Capacitor's energy required for a predetermined drop in its voltage - with LDO and constant current elements as load

Question

I am trying to understand how much energy a circuit (composed by some passives here called analog, and an MCU) will require. The system is powered through a capacitor, and the "passives" are actually some low power op amps attached directly to the capacitor, like the following schematic:

^{simulate this circuit – Schematic created using CircuitLab}

The MCU is powered through an LDO, and has a determined active time while otherwise is in sleep (now for simplicity I consider sleep equal to 0mA), but the active part is always on.

I need to understand how much energy the system requires to have a predermined drop in the capacitor voltage. From such results, I will define the next design steps or various tunings.

The approach was to compare the total energy required, given the capacitance in consideration, with $$ E = \frac{1}{2} C (V_{0ms}^2 - V_{10ms}^2) $$

The MCU activates with a given duty cycle, and I need to calculate the energy balance only in this period of time, with a given MCU duty cycle. Example, the MCU wakes for 1ms every 10ms, so duty is 1ms and period is 10ms. Duty cycle is 10%. I only want to know the power balance over 10ms, because afterwards the capacitor will gets fully recharged before the next cycle.

But here the problem:

• I can calculate how much drop in the capacitor voltage I have with a given energy, but because the energy can be rewritten in $$ E = P t = V I t $$, I can use this form to check, in the duty time activity, how much energy is drawn form the MCU, LDO and active components and check it against the capacitor's energy calculated in the initial formula (related to a given drop).
• Despite this assumption, because the energy is th eintegration of the power over time, I am not sure if makes sense to calculate the energy of the active part in this way: $$ E = (V_{0ms} - V_{10ms}) I_{active} t_{10ms} + (V_{0ms} - V_{10ms}) I_{MCU} t_{1ms} + (V_{0ms} - V_{10ms}) I_{MCU sleep} t_{9ms}$$ What would be the right approach?

Answer 1

I'm now sure if I understand your question right. I think you want to know (based on the capacitor voltage drop) how much energy is absorbed by the "always on" (static) circuit and how much by the MCU (dynamic) circuit? If you only know the voltage difference of the capacitor (dV) from t=0ms to t=10ms, then this is not possible to calculate, because you have two degrees of freedom in the equations. You would need to measure the capacitor voltage drops of the dynamic or the static circuits each alone.

Also, like Andy aka wrote, your formula is not right: The right one would be $$dE=\frac{1}{2}CV_{0ms}^2-\frac{1}{2}CV_{10ms}^2$$

Answer 2

So, rephrasing what I found so far, also thanks to discussing in the comments, I think it might help to write a new answer to myself, as I need to put also some graphics in it.

My problem is that I have a theoretical capacitance, and naturally it loses potential as long as the energy is taken from it. This capacitance is used to power some devices, which in the end are not important, it just matters that such devices are taking a known amount of constant current for a known period of time. The capacitor has a known initial voltage, and it is known the minimum allowable voltage we need.

The energy the capacitor can provide, is $$ E_{c} = \frac{1}{2} C (V_{init}^2 - V_{final}^2) $$.

The energy the devices are using can be found through the usage time and their current, but the voltage is dropping. So is NOT possible to assume $$ E_{used} = (V_{init} - V_{final}) I_{active} t $$, because I am interested in finding the total energy used to make such drop in C, not just it's delta:

So in order to find the energy the devices are using (is the first area in the graph), I must simply do some geometry:

$$ E = P_{min} t_{max} + \frac{(P_{max} - P_{min}) t_{max}}{2} $$

And finally I can compare this to the original energy the capacitor can provide to see if the circuit will stay within specifications.

Any thoughts?