ELECTRIC SHOCK : Sinusoids and Phasors

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 pos-
itive and negative values. Circuits driven by sinusoidal current or volt-
age sources are called ac circuits.

A sinusoidal forcing function produces both a transient response
and a steady-state response, much like the step function, which we stud-
ied in Chapters 7 and 8. The transient response dies out with time so
that only the steady-state response remains. When the transient response
has become negligibly small compared with the steady-state response,
we say that the circuit is operating at sinusoidal steady state. It is this
sinusoidal steady-state response.

A sinusoid having period T and angular frequency

can be represented as

A periodic function is one that satisﬁes f (t) f (tnT), for all t and
for all integers n.

Basically a rotating vector, simply called a “Phasor” is a scaled line whose length represents an AC quantity that has both magnitude (“peak amplitude”) and direction (“phase”) which is “frozen” at some point in time.

A phasor is a vector that has an arrow head at one end which signifies partly the maximum value of the vector quantity ( V or I ) and partly the end of the vector that rotates.

Polar representation: $z = \vert z\vert\;e^{j\angle z}$ , or simply $z=\vert z\vert\;\angle z$ , where $\vert z\vert$ and $\angle z$ are the magnitude and phase angle, respectively.

The two representations can be converted from one to the other:

From to $(\vert z\vert,\angle z)$ :

$\begin{displaymath}\left\{ \begin{array}{ll} \vert z\vert=\sqrt{x^2+y^2} & \mbox... ...gle z=tan^{-1} (y/x) & \mbox{phase angle} \end{array} \right. \end{displaymath}$
From $(\vert z\vert,\angle z)$ to :

$\begin{displaymath}z=\vert z\vert\;e^{j\angle z}=\vert z\vert(\cos\angle z+j\;\sin\angle z)=x+jy \end{displaymath}$

due to Euler identity, i.e.,

$\begin{displaymath}\left\{ \begin{array}{ll} x=\vert z\vert\;\cos\angle z & \mbo... ...ert\;\sin\angle z & \mbox{imaginary part} \end{array} \right. \end{displaymath}$

The arithmetic operations of two complex numbers $z=x+jy=\vert z\vert\;e^{j\angle z}$ and $w=u+jv=\vert w\vert\;e^{j\psi}$ are listed below:

Add/Subtract:

$\begin{displaymath}z+w=(x+u)+j(y+v),\;\;\;\;z-w=(x-u)+j(y-v) \end{displaymath}$
Multiply:

$\begin{displaymath}z\;w=(x+jy)(u+jv)=(xu-yv)+j(xv+yu)=\vert z\vert\;\vert w\vert\;e^{j(\angle z+\angle w)} \end{displaymath}$
Divide:

$\begin{displaymath}\frac{z}{w}=\frac{x+jy}{u+jv}=\frac{(x+jy)(u-jv)}{(u+jv)(u-jv... ... =\frac{\vert z\vert}{\vert w\vert}\;e^{j(\angle z-\angle w)} \end{displaymath}$

ELECTRIC SHOCK

Sabado, Enero 3, 2015

Sinusoids and Phasors

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