Analysis for Challenge 3

This is a resonant circuit, and will therefore exhibit sinusoidal behavior.

1.  The peak voltage on the capacitor will be 2VBATT (the circuit is a voltage doubler).
2.  The peak voltage will occur at t = πSQRT(LC), at t = 3πSQRT(LC), at t = 5πSQRT(LC), …

3.  The maximum current will be VBATT(SQRT(C/L)), + or – depending on t.
4.  This maximum current occurs at t = π(SQRT(LC))/2, at t = 3π(SQRT(LC))/2, at t = 5π(SQRT(LC))/2, …

Note 1:  A complete derivation is found at
Circuit 1).

Note 2:  Assuming the switch does not bounce, and assuming it never opens, the
diode will remain completely out of the picture, and will not affect this
circuit’s behavior.  Commentary regarding the need for this diode is found
in the reference cited in Note 1 above.

Note 3:  With the possible exception of circuit operation near Absolute Zero (0 °K),
there are no known zero-resistance circuits. The switch (usually a MOSFET)
and the inductor would be likely candidates for introducing enough series
resistance to merit some investigation.  Interested readers are referred to
to an analytical treatment of Series RLC Circuitry.

Note 4:  A series diode could be placed in this circuit, and would guarantee a DC
steady state.  After considering circuit behavior with an ideal series diode
(see the declared circuit conditions), a circuit enthusiast would then need
to consider that actual diodes have multiple parasitics:

a. Forward voltage drop
b. Bulk resistance
c. Dynamic impedance that varies non-linearly with current
d. Capacitance (particularly when reverse-biased)

Note 5:  If the actual switch used in Figure 1 is a relay (rather than a MOSFET),
there is a danger if the switch opens while current is negative (i.e., flowing
from right-to-left in the inductor).  Current cannot change instantaneously
in an inductor.  If the switch opens, and if there is no path into which this
reverse current can easily flow, the inductor will force a path to exist (via
arcing, and/or reverse breakdown of the diode that is now drawn in the
circuit schematic).

Note 6:  If the actual switch used in Figure 1 is a MOSFET, a circuit designer then
usually has a built-in (but not usually drawn) safety valve:  A bulk (body)
diode that will provide a low-voltage path to the battery in the event of a
surprise reverse-current event.  For those who want to know more, please
see
Reference 1 or Reference 2.

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