Volterra Series Analysis of Time-Varying Circuits

3.2 Frequency-Domain Techniques for Small-Signal Distortion

3.2.3 Volterra Series Analysis of Time-Varying Circuits

In the same way as Volterra series analysis of mildly nonlinear time-invariant circuits was an extension, to nth order, of the traditional linear (or first-order) circuit analysis methods, Volterra series analysis of mildly nonlinear time-varying circuits is an extension of linear time-varying ones. Its development is thus devoted to intermodulation phenomena in many types of analog, RF and microwave circuits, switched capacitor filters, sampler circuits, frequency converters, analog switches, parametric amplifiers, and modulators. The method that follows is undertaken in the frequency-domain and is an extension of the Conversion Matrix formalism [2, 5]. So, for the reader who is not familiar with that framework, we will begin by an introductory explanation of linear mixer analysis techniques.

Let us recall the illustrative nonlinear circuit of Figure 3.1. Since we want to perform a small-signal analysis of this circuit, we assume we have already calculated the quiescent point, so that the signal node voltages or branch currents behave as small perturbations to that fixed point. In the mixer case, for example, this quiescent point is composed by the voltages and currents forced by the large local oscillator, or pumping signal, plus any possible dc value. In previous sections, we have seen that the quiescent point could be calculated by a suitable nonlinear numerical method like the Newton-Raphson iteration. Thus, a similar task has to be done here, with the only difference that, now, the quiescent point is no longer a constant dc, but a time-varying (usually periodic) generalized one. An appropriate method for this task is the harmonic balance algorithm, explained in detail in Section 3.3.2.

If we have a time-dependent quiescent point, the Taylor series expansions of the current and charge nonlinearities should be given by

qNL(t)=QNL(t)+c1(t) [vO(t) −VO(t)]

+c2(t) [vO(t)−VO(t)]2

+c3(t) [vO(t)−VO(t)]3+. . . (3.123)

where cn(t) (n = 1, 2, 3, . . .) is the nth-order derivative, now evaluated in the time-varying quiescent point VO(t):

cn(t)≡ 1 n!

dnqNL(vO)

dvnO |vO=VO(t) (3.124) and

iNL(t)=INL(t) +g1(t) [vO(t)−VO(t)]+g2(t) [vO(t)−VO(t)]2

+g3(t) [vO(t) −VO(t)]3+. . . (3.125) where, accordingly,

gn(t) ≡ 1 n!

dniNL(vO)

dvnO |vO=VO(t) (3.126) The mildly nonlinear differential equation that models the dynamic circuit then becomes

d

dt[c1(t)vo(t)+c2(t)vo(t)2+c3(t)vo(t)3]+[G+g1(t)]vo(t)

+g2(t)vo(t)2+g3(t)vo(t)3=is(t) (3.127) and is supposed to admit a solution,vo(t), described by the following time-varying Volterra series:

vo(t)= ∑∞

n=1

von(t) (3.128a)

von(t)= 冕∞

−∞

. . . 冕∞

−∞

hn(t,␶1, . . . ,␶n)is(t−␶1) . . .is(t−␶n)d␶1. . . d␶n

(3.128b) For expressing (3.128) in frequency-domain, we again consideris(t) as a sum of Qsinusoids:

is(t)=1 2 ∑Q

q=−Q (q≠0)

Isqej␻qt (3.129)

which leads to von(t)= 1

2n ∑Q

q1=−Q . . . ∑Q

qn=−Q Isq

1 . . . Isq

nHn(t, ␻q1, . . . ,␻qn)ej(␻q1+. . .+␻qn)t (3.130) For further describing Hn(t, ␻1, . . . ,␻n) entirely in the frequency-domain, we assume that the large pumping is a sinusoid of frequency␻p, or, more generally, a periodic signal of fundamental frequency ␻p. In any case, everycn(t) orgn(t) will be periodic functions of the same fundamental frequency, which may then be represented by the Fourier series:2

cn(t) = ∑K

k=−K

Cnkejk␻pt (3.131)

and

gn(t)= ∑K

k=−K

Gnkejk␻pt (3.132)

The product of any of these time-varying Taylor coefficients by a control voltage composed of Qsinusoids generates a current whose components have all possible frequency mixing products: ␻q,k=k␻p +␻q. That current will be then converted again into voltage in any circuit impedance. So, the pumping signal translates the input spectrum at␻qinto a large number of frequency clusters located around every pumping harmonic of ␻pplus dc. An example of such a spectrum, whereQ =2, is depicted in Figure 3.13.

