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GOAL: Reduce the Bulk Cap to the lowest capacitance allowing for proper functionality of the ESC (120F3[X]v2) used in the pegasus drone.

Due to the inductance of the conductor path from the battery to the ESC, load transients presented by the ESC will cause voltage to fluctuate at the ESC terminals. If this voltage drops below a certain value (need to find the optimal nominal voltage range, and characteristics of the motor’s load transients), also known as rail droop/collapse, for a long enough time, the motor/ESC functionality will be negatively affected. A Bulk capacitor can be used to compensate for this, and provide the demanded transient current.

Specifications:

The length of the conductor path (8 gauge wire) from the positive battery terminal to the ESC is ~117cm. This might be useful for finding the inductance of the PDN (NOTE: Inductance strongly depends on the loop area. Since this loop area is not well defined, it might be fine to approximate it with straight wire inductance… need more research). From this resistance calculation tool, this segment of wire results in ~2.5 Ohms of resistance (though the PDB and other interconnects will increase this valueits best to determine it through measurement).

The measured inductance of the harness (+ PDB, current measurement board) is ______(need to find).

The measured resistance of the harness is ______(need to find).

From the Pegasus Overview, the maximum required current from all 4 motors “is anticipated to be 90Amps… Any individual motor will not draw more than 23 amps at a time, not including the path”. Can . Another section of the doc says the max is 25 amps. I will use that, and neglect the, much smaller, ESC control circuitry (3 Half bridge drivers, MCU, MOSFET gates, …) current be neglected?
(But the doc section below seems to indicate a max of 25A).

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current (gate driver quiescent current, LEDs, etc.).

Upper Bound Calculations:

Our “to-ESC” ESC’s PDN can be abstracted to:

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In this scenario, the only source of current for the ESC is the capacitor, and given some general ESC ratings/requirements for good performance (need to figure out), an upper limit for the necessary capacitance can be found.

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• Modelling in terms of load transient current as the input and capacitor voltage as the output results in a non-linear system. Can try to use Numerical methods to come up with a formula for finding capacitance.

• It is an LC circuit, so current transients cause it to resonate. The higher the cap’s ESR, the more these oscillations are damped. Low ESR is better for efficiency and reducing voltage droop magnitude but worse for dampening oscillations!

Closer Upper-Bound Calculations

A slightly more accurate approximation of the ESC’s PDN is the following:

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or

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During a load transient, when the ESC suddenly sinks some current, we can employ AC analysis to see what happens at the input of the ESC (what will the voltage differebcedifference/ripple at the ESC, caused by the transient look like?) and from there, we can add proper decoupling to compensate:

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To elaborate, we want to design our PDN such that, for the worst case load transient (at a rate/frequency of our concern; i.e. the fastest rate at which current can change through the BLDC motor coils), the rail collapse / voltage ripple doesn’t go beyond the ESC’s rated tolerance (need to find).

This means, for the maximum current I_MAX (which gets drawn during the transient), the voltage difference across the PDN’s equivalent impedance must be below V_ripple (maximum voltage ripple tolerable by the ESC).

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This maximum allowable equivalent impedance Z is called the target impedance.

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The results of this expression have been found to match those of this PDN analysis tool.

Now, if If we know Desmos graph parameters:

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we can find an appropriate bulk capacitor, with parameters

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V_ripple and the frequency range of our concern has yet to be identified.