Weakly Nonlinear Time Invariant Circuits

Abstract

The aim of this chapter is to show the utility of the theory that we developed. This is done by applying it to the analysis of nonlinear effects, that is of deviation from linear behaviour, in analog circuits. The vast majority of analog circuits are limited by noise on the bottom end of their dynamic range and by nonlinear effects on the upper end.

The aim of this chapter is to show the utility of the theory that we developed. This is done by applying it to the analysis of nonlinear effects, that is of deviation from linear behaviour, in analog circuits. The vast majority of analog circuits are limited by noise on the bottom end of their dynamic range and by nonlinear effects on the upper end. While the analysis of noise is well understood by practising engineers, the analysis of nonlinear effects is much less so, and their minimisation poses great practical challenges. The applications presented in this chapter are therefore of practical utility.

The components serving as the building blocks of analog circuits can be represented by linear elements and controlled sources representing nonlinear behaviour. The total response of the circuit can be calculated from a hierarchy of electrical networks with the familiar small-signal linear network forming its core. The hierarchy of networks is constituted by the linear core driven by sources of increasing order. This can be seen as a specialisation to electrical networks of the signal-flow graph method that we saw in Sect. 10.2.

Analog electrical circuits are operated around a stable equilibrium point called the (quiescent) operating point of the circuit. The dynamic variables of interest in the theory of weakly nonlinear systems are the ones describing the deviation from the operating point (see Sect. 9.1). We call such variables small-signal (or incremental) variables. In the following, to distinguish the incremental part of a quantity from the total quantity, we will adopt the notational conventions summarised in Table 11.1.

Table 11.1 Definition of symbols used for various quantities

In Sects. 11.2 and 11.3 of this chapter we develop equivalent circuits for electronic components allowing us to model arbitrary weakly nonlinear analog circuits. In the remaining sections we study concrete circuits used in many types of systems and in particular in communication systems. Before that, in the following section we review a few standard metrics used to characterise the nonlinear behaviour of weakly nonlinear analog circuits.

1 Metrics for Nonlinear Effects

It’s common to distinguish between two classes of nonlinear effects. The first is characterised with input signals of large magnitude, the compression characteristics being the archetypal example. The second is characterised using small signals with intermodulation as the archetypal example. In the following we analyse these and related effects.

Fig. 11.1

Output signal magnitude versus input signal one of a typical weakly nonlinear system

1.1 Gain Compression and Expansion

Gain compression and gain expansion refer to the change in the gain experienced by a signal passing through a weakly nonlinear system as the amplitude of the input signal changes. At sufficiently large input signal levels all electronic circuits exhibit saturation. However, at the onset of deviation of gain from the small signal value, we may observe a gradual gain reduction, referred to as gain compression; or some gain increase, referred to as gain expansion (see Fig. 11.1). Which of these effects occurs and at which signal level depends on the nonlinear characteristics of the system.

Consider a weakly nonlinear system ${\mathcal {H}}$ driven by a sinusoidal signal

$$\begin{aligned} x(t) = \left|A_i \right| \cos (\omega _1 t + \varphi _1) = \Re \big \{A_i e^{\jmath \omega _1t}\big \}\,. \end{aligned}$$

As discussed in Sect. 9.8.2 its output is composed by tones at $\omega _1$ and at integer multiples of it, the harmonics. Let’s denote by $y_{\omega _1}$ the sum of all the terms at $\omega _1$

$$\begin{aligned} \begin{aligned} y_{\omega _1}(t) &:=\left|A_o \right| \cos (\omega _1 t + \psi _1) = \Re \big \{A_o e^{\jmath \omega _1t}\big \}\\ &= y_{1,(0,1)}^c(t) + y_{3,(1,2)}^c(t) + y_{5,(2,3)}^c(t) + \cdots \\ &= \Re \biggl \{ \Bigl [ \hat{h}_{1,(0,1)} + \frac{3}{4} \left|A_i \right|^2 \hat{h}_{3,(1,2)} + \frac{5}{8} \left|A_i \right|^4 \hat{h}_{5,(2,3)} + \cdots \Bigr ] A_i e^{\jmath \omega _1t} \biggr \}\,. \end{aligned} \end{aligned}$$

From this expression we see that $y_{\omega _1}$ is proportional to the input signal. Therefore, in a similar way as we do with LTI systems, we can consider the ratio of the output signal phasor to the one of the input signal and obtain a sort of frequency response. However, differently from the frequency response of linear systems, the obtained ratio is a function of the input signal amplitude and is called the describing function

$$\begin{aligned} K(\left|A_i \right|, \omega _1) :=\frac{A_o}{A_i} = \hat{h}_{1,(0,1)} + \frac{3}{4} \left|A_i \right|^2 \hat{h}_{3,(1,2)} + \frac{5}{8} \left|A_i \right|^4 \hat{h}_{5,(2,3)} + \cdots \,. \end{aligned}$$

(11.1)

Its magnitude is called the gain of the system

$$\begin{aligned} G(\left|A_i \right|, \omega _1) :=\left|K(\left|A_i \right|, \omega _1) \right| = \frac{\left|A_o \right|}{\left|A_i \right|}\,. \end{aligned}$$

At sufficiently small input signal levels, at the onset of nonlinear behaviour, the third order nonlinearity usually dominates and the describing function can be approximated by

$$\begin{aligned} K(\left|A_i \right|, \omega _1) \approx \hat{h}_{1,(0,1)} \cdot \Bigl ( 1 + \frac{3}{4} \left|A_i \right|^2 \frac{\hat{h}_{3,(1,2)}}{\hat{h}_{1,(0,1)}} \Bigr )\,. \end{aligned}$$

(11.2)

Note that we have factored the linear frequency response to obtain an explicit factor representing the deviation of the system’s behaviour from the one of a perfectly linear system. This factor can be visualised in the complex plane as the sum of the vector

$$\begin{aligned} \Xi = \Xi _r + \jmath \Xi _i :=\frac{3}{4} \left|A_i \right|^2 \frac{\hat{h}_{3,(1,2)}}{\hat{h}_{1,(0,1)}}; \qquad \Xi _r, \Xi _i \in {\mathbb {R}}\end{aligned}$$

and the unit vector 1 (see Fig. 11.2). If the angle of $\Xi$ is around 0° then the two vectors point approximately in the same direction. Therefore, as the amplitude of the input signal grows, the magnitude of the output signal grows faster than linearly and the system exhibits gain expansion. If the angle of $\Xi$ is around 180$^\circ$ then the two vectors point approximately in opposite directions and the system exhibits gain compression. If the angle is around $\pm 90$° then the vectors are approximately perpendicular and the gain of the system is less sensitive to variations of the input signal (terms of order higher than third will become important). However, in this case it is the angle of the output signal that is sensitive to changes in the input signal magnitude. Such a system is said to exhibit amplitude-modulation (AM) tophase-modulation (PM) conversion.

Fig. 11.2

Visualisation of $K(\left|A_i \right|,\omega _1) / \hat{h}_{1,(0,1)}$ as the sum in the complex plane of $\Xi$ and the unit vector

Let’s have a closer look at the gain of the system. The ratio of the system gain to the one of the system if it would be perfectly linear is called the gain compression/expansion ratio and denoted by $\textrm{GCER}$

$$\begin{aligned} \textrm{GCER}:=\frac{G(\left|A_i \right|, \omega _1)}{\left|\hat{h}_{1,(0,1)} \right|}. \end{aligned}$$

(11.3)

Using (11.2), at the onset of deviation from linear behaviour, it is given by

$$\begin{aligned} \begin{aligned} \textrm{GCER}&\approx \sqrt{(1 + \Xi _r)^2 + \Xi _i^2} \\ & = (1 + \Xi _r) \sqrt{1 + \frac{\Xi _i^2}{(1 + \Xi _r)^2}}\,. \end{aligned} \end{aligned}$$

If we expand the square root in a Taylor series

$$\begin{aligned} \textrm{GCER}\approx (1 + \Xi _r) (1 + \frac{1}{2}\frac{\Xi _i^2}{(1 + \Xi _r)^2} + \cdots ) \end{aligned}$$

we see that, to first order, the $\textrm{GCER}$ can be estimated by

$$\begin{aligned} \textrm{GCER}\approx 1 + \frac{3}{4} \Re \Bigl \{\frac{\hat{h}_{3,(1,2)}}{\hat{h}_{1,(0,1)}} \Bigr \} \left|A_i \right|^2\,. \end{aligned}$$

(11.4)

Given our small signal assumption, this expression should only be used to estimate gain compression or expansion up to ca. 1 dB.

A standard linearity metric used to test analog circuits is the 1  dB compression pointwhich is the signal magnitude causing the system gain to decrease by 1 dB. Equation (11.4) allows estimating the magnitude of the input signal producing a given gain compression or expansion

$$\begin{aligned} \left|A_i \right| \approx \sqrt{ \frac{4}{3} \Biggl | \frac{(\textrm{GCER}- 1)}{\Re \Bigl \{\frac{\hat{h}_{3,(1,2)}}{\hat{h}_{1,(0,1)}} \Bigr \}} \Biggr | }\,. \end{aligned}$$

(11.5)

If $\Re \{\hat{h}_{3,(1,2)} / \hat{h}_{1,(0,1)}\}$ is negative the 1 dB compression point can thus be estimated by

$$\begin{aligned} A_{1\text {dB}} :=\frac{0.381}{\sqrt{\Bigl | \Re \Bigl \{\frac{\hat{h}_{3,(1,2)}}{\hat{h}_{1,(0,1)}} \Bigr \} \Bigr |}}\,. \end{aligned}$$

(11.6)

For small input signals the phase change can also be calculated from the ratio $K(\left|A_o \right|,\omega _1) / \hat{h}_{1,(0,1)}$

$$\begin{aligned} \Delta \psi _1 = \arctan \frac{\Xi _i}{1 + \Xi _r} \approx \frac{\Xi _i}{1 + \Xi _r} \approx \Xi _i\,. \end{aligned}$$

From this we can estimate the input signal magnitude producing a phase change of $\Delta \psi _1$ radiants by

$$\begin{aligned} \left|A_i \right| \approx \sqrt{ \frac{4}{3} \Biggl | \frac{\Delta \psi _1}{\Im \Bigl \{\frac{\hat{h}_{3,(1,2)}}{\hat{h}_{1,(0,1)}} \Bigr \}} \Biggr |}\,. \end{aligned}$$

(11.7)

1.2 Intermodulation

In Example 9.8 we analyzed the response of a weakly nonlinear system to a two tones input signal and found that it is composed by several tones at various frequencies. In the context of communication systems and analog circuit design all signal tones at a frequency that is not a multiple of one of the input frequencies are referred to asintermodulation products. An intermodulation product is said to be of order k and denoted by  IMk if k is the lowest order nonlinearity able to produce it (see Fig. 9.8). For example, given input tones $\omega _1$ and $\omega _2$, the tones at $2\omega _1 - \omega _2$ and $2\omega _2 - \omega _1$ are intermodulation products of third order (IM3); the ones at $3\omega _1 - 2\omega _2$ and $3\omega _2 - 2\omega _1$ of fifth order (IM5).

As an example showing the importance of controlling and limiting the strength of intermodulation products, consider a communication receiver designed for a specific service. Most communication services divide the allocated frequency band in equally spaced channels. Suppose that we are interested in receiving a signal transmitted by a distant transmitter on channel j. Suppose further that the receiver also receives relatively strong interfering signals on channels $j+m$ and $j+2m$ destined to other users. If the receiver is not sufficiently linear then the two interfering signals will produce intermodulation products degrading and possibly completely masking the wanted signal. While the modulation of the involved signals plays a role, due to its simplicity, communication receivers are also invariably benchmarked and tested with tones as shown in Fig. 11.3.

Fig. 11.3

Interfering signals causing the IM3 product to mask the wanted signal

Consider the two tones input signal

$$\begin{aligned} x(t) = \left|A_1 \right|\cos (\omega _1 + \varphi _1) + \left|A_2 \right|\cos (\omega _2 + \varphi _2) = \Re \{A_1 e^{\jmath \omega _1} + A_2 e^{\jmath \omega _2}\} \end{aligned}$$

where we assume $\omega _2 > \omega _1 > 0$. The intermodulation product of order k characterised by the frequency mix m is

$$\begin{aligned} y_{k,m}^c(t) = \frac{1}{2^{k-1}}\frac{k!}{m!} \Re \{ A_1^{m_1} \overline{A_1}^{m_{-1}} A_2^{m_2}\overline{A_2}^{m_{-2}} \hat{h}_{k,m} e^{\omega _m t} \}\,. \end{aligned}$$

At relatively low input signal levels the strongest intermodulation products are the second and the third order ones with amplitudes

$$\begin{aligned} A_{\text {IM2}L} &:=\left|y_{2,(0,1,0,1)}^c(t) \right| = \left|A_1 \right| \left|A_2 \right| \left|\hat{h}_{2,(0,1,0,1)} \right| \\ A_{\text {IM2}H} &:=\left|y_{2,(0,0,1,1)}^c(t) \right| = \left|A_1 \right| \left|A_2 \right| \left|\hat{h}_{2,(0,0,1,1)} \right| \\ A_{\text {IM3}L} &:=\left|y_{3,(0,2,0,1)}^c(t) \right| = \frac{3}{4} \left|A_1 \right|^2 \left|A_2 \right| \left|\hat{h}_{3,(0,2,0,1)} \right| \\ A_{\text {IM3}H} &:=\left|y_{3,(0,1,0,2)}^c(t) \right| = \frac{3}{4} \left|A_1 \right| \left|A_2 \right|^2 \left|\hat{h}_{3,(0,1,0,2)} \right|\,. \end{aligned}$$

These expressions show that the IM2 products are proportional to the amplitudes of each of the two input tones while the IM3 products are proportional to the square of the magnitude of the closest tone and proportional to the magnitude of the more distant one (see Fig. 11.3).

