3.3 Membrane model and circuit simulation
3.3.3 Time domain simulation
For the transient analysis, the input signal has been set in the same way as in the previous case (see Figure 3.3). In Figure 3.15 it can be seen the behavior of both membranes that is coherent with the one expected. When a 50% DC input signal is applied, the output, after a rising edge of about 10ms, is equal to 1.5V;
at a 33% DC, it becomes 1V; whereas, at 17%, it is equivalent to 0.5V and the same for negative potentials. The same voltage is generated for both positive and negative tensions at the same duty cycle and the difference between the schemes lies in the models’ components values.
In the time domain, by taking a deeper look at the output signals, it is possible to notice a slight difference between the two membranes as SiO2 presents higher oscillations around the average value. This phenomenon is probably associated with the capacitor’s size since probably a higher capacitance entails a more significant transient that manifests itself in terms of smaller oscillations. To analyze this aspect, applied voltages equal to -0.5V and 1.5V have been considered and are shown
0 0.05 0.1 0.15 0.2 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 0.05 0.1 0.15 0.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 0.05 0.1 0.15 0.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 0.05 0.1 0.15 0.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Figure 3.15: Output voltages corresponding to different input applied with duty cycles equal to 16.7%, 33.3%, 50%. The model of the membrane depends on the potential. The graphs show a behaviour coherent with the PWM signal duty cycle.
1.0282 1.02821 1.02822 1.02823 0.492
0.493 0.494 0.495 0.496
(a)
1.01691 1.01692 1.01693 1.01694 1.502
1.503 1.504 1.505 1.506 1.507
(b)
Figure 3.16: Comparison of the oscillations for SiC and SiO2 membranes at different voltages. (a) shows the fluctuations at 0.5V and (b) at 1.5V. The figure highlights the dependency from the value of the capacitance. Concerning the LSB, however, the oscillations are all negligible.
in Figure3.16. Their models, indeed, have either the least different capacitances or the most marked gap. The maximum oscillations at -0.5V are 5 mV and 2.9 mV
Some paragraphs earlier, the Fourier transform of the converter without the membrane has been evaluated. Following an ilk process, the same has been done for SiC and SiO2 membranes, thus allowing the comparison with each other and the no-load case. Therefore, the transforms have been computed for a simulation with the PWM signal working at a 50% duty cycle and it is shown in Figure 3.17.
100 102 104 106
10-12 10-10 10-8 10-6 10-4 10-2
100 102 104 106
10-12 10-10 10-8 10-6 10-4 10-2
100 102 104 106
10-12 10-10 10-8 10-6 10-4 10-2
Figure 3.17: Comparison of the Fourier Transform of the output voltage without load and with SiC and SiO2 membranes. They all show the same behaviour with a first peak at the input frequency and the following at the successive harmonics.
The three Fourier transforms show assimilable characteristics: the first peaks are all localized at 188700 Hz that corresponds to the simulated period of 5.3µs, whereas the successives correspond to the harmonics of the signal frequency. They behave similarly in terms of magnitude, even though SiO2 higher peak coincides with the second harmonics, whereas for the SiC, it is the fundamental frequency. At 0Hz, all these simulations assume the constant output value of 1.5V, as shown in Table3.4.
To evaluate the filter’s effect on the fundamental frequency and the successive harmonics, for each case it has been compared the input, and the output Fourier transform at the most significant frequency values.
Amplitude (V)
SiO2 1.5043
SiC 1.5043
No Load 1.5043
Table 3.4: Magnitude of the Fourier Transforms at 0Hz without the load and with the two different membranes.
rms(harmonicsout) rms(harmonisin)
rms(Vout(0Hz)) rms(Vin(0Hz))
SiO2 -65.22 dB -0.004 dB
SiC -68.25 dB -0.008 dB
No Load -65.32 dB -0.008 dB
Table 3.5: Ratio between the magnitud of the Fourier transform at the harmonics and 0Hz. As desired the DC output remains almost unchanged, whereas the other components are significantly reduced.
0 200 400 600 800 1000 1200 1400 1600 -6
-5 -4 -3 -2 -1 0 1
0 200 400 600 800 1000 1200 1400 1600 -6
-5 -4 -3 -2 -1 0 1
Figure 3.18: Comparison of the magnitude of the Fourier Transform at precise frequency values between the input and the output signals.
In Figure3.18it is shown the diminishing of the magnitude of the principal frequency components except for the one at 0Hz, which is almost unchanged. Starting from the Fourier transform of the signal, it is possible to evaluate some parameters such as SNR (signal to noise ratio), SFDR (spurious free dynamic range), and ENOB (effective number of bits), which, altogether, give essential information about how much noise is introduced and in what magnitude the harmonics influence the performance of the converter.
The SNR assumes very high values for the three conditions, also considering that an 8-bit DAC is in use; moreover, the load does not influence enough to induce an evident variation. A noticeable aspect is that the best performance is witnessed for the SiC membrane, probably due to the model itself and also to the fact that
sharply and some of them could get lost. In a real DAC the effective number of bits is always smaller than the designed value, since there are always disturbances and noises due to many different sources. In this case, since the device under test is not properly a DAC and the input is a square wave, it has been decided to evaluate how the highest income of interference influences the performance, by adding to the least significant bit value, the amplitude of the most prominent peak in the Fourier Transform and then it has been translated into a binary information. The results in Table 3.6 show very high ENOBs with equal values up to the second decimal, this means that, in all these cases, the oscillations around the analog voltage are not high enough to compromise the DAC resolution.
N = 8 (Number of bits) Total uncertainty (TU) = 1
2N + Amplitude of the highest spurious peak Full range
EN OB = − log10(T U )
log10(2) (Change of base to obtain a value in bits)
(3.1)
SNR (dB) SFDR (dB) ENOB (bits)
SiO2 110.50 65.98 7.98
SiC 112.80 67.27 7.98
No Load 112.05 68.49 7.98
Table 3.6: Table with the parameters calculated on the Fourier transforms for each considered case.