Question

In symmetrical-component theory, express the complex-number operator $\mathit{a}\mathbf{=}\mathbf{1}\mathbf{\angle }{\mathbf{120}}^{\mathbf{°}}$in exponential and rectangular forms.

Short answer

Answer

The exponential and rectangular form of the operator $a$are $1e^{j120}$ and $0.5+j0.866$respectively.

Step-by-step solution

concept of conversion from the polar form to rectangle and exponential.

The equation to convert the from polar form to rectangular form is given as follows.

$\mathit{A}\mathbf{\angle }\mathit{\varphi }\mathbf{=}\mathit{A}\mathit{c}\mathit{o}\mathit{s}\left(\varphi \right)\mathbf{+}\mathit{j}\mathit{s}\mathit{i}\mathit{n}\left(\varphi \right)$ …… (1)

Here $\mathit{A}$is the amplitude and $\mathit{\varphi }$is the angle.

The equation to convert from polar to exponent form is given as follows.

$\mathit{A}\mathit{e}\mathbf{\angle }\mathit{\varphi }\mathbf{=}\mathit{A}{\mathit{e}}^{\mathbf{j}\mathbf{\varphi }}$ …… (2)

Express the operator into exponent and rectangular forms.

Express the operator $a$into exponent form.

$1\angle {120}^{°}=1e^{j120}$

Express the operator $a$into rectangular form.

$$\begin{gathered} 1\angle {120}^{°}=1\left[\cos \left({120}^{°}\right)+j\sin \left({120}^{°}\right)\right] \\ 1\angle {120}^{°}=1\left(\frac{1}{2}+j\frac{\sqrt{3}}{2}\right) \\ 1\angle {120}^{°}=0.5+j0.866 \end{gathered}$$

Hence, the exponential and rectangular form of the operator $a$are $1e^{j120}$ and $0.5+j0.866$respectively.