6. Estimating the MIMO BLA in weakly non-linear circuits

In weakly non-linear circuits, the BLA doesn’t deviate a lot from the small-signal frequency response of the circuits, so the small-signal behaviour can be used to replace the BLA. When the MIMO identification scheme is avoided, a significant reduction in simulation time for the DCA is obtained. The small-signal S-parameters can be obtained quickly in modern simulators combining several AC or S-parameter simulations.

Combining small-signal and large-signal results is trivial when the response of the circuit to the multisine excitations is obtained with HB. If the response is obtained with a time-domain simulation, frequency warping should be properly taken into account:

• A trapezoidal integration method should always be used to avoid artificial damping of the circuit poles [30].
• A fixed time step $T_{s}$ should be used to allow calculating the spectrum easily with the DFT.
• The remaining frequency warping introduced by going to the discrete time domain should be taken into account.

When the trapezoidal integration method is used, each frequency bin of the large-signal simulation $kf_{0}$ is warped to a frequency $f_{\mathrm{warp},k}$ according to the following relationship:

It is as if the circuit is working on the frequency grid determined by $f_{\mathrm{warp},k}$. Hence, the small-signal behaviour should preferably be determined on the frequency grid $f_{\mathrm{warp},k}$, or alternatively should be interpolated to the warped frequency grid to minimise any errors.

When the BLA deviates too far from the small-signal S-parameters due to strongly non-linear behaviour of the circuit, one cannot replace the BLA by small-signal S-parameters without introducing errors. It is therefore important to assess the quality of the small-signal behaviour when using it to predict the distortion in the circuit. The perfect assessment could be obtained by comparing the small-signal behaviour to the estimated MIMO BLA of each sub-circuit, but calculating the MIMO BLA beats the purpose of using the small-signal behaviour in the first place. Instead, the BLA from the reference signal to the input and output waves ($\mathbf{G}_{R_{1}\rightarrow\mathbf{Z}_{\left[n\right]}}^{\mathrm{BLA}}$ as defined in (5.2)) can be compared to the frequency responses obtained with an AC simulation. When the difference between $\mathbf{G}_{R_{1}\rightarrow\mathbf{Z}_{\left[n\right]}}^{\mathrm{BLA}}$ and the AC result lies significantly above the distortion level, the small-signal behaviour deviates too far from the correct BLA and should not be used in the DCA.

As an example, this test is applied to the class-C amplifier from before. The FRF from the main reference signal to the output wave is calculated in three different ways: First, the BLA from the reference to the output wave is calculated using the SISO techniques described in Section 2. Second, the small-signal frequency response from the reference to the output wave was calculated with an AC simulation. Finally, the MIMO BLA of the transistor was used to predict the same frequency response (Appendix 4). The three frequency responses are shown in Figure 6.1. The frequency response obtained with the AC simulation deviates strongly from the BLA from the reference to the output wave. When the estimated MIMO BLA is used to predict the frequency response, the correct result is obtained.

In this class-C amplifier, it is therefore not possible to use the small-signal S-parameters in the DCA, which is to be expected for such a strongly non-linear circuit. In the examples shown in the next section, two circuits are shown where this small-signal assumption is valid.

Figure 6.1 If the small-signal behaviour deviates too far from the BLA, the predicted BLA from the main reference to the output waves is incorrect. The correct BLA from reference to the output wave is shown with green +). Its prediction using the small-signal behaviour is shown with orange -. When the MIMO BLA is used (cyan -), a correct prediction is obtained.