On 3/10/2009 3:38:45 PM, Tom_Gutman wrote:
>I believe you are talking
>about a different use of
>complex numbers to represent
>voltages and currents, what I
>have previously seen referred
>to as phasors. As you say,
>with this representation the
>phasor for a simple sinusoid
>is a constant, representing
>the amplitude and phase of the
>sinusoid (or, if you prefer,
>the in phase and quadrature
>components relative to some
>arbitrary phase reference).
>Phasors are used only when
>analyzing systems with a
>single frequency, and I think
>they work only if the
>variations in the phasor are
>slow relative to the
>frequency.
By definition, the complex amplitudes (phasors) are constants, and have no "slow variations."
>The factor of � in the
>power equation is from the
>integral of the square of the
>sign, the same thing that
>gives the factor of half the
>square root of two for the RMS
>value. You can see this if
>you use a pure resistive
>system and the voltage as the
>phase reference. Then both I
>and E are real. But I and E,
>as phasors, are the peak
>current and voltage, not the
>RMS values, hence the power is
>�IE.
In order to give the correct real power (and independent of phase), the formula needs to be (1/2)Iconj*E. The alternate choice (1/2)I*Econj gives the same real power, but changes the sign of the imaginary part. The choice is by convention, and makes inductive stored enrgy positive).
>As you say, what I have done
>is quite different. I've used
>a complex time varying signal,
>where the real part is the
>observed signal (voltage,
>typically), and the inaginary
>part is, well, imaginary -- or
>something, I'm not an EE and
>don't fully understand it.
>Here we are dealing with
>signal processing concepts (as
>discussed earlier), there is
>no specific circuit in mind
>and no actual current. Power
>is apparently just nominal,
>the power that would be
>dissipated if the signal were
>measured across a one ohm
>resistor. I am not quite sure
>of the terminology that goes
>with this procedure, ISTR the
>term "complex envelop", and
>that may be what applies.
You created the analytic signal corresponding to r(t). In general, the complex analytic signal of x(t) is xa(t) = x(t) +jHilbert[x(t)]. It's defining characteristic is that its FT(xa) is zero for f<0, and FT(xa) = 2*FT(x) for f>0. (The scale factor of 2 gives x=Re(xa), assuming x is real). This complex xa(t) is the generalization of the single freq complex signal x1(t)=X0*exp(2*pi*f*t). |X0| is the (constant) complex envelope of x1(t). |xa(t)| is the time varying complex envelope of x(t). As a vector in the complex plane, xa(t) is the generalization of the fixed amplitude single freq rotating "phasor."
>IAC, the request, as clarified
>in one of the requestors early
>post, was for the envelop of
>the instaneous power (again,
>taken as simply the square of
>the instaneous amplitude), and
>this method produces that.
>The instaneous average power
>would be half of that. So
>this answers the original
>question. I believe, without
>evidence or proof, that it is
>reasonably general and would
>work for a more complicated
>system (such as having
>different amplitudes for the
>two sine waves, where the sum
>to product transformation does
>not work).
To be pedantic about the use of words, the envelope (see above comments) of instantaneous power can be defined, but is not the function expected, or of use. It has no physical (nor a signal prcoessing equivalent), and is similar to "RMS power." Average power is the quantity of interest, and RMS voltage (by def) is the signal domain quantity whose square gives this average. RMS power squared will give avg(Pwr^2) - by def, not usually of interest. The power envelope is in the same category, and its avg does not give the avg power of the complex signal. However, as you noted, the square of the complex envelope of the signal does have the desired properties. If EnvX(t) is the envelope of the analytic signal xa(t) derived from real x(t), then avg power of x = avg{x(t)^2] = (1/2)avg[EnvX(t)^2], and EnvX(t)^2 may be interpreted as the "instantaneous," time varying envelope power of the complex signal.
Keeping track of the domains, the facotr of 1/2, etc. can be a full time job in signal analysis:-).
Lou
