using a higher order modulator• using random dither signal• limiting the signal levels, as overloading creates

deterministic spurs

Also, the D/A converters can be linearized by

• using RZ coding• differential structures (minimises even order nonlin)

Karema et al: Intermodulation in sigma-delta D/A con-verters. IEEE

fs

limit cycles

quantization noisegets mixed down

42


AM/AM AND AM/PM CUR VES OFAMPLIFIERS

• Gain compression or expansion in a memorylessamplifier can be modelled by AM/AM curves

• If the amplifier exhibits some (not severe) memoryeffects, these can be modeled as amplitude depen-dent phase variations, i.e. AM/PM

• These can simply be measured or simulated with asingle-tone test using a vector analyzer or 1-toneharmonic balance.

Next, AM-AM and AM-PM curves for the amplifiershown left and biased in class A are simulated.

OUT R A

NETWORK ANAL. fr eq = const.power is swept

43


10.0m 30.0m 0.1 0.310m

30m

0.1

0.3

1

3

-1

-0.5

171.

99

+0.5

+1

Power sweepAPLAC 7.30 User: EE Thu Nov 11 1999

Uout

V

Uin/V

Uout H3

AM/PM

0 0.15 0.3 0.45 0.60.00

0.75

1.50

2.25

3.00

-1

-0.5

171.

99

+0.5

+1

Power sweepAPLAC 7.30 User: Thu Nov 11 1999

UoutV

Uin/V

Uout H3

AM/PM

Class A amplifier on lin and log scales

44


BANDPASS NONLINEAR MODELS

Principle

Because time-domain analysis of modulated signals is time-consuming, it is customary to simulate only themodulation, i.e. stripping the carrier away from the signal phasor

Implementations in simulators

• Measured AM-AM and AM-PM curves can be interpolated. Because of errors and noise in vector analyser,the data must be stored in Re-Im pairs, filtered with non-delaying averaging filter and converted to polar cor-dinates.

• Alternatively, analytical functions (Rapp’s limiter, Honkanen model,..) can be used

Problems

• Effect of baseband or harmonic impedances are ‘built-in’. For new biasing and matching structures, newAM-AM and AM-PM curves must be measured

• Effects caused by signal-dependent DC bias (frequency dependent compression) are not modeled.• Inherently odd order model. As in general in describing function analysis, this is not suited for modelling

cascades.

A t( ) jφ t( )( )exp jωct( )exp⋅ ⋅

45


PA MODEL FR OM MEASURED DATA

Measured AM-AM and AM-PM curves of an existingamplifier can be measured and used in behaviouralsimulations.

1-tone power sweep is easy to measure, although CWmeasurement leads to thermal problems, and usually, apulsed measurement is required. Power steps may notbe accurate, why some form of filtering of data isrequired. Matlab filtfilt() for Re-Im vectors separatelybefore calculating phase has been found usefull.

Higher order interpolation formulas are slow and causebumpy response between data points. Linear interpola-tion causes corner points and discontinuous derivati-ves, which appears as very high order intermodulationtones. To keep these down, at least 100 data points areneeded. This, then makes interpolation formula slow ...

Uout

Uin

Uin

φout

Linear inter polation in Matlab

x = abs(in);r = interp1(Uin,Uout,x,'linear');fi = interp1(Uin,fout,x,'linear');out= r.*exp(j*(angle(in)+fi));

46


ANALYTICAL MODELS

Various analytical equations can be fitted to measureddata. To minimise excess distortion, single equationswith continuous derivatives are preferred.

Common equations:

Rapp’s limiter (smooth limiting)Honkanen model (incl. crossover distortion)TWA models

47


CADENCE K-MODEL

One of the main drawbacks of bandpass nonlinearmodels are that they do not model the amplitude-dependent frequency response of the modulation.

These effects can be characterized by driving the amp-lifier with a 2-tone test, where the amplitude of the car-rier at wc is swept, and for each amplitude, the offsetfrequency of a small interferer at wc+woffset is sweptover the modulation bandwidth.

(Moult, Chen: )

f(mod)

amplitude

w

wc

A

48


MEMOR Y AND AM/AM CONT ..

Bösch and Gatti (IEEE Trans. on MTT, Dec.89) haveused two-tone test and two vector analyzers to find theeffect of modulation frquency: whenωο+∆ωis swept,envelope frequency ∆ω increases and memory effectsbegin to appear. This is easy to do with 2-tone simula-tions, too.

They also used a simulation model, where AM-AMand AM-PM curves where characterized at differentoperating points and envelope information was filteredto find out the present operating point, includingmemory effects. Memory was shown to cause hystere-sis in AM-AM curves, making memoryless predistor-tion inefficient.