A spectrum arrangement like the one of Figure 3.13 can be either created by mixing ␻pand its harmonics with ±␻1and ±␻2, or by mixing the same␻pwith an input composed by the base-band, ±␻b1 and ±␻b2. In this case, the mixing components would be given by ␻b,k′ = k′␻p +␻b(b= −2, −1, 1, 2), where the new k′ can be related to the previous k by k′ = k + 1 or k′ = k − 1, whether

␻b > 0 or ␻b < 0, respectively. This alternative description of the frequency components has a much more intuitive indexk′distribution, as can be verified in Figure 3.14, and is thus preferred against the original␻k,p.

2. Note that although (3.129) and (3.131) look like similar only in (3.131) we are dealing with a periodic function. Therefore, (3.131) is, indeed, a Fourier series, whereas (3.129) is not.

Figure 3.13 Output spectrum components of a mixer driven by a local oscillator and two RF tones.

Figure 3.14 Alternative mixer output frequency components’ indexing scheme.

Therefore, from now on we will adopt this new indexing scheme, in a way that the mixing product referred to as ␻b,khas a frequency component of ␻b,k= k␻p+␻b. This component corresponds tok␻p+␻bif␻bare the excitation signals, or to (k−1)␻p+␻qin case the inputs are at␻q=␻p+␻b,␻q>␻p, or (k+1)␻p

−␻qif the inputs are located at␻q=␻p−␻b,␻q<␻p. Ifkis a negative integer,

␻b,krepresents a frequency whose value is symmetric to the one given.

The product of this voltage by another Taylor series coefficient likeg1(t) in (3.132) produces a current i1(t) with a set of newly generated components given by: ␻b,k1+k2=(k1+k2)␻p+␻b. Since k1and k2are integers varying fromk1, 2

= −K, . . . , −1, 0, 1, . . . , K, k1+k2spans from −2K to+2K. Another product sequence like this one would lead to frequencies ranging from −3K to +3K, and so on. So, in practical terms, it is necessary to truncate this spectral regrowth to a certain pumping harmonicK␻p, determined by desired results’ accuracy criteria.3 The circuit’s first-order voltage can then be represented by

vo1(t) = ∑K

k=−K ∑B

b=−B

Vo1b,kej(k␻p+␻b)t (B =Q) (3.133)

3. The problem of selecting the highest harmonic orderK␻pis a very important issue in frequency-domain CAD, generally referred asspectrum truncation.It will be addressed in more detail in Section 3.3.

while the first-order current resulting from the product of g1(t) by vo1(t) is inl1(t) ≡g1(t)vo1(t)= ∑K

k1=−K ∑K

k2=−K ∑B

b=−B

G1k

1Vo1b,k

2ej[(k1+k2)␻p+␻b)]t (3.134) If the components of inl1(t) are to be truncated at K␻p (i.e., such that

|k1+k2|≤K), then (3.134) represents the following matrix product:

冤IIInlnlnl1⯗⯗11−−−B,B,B,0−KK ……… IIInlnlnl1⯗⯗11−−−1,1,1,0−KK IIInlnlnl11⯗⯗11,−1,K1,0K ……… IIInlnlnl11⯗⯗1B,−B,KB,0K冥

=冤GG00⯗⯗⯗11K0 … …… …… 0 GGG⯗⯗⯗⯗111−K0K … …… …0 … GG0⯗0⯗⯗11−0K冥 (3.135a)

冤VVVo1o1o1⯗⯗−−−B,B,0B,K−K ……… VVVo1o1o1⯗⯗−−−1,1,K1,0−K VVVo1o1o1⯗⯗−1,1,01,K−K ……… VVVo1o1o1⯗⯗B,−B,0B,KK冥

or

Inl1=G1Vo1 (3.135b)

G1in (3.135) is the so-calledconversion matrixof the first-order time-varying conductance g1(t).

Before moving forward with the explanation, it is useful to comment on the form herein adopted for (3.135).

