I have implemented a digital SMPS in a Cortex M3 controller. The feedback of the SMPS is done using the TL431 and opto coupler feedback.
The feedback is sampled by the ADC at 10kSa and the sampled values controls the duty-cycle of the SMPS.
It is stable, but I would like to simulate the loop in PSPice. The ADC has sample and hold along with quantiziation that affects the gain/phase. The timer generating the PWM has finite number of different dutycycles and that has a small scale effect also.
I would like to have a model both for transient and ac sweeps.
The ac sweep can be modelled by a sinc function, but how to model the quantization and PWM stage (discrete duty cycles)?
Also, for the transient simulation, the ADC can be modelled simple by a S/H. The quantization can be modelled by a floor function, and for the PWM I need another floor function and a couple of ABMs to model the PWM. But, is there a better method for this?
I think you're taking the right approach for the sample & hold and quantization in the transient simulation.
You're almost right with the sampling in the frequency domain -- you're not taking aliasing into account, but as long as your output filter doesn't have appreciable gain above Nyquist then you're all set there.
Traditionally, the way to account for the quantization in the frequency domain is to model it as noise injection. This makes sense if you think about it right: the output of a quantization step is your desired signal plus an error, so you just call that error "noise" and do some analysis.
You would like to be able to model it as being random, which would make it white in the sampled-time domain, and roughly white but perfectly band- limited to the sampling rate in the continuous-time domain. In reality, this only happens if you have enough other noise in the circuit to keep the quantization agitated -- the more pessimistic approach is to model the quantization noise as being a sine wave that is 1.09 a quantization step peak to peak (to account for the fact that it's a square wave), and at the worst possible frequency. This accounts for the fact that -- if all else is quiet -- your circuit will oscillate around a quantization step.
Regardless of whether you assume random or worst-case noise, the way to analyze it is to treat the quantization gain as a signal that's injected at the quantizer, do a sweep to find the transfer function from there to your output, then plug in one of the above two assumptions and see what the noise is on your output.
Shameless plug: this method of treating quantization is covered briefly in my book "Applied Control Theory for Embedded Systems". You'll find it mentioned briefly in chapter 8 (Nonlinear Systems), and in depth in chapter 10 (Software Implications).
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If you find that your PWM is too coarse to give good quantization behavior, you may also find this article to be of benefit:
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or
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if you are long-URL challenged.
Note that you need to pay attention to the output filter bandwidth, and that going to a 2nd-order sigma-delta modulator may be beneficial if you can stand the code complexity.
--
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?
Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com
The sampling delay in the band of interest could be approximated by allpass filter section. PWM is done with sawtooth waveform and comparator. Quantization error could be modelled by adding noise; however that won't tell about possible limit cycles.
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