3. BASICS OF VOLTERRA ANALYSIS - University of Oulutimor/DCourse/Chp3.pdf · 3. BASICS OF VOLTERRA...

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

3. BASICS OF VOLTERRA ANALYSIS

Volterra analysis is a way to extend small-signal analysis and the use of integral transforms to nonlinear dyna-mical systems. Volterra functionals (i.e. functional series consisting of sum of i’th order convolution integrals)were first developed by Volterra 1930, applied to electrical systems by Wiener 1942 and systematically usedto analysis of distortion since 1970’s, starting from Bell Labs. Currently Volterra analysis is implemented atleast in Berkeley SPICE (.DISTO analysis), some RF simulators, and symbolic analysis tool ISAAC.

Volterra digital filters have been used for years e.g. to linearise the response of high-power loudspeakers. IEEEOnline search engine finds ca. 650 hits with search word ‘Volterra’. References to basic litterature can befound from Wambacq and Maas.

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NOTATIONS

As remembered, the time response y(t) of alinear system can be calculated as a convolution integral of theimpulse response h(t) and input signal x(t)

which can be either Fourier or Laplace transformed into frequency domain, where output and input spectrumsare related by

.

where Y(s) and X(s) are output and input spectrums and H(s) is the integral transform of h(t).

In a nonlinear system, to model the effect of signal dependent impulse response, the convolution integral mustbe extended to series of 1 to N-dimensional convolutions.

y t( ) h τ( )x t τ–( ) τd∞–

∞∫=

Y jω( ) H jω( ) X jω( )⋅=

y t( ) u1d∞–

∞∫ ... unhn u1 .. un, ,( ) x t ur–( )

r 1=

n

∏d∞–

∞∫

k 1=

∞

∑=

h1 u1( )x t u1–( ) u1d∞–

∞∫=

∞–

∞∫ h

2u1 u2,( )x t u1–( )x t u2–( ) u1d u2d

∞–

∞∫+

...+

3


herehn(t1,t2,..tn)

are n-dimensional impulse responses or kernels. Forcausality,

hn(t1,t2,..tn)=0 when any ti < 0.

In many analysis it is required that the kernels are sym-metric, i.e.(for a 2nd order kernel)

h2(t1,t2) = h2(t2,t1)

An asymmetric kernel h2+(t1,t2) can be symmetrised

by

h2(t1,t2) = 0.5*( h2+(t1,t2) + h2

+(t2,t1) )

If each convolution integral is marked as Hi(x(t)), aVolterra series can be drawn as system of paralleltransfer functions of different order. Here, H1 is thenormal linear transfer function, and H2 is a secondorder term and so forth.

H1

H2

Hn

...

4


INTEGRAL TRANSFORMS

The basic integral transforms, Fourier and Laplacetransforms can be extended to Volterra series represen-tation. In the following, main interest is in Fouriertransform and discrete line spectrums, and followingrules are usefull:

H2(-jω1,-jω2) = H2*(jω1,jω2)

(negative frequencies mean complex conjugate) and ifthe impulse response h() is symmetric, then also

H2(jω1,jω2) = H2(jω2,jω1)

Due to nonlinear nature, output spectrum is not merelya product of transfer function and input spectrum X(f),but convolution integral is needed to calculate higherorder spectrums.

H2 jω1 jω2,( ) h2 u1 u2,( )ej ω1u1 ω2u2+( )–

u1d u2d∞–

∞∫=

h2 u1 u2,( ) 1

2π( )2-------------- H2 jω1 jω2,( )e

j ω1u1 ω2u2+( )ω1d ω2d

∞–

∞∫⋅=

H2 s1 s2,( ) h2 u1 u2,( )es1u1 s2u2+( )–

u1d u2d∞–

∞∫=

2nd order Fourier and inverse Fourier transforms

2nd order Laplace transform

Y f( ) H1 f( )X f( )=

f 1H2 f 1 f f 1–,( )X f 1( )X f f 1–( )d∞–

∞∫+

f 2 f 1d∞–

∞∫ H3 f 1 f 2 f f 2– f 2–, ,( )X f 1( )X f 2( )X f f 2– f 2–( )d

∞–

∞∫+

...+

Calculation of output spectrum

5


THE MEANING OF H N(S1,..SN)

In an nth order transfer function

is a general transfer function that describes the res-ponse to any of the nth order output frequencies. Ithas n frequency arguments s1,s2,..sn, because there a

2n possible output frequencies that are of form

In a 2-tone test, only frequencies wh and wl are used,but any of their permutation may appear in the trans-fer function. Thus, e.g. in a 3rd order transfer func-tion the possible positive output frequencies are

Hn s1 ... sn, ,( )

s s1 s2±± ...± sn±=

Table 1:

Freq. H3(x) type

wl+wl-wh H3(wl,wl,-wh) IM3L

wl+wl-wl H3(wl,wl,-wl) FUND

wh+wh-wh H3(wh,wh,-wh) FUND

-wl+wh+wh H3(-wl,wh,wh) IM3H

wl+wl+wl H3(wl,wl,wl) HD3

wl+wl+wh H3(wl,wl,wh)

wl+wh+wh H3(wl,wh,wh)

wh+wh+wh H3(wh,wh,wh) HD3

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RESPONSE TO SINUSOIDS

Suppose a second order system with a sinusoidal exci-tation

Now, when x(t) = exp(jωt), the double convolution ofthe impulse response is the same as the 2-D Fouriertransform H2(jω,jω) of the impulse response, and thetime domain output y(t) is

x t( ) A2--- e

jωte

j– ωt+( )⋅ xa t( ) xb t( )+= =

y t( ) H2 xa t( )[ ] H2 xa t( )[ ]+=

H2 xa t( ) xb t( ),( ) H2 xb t( ) xa t( ),( ) + +

A2

4------ H2 jω jω,( )⋅ e

j2ωt⋅=

A2

4------ H2 j– ω j– ω,( )⋅ e

j– 2ωt⋅+

2A2

4---------- H2 jω j– ω,( ) 1⋅ ⋅+

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RESPONSE TO A 2-TONE TEST

A sum of two sinusoids is

In a second order nonlinearity, this produces the following outputs

x t( )A12

------ ejω1t

⋅A1

*

2------ e

j– ω1t⋅+

A2

2------ e

jω2t⋅

A2*

2------ e

j– ω2t⋅+

+ xa t( ) xb t( ) xc t( ) xd t( )+ + += =

y t( ) 2A1

2

4------------ H2 jω1 j– ω1,( )⋅

A22

4------------ H2 jω2 j– ω2,( )⋅+

⋅=

A12

4------ H2 jω1 jω1,( ) e

j2ω1t⋅ ⋅

A1* 2

4--------- H2 j– ω1 j– ω1,( ) e

j– 2ω1t⋅ ⋅+ +

A22

4------ H2 jω2 jω2,( ) e

j2ω2t⋅ ⋅

A2* 2

4--------- H2 j– ω2 j– ω2,( ) e

j– 2ω2t⋅ ⋅+ +

A1A22

------------- H2 jω1 jω2,( ) ej ω1 ω2+( )t

⋅ ⋅A1

*A2

*

2------------- H2 j– ω1 j– ω2,( ) e

j– ω1 ω2+( )t⋅ ⋅+ +

A1A2*

2------------- H2 jω1 j– ω2,( ) e

j ω1 ω2–( )t⋅ ⋅

A1*

A22

------------- H2 j– ω1 jω2,( ) ej– ω1 ω2–( )t

⋅ ⋅+ +

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Note that amplitudes contain also phase information, and for negative frequencies a complex conjugate hasbeen used. The relative amplitudes of the terms depends on how many terms at that frequency is produced,when the input signal is multiplied by itself. For reference, all frequencies produced in a 2nd order system arelisted below.

Table 1: Output frequencies in a 2nd order nonlinearity

+jw1 -jw1 +jw2 -jw2

+jw1 j2w1 0 +j(w1+w2) +j(w1-w2)

-jw1 0 -j2w1 +j(w2-w1) -j(w1+w2)

+jw2 +j(w1+w2) +j(w2-w1) j2w2 0

-jw2 +j(w1-w2) -j(w1+w2) 0 -j2w2

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AMPLITUDES A T DIFFERENTFREQUENCIES

As shown before, Volterra kernels are general Nth-order transfer functions valid at all frequencies. In a2-tone test the relative amplitudes of different termsdepends on how many times these frequenciesappear in the convolution of the original spectrum.The different outputs are shown below .

(Wambacq, p. 76)

Table 1:

DC 0.5|A1|2H2(jω1,-jω1)

0.5|A2|2H2(jω2,-jω2)

ω2−ω1 A1*A2 H2(jω2,-jω1)

2ω1−ω2 0.75A12 A2

*H3(jω1,jω1,-jω2)

ω1(compr)(desens)

A1 H1(jω1)

0.75A1|A1|2H3(jω1,jω1,-jω1)

1.5A1|A2|2 H3(jω1,jω2,-jω2)

ω2(compr)(desens)

A2 H1(jω2)

0.75|A2|2A2H3(jω2,jω2,-jω2)

1.5|A1|2A2H3(jω1,-jω1,jω2)

2ω2−ω1 0.75A1*A2

2H3(jω2,jω2,-jω1)

2ω1 0.5A12 H2(jω1,jω1)

ω1+ω2 A1A2 H2(jω1,jω2)

2ω2 0.5A22 H2(jω2,jω2)

3ω1 0.25A13H3(jω1,jω1,jω1)

2ω1+ω2 0.75A12A2 H3(jω1,jω1,jω2)

ω1+2ω2 0.75A1A22 H3(jω1,jω2,jω2)

3ω2 0.25A23H3(jω2,jω2,jω2)

Table 1:

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MEMOR Y BEFORE NONLINEARITY

An important simplification is a sort of block diagrampresentation, where frequency dependent signal pathsare multiplied so that the multiplication does not affectthe time constants of the filters.