The standard intermodulation test is performed with two tones of equal amplitude

$$\begin{aligned} \left|A_1 \right| = \left|A_2 \right| = A\,. \end{aligned}$$

In this case the magnitude of the IM product of order k is proportional to $A^k$ (remember that $\left|m \right| = k$)

$$\begin{aligned} \frac{1}{2^{k-1}}\frac{k!}{m!} A^{k} \left|\hat{h}_{k,m} \right|\,. \end{aligned}$$

Fig. 11.4

Second and third order intermodulation intercept points

Thus knowing the IMk product level at one value of A is enough to compute its value at a different value of A. This is of course only true at sufficiently small input signals, when the contributions to the IMk product of nonlinearities of order higher than k can be neglected. Instead of specifying the IMk at a specific value of A it is common practice to specify the  intermodulation intercept point of order k ( IPk). This is the level, extrapolated from sufficiently small values of A, at which the IMk reaches the same magnitude as the (linear) output of the system at $\omega _m$ when driven by a single tone of magnitude A and frequency $\omega _m$ (see Fig. 11.4). The kth order intercept point is thus defined by the equation

$$\begin{aligned} \frac{1}{2^{k-1}}\frac{k!}{m!} A^{k} \left|\hat{h}_{k,m} \right| = A \left|\hat{h}_1(\omega _m) \right|\,. \end{aligned}$$

Solving for the amplitude we find

$$\begin{aligned} A_{\text {IIP}k} :=\root k-1 \of { \frac{2^{k-1} m!}{k!} \biggl |\frac{\hat{h}_1(\omega _m)}{\hat{h}_{k,m}}\biggr | }\,. \end{aligned}$$

(11.8)

This quantity is also called the input referred IPk and denoted by  IIPk. Sometimes it is more convenient to refer this quantity to the output of the circuit in which case it is called the output referred IPk and denoted by  OIPk. Its value is found by multiplying the IIPk by the linear gain at $\omega _m$

$$\begin{aligned} A_{\text {OIP}k} :=\left|\hat{h}_1(\omega _m) \right| A_{\text {IIP}k}\,. \end{aligned}$$

(11.9)

The second and third order intercept points are the most important ones and can be estimated by

$$\begin{aligned} A_{\text {IIP}2} &= \biggl |\frac{\hat{h}_1(\omega _m)}{\hat{h}_{2,m}}\biggr | \end{aligned}$$

(11.10)

$$\begin{aligned} A_{\text {IIP}3} &= \sqrt{ \frac{4}{3} \biggl |\frac{\hat{h}_1(\omega _m)}{\hat{h}_{3,m}}\biggr | }\,. \end{aligned}$$

(11.11)

Expressed in decibels the IPk assumes a particularly simple form. To that end, let’s first rewrite the output referred IP k as

$$\begin{aligned} \begin{aligned} A_{\text {OIP}k} &= \left|\hat{h}_1(\omega _m) \right| \root k-1 \of { \frac{2^{k-1} m!}{k!} \biggl | \frac{A^k}{A^k}\frac{\hat{h}_1(\omega _m)}{\hat{h}_{k,m}} \biggr | }\\ &= A \left|\hat{h}_1(\omega _m) \right| \root k-1 \of { \frac{A \left|\hat{h}_1(\omega _m) \right|}{A^k \frac{k!}{2^{k-1} m!}\left|\hat{h}_{k,m} \right|}}\,. \end{aligned} \end{aligned}$$

Then note that

$$\begin{aligned} \bigl ( A \left|\hat{h}_1(\omega _m) \right| \bigr )^2 \end{aligned}$$

is the output power of the fundamental tone normalised to a load of $1/2~\Omega$. Similarly,

$$\begin{aligned} \Bigl ( A^k \frac{k!}{2^{k-1} m!}\left|\hat{h}_{k,m} \right| \Bigr )^2 \end{aligned}$$

is the one of the IMk product. Thus, if for a fixed and sufficiently small value of A we denote by $P_o$ the output power of the fundamental expressed in dB relative to some reference power and by $P_{\text {IM}k}$ the one of the IMk product relative to the same reference level, then we can express the OIPk by

$$\begin{aligned} \text {OIP}k = P_o + \frac{P_o - P_{\text {IM}k}}{k-1}\,. \end{aligned}$$

(11.12)

Similarly, by denoting the normalised power of an input tone by $P_t$, the IIPk can be expressed by

$$\begin{aligned} \text {IIP}k = P_t + \frac{P_o - P_{\text {IM}k}}{k-1}\,. \end{aligned}$$

(11.13)

These relationships are easily checked geometrically for the IP2 and IP3 in Fig. 11.4.

In  memory-less weakly nonlinear systems for which, for every k, ${\hat{h}_{k,m}}$ is a real number $c_k$ independent of m, the IP3 and the input signal level producing a gain compression/expansion of $\textrm{GCER}$ are both proportional to (see (11.5))

$$\begin{aligned} \sqrt{\Bigl | \frac{c_1}{c_3} \Bigr |}\,. \end{aligned}$$

Therefore, in this type of systems, these two quantities are proportional to each other

$$\begin{aligned} 20\log \left( \frac{A_{\text {1dB}}}{A_{\text {IIP3}}} \right) = 20\log \sqrt{\left|\textrm{GCER}- 1 \right|}\,. \end{aligned}$$

For a memory-less system exhibiting gain compression, the difference between the IP3 and the 1 dB compression point is

$$\begin{aligned} 20\log \left( \frac{A_{\text {1dB}}}{A_{\text {IIP3}}} \right) = 20\log \sqrt{1 - 10^{-1/20}} \approx -9.6\ \text {dB}\,. \end{aligned}$$

1.3 Desensitisation

The response of an LTI system to a signal is unaffected by the presence of a second signal. As long as we have a way of distinguishing the two signals, for example by separating them in frequency, we can ignore the presence of the second one. This is not the case in nonlinear systems where the response to one signal is affected by the presence of other ones. The effect is again most easily illustrated using a two tones input signal.

Let ${\mathcal {H}}$ be a weakly nonlinear system driven by the two tones input signal

$$\begin{aligned} x(t) = \left|A_1 \right|\cos (\omega _1 + \varphi _1) + \left|A_2 \right|\cos (\omega _2 + \varphi _2) = \Re \{A_1 e^{\jmath \omega _1} + A_2 e^{\jmath \omega _2}\}\,. \end{aligned}$$

The first tone represents the signal of interest, while the second one is an undesired signal that is referred to as a blocking signal or a jammer. As discussed, the response of the system is composed by several tones at various frequencies, among which several at $\omega _1$. As in Sect. 11.1.1, we denote by $y_{\omega _1}$ the sum of all terms at $\omega _1$

$$\begin{aligned} \begin{aligned} y_{\omega _1}(t) &:=\left|A_o \right| \cos (\omega _1 t + \psi _1) = \Re \big \{A_o e^{\jmath \omega _1t}\big \}\\ &= y_{1,(0,0,1,0)}^c(t) + y_{3,(0,1,2,0)}^c(t) + y_{3,(1,0,1,1)}^c(t) + \cdots \\ &= \Re \biggl \{ \Bigl [ \hat{h}_{1,(0,0,1,0)} + \frac{3}{4} \left|A_1 \right|^2 \hat{h}_{3,(0,1,2,0)} + \frac{3}{2} \left|A_2 \right|^2 \hat{h}_{3,(1,0,1,1)} + \cdots \Bigr ] A_1 e^{\jmath \omega _1 t} \biggr \}\,. \end{aligned} \end{aligned}$$

and find again an expression that is proportional to the phasor of the first input tone. At relatively small input signal levels the contributions of order higher than third can usually be neglected. In addition, we assume that the magnitude of the blocking signal is much larger than the one of the desired signal

$$\begin{aligned} \left|A_1 \right| \ll \left|A_2 \right|\,. \end{aligned}$$

Under these assumptions $y_{\omega _1}$ can be simplified to

$$\begin{aligned} y_{\omega _1}(t) \approx \Re \biggl \{ \Bigl [ \hat{h}_{1,(0,0,1,0)} + \frac{3}{2} \left|A_2 \right|^2 \hat{h}_{3,(1,0,1,1)} \Bigr ] A_1 e^{\jmath \omega _1 t} \biggr \}\,. \end{aligned}$$

Following a procedure similar to the one that we used to analyse gain compression and expansion, we build the ratio of the output phasor to the one of the first tone

$$\begin{aligned} XM(\left|A_2 \right|,\omega _1) :=\frac{A_o}{A_1} = \hat{h}_{1,(0,0,1,0)} \cdot \Bigl ( 1 + \frac{3}{2} \left|A_2 \right|^2 \frac{\hat{h}_{3,(1,0,1,1)}}{\hat{h}_{1,(0,0,1,0)}} \Bigr ) \end{aligned}$$

to obtain a sort of frequency response. Similarly to the approximation of the describing function (11.2), it is the product of the linear frequency response of the system and a factor that characterises the deviation from linear behaviour. Differently from the describing function, however, this second factor depends on the amplitude of the second tone, the blocking signal.

The ratio $XM(\left|A_2 \right|,\omega _1)/\hat{h}_{1,(0,0,1.0)}$ can again be visualised in the complex plane as the sum of the unit vector and the vector

$$\begin{aligned} \frac{3}{2} \left|A_2 \right|^2 \frac{\hat{h}_{3,(1,0,1,1)}}{\hat{h}_{1,(0,0,1,0)}}\,. \end{aligned}$$

If the angle of the latter is close to 180° then the second tone will induce a reduction in the gain experienced by the first one. If the angle is close to 0° it will induce a gain expansion and, if the angle is close to $\pm 90$° it will induce mostly a change in the phase of the first tone. The change in gain can be characterised by the magnitude of the above ratio, the desensitisation ratio

$$\begin{aligned} DR :=\biggl |\frac{XM(\left|A_2 \right|,\omega _1)}{\hat{h}_{1,(0,0,1.0)}}\biggr | = \Bigl | 1 + \frac{3}{2} \left|A_2 \right|^2 \frac{\hat{h}_{3,(1,0,1,1)}}{\hat{h}_{1,(0,0,1,0)}} \Bigr | \end{aligned}$$

(11.14)

and to second order in $\left|A_2 \right|$ can be estimated by

$$\begin{aligned} DR \approx 1 + \frac{3}{2} \left|A_2 \right|^2 \Re \Bigl \{ \frac{\hat{h}_{3,(1,0,1,1)}}{\hat{h}_{1,(0,0,1,0)}} \Bigr \}\,. \end{aligned}$$

(11.15)

From this expression we can estimate the magnitude of the blocker causing a certain wanted signal gain change

$$\begin{aligned} \left|A_2 \right| \approx \sqrt{ \frac{2}{3} \Biggl | \frac{(DR - 1)}{\Re \Bigl \{\frac{\hat{h}_{3,(1,0,1,1)}}{\hat{h}_{1,(0,0,1,0)}} \Bigr \}} \Biggr | }\,. \end{aligned}$$

(11.16)

If $\Re \{\hat{h}_{3,(1,0,1,1)} / \hat{h}_{1,(0,0,1,0)}\}$ is negative, a desensitisation of 1 dB is produced by a blocker at the 1 dB blocking level

$$\begin{aligned} A_{\text {B1dB}} :=\frac{0.269}{\sqrt{\Bigl | \Re \Bigl \{\frac{\hat{h}_{3,(1,0,1,1)}}{\hat{h}_{1,(0,0,1,0)}} \Bigr \} \Bigr |}}\,. \end{aligned}$$

(11.17)

The change in phase of the first tone caused by the presence of the blocker can also be estimated from $XM(\left|A_2 \right|,\omega _1) / \hat{h}_{1,(0,0,1.0)}$. To first order a phase change of $\Delta \psi _1$ radiants is produced by a blocker of magnitude

$$\begin{aligned} \left|A_2 \right| \approx \sqrt{ \frac{2}{3} \Biggl | \frac{\Delta \psi _1}{\Im \Bigl \{\frac{\hat{h}_{3,(1,0,1,1)}}{\hat{h}_{1,(0,0,1,0)}} \Bigr \}} \Biggr |}\,. \end{aligned}$$

(11.18)

Note that if the blocker is modulated, then the modulation will be transferred from it to the wanted signal. For example, if the blocker is amplitude modulated (AM) and the angle of $\hat{h}_{3,(1,0,1,1)} / \hat{h}_{1,(0,0,1,0)}$ is close to either 180°or 0° then the gain experienced by the wanted signal is modulated and, as a result, its output amplitude will also be modulated. If the angle of $\hat{h}_{3,(1,0,1,1)} / \hat{h}_{1,(0,0,1,0)}$ is close to $\pm 90$° then an amplitude modulation of the blocker will produce a phase modulation of the wanted signal. This effect of transferring the modulation of one signal to another one is called cross-modulation.

2 Nonlinear Two-Terminal Elements

In this section we investigate two-terminal electrical components that can be characterised by two quantities $x_E$ and $y_E$, related by an equation of the form

$$\begin{aligned} f(x_E, y_E) = 0 \end{aligned}$$

with f a function called the element x-y characteristic (see Fig. 11.5). If the equation can be expressed as a function of $x_E$, $y_E = \tilde{f}(x_E)$ then the element is called an x-controlled device. Similarly, if it can be expressed as a function of $y_E$, $x_E = \tilde{f}(y_E)$ then it is called a y-controlled device.

The devices that interest us are the ones that, in a region of interest around a quiescent operating point $(X_E,Y_E)$, are either x- or y-controlled and whose function $\tilde{f}$ can be approximated to any desired accuracy by a power series

$$\begin{aligned} y_e = \sum _{k=1}^\infty \tilde{f}_k x_e^k \qquad (x\text {-controlled}) \end{aligned}$$

or

$$\begin{aligned} x_e = \sum _{k=1}^\infty \tilde{f}_k y_e^k \qquad (y\text {-controlled}) \end{aligned}$$

with

$$\begin{aligned} y_e = y_E - Y_E, \qquad x_e = x_E - X_E. \end{aligned}$$

Fig. 11.5

Characteristic of an x-controlled two-terminal element

2.1 Nonlinear Resistors

A nonlinear resistor is a device characterised by the current $i_R$ flowing through it, the voltage $v_R$ across its terminals and by an i-v characteristic $f_R(i_R, v_R) = 0$. In the following we are going to represent a nonlinear resistor by the symbol shown in Fig. 11.6. A current controlled resistor can be characterised by a function

$$\begin{aligned} v_R = r(i_R) \end{aligned}$$

which, by assumption, around the operating point $(I_R,V_R)$, can be approximated by a power series

$$\begin{aligned} v = \sum _{k=1}^\infty r_k i^k\,, \qquad v = v_r = v_R - V_R\,,\quad i = i_r = i_R - I_R\,. \end{aligned}$$

If we consider the current i and the voltage v as signals, or, more precisely, elements of ${\mathcal {D}}'_{\oplus ,\text {sym}}$, then a nonlinear resistor can be regarded as a weakly nonlinear system and the components of v can be expressed in terms of the ones of i using (10.1) and Table 10.2

$$\begin{aligned} \begin{aligned} v_1 &= r_1 i_1\\ v_2 &= r_1 i_2 + r_2 i_1^{\otimes 2}\\ v_3 &= r_1 i_3 + 2 r_2 \left[ i_1 \otimes i_2\right] _{\text {sym}} + r_3 i_1^{\otimes 3}\\ \cdots \end{aligned} \end{aligned}$$

(11.19)

From this representation we observe that each voltage component $v_k$ is determined (i) by a term proportional to the kth current component $i_k$ and (ii) by other terms proportional to current components of order lower than k. In an electric network, the former can be represented by a linear resistor of value $r_1$, the latter by a voltage source $\tilde{v}_{R,k}$ whose value is determined by the current components $i_n, n=1,\dotsc ,k-1$ (see Fig. 11.7a)

$$\begin{aligned} v_k = r_1 i_k + \tilde{v}_{R,k}(i_1,\dotsc ,i_{k-1})\,. \end{aligned}$$

Fig. 11.6

Symbols used to represent nonlinear two-terminal devices. a resistor. b capacitor. c inductor

The various current and voltage components can therefore be calculated using a hierarchy of linear networks. First, we find the linear current $i_1$ using linearised components and the sources representing the system input. Once $i_1$ is found, $\tilde{v}_{R,2}$ can be determined. With it we can draw the second order network. It is obtained from the linearised network by removing the system input sources (since they are of first order), by adding the second order source $\tilde{v}_{R,2}$ and, if the case, the ones of other nonlinear components. With this network we compute $i_2$. Having found the first two components of i, the third order source $\tilde{v}_{R,3}$ can be calculated. We then proceed to draw the third order network which is again composed by the linearised network with the addition of independent sources of third order only. With it, we find $i_3$ and so on.

Fig. 11.7

a Weakly nonlinear resistor current-controlled equivalent model b Weakly nonlinear resistor voltage-controlled equivalent model

If the nonlinear resistor is voltage controlled, then its characteristic around the operating point can be described by a power series where the role of the independent variable is played by the voltage v

$$\begin{aligned} i = \sum _{k=1}^\infty g_k v^k\,. \end{aligned}$$

Proceeding as for the case of a current controlled nonlinear resistor, but with the roles of the signals i and v exchanged, we can express the first few components of the current i in terms of the ones of the voltage

$$\begin{aligned} \begin{aligned} i_1 &= g_1 v_1\\ i_2 &= g_1 v_2 + g_2 v_1^{\otimes 2}\\ i_3 &= g_1 v_3 + 2 g_2 \left[ v_1 \otimes v_2\right] _{\text {sym}} + g_3 v_1^{\otimes 3}\\ &\cdots \end{aligned} \end{aligned}$$

(11.20)

As before, each current component $i_k$ is the sum of a term linear in $v_k$ and other terms only depending on components of v of order lower than k

$$\begin{aligned} i_k = g_1 v_k + \tilde{i}_{R,k}(v_1,\dotsc ,v_{k-1})\,. \end{aligned}$$

From this representation we deduce the equivalent circuit shown in Fig. 11.7b.

