Time, Frequency And Complex Frequency Domains Of A Sinusoidal Excitation

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Time, Frequency And Complex Frequency Domains Of A Sinusoidal Excitation

Analyses of sinusoidal excitations of orders greater than two, are not simple. There are various computer programs that aid in the analyses. Nonetheless, it is very important that one grasps the mathematical tools used to simplify the analyses of sinusiods in the time domain where they exist.
**(a)** Given that the time, frequency and complex frequency domains are associated with the *containerships* of sinusoidal excitations; indicate the general *identity* of a sinusoidal excitation in each of the domains.
**(b)** What mathematical tools are used to arrive at the general identities of sinusoidal excitations in the frequency and complex frequency domains?
**(c)** Given an arbitrary sinusoidal excitation in the time domain, derive its representation in the frequency and complex frequency domains. Thus establish the PJProblemStrings Sequence of the *domain switch* of a sinusoidal excitation.
**(d)** Determine the complex frequencies that are associated with:

(i) 5e^{-4t} (ii) cos2ωt (iii) sin (ωt + 2θ)

(iv) 4e^{-2t}sin(3t - 50^{o}) (v) e^{-3t}(2 + cos4t)
**(e)** Determine the complex frequency s and **V**(s) associated with:

(i) 5e^{-2t} (ii) 5e^{-2t}cos(4t + 10^{o}) (iii) 4cos(2t - 20^{o})
**(f)** Determine v(t) if:

(i) s = -2, **V** = 2<0^{o} (ii) s = j2, **V** = 12<-30^{o} (iii) s = -4 + j3, **V** = 6<10^{o}
**(g)** Determine the Laplace transform of the following functions:

(i) e^{-at}sin(ωt)u(t) (ii) e^{-at}cos(ωt)u(t)

**The strings**: all PjProblems are at play. However, S_{7}P_{1}A_{17} (*containership - location*), S_{7}P_{2}A_{21} (*identity - physical*), S_{7}P_{3}A_{32} (*force - push*) and S_{7}P_{5}A_{51} (*change - physical*) are in focus.
**(a)** General representation of a sinusoidal excitation in the *time domain*: v(t) = Acos(ωt + θ) -------(1)

Where A is the amplitude of the sinusoid; ω is the radian frequency (angular velocity) and θ is the phase angle.

General representation of a sinusoidal excitation in *frequency domain*: **V**(jω) = Ae^{jθ} = A<θ (phasor form) -------(2)

The *frequency domain* is also called the *phasor domain*. It is a mathematical space formulated for analytical simplicity. In other words there are no *real phasor sinusoids*. It should be remembered that the radian frequency ω is implicit in (2) even though it is not explicitly represented.

General representation of a sinusoidal excitation in *complex frequency domain * (also called the **s** domain): **V**(s) = A<θ

Where s = σ + jω -------(3)

In essence the *complex frequency* domain is an extension of the *frequency domain* which takes into account the *damping* of sinusoids. A sinusoid can be *negatively damped* (σ is negative, figure 7.1(b), exponential decay); or *positively damped* (σ is positive, figure 7.1(c), exponential growth); or *undamped* (σ = 0, figure 7.1 (a), steady state sinusoid). There is *overdamped*, *critically damped* and *under damped* within the *damped* category. All of which has their respective formulas. It is sufficient to identify *s* and replace *jw* with *s* when *switching* from the *frequency domain* to the *complex frequency domain*.

A *damped* sinusoid in the time domain is represented as: v(t) = Ae^{σt}cos(ωt + θ) -------(4)
**(b)** Mathematical tools used to arrive at the general identities of sinusoidal excitations in the frequency and complex frequency domains are:

**(c)** Consider the complex exponential Ae^{j(ωt + θ)}.

By Euler's Identity, Ae^{(jωt + θ)} = Acos(ωt + θ) + jsin(ωt + θ)

So, v(t) = Acos(ωt + θ) = Re(Ae^{j(ωt + θ)}) = Re(Ae^{jθ}e^{jωt}) -------(5)

Equation(5) indicates that the general representation of a sinusoid in the time domain can be represented by the *real* component of a *complex vector* having *argument* (angle) = (ωt + θ) and *magnitude *= peak amplitude of the sinusoid.

