Sinusoids

A sinusoid is a signal that has the form of the sine or cosine function.

A sinusoidal current is usually referred to as alternating current (ac). Such a current reverses at regular time intervals and has alternately positive and negative values. Circuits driven by sinusoidal current or voltage sources are called ac circuits.

Reasons:

Nature itself is characteristically sinusoidal.
Sinusoidal signal is easy to generate and transmit.
Through fourier analysis, any practical periodic signal can be represented by a sum of sinusoids.
A sinusoid is easy to handle mathematically. The derivative and integral of a sinusoid are themselves sinusoids.

A periodic function is one that satisfies \(f\left(t\right) = f\left(t + nT\right)\), for all \(t\) and for all integers \(n\).

\(\boxed{f\left(t\right) = f\left(t + nT\right)}\)

The period \(T\) of the periodic function is the time of one complete cycle or the number of seconds per cycle.

\(\boxed{T = \frac{2\pi}{\omega}}\)

where \(\omega\) is the angular frequency, measured in radians per second.

The reciprocal of the period \(T\) is the number of cycles per second, known as the cyclic frequency \(f\) of the sinusoid, measured in hertz (Hz).

\(\boxed{f = \frac{1}{T}}\)

The angular frequency \(w\) from the equations of period \(T\) and cyclic frequency \(f\), measured in radians per second.

\(\displaystyle T = \frac{2\pi}{\omega}\)

\(\displaystyle \frac{1}{f} = \frac{2\pi}{\omega}\)

\(\boxed{\omega = 2\pi f}\)

General expression for the sinusoid.

\(\boxed{v\left(t\right) = V_m \sin{\left(\omega t + \phi\right)}}\)

where:

\(V_m\) is the amplitude of the sinusoid.

\(\omega\) is the angular frequency, measured in radians per second.

\(\phi\) is the phase.

Using the two sinusoids \(v_1\left(t\right) = V_m \sin\left(\omega t\right)\) and \(v_2\left(t\right) = V_m \sin\left(\omega t + \phi\right)\).

If \(\phi \ne 0\), then \(v_1\) and \(v_2\) are out of phase. If \(\phi = 0\), then \(v_1\) and \(v_2\) are said to be in phase.

A sinusoid can be expressed in either sine or cosine form. When comparing two sinusoids, it is expedient to express both as either sine or cosine with with positive amplitudes. Using the trigonometric identities:

\(\sin\left(A \pm B\right) = \sin\left(A\right)\cos\left(B\right) \pm \cos\left(A\right)\sin\left(B\right)\)

\(\cos\left(A \pm B\right) = \cos\left(A\right)\cos\left(B\right) \mp \sin\left(A\right)\sin\left(B\right)\)

With these identities

\(\boxed{\sin\left(\omega t \pm 180^{\circ}\right) = -\sin\left(\omega t\right)}\)

\(\boxed{\cos\left(\omega t \pm 180^{\circ}\right) = -\cos\left(\omega t\right)}\)

\(\boxed{\sin\left(\omega t \pm 90^{\circ}\right) = \pm\cos\left(\omega t\right)}\)

\(\boxed{\cos\left(\omega t \pm 90^{\circ}\right) = \mp\sin\left(\omega t\right)}\)

Use these relationships to transform a sinusoid from sine form to cosine form or vice versa.

A graphical approach may be used to relate or compare sinusoids as an alternative to using trigonometric identities. The horizontal axis represents the magnitude of cosine, while the vertical axis (pointing down) denotes the magnitude of sine. Angles are measured positively counter-clockwise.

The graphical technique can also be used to add two sinusoids of the same frequency when one is in sine form and the other is in cosine form.

\(\boxed{A\cos\left(\omega t\right) + B\sin\left(\omega t\right) = C\cos\left(\omega t - \theta\right)}\)

where:

\(\displaystyle C = \sqrt{A^2 + B^2}\)

\(\displaystyle \theta = \tan^{-1}\left(\frac{B}{A}\right)\)

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Phasors

Adding sinusoids of the same frequency is equivalent to adding their corresponding phasors.

Sinusoids are easily expressed in terms of phasors which are more convenient to work with than sine and cosine functions.
Fourier Transform

Fourier series enable us to represent a periodic function as sum of sinusoids and to obtain the frequency spectrum from the series. The Fourier transform allows us to extend the concept of a frequency spectrum to non-periodic functions.
Fourier Series

Since the Fourier series describes the periodic signal \(f(t)\) as a sum of sinusoids at different frequencies, its signal power also equals the sum of the power in each sinusoid.

The (trigonometric) Fourier series of a periodic function \(f(t)\) is a representation that resolves \(f(t)\) into a DC component and an AC comprising an infinite series of harmonic sinusoids.

where \(\omega_0 = 2\pi / T\) is called the fundamental frequency in radians per second. The sinusoid \(\sin(n\omega_{0}t)\) or \(\cos(n\omega_{0}t)\) is called the \(n\mathrm{th}\) harmonic of \(f(t)\); it is an odd harmonic if \(n\) is odd and an even harmonic if \(n\) is even. The constants \(a_n\) and \(b_n\) are the Fourier coefficients. The coefficient \(a_0\) is the DC component or the average value of \(f(t)\) (since sinusoids have zero average values). For \(n \ne 0\), the coefficients \(a_n\) and \(b_n\) are the amplitudes of the sinusoids in the AC component.