Nonlinearity

Most of the time, we want our amplifiers to be linear. However, since the transistors that we use are inherently nonlinear, our circuits are also inherently nonlinear. One way to analyze nonlinear circuits is to use small-signal analysis, where we assume the circuit is linear for very small signals. In this module, we will look at the effects of nonlinearity that are not seen when performing small-signal analysis.

In this module, we will assume that we have memoryless systems, i.e. systems whose output is an instantaneous function of the input. Thus, we neglect the contribution of energy-storage elements like capacitors and inductors.

Harmonic Distortion (HD)

Consider a memoryless system with input $x\left(t\right)$ and output $y\left(t\right)$. In general, we can express the output as a power series:

$y\left(t\right)\approx \alpha _{1}x\left(t\right)+\alpha _{2}x^{2}\left(t\right)+\alpha _{3}x^{3}\left(t\right)+\ldots$

(1)

Note that for a linear system, $\alpha _{1}$ is the gain, and $\alpha _{i}=0$ for $i\neq 1$. Let us consider the case when the input is a sinusoid, $x\left(t\right)=A\cos \omega t$. Let us further assume that the higher-order terms ($\alpha _{i}$ for $i>3$) are negligible. We can then express the output as:

${\begin{aligned}y\left(t\right)&=\alpha _{1}A\cos \omega t+\alpha _{2}A^{2}\cos ^{2}\omega t+\alpha _{3}A^{3}\cos ^{3}\omega t\\&=\alpha _{1}A\cos \omega t+{\frac {\alpha _{2}A^{2}}{2}}\left(1+\cos 2\omega t\right)+{\frac {\alpha _{3}A^{3}}{4}}\left(3\cos \omega t+\cos 3\omega t\right)\\&={\frac {\alpha _{2}A^{2}}{2}}+\left(\alpha _{1}A+{\frac {3\alpha _{3}A^{3}}{4}}\right)\cos \omega t+{\frac {\alpha _{2}A^{2}}{2}}\cos 2\omega t+{\frac {\alpha _{3}A^{3}}{4}}\cos 3\omega t\\\end{aligned}}$

(2)

Note that the output contains:

• A DC term
• A component with frequency equal to the input frequency, or the fundamental frequency.
• A component at twice the input frequency, or the second harmonic frequency.
• A component at thrice the input frequency, or the third harmonic frequency.

General Distortion Terms

We can use the Binomial Theorem to get $\cos ^{n}\theta$ for any $n$:

${\begin{aligned}\cos ^{n}\theta &={\frac {1}{2^{n}}}\left(e^{j\theta }+e^{-j\theta }\right)^{n}\\&={\frac {1}{2^{n}}}\sum _{k=0}^{n}{n \choose k}e^{jk\theta }\cdot e^{-j\left(n-k\right)\theta }\\\end{aligned}}$

(3)

Recall that we can express any term in the binomial expansion of $\left(x+x^{-1}\right)^{n}$ as:

${n \choose k}x^{n-k}x^{-k}={n \choose k}x^{n-2k}$

(4)

Thus, for odd $n$, we get odd powers on of $x$, and they appear as pairs of positive and negative exponentials. This means that we only generate the odd harmonic frequencies. On the other hand, for even $n$, we get the even powers of $x$, generating only the even harmonics. Also, we get an unpaired middle term that corresponds to the DC component:

${2k \choose k}e^{jk\theta }\cdot e^{-jk\theta }={2k \choose k}$

(5)

Note that a function is odd if $f\left(-x\right)=-f\left(x\right)$, and it will have an odd power series expansion:

$f\left(x\right)=\alpha _{1}x+\alpha _{3}x^{3}+\alpha _{5}x^{5}+\ldots$

(6)

In contrast, a function is even if $g\left(-x\right)=g\left(x\right)$, and it will have an even power series expansion:

$g\left(x\right)=\alpha _{0}+\alpha _{2}x^{2}+\alpha _{4}x^{4}+\alpha _{6}x^{6}+\ldots$

(7)

Gain Compression

Aside from producing terms at multiples of the input or fundamental frequency, we can see that the gain of the fundamental component is also affected by the third-order coefficient, $\alpha _{3}$, giving the fundamental component:

$\left(\alpha _{1}+{\frac {3\alpha _{3}A^{2}}{4}}\right)\cdot A\cos \omega t$

(8)

Note that the gain now becomes a function of the input amplitude $A$. Thus, if $\alpha _{1}$ and $\alpha _{3}$ have the same sign, i.e. both positive or both negative, then the gain increases, and we have gain expansion. On the other hand, if $\alpha _{1}$ and $\alpha _{3}$ have different signs, the gain decreases, and we have gain compression. In most amplifiers, we normally get gain compression since the output swing is limited by the supply voltage, and thus saturates. This indicates that $\alpha _{1}$ and $\alpha _{3}$ have opposite signs.

