Question

If a phasor representation of a current is given by $\mathbf{I}\mathbf{=}\mathbf{70}\mathbf{.}\mathbf{7}\mathbf{\angle }\mathbf{45}\mathbf{°}$, it is equivalent to

(a)localid="1652440498932" $\mathbf{100}{\mathbf{e}}^{\mathbf{J}\mathbf{45}\mathbf{°}}$

(b)localid="1652440503756" $\mathbf{100}\mathbf{+}\mathbf{j}\mathbf{100}$

(c) localid="1652440508375" $\mathbf{50}\mathbf{+}\mathbf{j}\mathbf{50}$

Short answer

Answer

Therefore, the correct option is (c); that is, $50+\mathrm{j}50$

Step-by-step solution

Define phasor form.

In the phasor form $\mathbf{70}\mathbf{.}\mathbf{7}\mathbf{\angle }\mathbf{45}\mathbf{°}$, the magnitude is presented in the rms form. Thus, 70.7 is the rms current and$\mathbf{45}\mathbf{°}$is the phase angle.

Obtain the rectangular form.

The rectangular form of$70.7\angle 45°$is obtained by multiplying the rms value by the cosine and the sine of the phase angle:

$70.7\left(\cos 45°+j\sin 45°\right)$ …… (1)

Simplify Equation (1)

Equation (1) is simplified as follows:

$$\begin{gathered} 70.7\left(\cos 45°+\mathrm{jsin}45°\right)=70.7\left(\frac{1}{\sqrt{2}}+\mathrm{j}\frac{1}{\sqrt{2}}\right) \\ =50+\mathrm{j}50 \end{gathered}$$

Therefore, the rectangular form of $70.7\angle 45°$is $50+\mathrm{j}50$. Hence (c) is the correct option.

Explanation for Option (a)

In Option (a), the given polar form has the rms voltage of 100 V while the correct rms voltage is 70.7 V and the rectangular form of $70.7\angle 45°$is $50+\mathrm{j}50$.

Thus (a) is an incorrect option.

Explanation for Option (b)

In Option (b), the rectangular form is $100+\mathrm{j}100$. But the correct rectangular form of $70.7\angle 45°$is $50+\mathrm{j}50$and the maximum voltage is 100 V. Thus (b) is an incorrect option.