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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. 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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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] [Induced-Charge Effects] [DEP] [Solution Chemistry]

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 = V0cosmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell, we would write the voltage’s analytic representation microfluidics textbook nanofluidics textbook Brian Kirby Cornell as

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The phasor microfluidics textbook nanofluidics textbook Brian Kirby Cornell0 = V0expis 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell. The phasor is always denoted with a subscript zero and has an undertilde: p0. The phasor magnitude is real, and is denoted with subscript zero but no undertilde. For example, the analytic representation of p = p0cosωt +α is written as microfluidics textbook nanofluidics textbook Brian Kirby Cornell and given by p0 = p0expjωt, where p0 = p0exp. The phasor is p0, and the phasor magnitude is p0. For the pressure signal p = p0cosωt, 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell of a function f, the function can be determined by simply taking the real part of microfluidics textbook nanofluidics textbook Brian Kirby Cornell or, equivalently, taking the average of microfluidics textbook nanofluidics textbook Brian Kirby Cornell and its complex conjugate:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The use of the average of microfluidics textbook nanofluidics textbook Brian Kirby Cornell 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell = εmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell.

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:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.33)

For example, consider a harmonic electric field E given by E = E0cosωt and let its analytic representation is microfluidics textbook nanofluidics textbook Brian Kirby Cornell = E0expjωt. The parameter E2 is given by E2 = E02cos2ωt = E02microfluidics textbook nanofluidics textbook Brian Kirby Cornell. The square of the analytic representation of E is microfluidics textbook nanofluidics textbook Brian Kirby Cornell2 = E0exp2jωt. Unfortunately, microfluidics textbook nanofluidics textbook Brian Kirby Cornell is clearly not equal to microfluidics textbook nanofluidics textbook Brian Kirby Cornell2. 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:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.34)

and since f = (12)microfluidics textbook nanofluidics textbook Brian Kirby Cornell, this becomes
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.35)

which finally leads to the identity

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

From this identity, we can derive useful relations. For example, for DEP analysis (Chapter 17), we use Equation G.36 to show that the time-average of the product of two harmonic functions of the same frequency is given by:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

With Equations G.36 (in general) and G.37 (for time-averaged values) we can directly calculate nonlinear quantities from the analytic representations without the need to return to the real representation of the harmonic function.

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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] [Induced-Charge 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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