It can be shown that from x(t) to output y(t)

which results in frequency domain into second ordertransfer function of

Thus, Hc() is evaluated at the frequency componentsproduced by the multiplication.

htot τ1 τ2,( ) hc σ( )ha τ1 σ–( )hb τ2 σ–( ) σd∞–

∞∫=

H2 jω1 jω2,( ) Ha jω1( ) Hb jω2( ) Hc jω1 jω2+( )⋅ ⋅=

ha(t)

hb(t)

hc(t)x(t)

y(t)xt(t)

xt t( ) ha τ( )x t τ–( ) τd∞–

∞∫

hb τ( )x t τ–( ) τd∞–

∞∫

⋅=

∞–

∞∫ ha τ1( )hb τ2( )x t τ1–( )x t τ2–( ) τ1d τ2d

∞–

∞∫

=

...=

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Example 1. Amp with input and output filters

(see Wambacq p.71-)

First order r esponse

2nd order response

2nd order response is shown left: input filter is evalua-ted at different frequencies, reponses are multipliedand further multiplied by 2nd order gain and output fil-ter response at the output frequency.

Example: input freqs are w1=5 and w2=12 rad/s. thenresponses at following frequences are2w1: H2(5,5)w2-w1: H2(12,-5)w1+w2:H2(5,12)2w2: H2(12,12)

H1 jω( )K1

1 jωτ1+( ) 1 jωτ2+( )⋅----------------------------------------------------------=

11 jωτ1+---------------------- A(x)

11 jωτ2+----------------------

A(x) = K1x + K2x2 + K3x

3

11 jω1τ1+------------------------

11 jω2τ1+------------------------

K21

1 j ω1 ω2+( )τ2+-------------------------------------------

H2 jω1 jω2,( )K2

1 jω1τ1+( ) 1 jω2τ1+( ) 1 j ω1 ω2+( )τ2+( )⋅ ⋅--------------------------------------------------------------------------------------------------------------------=

2nd order response

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Amp model cont ...

11 jω1τ1+------------------------

11 jω2τ1+------------------------

K31

1 j ω1 ω2 ω3+ +( )τ2+--------------------------------------------------------

H3 jω1 jω2 jω3, ,( )K3

1 jω1τ1+( ) 1 jω2τ1+( ) 1 jω3τ1+( ) 1 j ω1 ω2 ω3+ +( )τ2+( )⋅ ⋅ ⋅-------------------------------------------------------------------------------------------------------------------------------------------------------------------=

3rd order r esponse

11 jω3τ1+------------------------

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Example 2. Block level

In this simple 2nd order example, the nonlinearity is onthe output side and affects the gain of the circuit. Aswill be seen later, the frequency response of this circuitisHere

and

where K2 = 1 and a = g/C. After some algebra the 2-dimensional impulse response can be calculated to be

u t( ) ig jωC+-------------------- H1 jω( ) i⋅= =

H2 ω1 ω2,( ) K2 H1 jω1( ) H1 jω2( )⋅ ⋅=

1 C⁄( )2

jω1 a+( ) jω2 a+( )⋅----------------------------------------------------=

h2 t1 t2,( ) a t1 t2+( )–( )exp=

Cg

u(t)

i in

0

0.5

1

1.5 0

0.5

1

1.50

0.2

0.4

0.6

0.8

1

t2 st1 s

h2(t

1,t2

)

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Example 3.

In this example, a nonlinear conductance of type

causes the system to behave according to nonlinear dif-ferential equation

In this case, the nonlinearity affects also the systemtime constants, and compared to example 1, moreterms appear. The frequency response is

where wo= g/C. The 2-dimensional impulse responsegets complicated and can not be show in closed form:it consists of integral functions of the exponential func-

tion Ei(t,n) = Int(exp(t)/tn).

i g v K2′ v2⋅+( )⋅=

i t( ) Cdvdt------⋅ g v t( )⋅ K2g v t( )2⋅+ +=

H2 ω1 ω2,( )K2g C

3⁄jω1 ωo+( ) jω2 ωo+( ) j ω1 ω2+( ) ωo+( )⋅ ⋅

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

Cg(v)

v(t)

i in

Note: Volterra analysis still is small-signalanalysis. It does not take into account thatthe time constant of the circuit varies withu(t)

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Example 3 Cont ...

The previous system is described as a signal flowgraph left. This shows a very common way of analy-sing Volterra kernels by evaluating the terms by order,i.e. linear term first, then the 2nd order and so fort.Thus, denoting v = v1+v2+v3+... and supposing inputi(t) has linear term only,

where

i t( ) Cdv1dt

-------- g v1 t( )⋅+=

0 Cdv2dt

-------- g v2 t( )⋅ K2g v12 t( )⋅+ +=

0 Cdv3dt

-------- g v3 t( )⋅ K3 v13 t( )⋅ K2g v1 t( ) v2 t( )⋅ ⋅+( )+ +=

....

v1 h1 u1( )iin t u1–( ) u1d∞–

∞∫=

v2 ∞–

∞∫ h

2u1 u2,( )iin t u1–( )iin t u2–( ) u1d u2d

∞–

∞∫=

...

i in

1/C td∫v(t)

g v(t)

K2 v2(t)

K3 v3(t)

-

1 C⁄ td∫⋅

-i in(lin)

1 C⁄ td∫⋅

-0

v1

K2 v12

1 C⁄ td∫⋅

-0

K3 v13+2K2 v1v2

v2 v3

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CASCADE OF TWO STAGES

In a cascade of two stages, nonlinearities are caused inboth stages. Moreover, IM3 terms are produced by 3rdorder nonlinearities in both blocks, and also by up ordownmixed second order terms. Thus,

Q1 s1( ) H1 s1( )F1 s1( )=

Q2 s1 s2,( ) H2 s1 s2,( )F1 s1 s2+( ) +=

H1 s1( )H1 s2( )F2 s1 s2,( )

Q3 s1 s2 s3, ,( ) H3 s1 s2 s3, ,( )F1 s1 s2 s3+ +( ) +=

H1 s1( )H1 s2( )H1 s3( )F3

s1 s2 s3, ,( ) +

23--- H1 s1( )H2 s2 s3,( )F2 s1 s2 s3+,( )⋅ +

23--- H1 s2( )H2 s1 s3,( )F2 s2 s1 s3+,( )⋅ +

23--- H1 s3( )H2 s1 s2,( )F2 s3 s1 s2+,( )⋅

H1

H2

H3

F1

F2

F3

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CASCADE EQUATIONS IN SYSTEMSIMULA TIONS

IM3 level in receiver systems is easy to estimate usinge.g. spreadsheet calculators (Microsoft Excel). In mostsystem calculations, following simplifications areused:

• IM3 due to second order terms is not modelled. Thisdoes not cause errors, if interstage filters removeenvelope and 2nd harmonic terms.

• Usually, IM3 contributions from different stages issupposed to add coherently. This gives a worst caseestimate, and its justification is discussed in Maas95

• The effect of interstage filtering has amplitude res-ponse only, showing IIP3 improvement due to pas-sive filtering

Moreover, one should note that that input-output pairof an amplifier is also a cascade. Thus, the impedancesseen at envelope and 2nd harmonic affect the total IM3level.

-10 -8 -6 -4 -2 0 2 4 6 8 10-12

-10

-8

-6

-4

-2

0

IIP32 - ( G1 + IIP31 ) dB

IIP3(

TOT

) -

IIP31

coherentnoncoherent

G1

IIP31

G2

IIP32

y = a1*x + a2*x2 + a3*x3

z = b1*y + b2*y2 + b3*y3

z =+(b1*a1)*x 1. ord+(b1*a2+b2*(a1^2) 2. order

+( b1*a3+ b3*(a1^3+ b2*(2*a1*a2) )*x^3 3. order

+ ...

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ANALYSIS OF INVERSE FUNCTIONS

Suppose that we have function y = f(x) and its inverse

function x = g(y) = f-1(y) (e.g. i= f(v) and v = f-1(i)).Volterra series is based on nth order derivatives, so wewill use differentiating rules for inverse functions:

As a result of this, Volterra expansions for inverse fun-ctions below (here ac conductance and ac resistance)can be calculated as shown left:

f 1–( )′ a( ) 1

f′ f1–

a( )( )--------------------------=

f 1–( )″ a( ) f– ″ f1–

a( )( )

f′ f1–

a( )( )( )3----------------------------------=

f 1–( )′″ a( )

f′″ f1–

a( )( )–3 f″ f

1–a( )( )

f′ f1–

a( )( )-------------------------------+

f′ f1–

a( )( )( )4-----------------------------------------------------------------------=

v r i⋅ K2r i2⋅ K3r i3⋅+ +=

i g u⋅ K2g u2⋅ K3g u3⋅+ +=

v if(v)

v if-1(i)

a

f(v)

f-1(i)

v

i

f-1(a)

f ’(f -1(a))

g1r---=

K2g

K– 2r

r3

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

K3g

K– 3r

2 K2r( )2

r--------------------+

r4

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

r1g---=

K2r

K– 2g

g3

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

K3r

K– 3g

2 K2g( )2

g---------------------+

g4

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

r -> g g -> r

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ANALYSIS OF FEEDBACK

In a feedback system shown left, the linear response is

where R is the so-called gain-reduction function

Supposing both the amplifier H and the feedback F arenonlinear, total response can be derived to be as shownon the next slide.

Q1 s1( ) H1 s1( ) R s1( )⋅=

R s1( ) 11 H1 s1( ) F1 s1( )⋅+------------------------------------------------=

H

F

+

-

20


FEEDBACK SYSTEMS CONT...

where

Q1 s1( ) H1 s1( ) R s1( )⋅=

R s1( ) 11 H1 s1( ) F1 s1( )⋅+------------------------------------------------=

Q2 s1 s2,( ) R s1( ) R s2( ) H2 s1 s2,( ) H1 s1( )H1 s2( )F2 s1 s2,( )H1 s1 s2+( )–( ) R s1 s2+( )⋅ ⋅ ⋅=

Q3 s1 s2 s3, ,( ) R s1( ) R s2( ) R s3( ) R s1 s2 s3+ +( )⋅ ⋅ ⋅ [×=

H3 s1 s2 s3, ,( ) 2H2 s1 s2,( )F1 s1 s2+( )R s1 s2+( )H2 s3 s1 s2+,( ) -- amplifier non-lin–

H1 s1( ) H1 s2( ) H1 s3( ) H1 s1 s2 s3+ +( )⋅ ⋅ ⋅+ [×

F– 3 s1 s2 s3, ,( ) 2F2 s1 s2,( )H1 s1 s2+( )R s1 s2+( )F2 s3 s1 s2+,( ) ] -- feedback non-lin+

