Example 6. Gm-C filter

The final example is a fully differential Gm-C biquad [33] designed in the same commercial $0.18\mathrm{\mu m}$ CMOS technology as the other examples (Figure E6.1). Each Operational Transconductance Amplifier (OTA) in the biquad consists of an input pair and a cascode stage. The common-mode feedback in the OTA is active. The biquad is configured to create a resonant pole pair at $10\mathrm{MHz}$.

Figure E6.1The fully-differential Gm-C biquad under test (top) consists of four identical OTA (bottom). The first stage of the OTA is indicated in red, the second stage in green and the common-mode feedback in blue.

The differential mode of the biquad is excited by full lowpass RPM (all frequency lines are excited). The multisines have $f_{0} = f_{\mathrm{min}} = 200\mathrm{kHz}$ and $f_{\mathrm{max}} = 100\mathrm{MHz}$. In a resonant system like this, the frequency resolution of the multisines should be chosen to have several lines in the resonance [34]. If, for example, only a single spectral line is placed in a sharp resonance, the PDF of the internal signals will tend to that of a sine wave, instead of the wanted Gaussian PDF. The wanted noise-like properties of the internal signals in the circuit then disappear, which is unwanted if the results are to be valid for Gaussian input signals.

The RMS of the multisines was set to $50\mathrm{mV}$ and the steady-state response of the circuit to $50$ different-phase multisines was obtained with HB. The resulting spectrum at the differential output is shown in Figure E6.2. The output distortion lies $50\mathrm{dB}$ below the signal level, so the circuit is behaving close to linear. No even-order contributions are present due to the differential nature and perfect symmetry in the simulations of the circuit. Note that the obtained odd-order distortion at the output shows a strong frequency dependence around the resonance. The sub-circuits in this biquad are assumed to be weakly non-linear, so the 4-port S-parameters of each OTA were used in the DCA.

The small-signal assumption was verified by comparing the frequency response from the input of the total circuit to each of the waves in the circuit with the corresponding BLA. The largest difference was observed on the frequency response from the reference to the output waves of OTA4 (shown in Figure E6.3), but this difference is small enough to consider the small-signal assumption to be valid.

Figure E6.2The spectrum of the output wave $B_t$ shows only odd non-linear distortion (red stars), as can be expected from a differential circuit. The excited frequency lines are indicated with black stars while the RMS value of the distortion is shown with red dashed lines.

Figure E6.3The small-signal frequency response from the input voltage to one of the output waves of OTA4 (orange) doesn’t lie far from the corresponding BLA (green), so the small-signal S-parameters can be used to represent the sub-circuits instead of the MIMO BLA.

The first OTA is found to be the dominant source of distortion in the resonance peak of this circuit (Figure E6.4, left). The fourth OTA also introduces a considerable contribution. To find out which part of the OTA is mainly responsible, the first and fourth OTA were split into two parts and the DCA was applied again. With this hierarchical application of the BLA-based DCA, it is found that the first stage of both OTA 1 and 4 are the dominant contribution (Figure E6.4 right).

Figure E6.4The first OTA is found to be the dominant source of distortion in the resonant peak of the circuit (left). The contribution of the fourth OTA cannot be ignored however. By applying the DCA hierarchically (right), it is found that the first stage is the dominant contributor in both OTA.

With this final example, we have demonstrated how the BLA-based DCA can be used in larger circuits and how it can be used hierarchically to zoom in on certain sub-circuits to determine the actual source of non-linear distortion.