If a nonlinear resistor is voltage as well as current controlled, then we can choose the most convenient representation for the problem at hand. If one representation is known, then the other one can be obtained by power series inversion. For example, if we know the voltage-controlled representation, the current-controlled one is obtained by inserting the expression for the components given by (11.20) into (11.19) and by choosing the coefficients $r_k$ so that the equations are satisfied. Specifically, $r_1$ is found by solving

$$\begin{aligned} i_1 = g_1 v_1 = g_1 r_1 i_1 \end{aligned}$$

which gives

$$\begin{aligned} r_1 = \frac{1}{g_1}\,. \end{aligned}$$

$r_2$ is obtained by solving

$$\begin{aligned} i_2 = g_1 v_2 + g_2 v_1^{\otimes 2} = g_1 (r_1 i_2 + r_2 i_1^{\otimes 2}) + g_2 r_1^2 i_1^{\otimes 2}\,. \end{aligned}$$

Using the previously obtained value for $r_1$, the equation is satisfied if

$$\begin{aligned} g_1 r_2 + g_2 r_1^2 = 0 \end{aligned}$$

or

$$\begin{aligned} r_2 = -\frac{g_2}{g_1^3}\,. \end{aligned}$$

$r_3$ is found in a similar way to be

$$\begin{aligned} r_3 = \frac{2 g_2^2 - g_1 g_3}{g_1^5}\,. \end{aligned}$$

Higher order coefficients are easily calculated using the same procedure.

2.2 Nonlinear Capacitors

A nonlinear capacitor is a two-terminal device whose voltage $v_C$ across the terminals and the charge $q_C$ stored in it are related by a q-v characteristic $f_C(q_C,v_C) = 0$. In the following, we are going to represent a nonlinear capacitor by the symbol shown in Fig. 11.6. A voltage controlled capacitor is a capacitor whose charge is a function of the voltage $q_C = \tilde{f}_C(v_C)$. Since the electric current is the time derivative of electric charge, if the voltage $v_C$ is a differentiable function of time, the capacitor current is related to the voltage across its terminals by

$$\begin{aligned} i_C = \frac{\textrm{d}\tilde{f}_C(v_C)}{\textrm{d}v_C} \frac{\textrm{d}v_C}{\textrm{d}t}\,. \end{aligned}$$

The slope of the q-v characteristic is called the small signal (or incremental) capacitance of the nonlinear capacitor

$$\begin{aligned} C(v_C) :=\frac{\textrm{d}\tilde{f}_C(v_C)}{\textrm{d}v_C}\,. \end{aligned}$$

As before, we assume it to be expandable in a power series around the operating point $(Q_C, V_C)$

$$\begin{aligned} c(v) :=C(v + V_C) = \sum _{k=0}^\infty c_{k+1} v^k\,, \qquad v = v_C - V_C\,. \end{aligned}$$

Using this expression in the equation for the current, we can express the latter as the following power series

$$\begin{aligned} \begin{aligned} i_C = \sum _{k=0}^\infty c_{k+1} v^k \frac{\textrm{d}v}{\textrm{d}t} = \sum _{k=1}^\infty \frac{c_{k}}{k} \frac{\textrm{d}}{\textrm{d}t} v^{k}\,. \end{aligned} \end{aligned}$$

This last expression can be extended to currents and voltages represented by elements of ${\mathcal {D}}'_{\oplus ,\text {sym}}$, so that we take it as defining the relationship between current and voltage of a voltage-controlled weakly-nonlinear capacitor

$$\begin{aligned} i = i_C = \sum _{k=1}^\infty \frac{c_k}{k} Dv^k\,. \end{aligned}$$

(11.21)

The first few components of the current expressed in terms of the components of the voltage are given by

$$\begin{aligned} \begin{aligned} i_1 &= c_1 Dv_1\\ i_2 &= c_1 Dv_2 + \frac{c_2}{2} Dv_1^{\otimes 2}\\ i_3 &= c_1 Dv_3 + c_2 D\left[ v_1 \otimes v_2\right] _{\text {sym}} + \frac{c_3}{3} Dv_1^{\otimes 3}\\ &\cdots \end{aligned} \end{aligned}$$

(11.22)

Each current component $i_k$ is the sum of a term linear in $Dv_k$ and others that only depend on the voltage components of order lower than k. In an electric network the kth component of the current can therefore be represented by a linear capacitor of value $c_1$ and a current source (see Fig. 11.8b)

$$\begin{aligned} i_k = c_1 Dv_k + \tilde{i}_{C,k}(v_1,\dotsc ,v_{k-1})\,. \end{aligned}$$

The various components are calculated with the same hierarchy of networks that we described for nonlinear resistors.

An initial charge $q_0$ on the capacitor can be represented as usual by a current pulse $q_0 \delta$ applied across the capacitor in the linear convolution equation.

Note that the linearised current-voltage characteristic of a capacitor is not by itself an asymptotically stable differential equation. The nonlinear transfer function formalism is therefore only applicable when the nonlinear capacitor is embedded in a network whose linear approximation is asymptotically stable.

Fig. 11.8

a Weakly nonlinear capacitor current-controlled equivalent model b Weakly nonlinear capacitor charge-controlled equivalent model

A charge controlled nonlinear capacitor is a capacitor whose voltage is a function of the charge $v_C = \varsigma (q_C)$. Expanding this function around the operating point $(Q_C,V_C)$ we obtain

$$\begin{aligned} v = \sum _{k=1}^\infty \varsigma _k q^k\,, \qquad v = v_C - V_C\,,\quad q = q_C - Q_C\,. \end{aligned}$$

The electric charge is the integral of the current. In the convolution algebra of right sided distributions this can be expressed by the convolution product between current and the Heaviside step function

$$\begin{aligned} q(t) = \int _0^t i(\tau )\,d\tau = \textsf{1}_{+}(t) *i(t)\,. \end{aligned}$$

Substituting this equation in the preceding series we obtain a relation between current and voltage

$$\begin{aligned} v = \sum _{k=1}^\infty \varsigma _k (\textsf{1}_{+}*i)^k\,. \end{aligned}$$

The first few voltage components expressed as a function of the current components are given by

$$\begin{aligned} v_1 &= \varsigma _1 \textsf{1}_{+}*i_1\\ v_2 &= \varsigma _1 \textsf{1}_{+}*i_2 + \varsigma _2 (\textsf{1}_{+}*i_1)^{\otimes 2}\\ v_3 &= \varsigma _1 \textsf{1}_{+}*i_3 + \varsigma _2 2\left[ (\textsf{1}_{+}*i_1) \otimes (\textsf{1}_{+}*i_2)\right] _{\text {sym}} + \varsigma _3 (\textsf{1}_{+}*i_1)^{\otimes 3}\\ &\cdots \end{aligned}$$

As for the previous cases we see that each voltage component $v_k$ is composed by a term linear in the current $i_k$ and other ones that only depend on current components of order lower than k

$$\begin{aligned} v_k = \varsigma _1 \textsf{1}_{+}*i_k + \tilde{v}_{C,k}(i_1,\dotsc ,i_{k-1})\,. \end{aligned}$$

This expression can be represented in an electric network by the equivalent circuit shown in Fig. 11.8a.

If a capacitor is voltage controlled as well as charge controlled, then one can use either representation and one can be converted in the other one. The following equations give the first three coefficients of the charge controlled representation expressed in terms of the ones of the voltage controlled ones

$$\begin{aligned} \varsigma _1 &= \frac{1}{c_1}\\ \varsigma _2 &= -\frac{c_2}{2 c_1^3}\\ \varsigma _3 &= \frac{c_2^2}{2 c_1^5} - \frac{c_3}{3 c_1^4}\,. \end{aligned}$$

They were obtained by the same inversion procedure that we used to relate the two representations of nonlinear resistors.

2.3 Nonlinear Inductors

A nonlinear inductor is a two-terminal device whose current $i_L$ and magnetic flux $\phi _L$ are related by the $\phi$-i characteristic $f_L(\phi _L, i_L) = 0$. In the following we are going to represent a nonlinear inductor by the symbol shown in Fig. 11.6. A current controlled inductor is an inductor whose flux is a function of the current $\phi _L = \tilde{f}_L(i_L)$. The voltage across the terminals of an inductor is the time derivative of the flux. Thus, if the current is a differentiable function of time, the voltage is

$$\begin{aligned} v_L = \frac{\textrm{d}\tilde{f}_L(i_L)}{\textrm{d}i_L} \frac{\textrm{d}i_L}{\textrm{d}t}\,. \end{aligned}$$

The slope of the $\phi$-i characteristic is called the small signal (or incremental) Inductance of the inductor

$$\begin{aligned} L(i_L) :=\frac{\textrm{d}\tilde{f}_L(i_L)}{\textrm{d}i_L} \end{aligned}$$

(11.23)

that we assume, around the quiescent operating point $(\Phi _L, I_L)$, to be expandable in a power series

$$\begin{aligned} l(i) :=L(i + I_L) = \sum _{k=0}^\infty l_{k+1} i^k\,, \qquad i = i_L - I_L\,. \end{aligned}$$

It is apparent that inductors and capacitors are “dual” of each other, with the roles of current and voltage exchanged. We can therefore adapt previous results and define the voltage and current relationship of a current controlled weakly nonlinear inductor by

$$\begin{aligned} v = v_C = \sum _{k=1}^\infty \frac{l_k}{k} Di^k\,. \end{aligned}$$

(11.24)

The first few components of the voltage expressed in terms of the components of the current are given by

$$\begin{aligned} \begin{aligned} v_1 &= l_1 Di_1\\ v_2 &= l_1 Di_2 + \frac{l_2}{2} Di_1^{\otimes 2}\\ v_3 &= l_1 Di_3 + l_2 D\left[ i_1 \otimes i_2\right] _{\text {sym}} + \frac{l_3}{3} Di_1^{\otimes 3}\\ &\cdots \end{aligned} \end{aligned}$$

(11.25)

The component k has the form

$$\begin{aligned} v_k = l_1 Dv_k + \tilde{v}_{L,k}(i_1,\dotsc ,i_{k-1}) \end{aligned}$$

from which we read the equivalent circuit shown in Fig. 11.9a.

Fig. 11.9

a Weakly nonlinear inductor current-controlled equivalent model b Weakly nonlinear inductor flux-controlled equivalent model

Fig. 11.10

Current source representation in the Laplace domain of nonlinearities of weakly nonlinear elements

Similarly, the flux controlled representation of a weakly nonlinear inductor is

$$\begin{aligned} i = \sum _{k=1}^\infty \varrho _k (\textsf{1}_{+}*v)^k\,, \end{aligned}$$

with the first few voltage components expressed as a function of the current components given by

$$\begin{aligned} i_1 &= \varrho _1 \textsf{1}_{+}*v_1\\ i_2 &= \varrho _1 \textsf{1}_{+}*v_2 + \varrho _2 (\textsf{1}_{+}*v_1)^{\otimes 2}\\ i_3 &= \varrho _1 \textsf{1}_{+}*v_3 + \varrho _2 2\left[ (\textsf{1}_{+}*v_1) \otimes (\textsf{1}_{+}*v_2)\right] _{\text {sym}} + \varrho _3 (\textsf{1}_{+}*v_1)^{\otimes 3}\\ &\cdots \end{aligned}$$

The component k has the form

$$\begin{aligned} i_k = \varrho _1 \textsf{1}_{+}*v_k + \tilde{i}_{L,k}(v_1,\dotsc ,v_{k-1})\,. \end{aligned}$$

which leads to the equivalent circuit shown in Fig. 11.9b.

Fig. 11.11

Voltage source representation in the Laplace domain of nonlinearities of weakly nonlinear elements

As for nonlinear capacitors, the nonlinear impulse responses formalism can only be applied to circuits including nonlinear inductors when they are part of networks whose linear approximation is asymptotically stable.

3 Nonlinear Multi-port Elements

Weakly nonlinear multi-port and multi-terminal elements can be represented by two-terminal elements and controlled sources. Therefore, in this section we introduce weakly nonlinear controlled sources. With them, we will have at disposal all the necessary circuit elements necessary to model arbitrary weakly nonlinear electronic components.

A controlled source is a two terminal element whose voltage $v_{CS}$ or current $i_{CS}$ is controlled by a control voltage $v_X$ or current $i_X$, a quantity in another part of the electric network of which it is part. There are four types of controlled sources:

• the voltage-controlled voltage source (VCVS), characterised by the equation $v_{CS} = \mu (v_X)$;

• the voltage-controlled current source (VCCS), characterised by the equation $i_{CS} = g_m(v_X)$;

• the current-controlled voltage source (CCVS), characterised by the equation $v_{CS} = r_m(i_X)$, and

• the current-controlled current source (CCCS), characterised by the equation $i_{CS} = \alpha (i_X)$.

As before, we assume that, around a quiescent operating point, the characterising function can be approximated to any desired accuracy by a power series. The incremental quantities can then be represented by elements of ${\mathcal {D}}'_{\oplus ,\text {sym}}$. For example, we assume that a VCCS can be represented by

$$\begin{aligned} i = \sum _{k=1}^\infty g_{m k} v^k\,, \qquad i = i_{CS} - I_{CS}\,,\quad v = v_X - V_X\,. \end{aligned}$$

The first three components of the current expressed in terms of the components of the voltage can be derived in the same way as we did for a voltage-controlled weakly-nonlinear resistor and are

$$\begin{aligned} \begin{aligned} i_1 &= g_{m1} v_1\\ i_2 &= g_{m1} v_2 + g_{m2} v_1^{\otimes 2}\\ i_3 &= g_{m1} v_3 + 2 g_{m2} \left[ v_1 \otimes v_2\right] _{\text {sym}} + g_{m3} v_1^{\otimes 3}\\ &\cdots \end{aligned} \end{aligned}$$

(11.26)

Note that each current component $i_k$ is the sum of a term linear in the kth component of the incremental control voltage $v_k$ and other terms that only depend on components of the voltage v of order lower than k

$$\begin{aligned} i_k = g_{mk} v_k + \tilde{i}_{CS,k}(v_1,\dotsc ,v_{k-1})\,. \end{aligned}$$

In an electric network a VCCS can thus be represented by a linear VCCS and independent current sources only depending on control voltage components of order lower than k. As for the two terminal weakly-nonlinear elements considered in Sect. 11.2, the linear term of a VCCS plays a special role. For this reason the quantity $g_{m1}$ has been given a name: it is called the transconductance of the source. A two-port representation of a VCCS is shown in Fig. 11.12.

Fig. 11.12

Equivalent circuit of a weakly-nonlinear VCCS

The situation is entirely analogous for the other types of controlled sources. The coefficient of the linear term of a CCVS $r_{m1}$ is called the transresistance, the one of a CCCS $\alpha _1$ is called the current transfer ratio and the one of a VCVS $\mu _1$ the voltage transfer ratio.

4 Low-Pass Filter with Nonlinear Capacitor

In this section we investigate the low-pass filter (LPF) shown in Fig. 11.13a. When implemented in an integrated circuit technology, a considerable fraction of the circuit area is often occupied by the capacitor. Given that the price of integrated circuits is determined to a large extent by occupied area, to reduce the cost of the circuit, it is desirable to use a capacitor type with a high capacitance per unit area. The highest capacitance per unit area available in CMOS technologies is offered by MOS capacitors which however have a rather nonlinear characteristic. For this reason we investigate the effects introduced in the circuit by the use of a nonlinear capacitor. In particular, we are interested in the upper linearity limit set by the nonlinear capacitor and therefore assume the operational amplifier (OpAmp) to be ideal.