2– H2 s1 s2,( )R s1 s2+( )H1 s3( )F2 s1 s2 s3+,( )H1 s1 s2 s3+ +( ) -- cross-terms

2– H1 s1( )H1 s2( )F2 s1 s2,( )R s1 s2+( )H2 s3 s1 s2+,( ) ]

21


SPECIAL CASES: LINEAR FEEDB ACK

If the feedback portion is linear, the kernels reduce tothe ones shown left. Marking loop gain as

and supposing that T(s) >> 1, the transfer functions canbe estimated as

T s( ) H1 s( ) F1 s( )⋅=

Q1 s1( )H1 s1( )T s1( )

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

Q2 s1 s2,( )H2 s1 s2,( )

T s2( ) T s2( ) T s1 s2+( )⋅ ⋅--------------------------------------------------------------=

Q3 s1 s2 s3, ,( ) 1T s1( ) T s2( ) T s3( ) T s1 s2 s3+ +( )⋅ ⋅ ⋅--------------------------------------------------------------------------------------------- ×=

H3 s1 s2 s3, ,( )2H2 s1 s2,( )H2 s3 s1 s2+,( )

H1 s1 s2+( )--------------------------------------------------------------------–

Q1 s1( ) H1 s1( ) R s1( )⋅=

Q2 s1 s2,( ) R s1( ) R s2( ) R s1 s2+( ) H2 s1 s2,( )⋅ ⋅ ⋅=

Q3 s1 s2 s3, ,( ) R s1( ) R s2( ) R s3( ) R s1 s2 s3+ +( )⋅ ⋅ ⋅ ×=

H3 s1 s2 s3, ,( ) 2H2 s1 s2,( )F1 s1 s2+( )R s1 s2+( )H2 s3 s1 s2+,( )–[ ]

R s1( ) 11 H1 s1( ) F1 s1( )⋅+------------------------------------------------=

F

+

-

H

22


In wideband case the loop gain T can be consideredconstant. Then,

Now the distortion in the linear feedback case can becompared to what would be achieved in open loop,where for the same output amplitude, the amplifierwould be driven with attenuated amplitude Va/Tcompared to feedback amplitude. In these cases,linear and 2nd and 3rd harmonic amplitudes D2 andD3 are

Thus, in general the feedback causes 1/T reduction indistortion compared to open-loop configuration withsame output amplitude. However, also 2nd ordernonlinearity is mixed to 3rd order distortion, and itmay either increase or decrease the total distortion.

For example, in a purely exponential BJT this upcon-verted 2nd order term would be dominant (Q3 =

gm(K3’ - 2(K2’)2)/T4 = gm(250-800)/T4). Thus, thefeedback is seriously changing the shape of the dis-tortion, in this case from expanding to compressing.

Q1 s1( )H1 s1( )

T-----------------=

Q2 s1 s2,( )H2 s1 s2,( )

T 3-------------------------=

Q3 s1 s2 s3, ,( ) 1

T 4------ ×=

H3 s1 s2 s3, ,( )2H2 s1 s2,( )H2 s3 s1 s2+,( )

H1 s1 s2+( )--------------------------------------------------------------------–

Table 1:

Closed-loop Open-loop

Lin H1/T x Va H1/T x Va

D2 H2/T3 x Va2 H2/T

2 x Va2

D3 H3/T4 x Va3 x

(1 - 2H2H2/H1H3)H3/T

3 x Va3

23


Here, emitter degeneration is used in a BJT to changethe type of distortion. Originally, IM3 is dominated byexpanding 3rd order nonlinearity. When the strength ofthe feedback is increased, 2nd order nonlinearity atemitter will mix to a compressing 3rd order distortion.

Ut = 25e-3;gm0 = 1e-2;beta = 50;ft = 5e9;

tf = 1/(2*pi*ft);gpi0 = gm0/beta;cpi0 = tf*gm0;Kp = [1 1/2/Ut 1/6/Ut/Ut];

Y = [ Ys+Ypi+Ymu -Ymu -Ys -Ypi ; ...gm-Ymu go+Ymu 0 -gm ; ...0 0 1 0 ; ...-gm 0 0 Ye+gm ] ;

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-65

-60

-55

-50

-45

-40

-35

IM3

dBc

AT 2

RE (x 1/gm)

2nd order dominates

V

RE

100Ω

24


Left above, fundamental amplitudes are shown as asum of linear and expansion/compression term for twodifferent values of RE. Below, fundamental amplitudeand phase are shown as functions of RE.

This clearly shows that while the feedback resistor isincreased the expanding behaviour of open-loop BJTturns to compressing behaviour due to the feedback.

0 50 100 1502.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

0 50 100 150172.5

173

173.5

174

174.5

175

175.5

-3 -2.5 -2 -1.5 -1 -0.5 0-2.5

-2

-1.5

-1

-0.5

0

0.5

RE=100 ohm

-3 -2.5 -2 -1.5 -1 -0.5 0-2.5

-2

-1.5

-1

-0.5

0

0.5

RE=10 ohm

FUND OUT VS. RE OUTP PHASE VS. RE

RE ohm

linear appr.