The 1-dB Compression Point

We can quantify gain compression using the 1-dB Compression Point, or the the value of the input amplitude that causes the gain to drop by 1 dB, or approximately 10%, relative to the gain when $A\rightarrow 0$. Thus,

$20\log \left|\alpha _{1}+{\frac {3\alpha _{3}A_{\mathrm {in,1dB} }^{2}}{4}}\right|=20\log \left|\alpha _{1}\right|-1\mathrm {dB}$

(9)

Since $\alpha _{1}$ and $\alpha _{3}$ have opposite signs, this is equivalent to:

${\frac {\left|\alpha _{1}\right|-{\frac {3\left|\alpha _{3}\right|A_{\mathrm {in,1dB} }^{2}}{4}}}{\left|\alpha 1\right|}}=1-{\frac {3}{4}}\left|{\frac {\alpha _{3}}{\alpha _{1}}}\right|A_{\mathrm {in,1dB} }^{2}=10^{\frac {\left(-1\right)}{20}}=0.891$

(10)

Thus, we can get $A_{\mathrm {in,1dB} }$:

$A_{\mathrm {in,1dB} }={\sqrt {0.145\left|{\frac {\alpha _{1}}{\alpha _{3}}}\right|}}$

(11)

Note that the higher $A_{\mathrm {in,1dB} }$, the more linear the circuit.

Desensitization

Let the input $x\left(t\right)=A_{1}\cos \omega _{1}t+A_{2}\cos \omega _{2}t$, where the first term could be our signal of interest, and the second term could be a large interfering signal. If $\alpha _{1}$ and $\alpha _{3}$ have opposite signs, we can write the fundamental component as:

$\left(\alpha _{1}+{\frac {3\alpha _{3}A_{1}^{2}}{4}}+{\frac {3\alpha _{3}A_{2}^{2}}{2}}\right)\cdot A_{1}\cos \omega _{1}t$

(12)

Thus, for a sufficiently large $A_{2}$, the gain can drop to zero, and we say that our signal is blocked by the interfering signal or the blocker.

Cross Modulation

If an interfering signal is amplitude modulated with a modulation index $m$ and a modulation frequency $\omega _{m}$:

$x\left(t\right)=A_{1}\cos \omega _{1}t+A_{2}\left(1+m\cos \omega _{m}t\right)\cos \omega _{2}t$

(13)

We can write the fundamental component, when $A_{1}\ll A_{2}$, as:

$\left[\alpha _{1}+{\frac {3\alpha _{3}A_{2}^{2}}{2}}\left(1+{\frac {m^{2}}{2}}+{\frac {m^{2}}{2}}\cos 2\omega _{m}t+2m\cos \omega _{m}t\right)\right]\cdot A_{1}\cos \omega _{1}t$

(14)

Thus, the AM signal modulating the interfering signal also modulates, or cross modulates, our desired signal.

Intermodulation (IM)

Consider the case when two signals are applied to a nonlinear system, $x\left(t\right)=A_{1}\cos \omega _{1}t+A_{2}\cos \omega _{2}t$. Again if we assume that the higher-order terms ($\alpha _{i}$ for $i>3$) are negligible, the output can be written as:

$y\left(t\right)=\alpha _{1}\left(A_{1}\cos \omega _{1}t+A_{2}\cos \omega _{2}t\right)+\alpha _{2}\left(A_{1}\cos \omega _{1}t+A_{2}\cos \omega _{2}t\right)^{2}+\alpha _{3}\left(A_{1}\cos \omega _{1}t+A_{2}\cos \omega _{2}t\right)^{3}$

(15)

The Second IM Terms

We can write the second power term as:

