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G.3 Analytic representation of harmonic parameters [complex algebra top]
The complex representation or analytic representation of a real sinusoidal signal consists of the complex exponential
whose real part is equal to the sinusoidal signal. So, if V = V_{0}cos, we would write the voltage’s analytic
representation as
The phasor _{0} = V_{0}expjα is what remains after the analytic representation of the parameter is normalized by the
reference sinusoid expjωt. The phasor is complex, and its angle captures phase differences with respect to a
reference sinusoid. In the applications in this text, the reference sinusoid is usually an applied electric or pressure
field, and the phase lag is related to fluid or charge buildup. Symbolically, the analytic representation of a
harmonic parameter is denoted by placing an undertilde under the symbol, so the analytic representation of
p is given by . The phasor is always denoted with a subscript zero and has an undertilde: p_{0}. The
phasor magnitude is real, and is denoted with subscript zero but no undertilde. For example, the analytic
representation of p = p_{0}cosωt +α is written as and given by p_{0} = p_{0}expjωt, where p_{0} = p_{0}expjα.
The phasor is p_{0}, and the phasor magnitude is p_{0}. For the pressure signal p = p_{0}cosωt, there is no
phase lag and the phasor is real. The analytic representationis exponential, leading to straightforward
mathematical manipulation facilitated by the properties of complex algebra. The analytic representation is
particularly effective for handling linear equations, in which case all analysis can be done with the analytic
representation, and the real part of the analytic result corresponds to the parameters in the physical
system.
G.3.1 Mathematical rules for using the analytic representation of harmonic parameters
Generating the analytic representation from a real parameter
Given a real harmonic function f, the analytic representation is found by replacing cos(ωt +α) with
exp[j(ωt +α] and replacing sin(ωt +α) with exp[j(ωt π∕2+α].
Generating the real parameter from its analytic representation
Given an analytic representation of a function f, the function can be determined by simply taking the real part
of or, equivalently, taking the average of and its complex conjugate:
The use of the average of and its complex conjugate is useful below when discussing nonlinear
relations.
Using analytic representations directly in linear relations
The analytic representation of a harmonic function satisfies a linear relation directly. Examples of important
linear relations include V = IR and = ε.
Using analytic representations in nonlinear relations
The analytic representation cannot cavalierly be used with any nonlinear equation or to evaluate a nonlinear
parameter, because the analytical representation of the product of two functions is not equal to the product of the
analytical representations of the functions:
 (G.33) 
For example, consider a harmonic electric field E given by E = E_{0}cosωt and let its analytic representation is
= E_{0}expjωt. The parameter E^{2} is given by E^{2} = E_{0}^{2}cos^{2}ωt = E_{0}^{2}. The square of the
analytic representation of E is ^{2} = E_{0}exp2jωt. Unfortunately, is clearly not equal to ^{2}. Thus, while
analytic representations considerably simplify linear analysis, they tend to make nonlinear analysis
unwieldy.
When evaluating a nonlinear relation (for example, the product of two harmonic parameters, say, the power
generated in a circuit or the force on an induced dipole), we must either return to real quantities or we must use
specific relations designed for this purpose.
Since analytic representations work for linear relations, we can show that the analytic representation of the
product of two functions is equal to the product of one function with the analytical representation of the
other:
 (G.34) 
and since f = (1∕2), this becomes
 (G.35) 
which finally leads to the identity
From this identity, we can derive useful relations. For example, for DEP analysis (Chapter 17), we use
Equation G.36 to show that the timeaverage of the product of two harmonic functions of the same frequency is
given by:
With Equations G.36 (in general) and G.37 (for timeaveraged values) we can directly calculate nonlinear
quantities from the analytic representations without the need to return to the real representation of the harmonic
function.
[Return to Table of Contents]
Jump To:
[Kinematics]
[Couette/Poiseuille Flow]
[Fluid Circuits]
[Mixing]
[Electrodynamics]
[Electroosmosis]
[Potential Flow]
[Stokes Flow]
[Debye Layer]
[Zeta Potential]
[Species Transport]
[Separations]
[Particle Electrophoresis]
[DNA]
[Nanofluidics]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections.
This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.
This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby
here. Click here for the most recent version of the errata for the print version.
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