25


VOLTERRA ANALYSIS OF ACTIVECIRCUITS: TEST SIGN AL METHOD

General principle

A simple method to evaluate Volterra kernels of realcircuits is described next. All nonlinear conductancesand capacitances produce distortion currents. Thus,they are modelled by the component itself with paralleldistortion current sources. The value of the distortioncurrent depends on lower order products.

There ar two methos for calculating the responses:

• General method, based on Volterra transfer functions• Direct method, based on calculating node voltages of

different order.

VOLTERRA ANALYSIS

AC-ANALYSIS

DISTORTION ANALYSIS

Nonlinear 2nd order test currents

iNL1 iNL2

26


1. GENERAL METHOD

In the general method, general transfer functions (ker-nels) H1(s1) ... Hn(s1,..,sn) are calculated in symbolicform, and reponse at different at frequencies isobtained by evaluating the functions at these frequen-cies and by scaling with proper amplitudes and coeffi-cients, as shown before.

In the calculations, kernels Hn() are directly related tonth order voltages and are obtained by solving networkequations, where Hn() represents voltage, Y networkadmittance matrix at frequency s1+... , I1 linear inputcurrent and iNLn nonlinear currents of the devices:

Hn s1 ...,( ) Y s1 ...+( ) 1– INLn⋅=

Y s1( ) H1 s1( )⋅ I1=

Y s1 s2+( ) H2 s1 s2,( )⋅ INL2=

Y s1 s2 s3+ +( ) H3 s1 s2 s3, ,( )⋅ INL3=

INL2 f H1( )=

INL3 f H1 H2,( )=

27


2. DIRECT METHOD

In the direct method, entire kernels are not solved, butresponses at different frequencies are solved directly.First, calculate 1st order voltages. Using them, 2ndorder voltages can be calculated, and so fort. This isshown in more detail on the next page and also imple-mented in the Matlab program nli_main().

vi,+m,+n

Notation used in 2-tone test:

node mω1 nω2

Examples:

2nd harmonic 2ω1: vi,2,0

IM3 2ω1-ω2: vi,2,-1

fundamentalω2: vi,0,1

28


VOLTERRA EXPANSION FOR A 2-TONETEST SIGNAL

• Evaluate the fundamental (1st order) node voltagesv1 using linear ac analysis.

• Evaluate 2nd order distortion currents iNL2 usingfundamental voltage amplitudes and K2 terms.These will appear on five different sum and diffe-rence frequencies.

• Based on these distortion currents, calculate distor-tion voltages v2 in each node using linear analysis.

• Using 1st and 2nd order voltages v1 and v2 and K2

and K3 nonlinearities, calculate third order distortioncurrents iNL3 in nonlinear components. These willappear on 8 different frequencies. It is important tonote that also 2nd order nonlinearity can create 3rdorder distortion, if the exciting voltage contains 2ndorder terms.

• Again at the frequencies of iNL3 terms, perform acanalysis to find 3rd order node voltages v3.

v1

iNL2 -> v2

iNL3 -> v3

total = v1+v2+v3

29


EXAMPLE SOFTWARE

Soon, a couple of examples are presented. There are acouple of ways to get Volterra kernels:

• ISAAC is a symbolic circuit analysis tool for KULeuven, Belgium. It gives directly transfer functionscontaining nonlinear terms.

• Maple is symbolic mathematics program that (aswell as Mathematica) can be used to to form symbo-lic Volterra transfer functions.

• Matlab can be used to to plot vector plots of domi-nant distortion terms, when numerical values areknown.

In the latter two cases, the following procedure is used:

• the circuit is described as MNA (modified nodalanalysis) matrix and a source vector

• besides MNA, nonlinear components are describedas a list containing component value and nonlincoeffs, and controlling and output node pairs.

Cg(v)

v(t)

i in

Y V⋅ I=

Y g jωC+= I Iin=

gm_nli = vo+ vo- vc+ vc- K1 K2 K3

30


1-DIMENSION AL GM

A nonlinear conductance is described by

It will will be excited not only by the linear voltages,but also by the 2nd order voltages caused by its owndistortion. Thus, it can be considered as a V-I-V cas-cade, and the values of the nonlinear current sources-parallel to the conductance are:

Table 1: iNL for general analysis

order iNL

2 K2g H1(s1) H1(s2)

3 K3g H1(s1) H1(s2) H1(s3)+ 2/3 K2g ( H1(s1) H2(s2,s3) +

H1(s2) H2(s1,s3) +H1(s3) H2(s1,s2) )

i gm v K2′ v2⋅ K3′ v3⋅+ +( )⋅=iNL

gm

K2g12!-----

V2

2

∂

∂I V( )⋅ 1

2---

V∂∂

g V( )⋅= =

K3g13!-----

V3

3

∂

∂I V( )⋅ 1

3!-----

V2

2

∂

∂g V( )⋅= =

ig gm vk K2gm′ vk2⋅ K3gm′ vk

3⋅+ +( )⋅=

vk

31


Previous expressions can be applied for calculatingresponse at any frequency, when scaled with properamplitudes and coefficients. When calculatingdirectly with nth order node voltages, differentexpressions are needed for different frequencies. Themost usual ones for 1-dimensional gm are shown left.

