Question

Consider an ideal transformer with ${\mathit{N}}_{\mathbf{1}}\mathbf{=}\mathbf{3000}$turns and ${\mathit{N}}_{\mathbf{1}}\mathbf{=}\mathbf{1000}$turns. Let winding 1 be connected to a source whose voltage is ${\mathit{e}}_{\mathbf{1}}\left(e\right)\mathbf{=}\mathbf{100}\left(1-\left|t\right|\right)\mathbf{}\mathit{v}\mathit{o}\mathit{l}\mathit{t}\mathit{s}$for $\mathbf{-}\mathbf{1}\mathbf{\le }\mathit{t}\mathbf{\le }\mathbf{1}$and ${\mathit{e}}_{\mathbf{1}}\left(t\right)\mathbf{=}\mathbf{0}$for $\left|t\right|\mathbf{>}\mathbf{1}$. A 2-farad capacitor is connected across winding 2. Sketch ${\mathit{e}}_{\mathbf{1}}\left(t\right)\mathbf{}\mathbf{,}\mathbf{}{\mathit{e}}_{\mathbf{2}}\left(t\right)\mathbf{}\mathbf{,}\mathbf{}{\mathit{i}}_{\mathbf{1}}\left(t\right)\mathbf{}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathbf{}{\mathit{i}}_{\mathbf{2}}\left(t\right)$versus time $\mathit{t}$.

Short answer

Answer

Therefore, the plot of $e_1\left(t\right)$is drawn below:

The plot of $e_1\left(t\right)$is drawn below:

The plot of $i_1\left(t\right)$is drawn below:

The plot of $i_2\left(t\right)$is drawn below:

Step-by-step solution

Determine the formulas of transformer ratio and secondary current.

Write the formula of transformer ratio.

$\frac{{\mathbf{e}}_{\mathbf{2}}\left(t\right)}{{\mathbf{e}}_{\mathbf{1}}\left(t\right)}\mathbf{=}\frac{{\mathbf{N}}_{\mathbf{2}}}{{\mathbf{N}}_{\mathbf{1}}}$ ……. (1)

Write the formula of secondary current

${\mathit{i}}_{\mathbf{2}}\left(t\right)\mathbf{=}\mathit{C}\frac{\mathbf{d}\mathbf{e}\left(t\right)}{\mathbf{d}\mathbf{t}}$ ……. (2)

Write the formula of transformer ratio in terms of current.

$\frac{{\mathbf{i}}_{\mathbf{1}}\left(t\right)}{{\mathbf{i}}_{\mathbf{2}}\left(t\right)}\mathbf{=}\frac{{\mathbf{N}}_{\mathbf{2}}}{{\mathbf{N}}_{\mathbf{1}}}$ ……. (3)

Sketch graph for the source voltage, secondary voltage, primary current and secondary current.

From the given data, the source voltage $e_1\left(t\right)$is exists between $-1\le t\le 1$, and it is zero for $t\ge 1$and $t\le -1$.

For $t=0,e_1\left(t\right)-100V$hence sketch for source voltage is drawn below:

Determine the secondary voltage.

Substitute $100\left(1-\left|t\right|\right)Vfor{e}_1\left(t\right),3000for{N}_{1}and1000for{N}_2$ in equation (1).

$$\begin{gathered} i_2\left(t\right)=2\frac{d}{dt}\frac{100}{3}\left(1-\left|t\right|\right) \\ =\frac{200}{3}\frac{d}{dt}\left(1-\left|t\right|\right)A \end{gathered}$$

The sketch for $e_2\left(t\right)$is drawn below:

Determine the secondary current

Substitute $2F$for C and $\frac{100}{3}\left(1-\left|t\right|\right)$for $e_2\left(t\right)$in equation (2).

$$\begin{gathered} i_2\left(t\right)=2\frac{d}{dt}\frac{100}{3}\left(1-\left|t\right|\right) \\ =\frac{200}{3}\frac{d}{dt}\left(1-\left|t\right|\right)A \end{gathered}$$

The sketch for $i_2\left(t\right)$is drawn below:

Determine the primary current

Substitute 3000 for $N_1$and 1000 for $N_2$ in equation (3).

$$\begin{gathered} \frac{i_1\left(t\right)}{i_2\left(t\right)}=\frac{1000}{3000} \\ {i}_1\left(t\right)=\frac{1}{3}{i}_2\left(t\right)A \\ {i}_1\left(t\right)=\frac{1}{3}\times \frac{200}{3}\frac{d}{dt}\left(1-\left|t\right|\right) \\ {i}_1\left(t\right)=\frac{200}{9}\frac{d}{dt}\left(1-\left|t\right|\right)A \end{gathered}$$

The sketch for $i_1\left(t\right)$is drawn below: