2. Testing methodology

The linear system around a periodic solution is a Linear Time-Periodic (LTP) system. The behaviour of such a LTP system can be described perfectly by a infinite series of Frequency Response Functions (FRF) called Harmonic Transfer Function (HTF) [1]. The HTF which describes the frequency response from the excitation frequency $\omega$ to a translated frequency $\omega+l\omega_{\mathrm{ex}}$ will be denoted as $G_{U\rightarrow Y}^{\left[l\right]}(j\omega)$. $\omega_{\mathrm{ex}}$ is used to indicate the pulsation of the LTP system.

To test the LSSS simulator, a periodically time-varying differential equation is implemented in ADS. The HTFs of such differential equations can also be calculated in Matlab using techniques described in [1]. By comparing the Matlab results to the results from ADS, we can get an indication of the error made in the simulator. The following periodically time-varying first order differential equation will be used:

The time-variation $a(t)$ will be a random-phase multisine with a base pulsation of $\omega_{\mathrm{ex}}=2\pi f_{\mathrm{res}}=0.5\mathrm{Hz}$. The multisine excites all tones up to $f_\mathrm{max}=5\mathrm{Hz}$. The amplitude of each tone in the multisine is set to $1$ and random phases are used. A DC offset of $1$ was added to the multisine as well. The resulting $a(t)$ signal is shown in Figure 2.1.

Figure 2.1 Values of the $a$ parameter in the differential equation (2.1) as a function of time.

The implementation of this differential equation is done in ADS with a Symbolically Defined Device (SDD) block. The simulation set-up is shown in Figure 2.2. The SDD block in the schematic is configured such that the voltage at port 1 acts as the input signal $u(t)$ and the resulting current flowing into port 1 is the output signal $y(t)$. The control of the factor $a(t)$ is done through the second port of the SDD block. It is important to force a non-zero initial condition in that port, as setting the node to zero leads to errors in the simulation.

The whole circuit is simulated with a HB simulation with a base frequency of $0.5\mathrm{Hz}$ which is given an order of $100$. The LSSS was run on a logarithmic frequency grid going from $0.1\mathrm{Hz}$ to $100\mathrm{Hz}$ with $100$ steps per decade. With these settings, ADS returns $201$ HTFs sampled at $301$ frequency points.

Figure 2.2 Simulation set-up in ADS to implement the time-varying differential equation (2.1)