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

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, $a_{1}$ is the gain, and $a_{i}=0$ 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:

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

(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 $f\left(-x\right)=f\left(x\right)$, and it will have an even power series expansion:

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

(7)