91影视

1. Resistance,$R=200\text{惟}$
2. Capacitance,$C=70.0\text{渭贵}=70.0脳{10}^{-6}\text{F}$
3. Inductance,$L=230\text{mH}=230脳{10}^{-3}\text{H}$
4. Driving frequency,$f_d=60.0\text{Hz}$
5. Amplitude of emf${蔚}_m=36.\text{0}\text{V}$

An electrical circuit composed of a capacitor, a resistor and an inductor, connected in series, is known as anLCRcircuit. For a series LCR circuit, we have the relations mentioned below, to find out the inductive and capacitive reactance. By using these relations, we can find out the impedance of the circuit, phase angle, and the amplitude of the current.

The impedance of theLCRcircuit for the driving frequency,

$\mathit{Z}\mathbf{=}\sqrt{{\mathit{R}}^{\mathbf{2}}\mathbf{+}{\left({\mathit{X}}_{\mathit{L}}\mathbf{-}{\mathit{X}}_{\mathit{c}}\right)}^{\mathbf{2}}}$ ...(1)

The capacitive reactance of the capacitor,

${\mathit{X}}_{\mathit{C}}\mathbf{=}\frac{\mathbf{1}}{{\mathit{蝇}}_{\mathit{d}}\mathit{C}}$ ...(2)

The inductive reactance of the inductor,

${\mathit{X}}_{\mathit{L}}\mathbf{=}{\mathit{蝇}}_{\mathit{d}}\mathit{L}$ ...(3)

The angular frequency of the LC oscillation,

${\mathit{蝇}}_{\mathit{d}}\mathbf{=}\mathbf{2}\mathit{蟺}{\mathit{f}}_{\mathit{d}}$ ...(4)

The phase angle of RLC circuit,

$\mathit{t}\mathit{a}\mathit{n}蠒\mathbf{=}\left(\frac{{\mathit{X}}_{\mathit{L}}\mathbf{-}{\mathit{X}}_{\mathit{C}}}{\mathit{R}}\right)\mathbf{}$ ...(5)

The current equation using Ohm鈥檚 law,

$\mathit{I}\mathbf{=}\frac{{\mathit{蔚}}_{\mathit{m}}}{\mathit{Z}}$ ...(6)

Here, $\mathit{R}$is the resistance of the resistor,$\mathit{C}$is the capacitance of the capacitor, $\mathit{L}$is the inductance of the inductor and ${\mathit{f}}_{\mathit{d}}$is the frequency of the LC oscillation.

First, the driving angular frequency is calculated using equation (4) as follows:

Capacitive reactance is given using equation (2) as:

$$\begin{gathered} X_c=\frac{1}{\left(376.8\text{rad/s}\right)脳\left(70.0脳{10}^{-6}\text{F}\right)} \\ =37.9\text{惟} \end{gathered}$$

Inductive reactance is given using equation (3) as:

$$\begin{gathered} X_L=376.8\text{rad/s}脳\left(230脳{10}^{-3}\right)\text{H} \\ =86.7\text{惟} \end{gathered}$$

Thus, the impedance of the RLC circuit is given using the above values and given data in equation (1) as follows:

$$\begin{gathered} Z=\sqrt{{\left(200\text{惟}\right)}^2+{\left(86.7鈥�\text{惟}-37.9\text{惟}\right)}^2} \\ =\sqrt{40000{\text{惟}}^2+2381.5{\text{惟}}^2} \\ =205.86\text{惟} \\ 鈮�206.0\text{惟} \end{gathered}$$

Hence, the value of the impedance is $206.0\text{惟}$.

Now, using the calculated values from part (a) in equation (5), we can get the phase angle of the RLC circuit as follows:

$$\begin{gathered} 蠒={\tan }^{-1}\left(\frac{86.7\text{惟}-37.9\text{惟}}{200\text{惟}}\right) \\ ={\tan }^{-1}\left(0.244\right) \\ =13.7掳 \end{gathered}$$

Hence, the value of the phase is role="math" localid="1662801363373" $13.7掳$.

Amplitude of current is calculated using the given data and impedance value in equation (6) as follows:

$$\begin{gathered} I=\frac{{蔚}_m}{Z} \\ =\frac{36.0\text{V}}{206\text{惟}} \\ =0.175\text{A} \end{gathered}$$

Hence, the value of the current is $0.175\text{A}$.

From part (a), we have,$X_L>X_C$

Phase angle, $蠒=13.70掳>0,$i.e., positive.

The circuit is more inductive than capacitive, so the corresponding phasor diagram is shown below.

Hence, the phasor diagram is plotted.