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 = V₀cos, we would write the voltage’s analytic representation  as

The phasor ₀ = V₀expjα 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: 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 and given by p₀ = p₀expjωt, where p₀ = p₀expjα. 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  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 = E₀cosωt and let its analytic representation is = E₀expjωt. The parameter E² is given by E² = E₀²cos²ωt = E₀². The square of the analytic representation of E is ² = E₀exp2jωt. 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 time-average of the product of two harmonic functions of the same frequency is given by:

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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