Question

Consider the circuit shown in Figure 2.23 in time domain. Convert the entire circuit into phasor domain.

Short answer

Therefore, the circuit diagram in the time domain is shown in figure below.

Step-by-step solution

Determine the formulas to convert the values from the time domain to the phasor domain.

Consider the formula to convert the voltage from the time domain to the phasor domain as,

$\mathbf{V}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{max}}}{\sqrt{\mathbf{2}}}\mathbf{\angle }\mathbf{\delta }$ ……. (1)

Write the formula to convert the inductive reactance.

${\mathbf{X}}_{\mathbf{L}}\mathbf{=}\mathbf{j\omega L}$ ……. (2)

Write the formula to convert the capacitive reactance.

${\mathbf{X}}_{\mathbf{c}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{j\omega C}}$ ……. (3)

Determine the value of the voltage in the phasor domain.

Substitute the values in equation (1) and solve for the voltage.

$$\begin{gathered} \mathrm{V}=\frac{120\sqrt{2}}{\sqrt{2}}(\angle -{30}^0) \\ =120(\angle -{30}^0)\mathrm{V} \end{gathered}$$

Note that the value of the resistance remains same in the time domain and the phasor domain.

$$\begin{gathered} {\mathrm{R}}_{\mathrm{T}}=5.76\mathrm{m\Omega } \\ {\mathrm{R}}_{\mathrm{L}}=5\mathrm{\Omega } \end{gathered}$$

Determine the value of the reactance in the phasor domain.

Substitute the values into equation (2) and solve for the inductive reactance.

Solve for inductor for 5 mH.

$$\begin{gathered} {\mathrm{X}}_{\mathrm{L}}=\mathrm{j}\left(377\right)(5\times {10}^{-3}\mathrm{H}) \\ =\mathrm{j}1.885\mathrm{\Omega } \end{gathered}$$

Solve for inductor for $30.6\mathrm{\mu H}$.

$$\begin{gathered} {\mathrm{X}}_{\mathrm{L}}=\mathrm{j}\left(377\right)(5\times {10}^{-6}\mathrm{H}) \\ =\mathrm{j}11.536m\mathrm{\Omega } \end{gathered}$$

Substitute the values into equation (3) and solve for the capacitive reactance.

role="math" localid="1655112359545" $$\begin{gathered} {\mathrm{X}}_{\mathrm{c}}=\frac{1}{\mathrm{j}\left(377\right)(921\times {10}^{-6}) \\ =\mathrm{j}2.88\mathrm{\Omega }} \end{gathered}$$

Determine the value of the circuit in the phasor domain.

Substitute the transformed values in the circuit diagram.

The required circuit diagram is shown in figure below.