What I find surprising is how slow the sawtooth is. I can't imagine any switching power-supply doing one pulse every 50 seconds!
What it could be is aliasing of a switching-power-supply with the sampling process of your current measurement - assuming that your measurement might suffer from aliasing. (Which is a big assumption - can you expand on precisely how you are measuring the current e.g. what is the sampling-mode of your 'scope; are you measuring the voltage-drop across a resistor, or using a current-transformer or what.)
Now the stm32L0xx has, I think, an internal linear LDO voltage-regulator for the core - so I wouldn't expect that to be switching and causing current-gulps. But if your external circuitry has a switching regulator to supply Vdd, then that could be what your measurement sampling process is beating against.
Can you say precisely which stm32L0xx it is?
I have used stm32l496 which has built-in a linear regulator, with the option to attach an external switcher. I think TI pioneered built-in switchers for their msp430 but ST might have them as well now.
Or it could be something to do with your code. If your code does different things over time, it is possible that some of those things use more current than others.
Hope this helps,
Danish
