The following is only for the determined, perhaps overzealous, student, it is
not intended to be a requirement for this course, read on if you dare.
The constant little E, "e" is the base of the natural logarithms, used
extensively in population growth studies. The phenomena of charging
a capacitor in series with a resistor powered from a fixed voltage source,
is a lot like this. As the capacitor charges, it drops an ever increasing
percentage of the available voltage that the series resistor would otherwise
drop. Thus the current flowing through the circuit is continuously being
reduced toward a mathematical asymptote approaching zero. In truth it
never reaches zero, rather it follows a curve described by e raised to the
negative x, this is all very deep, but e, Pi, sinewaves, and a clever notation
scheme we humans invented called the square root of negative 1, have several
common threads. One way for me to demonstrate how this all ties together, is
be extrapolating on Euler's Equation.
e^(jx) = cos(x) + j*sin(x)
The above can be derived/proven using the series expansions for e^x, sin(x)
and cos(x).
hence:
e^(jwt) = cos(wt) + j*sin(wt)
and:
cos(wt) = [e^(jwt) + e^(-jwt)]/(2)
sin(wt) = [e^(jwt) - e^(-jwt)]/(j2)
j = sqrt(-1) (i is generally not used in electronics to avoid confusion with
current) More than the Euler relationships, the use of radian as opposed to
hertzian frequencies is driven by things like time constants such as RC being
equal to 1/omega. That just comes out in the math.