Here, Vi,m,n refers to controlling voltage Vi = Vk-V l

at frequency mω1+nω2.

Table 1: gm iNL for dir ect calculation

Freq. iNL

ω1 ω2± K2gV i 1 0, , V i 0 1±, ,

2ω1 0.5K2g V i 1 0, ,( )2

desenscomprharmsumenvdc

Table 1: gm iNL for dir ect calculation

Freq. iNL

2ω1 ω2± 0.75K3g V i 1 0, ,( )2V i 0 1±, ,

K2gV i 1 0, , V i 1 1±, ,+

K2gV i 0 1±, , V i 2 0, ,+

3ω1 0.25K3g V i 1 0, ,( )3

K2gV i 1 0, , V i 2 0, ,+

ω1 1.5K3gV i 1 0, , Vi 0 1, , V

i 0 1–, ,

0.75K3gV i 1 0, , Vi 1 0, , V

i 1– 0, ,+

K2gV i 2 0, , V i 1– 0, ,+

K2gV i 1 1, , V i 0 1–, ,+

K2gV i 1 1–, , V i 0 1, ,+

K2gV i 0 0, , V i 1 0, ,+

32


VERIFY THIS FOR 2W1-W2

General Dir ect

Starting from general kernels

, marking s1=s2 = w1, s3 = -w2 and notingthe relative amplitudes of 3/4, 1, and 0.5 ofresponses at IM3, envelope, and harmonic fre-quencies, respectively, one obtains

Table 1: iNL for general analysis

order iNL

2 K2g H1(s1) H1(s2)

3 K3g H1(s1) H1(s2) H1(s3)+ 2/3 K2g ( H1(s1) H2(s2,s3)+ H1(s2) H2(s1,s3)+ H1(s3) H2(s1,s2) )

34---

K3g V i 1 0, ,( )2V i 0 1–, ,

2 3⁄( )K2gV i 1 0, , V i 1 1–, , 1⁄( )+

2 3⁄( )K2gV i 1 0, , V i 1 1–, , 1⁄( )+

2 3⁄( )K2gV i 0 1–, , V i 2 0, , 0.5⁄( )+

0.75K3g V i 1 0, ,( )2V i 0 1–, ,

K2gV i 1 0, , V i 1 1–, ,+

K2gV i 0 1–, , V i 2 0, ,+

=

Response at IM3 = 0.75*H3(w1,w1,-w2)

33


EXAMPLE OF NONLIN G (E.G. G Π)

Suppose a conductance G having 2nd order nonlinea-rity only. When it is driven by a voltage, 2nd order dis-tortion current is generated.

If G is driven from low impedance (left), the distortioncurrent iNL2 is short circuited and does not cause 2ndorder voltage signal over G. However, if driving impe-dance Zs is high (right) , iNL2 developes 2nd order dis-tortion voltage across G and it, mixed with the linearvoltage components, creates also 3rd order distortioncurrent iNL3, even though the device itself has only 2ndorder nonlinearity.

In BJT this appears so that when driven by current (Zs>>), a 3rd order nonlinear voltage developes in thebase due to gπ and cancels the 3rd order nonlinearityof gm. When driven by voltage, the base voltageremains sinusoidal and gm nonlinearity appears in theoutput. Thus, Zs may linearize the response.

V

V

V

V

iNL2

iNL2

Zs

Zs

G G

iNL3

34


2-DIMENSION AL GM

A nonlinear 2-dimensional conductance is controlledby two voltages k and l (e.g. VBE,VBC). This will bereplaced by its small signal value and a parallel non-linear current source, the value of which is calculatedby supposing independent nonlinearities in the inputand output and the following crossterms:

0.51

1.52

2.53

0

1

2

3

40

0.1

0.2

0.3

0.4

0.5

mA

VGS

VDS

35


Table 1: Cross terms in a 2-dimensional conductance

order iNL (crossterms only)

2 0.5 ( K2g1&g2 H1k(s1) H1l(s2) +K2g1&g2 H1k(s2) H1l(s1)

3 1/3 K2 g1&g2 ( H1k(s1) H2l(s2,s3)+ H1k(s2) H2l(s1,s3) + H1k(s3)H2l(s1,s2)+ H2k(s1,s2) H1l(s3)+ H2k(s1,s3) H1l(s2) + H2k(s2,s3)H1l(s1) )

+ 1/3 K3 2g1&g2 ( H1k(s1) H1k(s2)H1l(s3)+ H1k(s1) H1k(s3)H1l(s2)+ H1k(s2) H1k(s3)H1l(s1) )

+ 1/3 K3 g1&2g2 ( H1k(s1) H1l(s2)H1l(s3)+ H1k(s2) H1l(s1)H1l(s3)+ H1k(s3) H1l(s1)H1l(s2) )

iNLgm

vk vl

gm goiNLgo iNLgx

i gm vk K2gm′ vk2⋅ K3gm′ vk

3⋅+ +( )⋅=

go vl K2go′ vl2⋅ K3go′ vl

3⋅+ +( )⋅+

K2k&l vk vl⋅ ⋅ K32k&l vk2 vl⋅ ⋅ K3k&2l vk vl

2⋅ ⋅+ + +

here gm = g1, go = g2

36


MEANING OF THE CR OSSTERMS

The meaning of various terms in 2-dimensional gm isthe following:

• gm, K2gm, K3gm describe input behaviour, i.e.what is the input-output characteristics at a fixed out-put voltage.

