Chapter 9.6, Problem 46E

a.

To determine

To convert:

The phasors to complex numbers in rectangular form.

Answer to Problem 46E

The complex numbers are given by $34.63+20j$ and $30+51.96j$ .

Explanation of Solution

Given information:

The phasors of two voltage sources are given by $40\angle {30}^{\circ }$ and $60\angle {60}^{\circ }$ .

Consider the phasors of two voltage sources $40\angle {30}^{\circ }$ and $60\angle {60}^{\circ }$ .

Converting these phasors to complex numbers in rectangular form:

  $\begin{array}{l}40\angle {30}^{\circ }=40\left(\cos 30+j\sin 30\right)\\ \Rightarrow 40\left(\frac{\sqrt{3}}{2}+j\left(\frac{1}{2}\right)\right)\\ \Rightarrow 34.63+20j\end{array}$

  $\begin{array}{l}60\angle {60}^{\circ }=60\left(\cos 60+j\sin 60\right)\\ \Rightarrow 60\left(\frac{1}{2}+j\left(\frac{\sqrt{3}}{2}\right)\right)\\ \Rightarrow 30+30\sqrt{3}j\\ \Rightarrow 30+51.96j\end{array}$

Therefore, the complex numbers are given by $34.63+20j$ and $30+51.96j$ .

b.

To determine

To find:

The total voltage in the circuit.

Answer to Problem 46E

The total voltage is given by $64.63+71.96j$ .

Explanation of Solution

Given information:

The phasors of two voltage sources are given by $40\angle {30}^{\circ }$ and $60\angle {60}^{\circ }$ .

From part (a) the complex numbers are given by $34.63+20j$ and $30+51.96j$ .

Now the total voltage is given by

  $34.63+20j+30+51.96j=64.63+71.96j$

Therefore, the total voltage is given by $64.63+71.96j$ .

c.

To determine

To find:

The sinusoidal function that gives the total voltage in the circuit.

Answer to Problem 46E

The sinusoidal function is given by $v\left(t\right)=96.73\sin \left(250t+48\right)$ .

Explanation of Solution

Given information:

The phasors of two voltage sources are given by $40\angle {30}^{\circ }$ and $60\angle {60}^{\circ }$ .

A sinusoidal function that gives the total voltage in the circuit is of the form,

  $v\left(t\right)=r\sin \left(250t+\theta \right)$

Now to determine the values of r and $\theta$ ,

  $r=\sqrt{{\left(64.63\right)}^2+{\left(71.96\right)}^2}=\sqrt{4177.37+5178.242}=\sqrt{\text{9355}\text{.612}}\approx \text{96}\text{.724}$

  $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)={\tan }^{-1}\left(\frac{71.96}{64.63}\right)=48$

  $v\left(t\right)=96.73\sin \left(250t+48\right)$

Therefore, the sinusoidal function is given by $v\left(t\right)=96.73\sin \left(250t+48\right)$ .