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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-2tsin(3t - 50o) (v) e-3t(2 + cos4t)
(e) Determine the complex frequency s and V(s) associated with:
(i) 5e-2t (ii) 5e-2tcos(4t + 10o) (iii) 4cos(2t - 20o)
(f) Determine v(t) if:
(i) s = -2, V = 2<0o (ii) s = j2, V = 12<-30o (iii) s = -4 + j3, V = 6<10o
(g) Determine the Laplace transform of the following functions:
(i) e-atsin(ωt)u(t) (ii) e-atcos(ωt)u(t)
The strings: all PjProblems are at play. However, S7P1A17 (containership - location), S7P2A21 (identity - physical), S7P3A32 (force - push) and S7P5A51 (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) = Aejθ = 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) = Aeσtcos(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 Aej(ωt + θ).
By Euler's Identity, Ae(jωt + θ) = Acos(ωt + θ) + jsin(ωt + θ)
So, v(t) = Acos(ωt + θ) = Re(Aej(ωt + θ)) = Re(Aejθejωt) -------(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 ejωt are removed from equation(5), we have equation (2), the complex phasor or frequency domain of v(t) = Acos(ωt + θ):
V(jω) = Aejθ = 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) = Aestejθ = Aeσtej(ωt + θ) ------(6)
Equation (6) is the result of substituting s for jω in equation (5)
So, Re(F(s)) = eσtRe(Aej(ωt + θ)) = Aeσtcos(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.
S7P1A17: establishes the time domain
S7P2A21: establishes the identity of v(t)
S7P3A32:mathematical action pushes v(t) into frequency domain
S7P1A17: frequency domain
S7P5A51: change in identity
S7P2A21: new mathematical identity in the frequency domain
Thus, PjProblemstrings Sequence is:
(di) complex frequency s = σ + jω
Examining 5e-4t in the context of Aestejθ, indicates σ = -4 and ω = 0
So, s = -4.
(ii) Examining cos2ωt in the context of Aestejθ, indicates s = j2ω
(iii) sin(ωt + 2θ) = cos(ωt + (2θ - 90o))
Examining cos(ωt + (2θ - 90o)) in the context of Aestejθ, indicates s = jω
(iv) Examining 4e-2tsin(3t - 50o), 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<0o
(ii) v(t) = 5e-2tcos(4t + 10o), implies s = -2 + j4, A (peak magnitude) = 5 phase angle = 10o.
So, s = -2 + j4 and V(s) = 5<10o
(iii) v(t) = 4cos(2t - 20o), implies s = j2, A (peak magnitude) = 4 phase angle = -20o.
So, s = j2 and V(s) = 4<-20o
(fi) s= -2, V = 2<0o; implies v(t) = 2e-2t
(ii) s= j2, V = 12<-30o; implies v(t) = 12cos(2t - 30o)
(iii) s= -4 + j3, V = 6<10o; implies v(t) = 6e-4tcos(3t + 10o)
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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