[Phys-l] phasor sign convention

[John Denker <jsd@av8n.com>]

Alas on 07/25/2011 09:01 AM, I wrote:

>       V = v1 cos(ωt) + v2 sin(ωt)           [2]

>       V = Rp    v^^ exp(-iωt)               [4]

The minus sign in equation [4] is a Bad Idea.  It's an unconventional 
and inconvenient sign convention.

Sane people write 

          V = Re exp(+iωt + φ)

to represent a wave with positive frequency (assuming ω is positive).

The phasor is still as I wrote before:

> If we define the phasor
>       v^^ := v1 + i v2                      [3]

but in terms of real sines and cosines we are then required
to write

       V = v1 cos(ωt) - v2 sin(ωt)            [2']

with a minus sign.  If you think the minus sign in equation
[2'] seems ugly and/or inconvenient, rest assured that the
laws of trigonometry require there to be a minus sign somewhere,
and putting it here is (a) conventional and (b) vastly more
convenient than any of the other possibilities.

This means that a wave with positive frequency rotates in the
conventional CCW direction in the complex plane, when plotted
in the usual way.  It means that the phasor (V1, V2) = (0, 1)
is 90 degrees /later/ than the phasor (1, 0).  This is consistent
with the more-or-less universal practice of writing the impedance
of a capacitor as Z = 1 / iωt.

There's a lot more that could be said about this sign convention
in particular, and phasors in general, if anybody is interested.

I hope nobody was too confused by the kooky sign convention I
used the other day.