Question

An L-R-C series circuit with L = 0.120 H, R = 240 Ω, and C = 7.30 mF carries an rms current of 0.450 A with a frequency of 400 Hz. (a) What are the phase angle and power factor for this circuit? (b) What is the impedance of the circuit? (c) What is the rms voltage of the source? (d) What average power is delivered by the source? (e) What is the average rate at which electrical energy is converted to thermal energy in the resistor? (f) What is the average rate at which electrical energy is dissipated (converted to other forms) in the capacitor? (g) In the inductor?

Short answer

The phase angle is $45.8°$,power factor is 0.697,impedance is 344Ω, rms voltage of the source is 155V, the average power delivered by the source is 48.6W, the average rate at which electrical energy is converted to thermal energy in the resistor is 48.6W, the average rate at which energy is dissipated in the capacitor is 0 and the average rate at which energy is dissipated in the inductor is also 0.

Step-by-step solution

Step-1: Formula and important concepts

The angle between the voltage and current phasors is given by the equation

$\mathbf{tan\varphi }\mathbf{=}\left(\frac{X_L-X_C}{R}\right)$

The inductive reactance of the coil can be calculated by ${\mathbf{X}}_{\mathbf{L}}\mathbf{=}\mathbf{\omega L}$ and capacitive reactance by ${\mathbf{X}}_{\mathbf{C}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\omega C}}$. Impedance is given by $\mathbf{Z}\mathbf{=}\sqrt{{\mathbf{R}}^{\mathbf{2}}\mathbf{+}{\left(X_L-X_C\right)}^{\mathbf{2}}}$. The expression for rms voltage is${\mathbf{V}}_{\mathbf{rms}}\mathbf{=}{\mathbf{I}}_{\mathbf{rms}}\mathbf{Z}$ and the expression for average power is${\mathbf{P}}_{\mathbf{avg}}\mathbf{=}{\mathbf{V}}_{\mathbf{rms}}{\mathbf{I}}_{\mathbf{rms}}\mathbf{cos\varphi }$ and also ${\mathbf{P}}_{\mathbf{avg}}\mathbf{=}{\left(I_{\mathrm{rms}}\right)}^{\mathbf{2}}\mathbf{R}$.

Step-2: Calculations for phase angle, power factor and impedance 

Plug the values for L and to get inductive reactance and also C and $\omega$to get capacitive reactance.

$$\begin{gathered} {\mathrm{X}}_{\mathrm{L}}=2\mathrm{\tau \tau fL} \\ =2\mathrm{\tau \tau }\left(400\mathrm{Hz}\right)\left(0.120\mathrm{H}\right) \\ =301.6\mathrm{\Omega } \end{gathered}$$

$$\begin{gathered} {\mathrm{X}}_{\mathrm{C}}=\frac{1}{2\mathrm{\tau \tau fC}} \\ =\frac{1}{2\mathrm{\tau \tau }\left(400\mathrm{Hz}\right)\left(7.3\times {10}^{-6}\mathrm{F}\right)} \\ =54.5\mathrm{\Omega } \end{gathered}$$

$$\begin{gathered} \tan \varphi =\left(\frac{301.6-54.5}{240}\right) \\ =1.028 \end{gathered}$$

Hence, the angle is $\varphi =+45.8°$and the power factor$\cos \varphi =0.697$

$$\begin{gathered} \mathrm{Z}=\sqrt{\left({240}^2\right)+{\left(301.6-54.5\right)}^2}\mathrm{\Omega } \\ =344\mathrm{\Omega } \end{gathered}$$

Hence, the value of impedance is 344Ω.

Step-3: Calculations for rms voltage of the source

$$\begin{gathered} {\mathrm{V}}_{\mathrm{rms}}=\left(0.450\mathrm{A}\right)\left(344\mathrm{\Omega }\right) \\ =155\mathrm{V} \end{gathered}$$

Therefore, the rms voltage is 155V.

Step-4: Calculations for powers and rate of dissipation

$$\begin{gathered} {\mathrm{P}}_{\mathrm{avg}}=\left(155\mathrm{V}\right)\left(0.450\mathrm{A}\right)\left(0.697\right) \\ =48.6\mathrm{W} \end{gathered}$$

Therefore, the average power delivered by the source is 48.6W.

The average rate here represents the average power dissipated into the resistor, so we use equation ${\mathrm{P}}_{\mathrm{avg}}={\left({\mathrm{I}}_{\mathrm{rms}}\right)}^2\mathrm{R}$.

$$\begin{gathered} {\mathrm{P}}_{\mathrm{avg}}=\left(155\mathrm{V}\right)\left(0.450\mathrm{A}\right)\left(0.697\right) \\ =48.6\mathrm{W} \end{gathered}$$

It is the same result as of part (d).

So, as shown in the parts (d) and (e), all generated power is dissipated in the resistor R only. Hence, no power is consumed in the capacitor and inductor.

This occurs because the capacitor charges and discharges the energy when it is connected to ac source.

Hence, the answers for all the parts are:

(a)$\varphi =+45.8^{°}$ and$\mathrm{cos\varphi }=0.697$

(b)$Z=344\Omega$

(c) ${\mathrm{V}}_{\mathrm{rms}}=155\mathrm{V}$

(d) ${\mathrm{P}}_{\mathrm{avg}}=48.6\mathrm{W}$,(e)${\mathrm{P}}_{\mathrm{avg}}=48.6\mathrm{W}$

(f)${\mathrm{P}}_{\mathrm{avg}}=0$,(g)${\mathrm{P}}_{\mathrm{avg}}=0$