5. Envelope simulators

The final simulator we consider is the Envelope simulator. An envelope simulator combines the concepts of the HB simulation and transient simulation [5]. An envelope simulation is only useful for bandpass multisines, as it becomes equal to a transient simulation for low-pass multisines. In an envelope simulation, the internal signals in the circuit are represented as modulated harmonically related sine waves.

each harmonic of the base frequency $f_{\mathrm{center}}$ is modulated with a complex time domain signal $z_{k}\left(t\right)$. The envelope simulation calculates the variation of the envelope $z_{k}\left(t\right)$ in the time domain, while the relation between the different harmonics is enforced using Harmonic Balance. This leads to two sets of simulation parameters in Envelope: $f_{\mathrm{center}}$ and $O_{f_{\mathrm{center}}}$ for the underlying HB simulation and $t_{\mathrm{stop}}$, $t_{\mathrm{step}}$ and the integration method for the time-domain part of the Envelope simulation.

Choosing the Envelope settings

The underlying HB simulation should be performed at the center frequency of the multisine $f_{c}$ with an order equal to $\aleph$, the non-linear order of the circuit.

The time-step is determined using the bandwidth of the multisine. $f_{\mathrm{sample}}$ should be at least $\aleph f_{\mathrm{bandwidth}}$ to allow correct sampling of all intermodulation products without aliasing. To prevent leakage, $f_{\mathrm{sample}}$ should be set to an integer multiple of $f_{\mathrm{res}}$. Like in the transient simulation, $f_{\mathrm{sample}}$ is then increased such that the corresponding $t_{\mathrm{sample}}$ can be written down to the netlist with a finite number of digits. The current implementation of the envelope simulator in ADS only allows for a fixed time-step, but since the fixed time-step should be used anyway, this doesn’t bother us much.

The stop time $t_{\mathrm{stop}}$ is determined in the same way as in the transient simulations. The circuit should reach steady-state, so several periods of the multisine are simulated and the difference to the final period is investigated.

Choosing the integration method

As in the transient simulation, a trapezoidal integration method is preferred to avoid artificial damping. When an Envelope simulation is run with a trapezoidal integration method in ADS, the circuit transients don’t damp out after a few periods of the multisine and spurious oscillations appear in the circuit response. Both effects are shown in Figure 5.1.

We choose to use Gear2 as an alternative integration method. When Gear2 is used, the problems disappear and the circuit reaches a steady-state. We’ll have to accept the artificial damping introduced by these alternative integration methods.

Figure 5.1 Envelope simulations on the test system reveal that a trapezoidal integration method introduces spurious oscillations. As a consequence, the circuit transient doesn’t damp out fully and steady-state is not reached.

Going to the Frequency domain

The envelope simulation returns the $z_{k}(t)$ for each of the harmonics $k=0..\aleph$. The $z_{k}(t)$ are the complex time-domain signals with which each harmonic of the underlying HB simulation is modulated. In a post-processing step, the different time domain signals are transformed into the frequency domain and combined to obtain a single spectrum.

In a first step, the FFT of the last period of each of the $z_{k}(t)$ signals is calculated. Then, the DC bin of the signals is placed in the middle of the resulting vector using the fftshift function in Matlab. Finally, the DC bins of the different spectra are placed around their correct harmonic frequency. The base-band signal $z_{0}\left(t\right)$ is real-valued and has a symmetric spectrum. For $z_{0}(t)$ only the positive frequencies are saved in the result.