Fig. 11.13

a Active RC low-pass filter circuit b Active RC low-pass filter with ideal OpAmp model

4.1 Nonlinear Transfer Functions

Under the assumption of an ideal OpAmp, the circuit of Fig. 11.13a can be represented by the small-signal equivalent circuit shown in Fig. 11.13b. Since MOS capacitors are voltage controlled, we represent the nonlinear capacitor as a voltage controlled device. Then, using Kirchhoff’s current law (KCL) the system equation is

$$\begin{aligned} \frac{v_o}{R} + c_1 Dv_o = -i_s - \sum _{k=2}^\infty \frac{c_k}{k}Dv_o^k\,. \end{aligned}$$

(11.27)

To highlight that the nonlinear part of the capacitor characteristic act as a source, we have moved that part of the characteristic to the right-hand side of the equation together with the source $i_s$ and collected all linear terms on the left-hand side.

We solve the equation in the Laplace domain using the equivalent circuits that we developed in Sects. 11.2 and 11.3. The first order output voltage component $V_{o.1}$ is obtained by replacing the nonlinear capacitor with an ideal capacitor with a value $c_1$ corresponding to the value of the nonlinear capacitor at the operating point

$$\begin{aligned} \frac{V_{o,1}(s_1)}{R} + c_1 s_1 V_{o,1}(s_1) = -I_s(s_1)\,. \end{aligned}$$

Using a Dirac impulse as input signal, the first order transfer function is found to be

$$\begin{aligned} H_1(s_1) = . V_{o.1}\bigg |_{I_s(s_1)=1} = \frac{-R}{1 + \frac{s_1}{\omega _{\text {3dB}}}} \end{aligned}$$

with

$$\begin{aligned} \omega _{\text {3dB}} :=\frac{1}{R c_1} \end{aligned}$$

the 3 dB cut-off frequency of the filter.

Having found the first order output component of the voltage, we can calculate the equivalent source representing the second order nonlinearity of the capacitor $\tilde{I}_{c,2}(s_1,s_2)$ (see Fig. 11.10). With a Dirac pulse as the first order input we find

$$\begin{aligned} \tilde{I}_{c,2}(s_1,s_2) = \frac{c_2}{2}(s_1 + s_2)H_1(s_1) H_1(s_2)\,. \end{aligned}$$

The second order transfer function is found with the help of the second order equivalent circuit. It is obtained from the first order one by removing the current source $I_s(s_1)$, which is of first order, and by inserting the source $\tilde{I}_{c,2}(s_1,s_2)$ representing the second order nonlinearity of the capacitor. The second order equivalent circuit is shown in Fig. 11.14. Note how the current generated by second (and higher) order nonlinearity is injected into the input node of the filter. For this reason, as discussed in Sect. 10.2, feedback is unable to suppress it. Since the variables in this network are of second order, we have to use the definition of the derivative for second order distributions (Eq. (9.14)). The second order transfer function is thus found to be

$$\begin{aligned} \begin{aligned} H_2(s_1,s_2) &= \frac{-R}{1 + (s_1 + s_2) R c_1} \tilde{I}_{c,2}(s_1,s_2)\\ &= \frac{c_2}{2} (s_1 + s_2) H_1(s_1 + s_2) H_1(s_1) H_1(s_2)\,. \end{aligned} \end{aligned}$$

Fig. 11.14

LPF $2{\text {nd}}$ order equivalent circuit

With the first two components of the output voltage we can compute the equivalent source representing the capacitor nonlinearity of third order (see Fig. 11.10)

$$\begin{aligned} & \tilde{I}_{c,3}(s_1,s_2,s_3) \\ & \qquad \quad = (s_1+s_2+s_3)\bigl \{ c_2\left[ H_1(s_1) H_2(s_2,s_3)\right] _{\text {sym}} + \frac{c_3}{3} H_1(s_1) H_1(s_2) H_1(s_3) \bigr \} \end{aligned}$$

and with it the equivalent circuit of third order. Using the definition of the derivative for third order distributions, the third order transfer function is found to be

$$\begin{aligned} \begin{aligned} H_3(s_1,s_2,s_3) &= \frac{-R}{1 + (s_1 + s_2 + s_3) R c_1} \tilde{I}_{c,3}(s_1,s_2,s_3)\\ &= (s_1+s_2+s_3) H_1(s_1 + s_2 + s_3)\\ &\quad \cdot \bigl \{ \frac{c_2^2}{2} \left[ H_1(s_1) H_1(s_2) H_1(s_3) (s_2 + s_3) H_1(s_2 + s_3)\right] _{\text {sym}}\\ &\qquad + \frac{c_3}{3} H_1(s_1) H_1(s_2) H_1(s_3) \bigr \}\,. \end{aligned} \end{aligned}$$

4.2 Second Order Intermodulation

Having found the first three transfer functions of the filter, we can evaluate the impact of the nonlinearities in concrete situations. As a first situation, suppose that there is a strong modulated signal in the stop-band of the filter. If the even order nonlinearities generate strong IM products masking the wanted signal in the pass-band, then the filter is of little use. To have a first indication of the strength of this effect, we only consider the nonlinearity of second order and calculate the IP2. We model the modulated signal in the stop-band with two tones at $\omega _1$ and $\omega _2$. We further assume $\omega _2 > \omega _1$ and

$$\begin{aligned} \Delta \omega :=\omega _2 - \omega _1 < \omega _{\text {3dB}} \end{aligned}$$

so that one of the IM2 products falls in the pass-band of the filter. The IM2 of interest is characterized by the frequency mix $m=(0,1,0,1)$ and thus by the frequency response

$$\begin{aligned} \begin{aligned} H_2(-\jmath \omega _1, \jmath \omega _2) &= \frac{c_2}{2} \jmath \Delta \omega H_1(\jmath \Delta \omega ) H_1(-\jmath \omega _1) H_1(\jmath \omega _2)\\ &\approx - \frac{c_2}{2} \jmath \Delta \omega R \frac{R \, \omega _{\text {3dB}}}{-\jmath \omega _1} \frac{R \, \omega _{\text {3dB}}}{\jmath \omega _2} \approx - \frac{c_2}{2} \jmath \Delta \omega \frac{R^3 \omega _{\text {3dB}}^2}{\omega _1^2}\,. \end{aligned} \end{aligned}$$

With it the IIP2 and OIP2 are obtained from (11.10)

$$\begin{aligned} I_{\text {IIP2}} &\approx \left|\frac{2}{c_2 \Delta \omega R^2 } \right| \Bigl (\frac{\omega _1}{\omega _{\text {3dB}}}\Bigr )^2 \end{aligned}$$

(11.28)

$$\begin{aligned} V_{\text {OIP2}} &\approx \left|\frac{2}{c_2 \Delta \omega R} \right| \Bigl (\frac{\omega _1}{\omega _{\text {3dB}}}\Bigr )^2\,. \end{aligned}$$

(11.29)

We have denoted the two intercept points by $I_{\text {IIP2}}$ and $V_{\text {OIP2}}$ to make it clear that the first characterizes the magnitude of the input current while the latter the magnitude of the output voltage.

These expressions reveal that the more the blocker is in the stop band, the lower the IM2. This makes intuitive sense as the voltage generated across the nonlinear capacitor by the interfering signal is the smaller, the lower the capacitor impedance. Since we have assumed a voltage-controlled capacitor, a small voltage will produce small intermodulation products. The above expressions also reveal that the IM2 is not homogeneous across the pass-band, but it is stronger when $\Delta \omega$ approaches the filter 3 dB cut-off frequency of the filter.

The IP2 can also be expressed in a slightly different form. If we replace one occurrence of $\omega _{\text {3dB}}$ by $1/(c_1R)$ in the above expression we obtain

$$\begin{aligned} V_{\text {OIP2}} \approx 2 \left|\frac{c_1}{c_2} \right| \frac{\omega _1^2}{\Delta \omega \, \omega _{\text {3dB}}}\,. \end{aligned}$$

(11.30)

This form highlights the value of the OIP2 as a function of the ratio of the linear capacitor coefficient to the coefficient of the second order nonlinearity.

Fig. 11.15

a Typical characteristic of an n-type accumulation-mode MOS varactor with a channel length of 0.2 $\upmu$m in a 40 nm CMOS technology b Small-signal model coefficients of an n-type accumulation-mode MOS varactor with a channel length of 0.2 $\upmu$m in a 40 nm CMOS technology

Figure 11.15a shows the typical characteristic of an n-type accumulation-mode MOS varactor [26] with a channel length of 0.2 $\upmu$m in a 40 nm CMOS technology. Figure 11.15b shows the small-signal model coefficients normalized to the linear capacitance $c_1$ as a function of the voltage across the capacitor. If we use such a capacitor biased at 0 V to implement a LPF with a cut-off frequency of 5 MHz to suppress a signal at 50 MHz modeled as two tones at 48.75 and 51.25 MHz respectively, (11.30) predicts that the filter will have an OIP2 of

$$\begin{aligned} \text {OIP2} \approx 44 \ \text {dBV}\,. \end{aligned}$$

For comparison we simulated the LPF IP2 numerically. To obtain a cut-off frequency of 5 MHz we used a resistor of 1 k$\Omega$ and a nominal capacitance of 31.58 pF. The results of the simulation are shown in Fig. 11.16. The value of the IP2 agrees very well with the predicted value. The IM2 starts to deviate from the ideal slope of 2 at a level of the input tones of ca. –55 dBA. This means that at that level the contribution of higher order nonlinearities to the IM2 become important. A -55 dBA tone at 50 MHz passing through a linear LPF with a transfer function equal to $H_1(\jmath \omega )$ and the above component values produces an output tone with a magnitude of approximately

$$\begin{aligned} \sqrt{2} \, 10^{-55/20} \, R \, \frac{\omega _{\text {3dB}}}{\omega } \approx 251 \ \text {mV}\,. \end{aligned}$$

The capacitor characteristic in Fig. 11.15a shows that a linear c-v approximation is only reasonably accurate up to this value. We thus see that a rough estimate of the range of validity of the approximation can be obtained by overlapping the approximation with the real characteristic. At larger positive and negative voltage levels the capacitor characteristic flattens out, and we can speculate that this is the reason for the slower increase of the IM2 at large signal levels.

Fig. 11.16

Simulated IM2 of the LPF with the capacitor having the characteristic shown in Fig. 11.15a and driven by two tones of equal magnitude at 48.75 and 51.25 MHz respectively

For many applications, such as in communication receivers, this IP2 is insufficient. One way to improve it is by using two equal nonlinear capacitors connected back-to-back as shown in Fig. 11.17a, each providing half of the required capacitance. In this way, when $v_o$ increases, the capacitance of one capacitor increases, while the one of the other capacitor decreases.

One way to analyze this circuit is to consider the combination of the two capacitors as a single nonlinear capacitor with the effective characteristic shown in Fig. 11.18. Figure 11.19 shows that, with identical devices, $c_2$ is identically zero. Hence, the IM2 is completely suppressed.

Fig. 11.17

a LPF with back-to-back capacitors ideal OpAmp model b LPF with back-to-back capacitors $2{\text {nd}}$ order equivalent circuit

Fig. 11.18

Characteristic of two back-to-back n-type accumulation-mode MOS varactor each with the characteristic shown in Fig. 11.15a

Fig. 11.19

Small-signal model coefficients of two back-to-back n-type accumulation-mode MOS varactor

Another way to analyze the circuit is to consider each capacitor individually. If the linear transfer function has to remain the same as the one of the original circuit with a single capacitor, then we must have $c_{p,1} + c_{n,1} = c_1$. The second order network is therefore composed by the same linear components as before, but now it includes two sources, each representing the second-order nonlinearity of one of the two capacitors (see Fig. 11.17b). The one of $c_p$ has the same reference direction as the one of the original circuit and has a value of

$$\begin{aligned} \tilde{I}_{c_p,2}(s_1,s_2) = \frac{c_{p,2}}{2}(s_1 + s_2)V_{o,1}(s_1) V_{o,1}(s_2)\,. \end{aligned}$$

The one of $c_n$ has the opposite reference direction and a value of

$$\begin{aligned} \begin{aligned} \tilde{I}_{c_n,2}(s_1,s_2) &= \frac{c_{n,2}}{2}(s_1 + s_2) \bigl (-V_{o,1}(s_1)\bigr ) \bigl (-V_{o,1}(s_2)\bigr )\\ &= \frac{c_{n,2}}{2}(s_1 + s_2) V_{o,1}(s_1) V_{o,1}(s_2)\,. \end{aligned} \end{aligned}$$

As the two negative signs coming from $v_n = -v_o$ cancel, the two currents flow in opposite directions and, if $c_{n,2} = c_{p,2}$ they cancel each other.

Note that this cancelling effect of even order responses is quite general. Given an arbitrary even order frequency response $\hat{h}_{k,m}$, the response of (even) order k to N input tones with phasors $A_1,\dotsc ,A_n$ is

$$\begin{aligned} y_{k,m}^c(t) = \Re \Bigl \{ \frac{1}{2^{k-1}} \frac{k!}{m!} A_{-N}^{m_{-N}} \cdots A_{-1}^{m_{-1}} A_1^{m_1} \cdots A_N^{m_N} \, \hat{h}_{k,m} \, \text {e}^{\jmath \omega _m t} \Bigr \}\,. \end{aligned}$$

If the sign of all input tones is reversed, every phasor will be multiplied by $\text {e}^{\jmath \pi }$. As k is assumed to be even and $\left|m \right|=k$ these factors will multiply to 1

$$\begin{aligned} \text {e}^{\jmath \pi k} = 1; \qquad k \text { even}. \end{aligned}$$

For this reason the response remains unchanged, but with opposite reference direction. Therefore, all even order harmonics and intermodulation products will be suppressed.

In reality there are two limitations to the amount of canceling that is practically achievable. The first one is due to the fact that small unavoidable manufacturing imperfections make nominally identical devices slightly different. This effect is called mismatch. For this reason the coefficients of $c_p$ will be slightly different from the one of $c_n$. Let’s represent the small variations due to mismatch in the following way

$$\begin{aligned} c_{p,1} &= c_{\text {nom},1} + \Delta c_{p,1} & c_{n,1} &= c_{\text {nom},1} + \Delta c_{n,1}\\ c_{p,2} &= c_{\text {nom},2} + \Delta c_{p,2} & c_{n,2} &= c_{\text {nom},2} + \Delta c_{n,2} \end{aligned}$$

with

$$\begin{aligned} c_{\text {nom},1} &= \frac{c_1}{2} & \Delta c_{p,1}, \Delta c_{n,1} &\ll c_{\text {nom},1}\\ c_{\text {nom},2} &= \frac{c_2}{2} & \Delta c_{p,2}, \Delta c_{n,2} &\ll c_{\text {nom},2}\,. \end{aligned}$$

Then, the two current sources $\tilde{I}_{c_p,2}(s_1,s_2)$ and $\tilde{I}_{c_n,2}(s_1,s_2)$ can be represented by a single source with the same reference direction of the former and a value of

$$\begin{aligned} \tilde{I}_{c_p,2}(s_1,s_2) - \tilde{I}_{c_n,2}(s_1,s_2) = \frac{\Delta c_{p,2} - \Delta c_{n,2}}{2} (s_1 + s_2)V_{o,1}(s_1) V_{o,1}(s_2)\,. \end{aligned}$$

The resulting network is similar to the one of the original circuit, the only difference being that the coefficient $c_2$ is replaced by $\Delta c_{p,2} - \Delta c_{n,2}$. The OIP2 is therefore

$$\begin{aligned} \begin{aligned} V_{\text {OIP2,B2B}} &\approx 2 \left| \frac{c_1 + \Delta c_{p,1} + \Delta c_{p,1}}{\Delta c_{p,2} - \Delta c_{n,2}} \right| \frac{\omega _1^2}{\Delta \omega \, \omega _{\text {3dB}}}\\ &\approx 2 \left|\frac{c_2}{\Delta c_{p,2} - \Delta c_{n,2}} \right| \left|\frac{c_1}{c_2} \right| \frac{\omega _1^2}{\Delta \omega \, \omega _{\text {3dB}}}\\ &= \left|\frac{c_2}{\Delta c_{p,2} - \Delta c_{n,2}} \right| V_{\text {OIP2}}\,. \end{aligned} \end{aligned}$$

(11.31)

Compared to the original circuit the IP2 has been improved by the mismatch limited factor

$$\begin{aligned} \left|\frac{c_2}{\Delta c_{p,2} - \Delta c_{n,2}} \right|\,. \end{aligned}$$

Fig. 11.20

Simulated IM2 of the LPF with the capacitor having the characteristic shown in Fig. 11.15a and driven by two tones of equal magnitude at 48.75 and 51.25 MHz respectively

Figure 11.20 shows the results of simulations with two identical nonlinear capacitors, and for the case where $\Delta c_{p,2}/c_{\text {nom},2} = -\Delta c_{n,2}/c_{\text {nom},2} = 0.01$. In the latter case we observe the expected improvement of

$$\begin{aligned} 20 \log \Bigl ( \frac{c_2}{0.02 c_2/2} \Bigr ) = 20 \log \Bigl ( \frac{1}{0.01} \Bigr ) = 40 \ \text {dB}\,. \end{aligned}$$

In the former, at input signal levels up to –65 dBA the value of the IM2 is limited by numerical noise. At larger input signal levels the simulation result is the product of the limited accuracy of the used numerical algorithm.

