Consider the phasors

V$1141$:$796$ j$3$:$852144$:$25$ff$115$ and V$2144$ j$3145$ff$143$

Determine V$1+$ V$2,$ V$1$ V$2$ and V$1$

V$2$

$.$

Solution

Using Eq$.10\mathrm{.}3-8$

V$1$ V$2141$:$796$ j$3$:$852144$ j$31$:$7964$ j$143$:$85235$:$796$ j$6$:$852$

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$43210.$ Sinusoidal Steady$-$State Analysis

Two phasors, V$1$ and V$2,$ are equal to each other if and only if one of the following two conditions

is satisfied:

$1.$ Both Re$\{$V$1\}=$ Re$\{$V$2\}$ and Im$\{$V$1\}=$ Im$\{$V$2\}.$

$2.$ Both $|$V$1|=|$V$2|$ and ffV$114$ ffV$2.$

$($Conditions $1$ and $2$ are not independent. If V$1=$ V$2,$ then both conditions are satisfied. If either

condition is satisfied, then V$1=$ V$2$ and the other condition is also satisfied.$)$

The use of phasors to represent sinusoids is based on Euler$$s formula. Euler$$s formula is

e j f $14$ cos f $$ j sin f $10$:$3-12$

Consequently,

Ae j f $14$ A cos f $$ jA sin f

Using Eqs. $10\mathrm{.}3-4$ and $10\mathrm{.}3-5,$ we have

A cos f $$ jA sin f $14$ Afff

Consequently, Ae j f $14$ Afff $10$:$3-13$

Ae jf is called the exponential form of a phasor. The conversion between the polar and exponential

forms is immediate. In both, A is the amplitude of the sinusoid and f is the phase angle of the sinusoid$.$

Next, consider

Ae j$$ o t$$y $14$ A cos$$ ot $$ y j A sin$$ ot $$ y $10$:$3-14$

Taking the real part of both sides of Eq$\mathrm{.}10\mathrm{.}3-14$ gives

Acos$14$ ot $$ y Re Ae j$$ ot$$y n o $14$ Re Ae jy e j ot $10$:$3-15$

Consider a sinusoid and corresponding phasor

v t$14$ A cos$$ ot $$ y V and V$14$ o Affy $14$ Ae j y V $10$:$3-16$

Substituting Eq$.10\mathrm{.}3-16$ into Eq$.10\mathrm{.}3-15$ gives

v t$14$ Re V$$ o e j o t $10$:$3-17$

Next, consider a KVL or KCL equation from an ac circuit, for example,

$014$ X

i

v i$$t $10$:$3-18$

Using Eq$.10\mathrm{.}3-17,$ we can write Eq$.10\mathrm{.}3-18$ as

$014$ X

i

Re Vi$$ o e j ot $14$ Re e j ot

X

i

Vi$$ o

$()10$:$3-19$

Next, using Eq$.10\mathrm{.}3-10$

V$1$ V$2144$:$25$ff$1155$ff$143144$:$255$ ff$1151431421$:$25$ff$2581421$:$25$ff$102$

Finally,

V$1$

V$2$

$144$:$25$ff$115$

$5$ff$143$

$144$:$25$

$5$

ff$115143140$:$85$ff$28$