Example 1.1: Determining the order $\aleph$ of a circuit

As a simple example, we consider a non-linear resistor driven by a voltage source with an output impedance of $1\Omega$.

Figure 1.1Circuit used to demonstrate nonlinear order selection

The non-linear resistor has a polynomial voltage-current relationship of third order. The finite output impedance of the voltage source introduces feedback around the non-linear resistor, which will cause an infinite amount of harmonics originating from the circuit. We perform two simulations with an excessive non-linear order of $20$ to show how $\aleph$ is to be selected. The amplitude of the input generator is set first to $0.1\mathrm{V}$ and then to $1\mathrm{V}$. The magnitude of the current below is plotted below:

Figure 1.2

In the first case ($V_{\mathrm{in}}=0.1\mathrm{V}$) the magnitude of the harmonics of the current lies below the numeric noise floor of $-350\mathrm{dB}$ starting from the harmonic number $8$, so setting $\aleph=8$ in this case seems the good option. When the input amplitude is raised to $1\mathrm{V}$, many more harmonics appear and $\aleph$ should be set to $12$.

Note that, in this purely static circuit, the harmonics alone can be used to determine the non-linear order. In a dynamic circuit, harmonics can be filtered, so also the intermodulation should be taken into account to obtain a proper choice for $\aleph$.

In this simple circuit, checking only one signal is sufficient. In a larger circuit, the amount of harmonics present in each signal differs, so many signals should be checked and the worst-case $\aleph$ should be used.