Example 5.1: Envelope simulation of the test system

The same test set-up as in the previous examples of this appendix is now simulated using the envelope simulator.

The underlying HB simulation in the envelope simulation was given a frequency of $10\mathrm{GHz}$ with an order equal to the non-linear order of the circuit $O_{f_{\mathrm{center}}}=12$. The minimum sampling frequency used in the envelope simulation is $\aleph f_{\mathrm{bandwidth}}\!=\!2.4\mathrm{GHz}$. This is rounded up to $2.5\mathrm{GHz}$ to obtain a round sampling time $t_{\mathrm{sample}}$ of $400\mathrm{ps}$.

$4$ periods of the multisine are simulated to allow the circuit to get into steady-state. This gives a stop time $t_{\mathrm{stop}}$ of $800\mathrm{ns}$. The steady-state condition was checked in the same way as is done in the transient simulations by subtracting the final period from the previous ones and by looking a the norm of the remaining signal. In an envelope simulation, this is done separately for each $z_{k}(t)$.

The envelope simulator returns $2000$ complex time samples around every harmonic of the harmonic balance simulation, which corresponds to $26000$ samples in total. The last $500$ points of each time series are used to calculate the spectrum of the steady-state response. The reconstructed spectrum at the output of the non-linear element in the circuit is shown below

Figure 5.2 Full spectrum of the envelope simulation

Like in the HB simulation, gaps are present in the spectrum, which results in reduced simulation time for these bandpass circuits and excitation signals compared to the transient simulation. The signals around $f_{\mathrm{center}}$ at the input and output of the filter are shown below:

Figure 5.3 Input output spectrum of the envelope simulation

These signals are used to estimate the FRF of the elliptic filter like we did before with the spectra obtained with the transient and HB simulations:

Figure 5.4 Estimated frequency response of the filter during envelope simulation

The artificial damping introduced by the Gear’s integration method is clearly visible when the FRF of the filter is compared with its true FRF: the transmission zeros in the stop band of the elliptic filter are shifted from their location on the imaginary axis, which reduces their attenuation. Instead of warping the whole frequency axis, the warping in an envelope simulation is performed on a per-harmonic basis: the filter seems to be squeezed inwards around each harmonic of $f_{\mathrm{center}}$.