• go, K2go, K3go describe output characteristics at afixed input voltage.

• Crossterms K2k&l, K32k&l, K3k&2l model the factthat ID-VDS curve has different shape at differentVGS voltages. They are essential in modelling e.g.Early effect.

VDS

ID

VDS

ID VGS

VGS

i gm vk K2gm′ vk2⋅ K3gm′ vk

3⋅+ +( )⋅=

go vl K2go′ vl2⋅ K3go′ vl

3⋅+ +( )⋅+

K2k&l vk vl⋅ ⋅ K32k&l vk2 vl⋅ ⋅ K3k&2l vk vl

2⋅ ⋅+ + +

no crossterms

with crossterms

37


1-DIMENSION AL CAP

Derivation of capacitance model is started from non-linear charge equation Q(V) which is differentiated toget current. This results in the following distortion cur-rents:

Table 1: CapiNL for general Hn()

order iNL

2 (s1+s2) K2c H1(s1) H1(s2)

3 (s1 + s2 + s3) x (K3c H1(s1) H1(s2) H1(s3)+ 2/3 K2c ( H1(s1) H2(s2,s3) +

H1(s2) H2(s1,s3) +H1(s3) H2(s1,s2) ))

Q Vo v+( ) Q Vo( ) 1k!----

V k

k

∂∂⋅ Q V( )( )

Vo

vk⋅k 1=

∞

∑+=

C, K2Q, K3Q, ..

CV∂∂

Q V( )=

K2C12!-----

V2

2

∂

∂Q V( )⋅ 1

2---

V∂∂

C V( )⋅= =

K3C13!-----

V3

3

∂

∂Q V( )⋅ 1

3!-----

V2

2

∂

∂C V( )⋅= =

iC jωC vk K2C ′ vk2⋅ K3C ′ vk

3⋅+ +( )⋅=

38


Table 1: cap iNL for dir ect calculation

Freq. iNL

ω1 ω2± j ω1 ω2±( )K2c

V i 1 0, , V i 0 1±, ,

2ω1 j 2ω1( )0.5K2c

V i 1 0, ,( )2

2ω1 ω2±

j 2ω1 ω2±( )

0.75K3c V i 1 0, ,( )2V i 0 1±, ,

K2cV i 1 0, , V i 1 1±, ,+

K2cV i 0 1±, , V i 2 0, ,+

⋅

3ω1j3ω1

0.25K3c V i 1 0, ,( )3

K2cV i 1 0, , V i 2 0, ,+

⋅

iC jωC vk K2gm′ vk2⋅ K3gm′ vk

3⋅+ +( )⋅=

vk

39


MEASURING V OLTERRA KERNELS

Numerical values for Volterra kernels of completeblocks or subcircuits can be measured using vectoranalyzers and AM/AM and AM/PM curves. However,different contributions can only be measured usingdevice level measurements.

40


IM3 ASYMMETR Y

As shown before, IM3 term consists partially of 2ndorder terms. In a nonlinear gm using a 2-tone test, IM3terms are as shown left. Supposing equal amplitudes ats1 and s2, terms caused by K3g match, but asymmetrymay be caused by

• unequal response at fundamental tones s1 and s2.Usually, this is not the case as flat passband is desi-red

• the fact that envelope s2-s1 term appears in oppositephase (complex conjugate) in the lower and higherIM3 terms

• the response of 2nd harmonics (that are already quitefar away from each other) may differ.

In more complex systems, partially compensating dis-tortion mechanisms may appear (e.g. in a current mir-ror), but the accuracy of the cancellation tends todeteriorate with increasing frequency.

s1 s2

s1 s2

IM3 2s1 s2–( ) 0.75K3g v s1( ) v s1( ) v s2( )∗⋅ ⋅ ⋅=

K2g v s1( ) v s2 s1–( )∗⋅ ⋅+

K2g v s2( )∗ v 2s1( )⋅ ⋅+

IM3 2s2 s1–( ) 0.75K3g v s2( ) v s2( ) v s1( )∗⋅ ⋅ ⋅=

K2g v s2( ) v s2 s1–( )⋅ ⋅+

K2g v s1( )∗ v 2s2( )⋅ ⋅+

2s1 2s2

41


EXTENSIONS: TIME-V ARYING V OLTERRA ANALYSIS

As such, Volterra analysis is suited for small signal, time-invariant systems. Thus, it is not well suited for ana-lysing circuits with large periodic excitations like MOS samplers and mixers, although some mixers consistingof linear multipliers have been analysed using Volterra analysis (Wambacq et al., IEEE TCAS-II, March 99).

Volterra analysis can be extended to periodically time-varying systems, and e.g. a MOS sampler has been ana-lysed in Yu, Sen & Leung: Time Varying Volterra Series and Its Application to the Distortion Analysis of aSampling Mixer.

42


Iin

gpi cpi gm/go gL

cmu

Ys

++

Zinsource

Iin

cpi gm/go gL

cmu

Ys

+

Zinsource

Cgs

Vgs

BJT

MOS

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