DUT

R A

VCO

R A

VCO

ωo

ωo+∆ω

Pin

G

∆ω

49


2.3 NONLINEAR CONTROL THEOR Y

State equations (x’s are state variables, u’s inputs and f(..)’s (nonlinear) functions

Phase plane portraits (nodes, focuses, centers, saddle points ,...)

Lyapunov stability analysis of

• equilibrium points• limit cycles and other periodic solutions

Perturbation Theory

• Sometimes the solution can be approximated by writing a Taylor expansion for a response for a small per-turbation in the initial state

Averaging

• Some nonlinear systems (e.g. pendulum standing in its high state) can be stabilised by driving the systemwith a small amplitude periodic perturbation or dither (ref. to dithering in sigma-delta converters to averagethe loop gain)

x1˙ f 1 t x1 .. xn u1 ... un, , , , , ,( )=

...

xn˙ f n t x1 .. xn u1 ... un, , , , , ,( )=

50


EXAMPLE SYSTEMS

Pendelum Equation

• One stable and one unstable equilibra and their multiples.

Tunnel diode circuits

Mass-spring systems with friction

• Non-linearity caused by the fact that friction always opposes the direction of movement• Results into piecewise linear solutions

Negative resistance Oscillator (van der Pol equation)

• also injection locking

Neural networks

Digital PLL with numerically contr olled oscillator (NCO)

• limit cycles due to finite amount of output frequencies (Kennedy)

51


VAN DER POL OSCILLA TOR

Van der Pol oscillator is a thoroughly studied model ofa nonlinear oscillator. It is described by state equations

whereε represents the amount of nonlinearity: whenε=0, the oscillator behaves as a linear sinusoidal oscil-lator, while whenε > 3 it behaves more like a relaxa-tion oscillator, as shown in the figures.

Left are shown phase trajectories for casesε=0.1 andε=3.0.

x1˙ x2=

x2˙ ε 1 x1

2–( ) x2⋅ ⋅ x1–=

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X1

X2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

X1

X2

ε=0.1

ε=3.0

52


-2-1

01

2

-2

-1.5-1

-0.50

0.51

1.52

X1

X2

05

1015

2025

3035

40-3-2-1

0123

X1

05

1015

2025

3035

40-3-2-1

0123

X2

eps = 0.1almost sinusoidal response

stable limit cycle

53


-2-1

01

2

-5

05

X1

X2

05

1015

2025

3035

40-3-2-1

0123

X10

510

1520

2530

3540

-6-4-2

0246

X2

eps = 3.0“relaxation oscillator”

54


COMPANDING FIL TERS

Recently, lot of interest has been paid to companding /externally linear, internally nonlinear (ELIN) / transli-near filters. They have following interesting features:

• due to companding, the SNR for a single signal ispractically independent of input signal level

• power consumption can be reduced• companding does not need to be continuous, resul-

ting in syllabic compression or so-called floating-point signal processing, where gain is varied e.g. inpowers of two

but• noise sources inside the filter appear now as multip-

licative, not additive noise sources. Thus,• an out-of-band interferer can boost the noise floor,

thus causing de-sensitation• mismatches cause internal distortion to appear in the

output, too.(see Tsividis, IEEE TCAS-II Febr. 97)

p Ap Bu+=

y Cp Du+=

p g q( )=

qAg q( )

q∂∂g

--------------- Bu

q∂∂g-------+=

y Cg q( ) Du+=

Conventional state equations are

Nonlinear mapping function

pq∂

∂gq⋅=

Because

the new state equations are

55


Example of companding filters

Using BJTs, obvious choise for p = g(v) is p=Isexp(v/Vt), in which case the state equations for a 1st order fil-ter are shown on the left.

In the schematic, Q1-Q3 form input compression

and Q4 output expansion. But now noise in2 is integra-ted in C and appears as multiplicative noise on top ofthe output signal

IQ3 IS

V B3 v–

V t-------------------

exp=

IS

V t u IS⁄( )log Io IS⁄( )log+( )⋅V t

--------------------------------------------------------------------------

exp

v V t⁄( )exp-----------------------------------------------------------------------------------------=

u Io⋅IS v V t⁄( )exp--------------------------------=

Iout Is

v vn+

V t--------------

exp Isv

V t-----

vn

V t-----

expexp= =

vAV t-----

BV t

IS v V t⁄( )exp-------------------------------- u⋅+=

y CIS q V t⁄( )exp=

Q1

Q2

Q3

Q4v

Iout

u

Io

I

in2

56


CHAOS IN COLPITTS OSCILLA TORS

Nonlinear systems whose order is at least three canexhibit chaotic behaviour. Also common Colpitts oscil-lator , described by the following state equations

has the following regions of operation:

• When resonator Q value is high and loop gain issmall, sinusoidal oscillation with low distortionappears. When Q is low (<3) and loop gain is exces-sive (>3), the circuit can exhibit