The second practical limitation is constituted by the fact that the terminals of real components are often coupled to other nodes of the circuit. This coupling can be modeled with parasitic components connected to the terminals. The parasitic components of the positive terminal are often different from the ones of the negative terminal. In addition, parasitic components are often nonlinear.

We conclude this section by noting that if the two tones are in the pass-band of the filter the OIP2 is

$$\begin{aligned} V_{\text {OIP2}} \approx \left|\frac{2}{c_2 \Delta \omega R} \right| = 2 \left|\frac{c_1}{c_2} \right| \frac{\omega _{\text {3dB}}}{\Delta \omega }\,. \end{aligned}$$

(11.32)

4.3 Third Order Intermodulation

In this section we investigate the situation where there are two interfering signals, one at $\omega _1$ and a second one close to twice this frequency $\omega _2 = 2\omega _1 - \Delta \omega$ so that the lower side-band IM3 falls in the pass-band of the filter

$$\begin{aligned} 2\omega _1 - \omega _2 = \Delta \omega < \omega _{\text {3dB}} \qquad \omega _1, \omega _2 > \omega _{\text {3dB}} > 0\,. \end{aligned}$$

To characterize this situation we compute the IP3.

The IM3 of interest is obtained from the third order transfer function

$$\begin{aligned} H_3(s_1,s_2,s_3) &= (s_1+s_2+s_3) H_1(s_1 + s_2 + s_3)\\ & \cdot \bigg \{ \frac{c_2^2}{2} \left[ H_1(s_1) H_1(s_2) H_1(s_3) (s_2 + s_3) H_1(s_2 + s_3)\right] _{\text {sym}}\\ & \qquad \qquad \qquad \quad + \frac{c_3}{3} H_1(s_1) H_1(s_2) H_1(s_3) \bigg \}. \end{aligned}$$

evaluated at the frequency mix $m = (1,0,2,0)$. Setting $s_1 = \jmath \omega _1, s_2 = \jmath \omega _1$ and $s_3 = -\jmath \omega _2$ the term enclosed in the symmetrization operator becomes

$$\begin{aligned} H_1(\jmath \omega _1) & H_1(\jmath \omega _1) H_1(-\jmath \omega _2)\\ {} & \cdot \frac{1}{6} \bigl [ 2 \jmath (2\omega _1) H_1(\jmath 2\omega _1) + 4 \jmath (-\omega _1 + \Delta \omega ) H_1(\jmath (-\omega _1 + \Delta \omega )) \bigr ]\,. \end{aligned}$$

If we assume $\left|\omega _1 - \Delta \omega \right| > \omega _{\text {3dB}}$ and use the approximation

$$\begin{aligned} \jmath \omega H_1(\jmath \omega ) \approx \jmath \omega \frac{-R}{\jmath \omega c_1 R} = \frac{-1}{c_1} \end{aligned}$$

we can simplify it to

$$\begin{aligned} H_1(\jmath \omega _1) H_1(\jmath \omega _1) H_1(-\jmath \omega _2) \frac{-1}{c_1}\,. \end{aligned}$$

Using these results we obtain

$$\begin{aligned} \begin{aligned} H_3(\jmath \omega _1, \jmath \omega _1, -\jmath \omega _2)\\ &\approx \jmath \Delta \omega H_1(\jmath \Delta \omega ) H_1(\jmath \omega _1) H_1(\jmath \omega _1) H_1(-\jmath \omega _2) \Bigl [- \frac{c_2^2}{2c_1} + \frac{c_3}{3}\Bigr ]\\ &\approx \jmath \Delta \omega (-R) \Bigl ( \frac{-1}{\jmath \omega _1 c_1} \Bigr )^2 \Bigl ( \frac{-1}{-2\jmath \omega _1 c_1} \Bigr ) \Bigl [- \frac{c_2^2}{2c_1} + \frac{c_3}{3}\Bigr ]\\ &= \frac{\Delta \omega R}{(\omega _1 c_1)^3} \Bigl [\frac{c_3}{6} - \frac{c_2^2}{4 c_1} \Bigr ]\,. \end{aligned} \end{aligned}$$

The IIP3 and OIP3 are obtained by inserting this result in (11.11)

$$\begin{aligned} I_{\text {IIP3}} &\approx \sqrt{\frac{4}{3} \left| \frac{(\omega _1 c_1)^3}{\Delta \omega \bigl [\frac{c_3}{6} - \frac{c_2^2}{4 c_1} \bigr ]} \right|} \end{aligned}$$

(11.33)

$$\begin{aligned} V_{\text {OIP3}} &\approx \sqrt{\frac{4}{3} \frac{\omega _1^3}{\Delta \omega \, \omega _{\text {3dB}}^2} \left| \frac{1}{\frac{c_3}{6 c_1} - \frac{1}{4}\bigl (\frac{c_2}{c_1}\bigr )^2} \right|}\,. \end{aligned}$$

(11.34)

These expressions reveal that the IP3 depends not only on the third order coefficient $c_3$, but also from the second order one $c_2$. The reason is the fact that second order intermodulation products are fed back to the input of the nonlinear component, where, in combination with the fundamental tones, they pass again through the second order nonlinearity. This is the effect that was discussed in Sect. 10.2 with the help of the signal-flow graph of Fig. 10.4 and the reason for $c_2$ being squared. The expression for the OIP3 highlights the fact that it is the ratio of the coefficients $c_2$ and $c_3$ to the linear capacitance $c_1$ that matters. The expressions also reveal that the IM3 generated by second order and third order nonlinearities have either the same or opposite phase and that, if

$$\begin{aligned} \frac{c_3}{c_1} = \frac{3}{2} \Bigl ( \frac{c_2}{c_1} \Bigr )^2\,, \end{aligned}$$

(11.35)

the two cancel each other.

In the previous section we discussed the fact that using equal nonlinear capacitors connected back-to-back eliminates even order components from the response of the system. This is not the case for odd order nonlinearities. To see this, we can draw the third order equivalent network of the filter with back-to-back capacitors. The equivalent sources representing the third order nonlinearities of $c_p$ and $c_n$ are

$$\begin{aligned} \tilde{I}_{c_p,3}(s_1,s_2, s_3) = \frac{c_{p,3}}{3} (s_1 + s_2 + s_3) V_{o,1}(s_1) V_{o,1}(s_2) V_{o,1}(s_3) \end{aligned}$$

and

$$\begin{aligned} \tilde{I}_{c_n,3}(s_1,s_2, s_3) = - \frac{c_{n,3}}{3} (s_1 + s_2 + s_3) V_{o,1}(s_1) V_{o,1}(s_2) V_{o,1}(s_3) \end{aligned}$$

respectively, where we have considered that $V_{o,2}(s_2,s_3)$ is zero. For equal capacitors $c_{p,3} = c_{n,3} = c_3/2$, therefore, having the sources opposite reference directions, they combine to form a single source equivalent to the one of a single nonlinear capacitor with $c_2 = 0$.

As an example, we consider again a filter with a cur-off frequency of 5 MHz implemented with a nonlinear MOS capacitor having the characteristic shown in Fig. 11.15a and biased at 0 V. At this bias point the ratios $c_2/c_1$ and $c_3/c_1$ are 1.73 and –0.94 respectively. If the filter is driven by a tone at 15 MHz and a second one at 27.5 MHz (11.34) predicts an OIP3 of

$$\begin{aligned} \text {OPI3} \approx 16.0 \text { dBV}\,. \end{aligned}$$

Note that in this example it is the second order nonlinearity that dominates the IM3 as

$$\begin{aligned} \left|\frac{c_3}{6 c_1} \right| \approx 0.16 < \frac{1}{4} \Bigl ( \frac{c_2}{c_1} \Bigr )^2 \approx 0.75\,. \end{aligned}$$

Thus, using back-to-back capacitors improves the OIP3 up to

$$\begin{aligned} \text {OIP3}_{\text {B2B}} \approx 23.6 \text { dBV}\,. \end{aligned}$$

Fig. 11.21

Simulated IP3 of the LPF with the capacitor having the characteristic shown in Fig. 11.15a and with two equal back-to-back (B2B) capacitors. The two input tones were of equal magnitude at 15 and 27.5 MHz respectively

Fig. 11.22

Simulated IM3 of the LPF with the capacitor having the characteristic shown in Fig. 11.15a as a function of the capacitor bias voltage $V_O$. The filter was driven by two equal tones of magnitude 0.1 at 15 and 27.5 MHz respectively

For comparison, we simulated the filter with the full nonlinear capacitor characteristic of Fig. 11.15a. The obtained IM3 as a function of the input tones magnitude is shown in Fig. 11.21. The figure also shows the IM3 obtained using back-to-back capacitors. In both cases the obtained IP3 is in good agreement with the above calculations. The IM3 starts to depart from a straight line with a slope of three at a level of the input tones of ca. –67 dBA. This corresponds to an output fundamental magnitude of ca. 0.2 V for the tone at $\omega _1$ and is close to the level at which the polynomial approximation starts to deviate significantly from the real characteristic of the capacitor.

Further, we verified the occurrence of canceling between the IM3 produced by the third order nonlinearity with the one produced by second order. Figure 11.22 shows the magnitude of the IM3 as a function of the bias voltage of the capacitor. The curve shows a clear notch at a bias voltage of ca. –0.19 V, the bias voltage at which the coefficient ratios $c_2/c_1$ and $c_3/c_1$ satisfy the canceling condition expressed by (11.35). This notch disappears at large signal levels, where contributions to the IM3 from higher order nonlinearities become important. The curve also suggest that, to obtain the best linearity, one should use a large bias voltage bringing the MOS capacitor in strong inversion, where its capacitance becomes almost constant.

Fig. 11.23

Simulated IP3 of the LPF with the capacitor having the characteristic shown in Fig. 11.15a and with two equal back-to-back (B2B) capacitors. The two input tones were of equal magnitude at 1 and 1.1 MHz respectively

Before concluding this section we investigate the case in which the two tones are in the pass-band of the filter. In this case the term in the symmetrization operator in $H_3(\jmath \omega _1,\jmath \omega _1,-\jmath \omega _2)$ is

$$\begin{aligned} & H_1(\jmath \omega _1) H_1(\jmath \omega _1) H_1(-\jmath \omega _2)\\ & \qquad \qquad \cdot \frac{1}{6} \bigl [ 2 \jmath (2\omega _1) H_1(\jmath 2\omega _1) + 4 \jmath (-\omega _1 + \Delta \omega ) (-R) \bigr ]\\ &\qquad = H_1(\jmath \omega _1) H_1(\jmath \omega _1) H_1(-\jmath \omega _2) \cdot \frac{2}{3} \jmath \bigl [ \omega _1 H_1(\jmath 2\omega _1) + (\omega _1 - \Delta \omega )R \bigr ]\,. \end{aligned}$$

If we further assume that $2\omega _1$ also falls in the pass-band of the filter it simplifies to

$$\begin{aligned} - H_1(\jmath \omega _1) H_1(\jmath \omega _1) H_1(-\jmath \omega _2) \cdot \frac{2}{3} \jmath \Delta \omega R\,. \end{aligned}$$

The third order nonlinear transfer function evaluated at the frequency mix $m = (1, 0, 2, 0)$ is therefore

$$\begin{aligned} \begin{aligned} H_3(\jmath \omega _1, \jmath \omega _1, -\jmath \omega _2) &\approx \jmath \Delta \omega R^4 \bigl [ \frac{c_3}{3} - \jmath \frac{c_2^2}{3} \Delta \omega R \bigr ]\\ &= \jmath \frac{\Delta \omega }{\omega _{\text {3dB}}} \frac{R^3}{3} \bigl [ \frac{c_3}{c_1} - \jmath \frac{c_2^2}{c_1} \Delta \omega R \bigr ]\,. \end{aligned} \end{aligned}$$

With this, the OIP3 is

$$\begin{aligned} V_{\text {OIP3,IB}} &\approx \sqrt{\frac{4 \omega _{\text {3dB}}}{\Delta \omega } \frac{1}{\left| \frac{c_3}{c_1} - \jmath \bigl (\frac{c_2}{c_1}\bigr )^2 \frac{\Delta \omega }{\omega _{\text {3dB}}} \right|}}\,. \end{aligned}$$

(11.36)

The results of a simulation with one tone at 1 MHz and the second one at 1.1 MHz is shown in Fig. 11.23. The results are again in good agreement with the OIP3 estimated with the help of the above equation which gives 10.1 and 10.7 dBV for a single capacitor and for back-to-back capacitors respectively.

4.4 Large Signal Effects

In this section we evaluate gain compression and amplitude-modulation to phase-modulation due to the nonlinear capacitor. The onset of both of these effects is governed by the third order transfer function evaluated at the frequency mix $m = (0,1,2,0)$ relative to the linear transfer function at the fundamental

$$\begin{aligned} \begin{aligned} \frac{H_3(\jmath \omega _1, \jmath \omega _1, -\jmath \omega _1)}{H_1(\jmath \omega _1)} &\approx -\jmath \omega _1 R^3 \bigl [ \frac{c_3}{3} - \jmath \frac{c_2^2}{3} \omega _1 R \bigr ]\\ &= - \frac{\omega _1}{\omega _{\text {3dB}}} \frac{R^2}{3} \Bigl [ \Bigl ( \frac{c_2}{c_1} \Bigr )^2 \frac{\omega _1}{\omega _{\text {3dB}}} + \jmath \frac{c_3}{c_1} \Bigr ]\,. \end{aligned} \end{aligned}$$

where we have assumed $2\omega _1 < \omega _{\text {3dB}}$. The phase of this expression determines the presence of gain compression or expansion and AM2PM.

As a concrete example, we consider again a low-pass filter with a cut-off frequency of 5 MHz, $R = 1$ k$\Omega$, the nonlinear capacitor with the characteristic shown in Fig. 11.15a and driven by a sinusoidal tone at 1 MHz. In this case the term in the square bracket in the above expression, multiplied by minus one, evaluates to $-0.6 + \jmath 0.94$. As the real part is negative we expect some gain compression. However, the imaginary part has a larger magnitude which implies that AM2PM should be somewhat more pronounced. If we use (11.7) to estimate the amplitude of the input tone producing a phase change of 1$^\circ$ we obtain a value of –67.3 dBA which corresponds to an output swing of 0.61 mV. A look at Fig. 11.15a shows that at these levels a second order approximation of the capacitor characteristic is a very poor approximation of the real characteristic. For this reason we can’t expect this estimate to be accurate.

