a) Using the complex exponential representation, evaluate the sum of the simple harmonic functions u1 = A cos(ωt) and u2 = B sin(ωt). b) Derive the same result using the phasor diagram.

Question

a) Using the complex exponential representation, evaluate the sum of the simple harmonic functions [latex] u_1 = A \cos (ωt) ext {and} u_2 = B \sin(ωt) [/latex] . b) Derive the same result using the phasor diagram.

Accepted Answer

a) The functions [latex] u_1 [/latex] and [latex] u_2 [/latex] are respectively the real parts of [latex] \underline{u}_1=A e^{iωt} ext { and } \underline{u}_2 =B e^{i(ωt-\pi /2)} =-iBe^{iωt} [/latex]. Their sum is [latex] \underline{u} =\underline{u}_1 + \underline{u}_2 = \underline{C} e^{iωt} [/latex] ,where the complex amplitude is [latex] \underline{C} [/latex] = A - iB which we may cast in the form [latex] C e^{i\phi } [/latex] with

[latex] C=\sqrt{A^2+B^2} \ , \ \ \ \ \ \ \ \cos\phi =A/C \ \ \ \ \ ext {and} \ \ \ \ \ \ \sin \phi = -B/C [/latex].

The complex function u can then be written as [latex] \underline{u} =Ce^{i(ωt+\phi )} [/latex] and, by taking its real part, we get u = C cos(ωt + Φ).

b) The functions [latex] u_1 ext {and} u_2 [/latex] are respectively represented by the phasors [latex]\overrightarrow{u_1} [/latex] and [latex] \overrightarrow{u_2} [/latex] of lengths A and B, and making the angles ωt and (ωt - π/2) with the x-axis (Figure 1.4a). Thus [latex] \overrightarrow{u_1} [/latex] is orthogonal to [latex] \overrightarrow{u_2} [/latex] . Their vector sum u is the hypotenuse of a right triangle. Pythagoras’ theorem gives its magnitude [latex] C = \sqrt{A^2+B^2} [/latex] The phasor [latex] \overrightarrow{u} [/latex] along with [latex] \overrightarrow{u_1} [/latex] makes an angle Φ such that tan Φ = -B/A. This angle is equal to the phase lead of u over [latex]u_{1}[/latex]. For instance, if A and B are positive, Φ lies between -π/2 and 0; then u effectively has a phase lag over [latex]u_{1}[/latex]. As [latex]u_{1}[/latex] and [latex]u_{2}[/latex] have the same frequency, we may draw the phasor diagram at t = 0 as can be seen in Figure 1.4b.

Question 1.2: a) Using the complex exponential representation, evaluate th...

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