Example 3.1: Transient simulation on a test circuit

The three different simulators discussed in this appendix will all be used to determine the steady-state response of a test circuit which is excited by a bandpass multisine. The used test circuit consists of a static non-linearity followed by a bandpass filter:

Figure 3.1 Example circuit used in the article

The static non-linear block is created with a Symbolically Defined Device (SDD) block from ADS. In an SDD block, the currents flowing into the ports can be specified as function of the voltage at the ports. In the configuration we used, no current flows into the input port of the SDD block while the output current is the $\tanh\left(\cdot\right)$ of the voltage at the first port.

The bandpass filter is an elliptic bandpass filter with its center frequency around $10\mathrm{GHz}$ and a bandwidth of $200\mathrm{MHz}$. Due to rounding errors in the components, the filter’s frequency response deviates from the ideal elliptic frequency response, but this imperfection in the filter response matters little for the goal of this example. The inductors in the filter have been given a series resistance of $1\mathrm{p\Omega}$ to avoid simulation errors due to the loop of shorts created by the inductors in the filter.

The whole circuit is excited by a bandpass multisine connected to the in node of the circuit. The multisine has a frequency resolution of $f_{\mathrm{res}}=5\mathrm{MHz}$ and a center frequency $f_{\mathrm{center}}\!=\!10\mathrm{GHz}$. The multisine excites $T\!=\!41$ frequencies in a $200\mathrm{MHz}$ band around the center frequency, which results in $f_{\mathrm{min}}\!=\!9.9\mathrm{GHz}$ and $f_{\mathrm{max}}\!=\!10.1\mathrm{GHz}$. The Root Mean Square (RMS) voltage of the multisine is set to $0.2\mathrm{V}$ and the multisine has a DC offset of $1\mathrm{V}$.

The non-linear order $\aleph$ of the circuit was set to $12$, so a minimum sampling frequency of $2\cdot12\cdot10.1\mathrm{GHz}\!=\!242\mathrm{GHz}$ is needed to be able to represent every harmonic up to the twelfth order in the circuit. This sampling frequency is rounded up to $250\mathrm{GHz}$ to obtain a time-step $t_{\mathrm{sample}}$ of $4\mathrm{ps}$ instead of the time step of $4.132...\mathrm{ps}$ which corresponds to a sample frequency of $242\mathrm{GHz}$.

$4$ periods of the multisine were simulated to obtain the steady-state response of the circuit, so $t_{\mathrm{stop}}$ of the simulation was set to $800\mathrm{ns}$. To verify that the circuit is in steady-state, the different periods were compared to the last period. The plot shown in Figure 3.1 shows the result of the steady-state check obtained for this transient simulation.

With these settings, the transient simulator returns $200001$ time samples for each of the measured signals. One period of the multisine corresponds to $50000$ points. The last $50000$ points in the waveforms were transformed to the frequency domain using the FFT. The result is shown below:

Figure 3.2 Output spectrum of the transient simulation

Looking at the harmonics in the spectrum of $V_{\mathrm{int}}$ indicates that the chosen $\aleph$ is still too low, although the error made at $\aleph=12$ can be assumed low enough. Below, we show the output signal of the non-linear block and the output signal of the filter around the center frequency of the multisine:

Figure 3.3 Input output spectrum of the transient simulation

The steady-state spectrum of $V_{\mathrm{out}}$ contains some indications of the frequency warping effect: The multisine is placed in the middle of the pass-band of the filter, but the transient simulation indicates that its highest frequencies are filtered out.

Another way to visualise the frequency warping warping present in the transient simulator is by determining the Frequency Response Function (FRF) of the elliptic filter starting from the obtained steady-state response. Once the steady-state spectra are calculated, calculating the frequency response of the filter boils down to a simple division in the frequency domain:

The results of this division are shown in green below. The FRF of the filter obtained with an AC simulation is shown with an orange dashed line.

Figure 3.4 Frequency response of the filter under transient excitation

When the two FRFs are compared we can again see the frequency shift introduced by the transient simulation: the whole frequency response of the filter is shifted to higher frequencies due to the warping. The reason for the noisy estimate of the filter far away from the passband is due to the very low amplitude input signal at those frequencies. Outside of the frequency band excited by the multisine, the intermodulation distortion created by the non-linear block is used as input signal for the estimation of the filter frequency response.