${\begin{aligned}\alpha _{2}x^{2}\left(t\right)&=\alpha _{2}\left(A_{1}\cos \omega _{1}t+A_{2}\cos \omega _{2}t\right)^{2}\\&=\alpha _{2}A_{1}^{2}\cos ^{2}\omega _{1}t+\alpha _{2}A_{2}^{2}\cos \omega _{2}t+2\alpha _{2}A_{1}A_{2}\cos \omega _{1}t\cdot \cos \omega _{2}t\\&=\alpha _{2}{\frac {A_{1}^{2}}{2}}\left(1+\cos 2\omega _{1}t\right)+\alpha _{2}{\frac {A_{2}^{2}}{2}}\left(1+\cos 2\omega _{2}t\right)+\alpha _{2}A_{1}A_{2}\left[\cos \left(\omega _{1}+\omega _{2}\right)t+\cos \left(\omega _{1}-\omega _{2}\right)t\right]\\\end{aligned}}$

(16)

Aside from generating our harmonic distortion terms at $2\omega _{1}$ and $2\omega _{2}$, we get our second order intermodulation terms at $\omega _{1}\pm \omega _{2}$.

The Third IM Terms

Similarly, for the cubic power terms, $\alpha _{3}x^{3}\left(t\right)=\alpha _{3}\left(A_{1}\cos \omega _{1}t+A_{2}\cos \omega _{2}t\right)^{3}$, we get the first and last terms:

${\frac {1}{4}}\alpha _{3}A_{1}^{3}\left(\cos 3\omega _{1}t+3\cos \omega _{1}t\right)+{\frac {1}{4}}\alpha _{3}A_{2}^{3}\left(\cos 3\omega _{2}t+3\cos \omega _{2}t\right)$

(17)

As expected, we get the gain compression/expansion terms, as well as the third harmonic terms.

The first cross term can be written as:

${\begin{aligned}{3 \choose 2}\alpha _{3}A_{1}A_{2}^{2}\cos \omega _{1}t\cdot \cos ^{2}\omega _{2}t&=\alpha _{3}A_{1}A_{2}^{2}\left(3\cos \omega _{1}t\cdot \cos ^{2}\omega _{2}t\right)\\&=\alpha _{3}A_{1}A_{2}^{2}{\frac {3}{2}}\cos \omega _{1}t\left(1+\cos 2\omega _{2}t\right)\\&=\alpha _{3}A_{1}A_{2}^{2}\left[{\frac {3}{2}}\cos \omega _{1}t+{\frac {3}{4}}\cos \left(2\omega _{2}\pm \omega _{1}\right)t\right]\\\end{aligned}}$

(18)

By symmetry, the other cross term is:

$\alpha _{3}A_{1}^{2}A_{2}\left[{\frac {3}{2}}\cos \omega _{2}t+{\frac {3}{4}}\cos \left(2\omega _{1}\pm \omega _{2}\right)t\right]$

(19)

Aside from the interference or blocker terms, we get the third order intermodulation terms at $2\omega _{2}\pm \omega _{1}$ and $2\omega _{1}\pm \omega _{2}$.

Fundamental Components

Collecting all the terms at $\omega _{1}$ and $\omega _{2}$, we get:

$\left(\alpha _{1}+{\frac {3}{4}}\alpha _{3}A_{1}^{2}+{\frac {3}{2}}\alpha _{3}A_{2}^{2}\right)\cdot A_{1}\cos \omega _{1}t+\left(\alpha _{1}+{\frac {3}{4}}\alpha _{3}A_{2}^{2}+{\frac {3}{2}}\alpha _{3}A_{1}^{2}\right)\cdot A_{2}\cos \omega _{2}t$

(20)

Third Intercept Point

If we set the $A_{1}=A_{2}=A$, then the ratio of the third intermodulation terms to the ideal output, or the relative third intermodulation, can be written as:

$\mathrm {IM_{3}} =20\log \left({\frac {{\frac {3}{4}}\alpha _{3}A^{3}}{\alpha _{1}A}}\right)=20\log \left({\frac {3}{4}}{\frac {\alpha _{3}}{\alpha _{1}}}A^{2}\right)\,\mathrm {dBc}$

(21)

Where the units dBc denote decibels with respect to the carrier to emphasize that this is a relative metric. We can then define the input third intercept point, $\mathrm {IIP_{3}}$ as the input amplitude where $\mathrm {IM_{3}} =0\,\mathrm {dBc}$, or equivalently, when the third order intermodulation is equal to the ideal fundamental output:

$\left|\alpha _{1}A_{\mathrm {IIP_{3}} }\right|=\left|{\frac {3}{4}}\alpha _{3}A_{\mathrm {IIP_{3}} }^{3}\right|$

(22)

Giving us:

$A_{\mathrm {IIP_{3}} }={\sqrt {{\frac {4}{3}}\left|{\frac {\alpha _{1}}{\alpha _{3}}}\right|}}$

