Understanding low current measurement
It's difficult to cover the dynamic range required to measure both peak current and sleep current at the same time.

When trying to verify very low sleep current requirements, I use the following circuit, with component values suited to the particular load.







simulate this circuit &ndash; Schematic created using CircuitLab

R1 is a reasonable value to develop a measurable drop at the expected uA levels. C1 is large enough to source all of the high current pulse demanded periodically by the load with negligible drop during the pulse, say &lt;100mV, verify this with your scope.

Of course a large electrolytic for C1 will have a large leakage current, and this current may vary with time, especially if your C1 has just come out of your junk box after sever years uncharged, but this can be measured, tracked, allowed for with a bit of care. Letting the capacitor sit at its maximum voltage for a while before using it will reform the insulating oxide, and should mitigate most of the high leakage and its variation before using.

Replace C1 and R1 with much smaller values to characterise your peak load current.