We can recognize a clear regularity in the position of the terms of the conversion matrix G1. They are all located as if the matrix was filled by simply horizontally shifting to the right the 4K+1 vector冋0 . . . 0G1K . . .G10. . .G1−K 0 . . . 0册and

retaining only the middle 2K+1 positions. A matrix in which the elements verify the relation aij= ti−j, where theti−j are the elements of a line vector, as is the case of G1, is called a Toeplitz matrix, and can be used to represent a linear convolution by a matrix-vector product. Noting also that the vector [ti−j] is nothing more than the inverted vector of the Fourier coefficients ofg1(t), we can conclude that (3.135) is, in fact, the matrix form of the frequency-domain convolution corresponding to the time-domain productinl1(t)=g1(t)vo1(t). In this sense, the null positions located next to the Fourier coefficients 冋G1−K. . .G10. . .G1K册

could be filled by nonzero values expanding the Fourier series from−2K␻pup to +2K␻p. In that case, G1would have the more common aspect of [5]

G1=冤GGG⯗⯗1112K0K ……… GGG⯗⯗111−K0K ……… GGG1⯗⯗11−−2K0K冥 (3.136)

[Because of the potential increased accuracy of this formulation in comparison to the one in (3.135), it will be adopted in the mixer studies carried on in Chapter 5.]

Continuing with the mixer analysis, if the intended current was now the one generated in the nonlinear charge,

ic1(t) = d

dt[c1(t)vo1(t)] (3.137)

then, in the frequency-domain it would be given by

ic1(t)= ∑K

k1=−K ∑K

k2=−K ∑B

b=−B

j[(k1+k2)␻p+␻b]C1k

1Vo1b,k

2ej[(k1+k2)␻p+␻b]t (3.138)

which can again be described in conversion matrix form as

冤IIIcc1c11⯗⯗−−−B,B,B,0−KK ……… IIIc1c1c1⯗⯗−−−1,1,01,K−K IIIc1c1c1⯗⯗1,−1,K1,0K ……… IIIc1c1c1⯗⯗B,B,KB,0−K冥

= j冤−KK␻␻−␻pp⯗⯗−−B␻␻BB ……… −KK␻␻−␻pp⯗⯗−−1␻␻11 −KK␻␻p␻p⯗⯗+1+␻␻11 ……… −KK␻␻p␻p⯗⯗+B+␻␻BB冥

.x冤CC00⯗⯗⯗11K0 … …… …… 0 CCC⯗⯗⯗⯗111−0KK … …… …0 … CC0⯗0⯗⯗11−0K冥

冤VVVo1o1o1⯗⯗−−−B,B,B,0−KK ……… VVVo1o1o1⯗⯗−−−1,1,01,K−K VVVo1o1o1⯗⯗1,−1,K1,0K ……… VVVo1o1o1⯗⯗B,−B,KB,0K冥 (3.139a)

or

Ic1= j⍀.xC1Vo1 (3.139b)

where the ‘‘.x’’ operator represents a matrix product on an element by element basis:

Z=X.xY: zij=xijyij

Finally, the first-order current passing through a time-invariant capacitance, C′, or conductance, G, (conventional linear circuit elements) would be equal to

冤IIIccc′−′−⯗′−⯗B,B,B,0−KK … I… I… Iccc′−⯗′−⯗′−1,1,01,K−K IIIccc′⯗⯗1,−′′1,K1,0K … I… I… Iccc′⯗⯗B,′′B,KB,0−K冥

= jC′冤−KK␻␻−␻pp⯗⯗−−B␻␻BB ……… K␻−K␻−␻pp⯗⯗−−1␻␻11 −KK␻␻p␻p⯗⯗+1+␻␻11 ……… K␻−K␻p␻p⯗⯗+B+␻␻BB冥

.x冤VVVo1o1o1⯗⯗−−−B,B,B,0−KK … V… V… Vo1o1o1⯗⯗−−−1,1,K1,0−K VVVo1o1o1⯗⯗1,−1,K1,0K … V… V… Vo1o1o1⯗⯗B,B,KB,0−K冥 (3.140a)

or

Ic′= j⍀C′.xVo1 (3.140b)

and

冤IIIGGG−⯗⯗−−B,B,KB,0−K … I… I… IGGG−⯗⯗−−1,1,1,0−KK IIIGGG⯗⯗1,−1,K1,0K … I… I… IGGG⯗⯗B,B,KB,0−K冥 (3.141a)

=G冤VVVo1o1o1⯗⯗−−−B,B,B,0−KK … V… V… Vo1o1o1⯗⯗−−−1,1,K1,0−K VVVo1o1o1⯗⯗1,−1,K1,0K … V… V… Vo1o1o1⯗⯗B,B,KB,0−K冥

or

IG=GVo1 (3.141b) respectively.