A believable prediction can be made for levels where the approximation is good. For example, a phase change of 0.1$^\circ$ is predicted to happen at an input signal level of –77.3 dBA which corresponds to an output swing of 0.193 mV. Similarly, (11.5) predicts a 10 mdB gain compression at an input level of –77.2 dBA. These levels compare quite favorably with the values obtained by a numerical simulation and shown in Fig. 11.24. The simulation shows that these effects remain very small up to the large output swing of 1 V RMS which is close to the reliability limit supported by these devices.

Fig. 11.24

Simulated AM2PM and gain compression of the LPF with a nonlinear capacitor having the characteristic shown in Fig. 11.15a and driven by a tone at 1 MHz

5 Class-AC Common-Source Stage

In this section we analyse the common-source stage shown in Fig. 11.25a for use as an RF amplifier. In particular, we are interested in the distortion introduced by the nonlinear $i-v$ characteristic of the transistor and in the influence on distortion of the choice of gate bias voltage $V_G$. For simplicity in this section we neglect the $C_{gd}$ capacitance. We will consider circuits with some form of local feedback in a later section.

Fig. 11.25

a Common-source amplifier AC schematic b Common-source amplifier small-signal model

The following is a simple large-signal MOSFET model presented in many textbooks [27, 28]

$$\begin{aligned} i_D = {\left\{ \begin{array}{ll} 0 &{} v_{GS} - V_T \le 0\\ K' \frac{W}{L} (v_{GS} - V_T - \frac{v_{DS}}{2}) v_{DS} (1 + \lambda v_{DS}) &{} 0 < v_{DS} \le (v_{GS} - V_T)\\ \frac{K'}{2} \frac{W}{L} (v_{GS} - V_T)^2 (1 + \lambda v_{DS}) &{} 0 \le (v_{GS} - V_T) \le v_{DS}\,. \end{array}\right. } \end{aligned}$$

The second equation describes the so-called linear regionof the characteristic. This is the region where the overdrive voltage$v_{GS} - V_T$ is sufficiently large to cause a conductive surface charge channel in the active area at the surface between source and drain of the transistor and $v_{DS}$ is sufficiently small that the channel extends all along from the source to the drain terminal of the transistor. In this region the transistor behaves essentially as a nonlinear resistor.

The third equation describes the saturation regionof the characteristic and is the one of interest for implementing amplifiers and most other analogue circuits. In this region $v_{GS} - V_T$ is sufficiently large to cause the formation of a conductive channel. However, $v_{DS}$ is larger than the saturation voltagewhich means that the channel is present close to the source side of the transistor, but doesn’t extend all along to the drain terminal. In this region the current through the transistor $i_D$ is almost independent of the drain voltage and the transistor behaves to a good approximation as a voltage-controlled current source with $v_{GS}$ the control voltage. The parameter $\lambda$ takes into account the fact that the length of the channel does depend on the drain voltage and makes $i_D$ a weak function of the drain voltage [28]. In this simple model the saturation voltage is equal to the overdrive voltage $v_{GS} - V_T$.

Fig. 11.26

a FinFET input side characteristic. $L = 22\,\text {nm}, n_{\text {fin}} = 10, n_f = 16, m = 1, W_{\text {eff}} = n_{\text {fin}} n_f m 71\,\text {nm}$ b FinFET output side characteristic. $L = 22\,\text {nm}, n_{\text {fin}} = 10, n_f = 16, m = 1, W_{\text {eff}} = n_{\text {fin}}\,n_f\,m\, 71\,\text {nm}$

The characteristic of real transistors depends on many effects not captured by this simple model. To enable the design of analogue circuits, very accurate transistor models have been developed and made available in circuit simulators. Unfortunately, most of those models depend on several dozens to hundreds of parameters making them unsuitable for analytical estimates. Figure 11.26a shows the characteristic of a FinFET with a channel length $L=22$ nm as given by the CMG-BSIM model [29] with parameters from [30]. Figure 11.26a shows $\sqrt{i_D}$ as a function of $v_G$ with the source connected to ground and $v_D$ at a fix potential of 0.4 V. It shows that between 0.35 and 0.65 V the deviation of the characteristic from a straight line as predicted by the above simple model is quite small. Figure 11.26b shows $i_D$ as a function of $v_D$ for a fix gate voltage of 0.5 V. Here as well, the simple model gives a fairly good approximation over an extended range of the characteristic. The pictures show the values of $K', V_T$ and $\lambda$ obtained by fitting the model to the curves.

Using the simple model the current $i_D$ can be split in two parts

$$\begin{aligned} i_D = i_{D,a} + i_{D.b} \end{aligned}$$

with

$$\begin{aligned} i_{D,a} &= \frac{K'}{2}\frac{W}{L}(v_{GS} - V_T)^2\\ i_{D,b} &= g_O(i_{D,a}) v_{DS} = i_{D,a}\lambda v_{DS}\,. \end{aligned}$$

The current $i_{D,a}$ can be interpreted as the output of an ideal voltage-controlled current-source, while the current $i_{D,b}$ can be interpreted as the current due to a nonlinear load resistance. Since an ideal current source is not affected by its load, from an analysis point of view, it is convenient to analyse the two parts separately and combine the effects with the results of Sect. 10.1. For this reason we lump the components to the right of line A in Fig. 11.25b into a nonlinear load. In this section we focus on $i_{D,a}$. A common nonlinear load will be considered in the next section. Similarly, for analysis purposes, the nonlinear $C_{gs}$ capacitance can be considered part of the driving circuit. In the case of a resistive source we can reuse the results of the previous section with minor modifications. Often however, the distortion introduced by $C_{ga}$ is small compared to the one introduced by the i-v characteristic. In the following we will simply write $i_D$ for $i_{D,a}$.

Fig. 11.27

Polynomial approximations of the transistor characteristic around $V_G$ = 0.5 V. $V_D$ = 0.4 V, same transistor size as in Fig. 11.26a

Fig. 11.28

First three coefficients of a polynomial approximation of the transistor characteristic as a function of the gate bias point. $V_D$ = 0.4 V, same transistor size as in Fig. 11.26a

While the above model can be used to obtain a relatively good approximation of the transconductance $g_m$ of the transistor, it doesn’t provide a good estimate of higher order distortion terms. Therefore, to analyse distortion we approximate the transistor characteristic around the operating point by a third order polynomial

$$\begin{aligned} i_d = g_m v_{gs} + g_2 v_{gs}^2 + g_3 v_{gs}^3 \end{aligned}$$

and extract the coefficients from simulation. Figure 11.27 compares first, second and third order polynomial approximations to the full characteristic at a bias level of $V_G$ = 0.5 V and $V_D$ = 0.4 V. At this bias level a third order approximation provides a good approximation up to a signal level of about 150 mV. Figure 11.28 shows the three coefficients $g_m, g_2$ and $g_3$ as a function of the gate bias voltage $V_G$ simulated using CMG-BSIM models. While the simple model predicts a vanishing third order coefficient $g_3$ the picture shows that it disappears only at a single gate bias point. For small gate bias voltages the $g_3$ coefficient is positive, while for large values it’s negative. We may try to minimize third order distortion by biasing the transistor at the bias point at which $g_3$ is zero. However, this strategy doesn’t lead to a robust design. In fact mismatch between the transistor and the bias devices introduces a statistical Gaussian bias error with a typical standard deviation of order [31]

$$\begin{aligned} \sigma _{V_T} \approx \frac{5\ \text {mV} \cdot \upmu m}{\sqrt{W L}} \end{aligned}$$

where L and W are the length respectively the width of the active channel. A more fruitful approach is to use two transistors connected in parallel, but biased at different bias levels. One biased at the minimum of $g_3$ and the second at its maximum. The relative size of the two transistors is chosen in such a way as to make the sum of the $g_3$s cancel. In this way the deviation of the bias point of each transistor due to mismatch has a smaller impact on the value of $g_3$. The resulting effective $g_3$ of the transistor couple, a so called Class-AC stage, is shown in Fig. 11.29.

Fig. 11.29

Second and third order coefficients of a polynomial approximation of the Class-AC stage characteristic as a function of the gate bias deviation from the nominal gate bias point $\Delta V_G$ for $V_D$ = 0.4 V

The IIP3 of the stage can be estimated from (11.11). For a single transistor biased at $V_G=0.46$ V we read from Fig. 11.28$g_m\approx 15$ mS, $g_3\approx -110\,\text {mA}/\text {V}^2$ giving an IIP3 of ca. –10.4 dBV. From Fig. 11.29 we see that a Class-AC stages reduces $g_3$ by ca. a factor of 10, while leaving $g_m$ almost unchanged. From this data we estimate that the IIP3 should be ca. 10 dB higher or approximately –0.4 dBV. The results obtained by numerical simulation with the full transistor models are shown in Fig. 11.30. To suppress the effect of the nonlinear output conductance $g_o$ the drain was held at 0.4 V using an ideal voltage source. The circuit was driven by a voltage source with a resistance of 50 $\Omega$ generating two tones of equal amplitude at $f_1= 1.01$ GHz and $f_2 = 1.02$ GHz. Note that the simulation does include the effect of a slightly nonlinear $C_{gs}$ as well as the one of $C_{gd}$. The results are in good agreement with our estimates up to a level of about –25 dBV ($\approx 80$ mV) per tone that translates in a peak input voltage of 160 mV. This is in line with expectation as beyond this level the third order approximation of the characteristic starts to break down as noted earlier.

Fig. 11.30

Simulated IM3 of a Class-AC stage compared to the one of a simple common-source stage consisting of the Class-A device only. Class-A device: $L=22$ nm, $n_{\text {fin}}=10$, $n_f=16$, $m=1$ biased at $V_G=0.453$ V. Class-C device: $L=22$ nm, $n_{\text {fin}}=10$, $n_f=8$, $m=1$ biased at $V_G=0.342$ V. $V_D$ = 0.4 V. $\left|v_t \right|$ is the magnitude of each of the two input tones

The Class-AC stage reduces $g_3$, but doesn’t reduce $g_2$. Therefore, if the second order transfer function of the preceding or following stage is also large, then the combined system will still produce third order distortion. If we call the first subsystem ${\mathcal {G}}$ and the second one ${\mathcal {H}}$ the combined third order impulse response is in fact (see Table 10.2)

$$\begin{aligned} (h \circ g)_3 = h_1 *g_3 + 2 \, h_2 *\left[ g_1 \otimes g_2\right] _{\text {sym}} + h_3 *g_1^{\otimes 3} \end{aligned}$$

which doesn’t disappear even if $g_3$ and $h_3$ are both zero. One approach to reduce $g_2$ (on top of $g_3$) is to use a complementary structure comprised of an nMOS Class-AC stage and a pMOS one as sketched in Fig. 11.31. Here we use common-gate stages (see the next section) to reduce the effects of $C_{gd}$ and combine the currents through a transformer. For good results one needs large coupling between the primary and secondary of the transformer. In a monolithic implementation this is best achieved using equal coils stacked one on top of the other. We can also directly connect the drains of the two stages. In this case the bias currents of the two stages must coincide and a mean of controlling the DC drain voltage is necessary.

Fig. 11.31

Complementary Class-AC stage suitable for RF applications

6 Common-Gate Stage

In this section we investigate the linearity properties of the common-gate stage shown in Fig. 11.32a. We first consider the case in which the stage is driven by a source with internal resistance $R_s$ and then specialise to the case in which the stage is used to form a Cascode. A basic variant of the Cascode stage suitable for use at RF frequencies is the combination of a common-source stage followed by a common-gate one. The combination of the two stages behaves as an improved common-source stage with much reduced $C_{gd}$ and output conductance $g_o$ [27]. In this section we will show that, under suitable conditions that we will work out, the addition of a common-gate stage does not degrade distortion either. Due to these very desirable benefits the Cascode stage is a widely used configuration.

Fig. 11.32

a Common-gate stage AC schematic b Common-gate stage Small-signal model

Consider the small-signal model shown in Fig. 11.32b. The input voltage $v_i$ corresponds to the source voltage. The input current is the current entering into the source terminal. The part of the input current that doesn’t flow through $C_{sg}$ is labeled $i_c$ and represents the current that flows through the transistor active channel to the drain. The current leaving the drain terminal must therefore have the same value. This is represented by the output side current-controlled current source with unit gain. For simplicity, we neglect the distortion introduced by the nonlinear capacitance $C_{sg}$ as well as the one introduced by the drain capacitance that in the figure was lumped together with the load $Z_L$. As before we characterise the linearity of the circuit by calculating the nonlinear terms present in the output current $i_c$.

6.1 Nonlinear Transfer Functions

According to the model presented in Sec. 11.5 (with $\lambda = 0$) the static characteristic of the transistor in saturation is given by

$$\begin{aligned} i_D = \frac{\beta }{2}{v_{OD}^2} \end{aligned}$$

with $v_{OD} = v_{GS} - V_T$ the overdrive voltage and $\beta = K' W/L$. In the present situation it is more convenient to express the input voltage as a function of the current. This is easily achieved by inverting the equation. If we further separate the DC bias terms from the small signal quantities we obtain

$$\begin{aligned} v_{gs} = \sqrt{\frac{2(I_D + i_d)}{\beta }} - V_{OD} \end{aligned}$$

which we approximate by a third order Taylor polynomial around the operating point

$$\begin{aligned} v_{gs} \approx V_{OD} \bigg [ \frac{1}{2} \frac{i_d}{I_D} - \frac{1}{8} \Big (\frac{i_d}{I_D}\Big )^2 + \frac{1}{16} \Big (\frac{i_d}{I_D}\Big )^3 \bigg ]\,. \end{aligned}$$

Using the relations $v_i = -v_{gs}$ and $i_c = -i_d$ we obtain that the input characteristic corresponds to the one of a nonlinear resistor

$$\begin{aligned} v_i = r_1 i_c + r_2 i_c^2 + r_3 i_c^3 \end{aligned}$$

(11.37)

with

$$\begin{aligned} r_1 = \frac{1}{g_m} = \frac{V_{OD}}{2 I_D}\,, \qquad r_2 = \frac{V_{OD}}{8 I_D^2}\,, \qquad r_3 = \frac{V_{OD}}{16 I_D^3}\,. \end{aligned}$$

(11.38)

Note that while the original expression giving $i_D$ as a function $v_{OD}$ doesn’t include any third order term, the inverted expression gives a well-defined term of third order. As a result, the latter is less sensitive to modeling inaccuracies than the former. We will therefore use the above values as estimates for $r_i, i=1,\dotsc ,3$.

Fig. 11.33

Equivalent circuit for the calculation of the nonlinear transfer functions of the common-gate circuit

Using the above third order polynomial to model the source-gate nonlinear characteristic we obtain the equivalent circuit shown in Fig. 11.33 with $V_2$ and $V_3$ the second resp. third order equivalent nonlinear source as given in Table 11.11.

The first order transfer function is calculated by discarding the contribution of all sources of order different from one. This amounts to calculating the contribution due to the input source and using a Dirac impulse as input signal. Working in the Laplace domain, the Kirchhoff’s voltage law gives

$$\begin{aligned} R_s (I_{c,1} + s C_{sg} r_1 I_{c,1}) + r_1 I_{c,1} = 1\,. \end{aligned}$$

Solving for the first order component of $I_c$ we find

$$\begin{aligned} I_{c,1}(s) = H_1(s) = \frac{1}{R_s + r_1 + s C_{sg} r_1 R_s} = \frac{1}{R_s + r_1} \frac{1}{1 + \frac{s}{\omega _0}} \end{aligned}$$

with $\omega _0 = (R_s + r_1) / (C_{sg} r_1 R_s)$.