(23)

Taking the ratio of $A_{\mathrm {IIP_{3}} }$ and $A_{\mathrm {in,1dB} }$, we get:

${\frac {A_{\mathrm {IIP_{3}} }}{A_{\mathrm {in,1dB} }}}={\sqrt {{\frac {4}{3}}{\frac {1}{0.145}}}}=3.03=9.64\,\mathrm {dB}$

(24)

Cascaded Nonlinear Stages

Consider two nonlinear cascaded stages, with the following transfer characteristics, and again assuming that the higher-order terms are negligible:

$y_{1}\left(t\right)=\alpha _{1}x\left(t\right)+\alpha _{2}x^{2}\left(t\right)+\alpha _{3}x^{3}\left(t\right)$

(25)

$y_{2}\left(t\right)=\beta _{1}y_{1}\left(t\right)+\beta _{2}y_{1}^{2}\left(t\right)+\beta _{3}y_{1}^{3}\left(t\right)$

(26)

Combining these two equations, we get:

$y_{2}\left(t\right)=\beta _{1}\left[\alpha _{1}x\left(t\right)+\alpha _{2}x^{2}\left(t\right)+\alpha _{3}x^{3}\left(t\right)\right]+\beta _{2}\left[\alpha _{1}x\left(t\right)+\alpha _{2}x^{2}\left(t\right)+\alpha _{3}x^{3}\left(t\right)\right]^{2}+\beta _{3}\left[\alpha _{1}x\left(t\right)+\alpha _{2}x^{2}\left(t\right)+\alpha _{3}x^{3}\left(t\right)\right]^{3}$

(27)

Looking at the first and third order terms:

$y_{2}\left(t\right)=\alpha _{1}\beta _{1}x\left(t\right)+\left(\alpha _{3}\beta _{1}+2\alpha _{1}\alpha _{2}\beta _{2}+\alpha _{1}^{3}\beta _{3}\right)x^{3}\left(t\right)+\ldots$

(28)

Let us write the overall transfer function as:

$y_{2}\left(t\right)=\gamma _{1}x\left(t\right)+\gamma _{2}x^{2}\left(t\right)+\gamma _{3}x^{3}\left(t\right)$

(29)

We can then see that $\gamma _{1}=\alpha _{1}\beta _{1}$, and $\gamma _{3}=\alpha _{3}\beta _{1}+2\alpha _{1}\alpha _{2}\beta _{2}+\alpha _{1}^{3}\beta _{3}$. Thus, the overall $\mathrm {IIP_{3}}$ is:

$A_{\mathrm {IIP_{3}} }={\sqrt {{\frac {4}{3}}\left|{\frac {\gamma _{1}}{\gamma _{3}}}\right|}}={\sqrt {{\frac {4}{3}}\left|{\frac {\alpha _{1}\beta _{1}}{\alpha _{3}\beta _{1}+2\alpha _{1}\alpha _{2}\beta _{2}+\alpha _{1}^{3}\beta _{3}}}\right|}}$

(30)

Rewriting the expression above in terms of ${\tfrac {1}{A_{\mathrm {IIP_{3}} }^{2}}}$, we get:

${\begin{aligned}{\frac {1}{A_{\mathrm {IIP_{3}} }^{2}}}&={\frac {3}{4}}\left|{\frac {\alpha _{3}\beta _{1}+2\alpha _{1}\alpha _{2}\beta _{2}+\alpha _{1}^{3}\beta _{3}}{\alpha _{1}\beta _{1}}}\right|\\&={\frac {3}{4}}\left|{\frac {\alpha _{3}}{\alpha _{1}}}+{\frac {2\alpha _{2}\beta _{2}}{\beta _{1}}}+{\frac {\alpha _{1}^{2}\beta _{3}}{\beta _{1}}}\right|\\&=\left|{\frac {1}{A_{\mathrm {IIP_{3},1} }^{2}}}+{\frac {3\alpha _{2}\beta _{2}}{2\beta _{1}}}+{\frac {\alpha _{1}^{2}}{A_{\mathrm {IIP_{3},2} }^{2}}}\right|\\\end{aligned}}$

(31)

Where $A_{\mathrm {IIP_{3},1} }$ and $A_{\mathrm {IIP_{3},2} }$ are the input third intercept points of the first and second stages, respectively.

Activity A2.1 Nonlinearity in Electronic Circuits -- This activity walks you through the analysis of nonlinear circuits, and its effects on circuit behavior.