Since a constant-matrix product can be substituted by a matrix-matrix product if the constant is replaced by a diagonal matrix in which all elements are equal to that constant, it is usually accepted that such a diagonal matrix is the conversion matrix representation of time-invariant elements. In this way, for instance, (3.141) can also be expressed by

IG=GVo1 (3.141c)

whereG:gij=0 ifi ≠j andgij=G ifi=j.

With the above definitions in mind, it is now possible to use a conversion matrix form for the Kirchoff laws, enabling the analysis of any time-varying linear circuit. For example, the linearized time-varying model equation of (3.127) can be written as

j⍀ .xC1Vo1+(G+G1)Vo1=Is (3.142) Isis a matrix representation of the input, where all elements are zeros except the line ofk=0, for the inputs at␻b(see Figure 3.14), or the lines ofk= −1 and k=1 for the excitation at␻q, respectively.

The first-order output voltagevo1(t) can now be derived from (3.142) as Vo1=[G+G1+ j⍀.xC1]−1Is=Z1Is (3.143) or

vo1(t)= ∑K

k1=−K ∑K

k2=−K ∑B

b=−B

Z1k

1Isb,k

2ej[(k1+k2)␻p+␻b]t

⇒ vo1(t)= ∑K

k=−K ∑B

b=−B

Vo1b,kej(k␻p+␻b)t (3.144) as was previously assumed by (3.143).

Using (3.143), it is obvious that the frequency-domain first-order transfer function can be expressed by

H1(␻) =Z1 (3.145)

Proceeding with the nonlinear currents method for next order components, von(t), we now calculatevo1(t)2:

vo1(t)2= ∑K

k1=−K ∑K

k2=−K ∑B

b1=−B ∑B

b2=−BVo1b

1,k1Vo1b

2,k2ej冋冠k1+k2冡␻p+␻b1+␻b2册t (3.146) which, again truncated to the Kth␻pharmonic, gives

vo1(t)2= ∑K

k=−K ∑B

b1=−B ∑B

b2=−B

Vo1(2)(b

1+b2),kej(k␻p+␻b1+␻b2)t (3.147)

= ∑K

k=−K ∑C

c=−C

Vo1(2)c,kej(k␻p+␻c)t

where Vo1(2)c,k stands for the frequency-domain representation of the second-order products generated from vo1(t).

Following the conversion matrix notation presented above, inl2(t) = g2(t)vo1(t)2is given by

冤IIInlnlnl22⯗2⯗−−−C,C,KC,0−K ……… IIInlnlnl22⯗⯗2C,−C,C,0KK冥

=冤GG⯗⯗0220K ……… GGG⯗⯗222−0KK ……… GG0⯗⯗22−0K冥 (3.148a) 冤VVVo1(2)o1(2)o1(2)⯗⯗−−C,C,C,0−KK ……… VVVo1(2)o1o1(2)(2)⯗⯗C,C,0C,K−K冥

or

Inl2=G2Vo1(2) (3.148b)

and

ic2(t)= d

dt[c2(t)vo1(t)2] (3.149)

= ∑K

k1=−K ∑K

k2=−K ∑C

c=−C

j[(k1+k2)␻p+␻c]C2k

1Vo1(2)c,k

2ej[(k1+k2)␻p+␻c]t or

冤IIIcc2c22⯗⯗−−−C,C,KC,0−K ……… IIIcc2c22⯗⯗C,−C,0C,KK冥= j冤−K␻K␻−␻pp⯗⯗−−C␻␻CC ……… −K␻K␻p␻p⯗⯗+C+␻␻CC冥

.x冤CC⯗⯗0220K ……… CCC⯗⯗222−K0K ……… CC0⯗⯗22−0K冥 冤VVVo1o1o1(2)(2)(2)⯗⯗−−−C,C,C,0−KK ……… VVVo1o1(2)o1(2)(2)⯗⯗C,C,KC,0−K冥 (3.150a)

or even

Ic2= j⍀.xC2Vo1(2) (3.150b) The second-order linear time-varying equation of our circuit can thus be expressed as

j⍀.xC1Vo2+(G+G1)Vo2= −Ic2−Inl2 (3.151) which gives the second-order output voltage:

Vo2= −[G+G1+ j⍀.xC1]−1(Ic2+Inl2)= −Z1(Ic2+Inl2) (3.152) Equation (3.152) can be used as a control voltage to determine third-order output components, or to calculate H2(␻1, ␻2). For that, an is(t) = ej␻1t +ej␻2texcitation is considered, and the output components at the converted fre- quency corresponding to␻1+␻2should be determined from (3.152):

vo2(t)= ∑K

k=−K ∑C

c=−CVo2c,kej(k␻p+␻c)t (3.153) Similarly to what was done to second order, third-order output components are determined fromvo1(t) andvo2(t) by first calculating the third-order nonlinear excitations:

inl3(t)=2g2(t)vo1(t)vo2(t) +g3(t)vo1(t)3

=2 ∑K

k1=−K ∑K

k2=−K ∑K

k3=−K ∑B

b1=−B ∑B

b2=−B ∑B

b3=−B

G2k

1Vo1b

1,k2Vo2(b

2+b3),k3

⭈ej冋冠k1+k2+k3冡␻p+␻b1+␻b2+␻b3册t + ∑K

k1=−K ∑K

k2=−K ∑K

k3=−K ∑K

k4=−K ∑B

b1=−B ∑B

b2=−B ∑B

b3=−BG3k

1Vo1b

1,k2Vo1b

2,k3Vo1b

3,k4

⭈ej冋冠k1+k2+k3+k4冡␻p+␻b1+␻b2+␻b3册t

(3.154) or

冤IIInlnlnl33⯗3⯗−−−D,D,KD,0−K ……… IIInlnlnl3⯗⯗33D,−D,D,0KK冥=

2冤GG0⯗⯗22K0 ……… GGG⯗⯗222−K0K ……… GG0⯗⯗22−0K冥 冤VVVo12o12(3)o12(3)(3)⯗⯗−−−D,D,0D,K−K ……… VVVo12o12(3)o12(3)(3)⯗⯗D,−D,KD,0K冥

+冤GG⯗⯗0330K ……… GGG⯗⯗333−K0K ……… GG0⯗⯗33−0K冥 冤VVVo1o1o1(3)(3)(3)⯗⯗−−−D,D,D,0−KK ……… VVVo1o1(3)o1(3)(3)⯗⯗D,−D,KD,0K冥 (3.155a)

or even

Inl3=2G2Vo12(3) +G3Vo1(3) (3.155b) and

ic3(t)= d

dt[2c2(t)vo1(t)vo2(t)+c3(t)vo1(t)3]

=2 ∑K

k1=−K ∑K

k2=−K ∑K

k3=−K ∑B

b1=−B ∑B

b2=−B ∑B

b3=−B

j[(k1+k2+k3)␻p+␻b1+␻b2+␻b3]

⭈C2k

1Vo1b

1,k2Vo2(b

2+b3),k3ej冋冠k1+k2+k3冡␻p+␻b1+␻b2+␻b3册t + ∑K

k1=−K ∑K

k2=−K ∑K

k3=−K ∑K

k4=−K ∑B

b1=−B ∑B

b2=−B ∑B

b3=−B

j[(k1+k2+k3+k4)␻p+␻b1+␻b2+␻b3]

⭈C3k

1Vo1b

1,k2Vo1b

2,k3Vo1b

3,k4ej冋冠k1+k2+k3+k4冡␻p+␻b1+␻b2+␻b3册t

(3.156) or

Ic3=2j⍀ .xC2Vo12(3) + j⍀ .xC3Vo1(3) (3.157) Therefore, vo3(t) can be obtained from

Vo3= −[G +G1+ j⍀.xC1]−1(Ic3+Inl3)= −Z1(Ic3+Inl3) (3.158) Following what was said for second-order Vo2, (3.158) can also be used to derive the third-order nonlinear transfer functionH3(␻1,␻2,␻3), if an appropriate elementary excitation is assumed foris(t).

The extension of the time-varying nonlinear currents method, just explained, to multiport networks is straightforward, although laboriously involved. So, it will not be further discussed.

The main conclusion one should keep in mind is that the formalism of conver- sion matrix enables the analysis of any linear time-varying circuit in matrix form, in much the same way linear time-invariant networks were already treated. And, since the Volterra series analysis of any weakly nonlinear circuit simply consists on repeatedly analyzing the linearized circuit with the appropriatenth-order excita- tion, the extension to time-varying networks simply requires the substitution of the circuit variables and elements by their conversion matrix counterparts.