If the *real notation* (Re) and the complex exponential e^{jωt} are removed from equation(5), we have equation (2), the *complex phasor* or *frequency domain* of v(t) = Acos(ωt + θ):
**V**(jω) = Ae^{jθ} = A<θ (phasor form). Phasors as previously noted, are mathematical formulations used to simplify analyses. Also the frequency component of equation(5) that was removed for simplification is not discarded. It is implicit in the phasor formulation

Now, define the complex frequency s = σ + jω, and the complex function **F**(s) = Ae^{st}e^{jθ} = Ae^{σt}e^{j(ωt + θ)} ------(6)

Equation (6) is the result of substituting s for jω in equation (5)

So, Re(F(s)) = e^{σt}Re(Ae^{j(ωt + θ)}) = Ae^{σt}cos(ωt + θ) = equation (4).

So, essentially the complex frequency domain consists of a *damping* component and a *phasor* component.

When σ = 0 (steady-state sinusoid), **F**(s) = complex frequency domain, **V**(s). Obtained by substituting s for jω in **V**(jω) = A<θ. This *phasor form* is similar to the Laplace transform with respect to circuit analysis.
**PjProblemstrings Sequence**

S_{7}P_{1}A_{17}: establishes the time domain

S_{7}P_{2}A_{21}: establishes the identity of v(t)

S_{7}P_{3}A_{32}:mathematical action pushes v(t) into frequency domain

S_{7}P_{1}A_{17}: frequency domain

S_{7}P_{5}A_{51}: change in identity

S_{7}P_{2}A_{21}: new mathematical identity in the frequency domain

Thus, **PjProblemstrings Sequence** is:

S_{7}P_{1}A_{17}S_{7}P_{2}A_{21}S_{7}P_{3}A_{32}S_{7}P_{1}A_{17}S_{7}P_{5}A_{51}S_{7}P_{2}A_{21}
**(di)** complex frequency s = σ + jω

Examining 5e^{-4t} in the context of Ae^{st}e^{jθ}, indicates σ = -4 and ω = 0

So, s = -4.

(ii) Examining cos2ωt in the context of Ae^{st}e^{jθ}, indicates s = j2ω

(iii) sin(ωt + 2θ) = cos(ωt + (2θ - 90^{o}))

Examining cos(ωt + (2θ - 90^{o})) in the context of Ae^{st}e^{jθ}, indicates s = jω

(iv) Examining 4e^{-2t}sin(3t - 50^{o}), indicates s = -2 + j3

(v) Examining 4e^{-3t}(2 + cos4t), indicates s = -3 + j4
**(ei)** v(t) = 5e^{-2t}, implies s = -2, A (peak magnitude) = 5 phase angle = 0.

So, s = -2 and V(s) = 5<0^{o}

(ii) v(t) = 5e^{-2t}cos(4t + 10^{o}), implies s = -2 + j4, A (peak magnitude) = 5 phase angle = 10^{o}.

So, s = -2 + j4 and V(s) = 5<10^{o}

(iii) v(t) = 4cos(2t - 20^{o}), implies s = j2, A (peak magnitude) = 4 phase angle = -20^{o}.

So, s = j2 and V(s) = 4<-20^{o}
**(fi)** s= -2, **V** = 2<0^{o}; implies v(t) = 2e^{-2t}

(ii) s= j2, **V** = 12<-30^{o}; implies v(t) = 12cos(2t - 30^{o})

(iii) s= -4 + j3, **V** = 6<10^{o}; implies v(t) = 6e^{-4t}cos(3t + 10^{o})
**(gi)**

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The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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