Question

Consider Figure 3.4. For an ideal phase-shifting transformer, the impedance is unchanged when it is referred from one side to the other.

(a) True

(b) False

Short answer

Therefore, the correct option is (a): True

Step-by-step solution

Use the basic relationship of the ideal transformer

The complex turn’s ratio is for the ideal phase-shifting transformer is given by,

$$\begin{gathered} \mathbf{a}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{jf}}}{\mathbf{1}} \\ \mathbf{a}\mathbf{=}{\mathbf{e}}^{\mathbf{jf}} \end{gathered}$$

Here,$\varphi$ is the phase shift angle.

The relationship between the primary and secondary voltage is given by,

role="math" localid="1655115051085" $$\begin{gathered} {\mathbf{E}}_{\mathbf{1}}\mathbf{=}{\mathbf{aE}}_{\mathbf{2}} \\ {\mathbf{E}}_{\mathbf{1}}\mathbf{=}{\mathbf{e}}^{\mathbf{J}\mathit{\varphi }}{\mathbf{E}}_{\mathbf{2}} \end{gathered}$$

The relationship between the primary and secondary current is given by

$$\begin{gathered} I_1=\frac{I_2}{a^{*}} \\ {I}_1=\frac{I_2}{{\left(e^{j\varphi }\right)}^{*}} \\ {I}_1=\frac{I_2}{e^{-j\varphi }} \\ {I}_1=e^{j\varphi }{I}_2 \end{gathered}$$

Calculate the referred impedance from secondary to primary.

Consider the expression for the secondary impedance referred to primary.

$Z_2^{\text{'}}=\frac{E_1}{I_1}$

Substitute the values from equations (1) and (2).

$$\begin{gathered} Z_2^{\text{'}}=\frac{e^{j\varphi }{E}_2}{e^{j\varphi }{I}_2} \\ =\frac{E_2}{I_2} \\ =Z_2 \end{gathered}$$

From the above, the impedance is unchanged when referred from one side to other side.

Therefore, the correct option is (a): True