With $I_{c,1}$ and referring to Table 11.11 we can now compute the equivalent source of second order $V_2 = r_2 I_{c,1}(s_1) I_{c,1}(s_2)$. The second order transfer function is the response to this source which is easily calculated to be

$$\begin{aligned} I_{c,2}(s_1,s_2) = H_2(s_1,s_2) = - \frac{1 + (s_1 + s_2) C_{sg} R_s}{R_s + r_1 + (s_1 + s_2) C_{sg} r_1 R_s} r_2 I_{c,1}(s_1) I_{c,1}(s_2) \end{aligned}$$

or, expressed in terms of $H_1$

$$\begin{aligned} H_2(s_1,s_2) = -r_2 [1 + (s_1 + s_2) C_{sg} R_s] H_1(s_1 + s_2) H_1(s_1) H_1(s_2)\,. \end{aligned}$$

With $I_{c,2}$ we can compute the equivalent source of third order

$$\begin{aligned} V_3 = 2r_2 \left[ I_{c,1}(s_1) I_{c,2}(s_2,s_3)\right] _{\text {sym}} + r_3 I_{c,1}(s_1) I_{c,1}(s_2) I_{c,1}(s_3)\,. \end{aligned}$$

The third order transfer function is the response to $V_3$ which is calculated in a similar way as $H_2$

$$\begin{aligned} H_3(s_1,s_2,s_3) = -[1 + (s_1 + s_2 + s_3) C_{sg} R_s] H_1(s_1 + s_2 + s_3) V_3(s_1,s_2,s_3)\,. \end{aligned}$$

6.2 Cascode

We now specialise to the case of a Cascode. Since the transfer functions are found by analysing a sequence of linear networks, we can use the Thévenin-Norton theorem [32] to transform the source into the parallel connection of an ideal current source and the internal resistor $R_s$ as shown in Fig. 11.34. The resistor $R_s$ corresponds now to the reciprocal of the output conductance $g_o$ of the driving common-source stage. The latter is usually much larger than $r_1$, so it has little effect on the operation of the circuit. For this reason and to obtain easier to interpret expressions we calculate the transfer functions in the limit as $R_s$ tends to infinity. Under this assumption and using the results of the previous section, the first, second and third order transfer functions from the ideal source $I_s$ to the output current $I_c$ are

$$\begin{aligned} H_{c1}(s) :=\lim _{R_s\rightarrow \infty } H_1(s) R_s = \frac{1}{1 + \frac{s}{\omega _0}}\,, \end{aligned}$$

(11.39)

$$\begin{aligned} \begin{aligned} H_{c2}(s_1, s_2) & :=\lim _{R_s\rightarrow \infty } H_2(s_1, s_2) R_s^2 \\ & = -r_2 (s_1 + s_2)C_{sg} H_{c1}(s_1) H_{c1}(s_2) H_{c1}(s_1 + s_2) \end{aligned} \end{aligned}$$

(11.40)

and

$$\begin{aligned} \begin{aligned} H_{c3}(s_1, s_2, s_3) & :=\lim _{R_s\rightarrow \infty } H_3(s_1, s_2, s_3) R_s^3 \\ & = (s_1 + s_2 + s_3)C_{sg} \Bigl \{ 2r_2^2 \left[ H_{c1}(s_1 + s_2) (s_1 + s_2)C_{sg}\right] _{\text {sym}} - r_3 \Bigr \} \\ & \quad \cdot H_{c1}(s_1) H_{c1}(s_2) H_{c1}(s_3) H_{c1}(s_1 + s_2 + s_3) \end{aligned} \end{aligned}$$

(11.41)

Fig. 11.34

Equivalent circuit for the calculation of the nonlinear transfer functions of the Cascode circuit

respectively, where now $\omega _0 = 1/(C_{sg}r_1)$. Note that the symmetrization in $H_{c3}$ is intended over all three Laplace variables $s_1, s_2$ and $s_3$

$$\begin{aligned} \begin{aligned} \left[ H_{c1}(s_1 + s_2) (s_1 + s_2)C_{sg}\right] _{\text {sym}} & = \frac{1}{3}\Bigl \{ H_{c1}(s_1 + s_2) (s_1 + s_2)C_{sg} \\ &\quad + H_{c1}(s_1 + s_3) (s_1 + s_3)C_{sg} \\ &\quad + H_{c1}(s_3 + s_2) (s_3 + s_2)C_{sg} \Bigr \}\,. \end{aligned} \end{aligned}$$

Consider now the classic two-tones third order intermodulation test with one tone at $\omega _1$ and the second one at $\omega _2 = \omega _1 + \Delta \omega$. In particular consider the IM3 tone characterised by $m=(1, 0, 2, 0)$. Assuming $\left|\Delta \omega \right| \ll \left|\omega _1 \right|$ the above symmetrised expression can be approximated by

$$\begin{aligned} \left[ H_{c1}(s_1 + s_2) (s_1 + s_2)C_{sg}\right] _{\text {sym}} \approx \frac{2}{3} \frac{\jmath \omega _1 C_{sg}}{1 + \jmath \frac{2\omega _1}{\omega _0}} \end{aligned}$$

and, with it, the third order transfer function by

$$\begin{aligned} H_{c3}&(\jmath \omega _1, \jmath \omega _1, -\jmath \omega _2) \\ & \approx \jmath \omega _1C_{sg} \Bigl \{ \frac{4}{3}r_2^2 \jmath \omega _1 C_{sg} H_{c1}(2\jmath \omega _1) - r_3 \Bigr \} H_{c1}(\jmath \omega _1) H_{c1}(\jmath \omega _1) H_{c1}(-\jmath \omega _1) H_{c1}(\jmath \omega _1)\,. \end{aligned}$$

If $\omega _1 \le \omega _0/5$ the value of $\left|H_{c1}(\jmath \omega _1) \right|$ can be approximated by 1 with an error of less than 2% and the magnitude of $\left|H_{c3} \right|$ becomes very nearly

$$\begin{aligned} \omega _1C_{sg} \Bigl |\frac{4}{3}r_2^2 \jmath \omega _1 C_{sg} H_{c1}(2\jmath \omega _1) - r_3 \Bigr |\,. \end{aligned}$$

Using (11.38) for the coefficients of the nonlinear characteristic of the transistor we thus obtain

$$\begin{aligned} \left|H_{c3}(\jmath \omega _1, \jmath \omega _1, -\jmath \omega _2) \right| \approx \frac{1}{8I_D^2}\frac{\omega _1C_{sg}}{g_m} \Bigl |\frac{2}{3}\frac{\jmath \omega _1 C_{sg}}{g_m} H_{c1}(2\jmath \omega _1) - 1 \Bigr |\,. \end{aligned}$$

(11.42)

The magnitude of the IM3 tone normalised to the DC current $I_D$ is therefore

$$\begin{aligned} \Bigl |\frac{I_{c3,m}}{I_D}\Bigr |\approx \frac{3}{32}\frac{\omega _1C_{sg}}{g_m} \Bigl |\frac{2}{3}\frac{\jmath \omega _1 C_{sg}}{g_m} H_{c1}(2\jmath \omega _1) - 1 \Bigr |\Bigl |\frac{I_s}{I_D}\Bigr |^3\,. \end{aligned}$$

From this expression we can read several interesting aspects. First, both the second and the third order nonlinearities of the transistor characteristic contribute to the IM3 tone. This is visible from the appearance of $r_3$ as well as $r_2$ in the expression for $H_{c3}$. The contribution to an intermodulation product of third order by second-order nonlinearities is due to the presence of (local) feedback. This can be appreciated graphically by looking at Fig. 11.34. The second order source $V_2$ creates a current that circulates again through the input of the circuit. Therefore, the generated second order tones pass again through the second order distortion where they can mix with the input tones to produce frequency mixes of third order.

The contribution to the IM3 tone from second-order distortion is approximately orthogonal to the one from third order distortion. Therefore, it’s not possible to size the transistor in such a way as to make the two cancel each other, not even at a specific frequency.

The IM3 is largely dominated by $r_3$ up to very high frequencies and for $\omega _1$ up to ca. $\omega _0/10$ it is proportional to $\omega _1$. The quantity $g_m/C_{sg}$ corresponds (neglecting $C_{gd}$) to the angular frequency at which a common-source stage has unity current gain. It is called transit frequency and denoted by

$$\begin{aligned} \omega _T = \frac{g_m}{C_{sg}}\,. \end{aligned}$$

(11.43)

It is one of the key parameters used to characterise the high-frequency capabilities of transistors. With it the magnitude of the IM3 up to ca. $\omega _1 \le \omega _T/10$ can be approximated by

$$\begin{aligned} \Bigl |\frac{I_{c3,m}}{I_D}\Bigr |\approx \frac{3}{32}\frac{\omega _1}{\omega _T} \Bigl |\frac{I_s}{I_D}\Bigr |^3\,. \end{aligned}$$

This shows that for low distortion one needs fast transistors. Looking again at Fig. 11.34 we can appreciate that in the limit as $\omega _1/\omega _T$ tends to zero (which means that $C_{sg}$ tends to zero) the nonlinear sources become floating and can’t generate any frequency mix current (remember that we also assume $R_s\rightarrow \infty$).

In general, distortion introduced by the input (common-source) stage of the Cascode configuration generates frequency mixes of second-order. These can mix with the fundamental tones in the second-order distortion of the output (common-gate) stage to produce other IM3 components. However, since $\left|H_{c2}(\jmath 2\omega _1, -\jmath \omega _2) \right|$ is also proportional to $\omega _1/\omega _T$ this does not substantially change the situation.

For simplicity in our discussion we assumed $R_s\rightarrow \infty$. From the gained insight we can appreciate that at low frequencies it is a finite value of $R_s$ which will limit IM3 and, the lower $R_s$, the higher the IM3. In general however, the common-gate stage of a Cascode is not the stage limiting low frequency linearity.

7 Degenerated Common-Source Stage

In this section we investigate the effect of local feedback on distortion and show that introduction of feedback may lead to degraded linearity. As a concrete example we analyse the degenerated common-source amplifier depicted in Fig. 11.35a. The impedance $Z_e$ is called the degeneration impedance. Its presence reduces the gate-source voltage across the transistor by an amount proportional to the output current. In other words it introduces feedback around the transistor. The impedance $Z_s$ represents a generic driving impedance.

Fig. 11.35

a Degenerated common-source stage AC schematic b Degenerated common-source stage small-signal model

One way to analyse the circuit is to model the transistor as a nonlinear voltage-controlled current-source characterised by a third (or higher) order polynomial

$$\begin{aligned} i_d = g_m v_{gs} + g_2 v_{gs}^2 + g_3 v_{gs}^3 \end{aligned}$$

and solve Kirchhoff’s equations for $v_{gs}$. Having found the voltage components $v_{gs,1}, \dotsc , v_{gs,k}$ up to some order of interest k, one then finds the output current components $i_{o,1}, \dotsc , i_{o,k}$ by use of the polynomial approximating the transistor characteristic. Instead of using this method we show how the use of a nullor allows the problem to be solved in a more direct way, by permitting to directly obtain an equation for the output current $i_o$.

Nullators and Norators are pathological network elements. A Nullatoris a two terminal element represented by the symbol shown in Fig. 11.36a and characterised by the two equations

$$\begin{aligned} V = 0 \qquad I=0\,. \end{aligned}$$

A Noratoris a two terminal element represented by the symbol shown in Fig. 11.36b whose current and voltage are arbitrary and completely determined by the surrounding network. In other words it is characterised by zero equations. For a linear network to have a well-defined solution a Nullator must therefore always appear alongside a Norator. Such a pair is called a Nullorand can be used to model several elements such as controlled sources, OpAmps and transistors. In particular, we can use it to represent the inverted series (see Sect. 11.6.2)

$$\begin{aligned} v_{gs} = r_1 i_d + r_2 i_d^2 + r_3 i_d^3 \end{aligned}$$

of the transistor characteristic. A nullor based small-signal model of the degenerated common-source stage using this transistor characteristic representation is shown in Fig. 11.37. Note that the transistor characteristic is represented by a nonlinear resistor.

Fig. 11.36

a Symbol of the Nullator reminding the shape of the zero digit 0 b Symbol of a Norator reminding the infinity symbol $\infty$

Fig. 11.37

Nullor based small-signal model of a degenerated common-source stage

7.1 Nonlinear Transfer Functions

From the model in Fig. 11.37 and Kirchhoff’s laws we obtain the following system of equations relating the output current $I_o$ to the input signal $V_s$

$$\begin{aligned} V_s & = (Z_s + \frac{1}{s C_{gs}}) I_s + Z_e (I_s + I_o)\\ V_{gs} &= \frac{1}{s C_{cs}} I_s \\ V_{gs} &= r_1 I_o + r_2 I_o^2 + r_3 I_o^3\,. \end{aligned}$$

After eliminating $V_{gs}$ and $I_s$ we obtain the single equation

$$\begin{aligned} \begin{aligned} V_s & = \big [ r_1 + Z_e + (Z_s + Z_e)r_1 C_{gs} s \big ] I_o\\ & \quad + \big [ 1 + (Z_s + Z_e) C_{gs} s \big ](r_2 I_o^2 + r_3 I_o^3)\,. \end{aligned} \end{aligned}$$

(11.44)

The first order transfer function is obtained by applying a Dirac impulse as input and discarding all terms of order higher than one in the equation. This is equivalent to removing the nonlinear sources from the equivalent circuit. Using the relation $r_1 = 1/g_m$ we obtain

$$\begin{aligned} H_1(s) = \frac{g_m}{L(s)} = \frac{g_m}{1 + g_mZ_e + s C_{gs}(Z_e + Z_s)}\,. \end{aligned}$$

(11.45)

To compute the second order nonlinear transfer function we first insert the first order solution into the nonlinear terms and retain only second order ones. Alternatively we can use Fig. 11.11 to read the value of the second order nonlinear source for a nonlinear resistor. In both cases, after adjusting the representation of the differential operator by replacing the variable s by $s_1+s_2$, we obtain

$$\begin{aligned} \begin{aligned} 0 & = \big [ r_1 + Z_e + (Z_s + Z_e)r_1 C_{gs} (s_1 + s_2) \big ] H_2(s_1,s_2)\\ & \quad + \big [ 1 + (Z_s + Z_e) C_{gs} (s_1+s_2)\big ] r_2 H_1(s_1) H_1(s_2)\,. \end{aligned} \end{aligned}$$

Note that for brevity we didn’t explicitly write the argument of impedances. Their value has of course to be evaluated at $s_1+s_2$. The second order nonlinear transfer function is thus

$$\begin{aligned} H_2(s_1,s_2) = -r_2 H_1(s_1) H_1(s_2) H_1(s_1+s_2) \big [ 1 + (Z_s + Z_e) C_{gs} (s_1+s_2)\big ]\,. \end{aligned}$$

(11.46)

To find the third order nonlinear transfer function we proceed in a similar way and obtain

$$\begin{aligned} H_3(s_1, s_2, s_3) = & -\big \{ 2 r_2 \left[ H_1(s_1) H_2(s_2, s_3)\right] _{\text {sym}} + r_3 H_1(s_1) H_1(s_2) H_1(s_3) \big \} \nonumber \\ & \qquad H_1(s_1 + s_2 + s_3) \big [ 1 + (Z_s + Z_e) C_{gs} (s_1 + s_2 + s_3)\big ]\,. \end{aligned}$$

(11.47)

7.2 Resistive Degeneration

We now specialise to the case of resistive degeneration $Z_e=R_e$ and a resistive driving impedance $Z_s=R_s$ and calculate the intermodulation products of third order when driven by two tones of equal amplitudes at frequency $\omega _1$ and $\omega _1 + \Delta \omega$ respectively. As usual we assume $\Delta \omega \ll \omega _1$.