• Subharmonic oscillation at fosc/N (2)• Shilnikov chaos (3)• Feigenbaum chaos (4)

C1 td

dVC1 IS– V– C2 VT⁄( )exp IL+=

C2 td

dVC2 IL IO–=

Ltd

dIL V– C1 VC2 RIL VCC+––=

log(Q)

log(T)

12

3 4

R

L

C1

C2

Io

IL

Kennedy: IEEE TCAS-II, 1999

57


JUMP RESONANCE

In active filters, the nonlinearity of the amplifiers (orgm elements) may cause interesting hysteresis in thefrequency response: the frequency response may havetwo values at certain frequency points, and this mayappear as a hysterisis and quick jump in frequencysweep as well as different responses with differentdirections of the sweep.

The analysis of this depends on the topology of the fil-ter. In Cherry94, this behaviour was analysed in a 2ndorder gm-C filter, resulting in forced Duffing equation :

These are time domain equations. Substituting sinusoi-dal input response, magnitude and phase responses canbe constructed.

C1

dv1dt

--------v1–

R1-------- gmivi εivi

3 gm1v2 ε1v23+ + + +=

C2

dv2dt

-------- gm2– v1 ε2– v13=

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

1

2

3

4

5

6

7

Frequency (Hz)

Mag

nitu

de (

V)

Duffing equation magnitude

A

B

C

D

EF

Left to right

Right to left

Cherry 94: Effect of sw eeping frequency in Duffing equation.

12

i

gm1

gmi

gm2

C1 1FC2 1F

V1V2

R1 5Ω

i gkv εkv3+=Vi

58


JUMP RESONANCE CONT..

Further, the maximum allowable Q value before theonset of jump resonance was analysed in Cherry94.For the topology given, it appears that

Related to Volterra analysis notations in Chp. 3,

For example, using a nonlinearised BJT differentialpair OTA with 1 mA bias current (gm = 20 mS), epsi-lon is 2.7 or +8 dB (K3’ = 130). Compared to thefigure, 10 mV signal level (-40 dBV, giving roughly -40 dBc IM3 level) would cause a jump resonancewhen the desired filter Q-value is larger than 30.

Qmax1

V in ε3⋅--------------------∝

ε gm K3′⋅=

-40

-35

-30

-25

-20 -40

-35

-30

-25

-20

20

25

30

35

40

Vin (dB)eps (dB)

Max

allo

wed

Q (

dB)

Jump resonance plane

Vin = 0.01 .. 0.1 V

ε = 0.01 .. 0.1

59


2.5 NUMERICAL AN ALYSIS METHODS

Transient analysis (solving non-lin differential equations in time domain)

• Time domain method, solves intial value problem• Efficient time step algorithms• Models signals by ramps or parabolas during each time step• Long transients need to decay to obtain steady-state response• Numerical noise• Integration algorithms may provide dampling or instability (pole position may move)• Modelling of frequency depedent circuits is not efficient, requires large FIR filters

Shooting methods

• Solves time domain, boundary value problem• Solves periodic solution by forcing response the same at t=0 and t=T• Converges quickly and reliably, but not effectvely (suited for small circuits)• Distributed components must be modelled expensively in time domain

Mixed Time-Frequency Method (MTF)

• One of the signals is chosen as sampling clock, and others are sampled with it• Sampled signal is presented as low order Fourier series• Well suited for sampling mixers and switched capacitor circuits

60


Harmonic balance

• Signals are modelled as sinusoids, aplitudes and phases of which are to be solved.• Suitable for N-signal periodic or almost periodic responses• Frequency dependent linear circuits are modelled efficiently in frequency domain• Non-linear components still analyzed in time domain - or by using Volterra models in frequency domain• Convergence problems. Number of harmonics causes tradeoff between accuracy and simulation time• Numerical noise from FFT and I-FFT. Sometimes almost period Fourier transform (APFT) is used, but then

timepoints need to be chosen carefully.• 1-, 2-, and 3-tone tests are easy, but presenting modulated signals is very difficult• Lots of numerical improvements lately (Krylov subspace methods, MRES, preconditioning, ...)

Envelope harmonic balance (HP MDS)

• A 1-tone harmonic balance at different time pointswith a built-in memory• Well suited for modulated narrowband signals• Because memory is modelledessentially astransient analysis, frequency domain models can not be used

for distributed elements

(see Kundert: j Solid-State Circuits Sept. 99)

61


T

i(t)

i Ctd

du( )=

u Ltd

di( )=

u A t( ) jωt( )exp⋅=

i C A' t( ) jωA+( ) jωt( )exp⋅ ⋅=

+ A(t) -

jωC

C

Initial transient

Interpolation error

Boundary value problem

Circuit envelope analysis

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