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 (10 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 value).
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 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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Upper Bound Calculations:
Our “to-ESC” PDN can be abstracted to:
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Because of the inductive impedance of this PDN, when the ESC’s equivalent load changes (gates switch, motor is introduced to source of friction, motor accelerates, etc.), rail droop will occur.
First, let’s approximate the circuit during a transient current change as such:
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This is an over-approximation, which neglects the current supplied through the inductor during the load transient.
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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We can plot this relationship…
The following graph assumes a 10% droop and a complete open circuit from the battery to the ESC for the duration of the droop.
The x-axis is how long it takes for the 10% droop to be reached given the bulk capacitance on the y-axis.
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Note: In desmos,
d is the fraction by which the supply voltage decreases (the droop),
I is the constant current drawn by the ESC
V is the battery voltage (initial Bulk cap voltage)
C is the calculated capacitance (Farads)
y is C*1000 (Milli-Farads)
t (x-axis) is the duration of the droop
The formula
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can give an upper bound on the bulk capacitance that should be used on the ESC.
Exact-value Calculations:
If we now consider the effects of the path inductance on the voltage droop, it gets a lot harder……
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!