As a first step, to calculate $H_3$ for the mix (1, 0, 2, 0) we evaluate

$$\begin{aligned} &\left[ H_1(\jmath \omega _1) H_2(\jmath \omega _1, -\jmath (\omega _1+\Delta \omega ))\right] _{\text {sym}} \\ & \qquad \qquad \qquad \approx -r_2 \frac{g_m^4}{3\,L(\jmath \omega _1)^2L(-\jmath \omega _1)} \bigg [ 2\frac{N(-\jmath \Delta \omega )}{L(-\jmath \Delta \omega )} + \frac{N(2\omega _1)}{L(2\omega _1)} \bigg ] \end{aligned}$$

with

$$\begin{aligned} N(s) :=1 + (Z_s + Z_e) s C_{gs}\,. \end{aligned}$$

Inserting this expression into $H_3$ we obtain

$$\begin{aligned} & H_3(\jmath \omega _1, \jmath \omega _1, -\jmath (\omega _1+\Delta \omega )) \\ & \qquad \qquad \qquad \approx \frac{g_m^4\,N(\jmath \omega _1)}{L(\jmath \omega _1)^3L(-\jmath \omega _1)} \bigg \{ \frac{2}{3} r_2^2 g_m \bigg [ 2\frac{N(-\jmath \Delta \omega )}{L(-\jmath \Delta \omega )} + \frac{N(2\omega _1)}{L(2\omega _1)} \bigg ] - r_3 \bigg \}\,. \end{aligned}$$

Since in Sect. 11.5 we characterised the transistor in terms of $g_m, g_2$ and $g_3$, we express the coefficients $r_2$ and $r_3$ in terms of them using the results of Sect. 11.2.1

$$\begin{aligned} r_2 &= \frac{-g_2}{g_m^3} & r_3 &= \frac{2g_2^2 - g_mg_3}{g_m^5}\,. \end{aligned}$$

Substituting these expressions leads finally to

$$\begin{aligned} & H_3(\jmath \omega _1, \jmath \omega _1, -\jmath (\omega _1+\Delta \omega )) \nonumber \\ & \qquad \quad \approx \frac{N(\jmath \omega _1)}{L(\jmath \omega _1)^3L(-\jmath \omega _1)} \bigg \{ 2 \frac{g_2^2}{g_m} \bigg [ \frac{2}{3} \frac{N(-\jmath \Delta \omega )}{L(-\jmath \Delta \omega )} + \frac{1}{3} \frac{N(2\omega _1)}{L(2\omega _1)} - 1 \bigg ] + g_3 \bigg \}\,. \end{aligned}$$

(11.48)

We can now discuss the effect of a small amount of feedback introduced by a small resistor $R_e$ on linearity. First note that, as expected, for $Z_e= 0~\Omega$ the term in square brackets vanishes making the IM3 depend only on $g_3$. As $R_e$ is increased the contribution of $g_2$ increases and at low to moderate frequencies there is some possibility of cancelling between the contribution due to $g_3$ and $g_2$. As $R_e$ increases beyond this value, the second order contribution starts to dominate. At high frequencies only imperfect cancelling is possible due to shift in phase of the $g_2$ contribution.

Fig. 11.38

a Weak feedback region of Fig. 11.38b b Moderate-frequency resistively degenerated common-source transconductance stage IIP3. The solid curves are calculated using (11.45), (11.47); the dashed line with (11.49). The transistor parameters are approximate values taken from Figs. 11.28 and 11.29: $g_m= 17$ mS, $g_2=70$ mS/V, $g_3=10$ mS/V$^2$, $R_s = 50\,\Omega$

Figure 11.38b shows the low to moderate frequency IIP3 of the Class-AC stage from Sect. 11.5. It shows that cancelling occurs for very small amounts of feedback and, as is typical for cancelling effects, the performance is very sensitive to small variations in component values. Due to the large value of $g_2$, a small to moderate amount of feedback with $g_m R_e$ in the range of 0.03–2.5 leads to an actual degradation in IIP3. Note that small values of $Z_e$ may be introduced unintentionally by parasitic effects due to the interconnections between components.

A linearity improvement can be obtained by using a large amount of feedback $g_m R_e \gg 1$. To simplify calculations let’s assume $\left|\omega _1C_{gs}(R_s + R_e) \right| \ll 1$, then

$$\begin{aligned} H_3(\jmath \omega _1, \jmath \omega _1, -\jmath (\omega _1+\Delta \omega )) \approx \frac{1}{L(\jmath \omega _1)^3L(-\jmath \omega _1)} \bigg \{ - 2 \frac{g_2^2}{g_m} + g_3 \bigg \} \end{aligned}$$

and

$$\begin{aligned} H_1(\jmath \omega _1) \approx \frac{1}{R_e}. \end{aligned}$$

Using (11.11) to calculate the IIP3 shows that under these conditions the latter does in fact increase with increasing $R_e$

$$\begin{aligned} \text {IIP3} \approx \sqrt{ \frac{4 (g_mR_e)^3}{ 3\left|\frac{g_3}{g_m} - 2\big (\frac{g_2}{g_m}\big )^2 \right|}}, \qquad g_m R_e \gg 1\,. \end{aligned}$$

(11.49)

The reason for the improvement is a substantially reduced amplitude of the voltage $V_{gs}$ controlling the nonlinear sources compared to the circuit input signal $V_s$. Linearity thus comes at the expenses of a much reduced signal transconductance which for RF circuits is often not acceptable.

7.3 Inductive Degeneration

A second type of degeneration widely used ar RF frequencies is the inductive one. This type of degeneration is often used in the input stage of low-noise RF amplifiers (LNAs), a basic small-signal model of which is shown in Fig. 11.39.

Fig. 11.39

Small-signal model of an inductively degenerated common-source stage

An important characteristic of RF amplifiers is the input impedance $Z_i$. In many situations it is required to be real and equal to the source impedance $R_s$, or some standard value. From our model a simple calculation shows that $Z_i$ is given by

$$\begin{aligned} Z_i = R_i + \jmath X_i = g_m \frac{L_e}{C_{gs}} + s (L_s + L_e) + \frac{1}{s C_{gs}}\,. \end{aligned}$$

A degeneration inductor $L_e$ thus allows a real part to be introduced to the input impedance without using resistors. Avoiding resistors at the input of LNAs is necessary to avoid limiting the achievable sensitivity. The reactive part of the impedance can be cancelled over some frequency band by resonating it, in our example using the inductor $L_s$. The input network thus consists of a series resonator tuned at the center frequency of the band of interest.

In this section we analyse the linearity characteristics of this stage and in particular its IP3. The nonlinear transfer functions are readily obtained from our previous results by setting $Z_e = s L_e$ and $Z_s = R_s + s L_s$. Doing so, the first order transfer function becomes

$$\begin{aligned} H_1(s) = \frac{g_m}{1 + s(g_mL_e + C_{gs} R_s) + s^2 C_{gs}(L_e + L_s)}\,. \end{aligned}$$

Note that $g_m L_e = C_{gs} R_i$. We can therefore write the denominator in the standard form

$$\begin{aligned} H_1(s) = \frac{g_m}{1 + \frac{s}{\omega _0}\frac{1}{q_t} + \big (\frac{s}{\omega _o}\big )^2} \end{aligned}$$

(11.50)

with

$$\begin{aligned} \omega _0^2 & = \frac{1}{C_{gs}(L_e + L_s)} & q_i &= \frac{1}{R_i \omega _0 C_{gs}}\\ \frac{1}{q_t} &= \frac{1}{q_i} + \frac{1}{q_s} & q_s &= \frac{1}{R_s \omega _0 C_{gs}}\,. \end{aligned}$$

The same parameters can also be used to put N(s) in standard form

$$\begin{aligned} N(s) = 1 + \frac{s}{\omega _0}\frac{1}{q_s} + \Big (\frac{s}{\omega _o}\Big )^2\,. \end{aligned}$$

The value of $H_3$ relevant for the two-tones IP3 test can then be obtained by substituting these expressions in (11.48). The resonance frequency of the input resonator is evidently set to the frequency of the input signal $\omega _0 = \omega _1$ so that

$$\begin{aligned} N(\jmath \omega _1) &= \frac{\jmath }{q_s}, & L(\jmath \omega _1) &= \frac{\jmath }{q_t}, & H_1(\jmath \omega _1) &= -\jmath q_t gm\,. \end{aligned}$$

and

$$\begin{aligned} H_3(\jmath \omega _1, \jmath \omega _1, -\jmath (\omega _1+\Delta \omega )) \approx \frac{-\jmath q_t^4)}{q_s} \bigg \{ 2 \frac{g_2^2}{g_m} \bigg [ \frac{2}{3} + \frac{1}{3} \frac{N(2\omega _1)}{L(2\omega _1)} - 1 \bigg ] + g_3 \bigg \}\,. \end{aligned}$$

With these results we can compute the IIP3 using Eq. (11.11) as before

$$\begin{aligned} \text {IIP3} \approx \frac{2}{q_t} \sqrt{\frac{q_s}{q_t} \frac{1}{\left| 2 \big (\frac{g_2}{g_m}\big )^2 \bigg [ \frac{2}{3} + \frac{1}{3} \frac{N(2\omega _1)}{L(2\omega _1)} - 1 \bigg ] + \frac{g_3}{g_m} \right|}}\,. \end{aligned}$$

(11.51)

In the common case in which the input resistance $R_i$ is equal to the source impedance $q_s/q_t = 2$. The IP3 of the circuit is thus approximately inversely proportional to the quality factor of the input resonance. This is due to the fact that at resonance the magnitude of the voltage across the reactive components is roughly $q_t$ times the one across the resistive part. In other words, the voltage $V_{gs}$ controlling the nonlinear sources is amplified by a factor of ca. $q_t$ compared to the input signal $V_s$. This very same characteristic is also the reason for the good noise characteristic of the circuit: the input network provides some voltage gain before the first noisy device.

The quality factor of the network also influences the relative contributions of $g_2$ and $g_3$ to distortion through the term

$$\begin{aligned} \frac{N(2\jmath \omega _1)}{L(2\jmath \omega _1)} = \frac{ -3 + \jmath \frac{2}{q_s}}{ -3 + \jmath \frac{2}{q_t}}. \end{aligned}$$

For large quality factors $q_t, q_s \gg 1$ the ratio approaches 1 which makes the IP3 essentially independent of $g_2$. For small quality factor values the contribution due to $g_2$ is not negligible, especially if $g_3/g_m$ is small compared to $(g_2/g_m)^2$ as is the case with Class-AC stages.

In practical implementations the component values are affected by manufacturing variations. For this reason and to avoid the need for tuning, the quality factor $q_t$ is most often chosen to have a value smaller than 5.

8 Pseudo-Differential Circuits

The analog signal path of many RF and mixes-signal integrated circuits is differential. This means that the signal of interest is transmitted on two equal lines carrying the same signal, but with opposite polarities. The main objective is to make the system insensitive to noise affecting both lines equally. This can be, for example, noise due to the activity of digital circuits propagating through the common substrate of the IC. A differential circuit is one that is designed to process the difference between the two input terminals sensing the two lines carrying the signal and rejecting the common component. Formally, if $v_i^+$ and $v_i^-$ are the two input voltages (relative to ground), the differential-modevoltage is defined as

$$\begin{aligned} v_{d} :=v_i^+ - v_i^- \end{aligned}$$

and the common-modevoltage as

$$\begin{aligned} v_{c} :=\frac{v_i^+ + v_i^-}{2}\,. \end{aligned}$$

Using this representation the two input voltages can be written as

$$\begin{aligned} v_i^+ & = v_c + \frac{v_d}{2}\,, & v_i^- & = v_c - \frac{v_d}{2}\,. \end{aligned}$$

The prototypical differential circuit is the differential-pairshown in Fig. 11.40. In the ideal drawn form the output currents are always $i_o^+ = i_o^- = I_0/2$ as long as $v_i^1 = v_i^-$. Any common-mode signal component is thus fully rejected.

Fig. 11.40

Differential pair

Differential circuits do also have disadvantages. A real current source is implemented with transistors and requires a certain voltage across its terminals to work properly. This reduces the headroom left for signal processing and in modern processes operating at supplies voltages below 1.0 V poses severe challenges. In addition, a current source does not only generate a DC current, but it also generates noise, reducing the sensitivity of the circuit to small signals.

Pseudo-differentialcircuits are a class of circuits that alleviate some of these problems while retaining some of the benefits of differential circuits. They are circuits composed by two equal single-ended sub-circuits each connected to one of the two lines carrying the differential signal. An example pseudo-differential transconductance is shown in Fig. 11.41.

In pseudo-differential circuits the input common-mode signal component is not rejected, but, if the circuit is sufficiently linear, the common-mode input appears as a common-mode signal at the output and remains separable from the wanted differential signal which appears at the output in differential form. The objective of this section is to quantify the conversion between common-mode and differential-mode in weakly nonlinear circuits.

Fig. 11.41

Pseudo-differential transconductance

We first show that weakly nonlinear circuits driven by a purely differential input signal produce a mixture of differential- and common-mode output signals. Let’s denote the input signals by $x^+, x^-$. the output signals by $y^+, y^-$, the relative common- and differential-mode components by the same letter with index c and d respectively; and the nonlinear transfer function of order k of the single-ended subsystems by $h_k$. By assumption the input signal is purely differential

$$\begin{aligned} x^+ & = \frac{x_d}{2} & x^- &= - \frac{x_d}{2}\,. \end{aligned}$$

The outputs of order k are therefore

$$\begin{aligned} y_k^+ & = \frac{1}{2^k} h_k *x_d^{\otimes k}\,, & y_k^- & = \frac{(-1)^k}{2^k} h_k *x_d^{\otimes k} \end{aligned}$$

from which we conclude that for k even the output is a common-mode signal, while for k odd it is differential.

Let’s now consider the response of a weakly nonlinear circuit to a mixture of differential- and common-mode signals

$$\begin{aligned} x^+ & = x_c + \frac{x_d}{2} & x^- &= x_c - \frac{x_d}{2}\,. \end{aligned}$$

Let’s first consider the second-order response of the two circuit halves. The positive and negative outputs are

$$\begin{aligned} \begin{aligned} y_2^+ & = h_2 *\bigg (x_c + \frac{x_d}{2}\bigg )^{\otimes 2}\\ &= h_2 *x_c^{\otimes 2} + \frac{1}{4} h_2 *x_d^{\otimes 2} + h_2 *\left[ x_d \otimes x_c\right] _{\text {sym}} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} y_2^- & = h_2 *\bigg (x_c - \frac{x_d}{2}\bigg )^{\otimes 2}\\ &= h_2 *x_c^{\otimes 2} + \frac{1}{4} h_2 *x_d^{\otimes 2} - h_2 *\left[ x_d \otimes x_c\right] _{\text {sym}} \end{aligned} \end{aligned}$$

respectively. The second-order differential output signal component is therefore

$$\begin{aligned} y_{d,2} = 2 h_2 *\left[ x_d \otimes x_c\right] _{\text {sym}} \end{aligned}$$

which includes the common-mode input signal. A similar calculation for the third order component gives

$$\begin{aligned} y_{d,3} = h_3 *\bigg ( \frac{x_d^{\otimes 3}}{4} + 3 \left[ x_d \otimes x_c^{\otimes 2}\right] _{\text {sym}} \bigg ) \end{aligned}$$

which also includes a term depending on the input common-mode. One can generalise the calculations and show that the differential- and common-mode input components are mixed by nonlinearities of all orders.

Consider now the cascade of two pseudo-differential weakly nonlinear circuits driven by a purely differential signal. If the two subsystems are optimised independently to maximise IP3 without paying attention to even order distortion components, then, when the two subsystems are put together, one may obtain a lower than expected total IP3. This is because the first stage produces second (and higher even) order mixes as common-mode signals which are also fed as input to the second subsystem. The second (and higher order) distortion components of the latter will then mix differential- and common-mode to produce differential output signal components at the IM3 frequencies.