Sinusoidal Signals#

Definition: Sinusoidal Signal

\[x(t) = X_{\text{m}} \cos(\omega t + \phi)\]

Phasors#

Definition: Phasors

\[x(t) = X_{\text{m}}\cos(\omega t + \phi)\]

is the complex number given by the multiplication of the amplitude \(X_{\text{m}}\) with the complex exponential \(\mathrm{e}^{\mathrm{j}\phi}\):

\[X \overset{\text{def}}{=} X_m \mathrm{e}^{\mathrm{j}\phi}\]

Theorem: Signal from Phasor

\[x(t) = X_{\text{m}}\cos(\omega t + \phi)\]

is equal to the real part of the multiplication of its phasor \(X\) by \(\mathrm{e}^{\mathrm{j}\omega t}\):

\[x(t) = \operatorname{Re} (X \mathrm{e}^{\mathrm{j} \omega t})\]

Proof

TODO

Theorem: Signal Equality \(\iff\) Phasor Equality

Two sinusoidal signals are identical if and only if their phasors are the same.

Proof

TODO

Theorem: Linearity of the Phasor Transform

If a sinusoidal signal \(x(t)\) can be expressed as a linear combination

\[x(t) = \sum_{k=1}^n \lambda_k x_k(t)\]

of sinusoidal signals \(x_1(t), \dotsc, x_n(t)\) with the same frequency, then its phasor can be expressed as a linear combination of their phasors \(X_1, \dotsc, X_n\) with the same coefficients:

\[X = \sum_{k=1}^n \lambda_k X_k\]

Proof

TODO

Theorem: Differentiation and Phasors

\[x(t) = X_{\text{m}} \cos(\omega t + \phi)\]

has the phasor \(X\), then its derivative has the following phasor:

\[\mathrm{j}\omega X\]

Proof

TODO

Theorem: Antidifferentiation and Phasors

\[x(t) = X_{\text{m}} \cos(\omega t + \phi)\]

has the phasor \(X\), then its antiderivative has the following phasor:

\[\frac{1}{\mathrm{j}\omega} X\]

Proof

TODO