Question

Question: A series RLC circuit has a resonant frequency of $\mathbf{600}\mathbf{}\mathbf{Hz}$. When it is driven at $\mathbf{800}\mathbf{}\mathbf{Hz}$, it has an impedance of and a phase constant of ${\mathbf{45}}^{\mathbf{°}}$. What are (a) R, (b) L, and (c) C for this circuit?

Short answer

1. Resistance of RLC circuit is $707\mathrm{\Omega }$.
2. The inductance of the RLC circuit is $32\mathrm{mH}$.
3. The capacitance of the RLC circuit is$21.9\mathrm{nF}$.

Step-by-step solution

Given

1. Driving frequency, ${\mathrm{f}}_{\mathrm{d}}=8\mathrm{kHz}=800\mathrm{Hz}$.
2. Resonant frequency, $\mathrm{f}=6\mathrm{kHz}=600\mathrm{Hz}$.
3. Impedance, $\mathrm{z}=1\mathrm{k\Omega }=1000\mathrm{\Omega }$.
4. Phase Constant,$\mathrm{\varphi }={45}^{°}$.

Determining the concept

From the formula for the phase angle, find the relation between the impedance and resistance. By substituting the given value of impedance, find the resistance. By using the relation between inductance and capacitance and substituting the given values, find the inductance and capacitance.

Formulae are as follows:

$$\begin{gathered} \text{tan}\mathrm{\varphi }=\left(\frac{{\mathrm{\omega }}_{\mathrm{d}}\mathrm{L}-\frac{1}{{\mathrm{\omega }}_{\mathrm{d}}\mathrm{C}}}{\mathrm{R}}\right) \\ \mathrm{z}=\sqrt{{\mathrm{R}}^2+{\left({\mathrm{\omega }}_{\mathrm{d}}\mathrm{L}-\frac{\text{1}}{{\mathrm{\omega }}_{\mathrm{d}}\mathrm{C}}\right)}^2}, \\ \mathrm{C}=\frac{\text{1}}{{\mathrm{\omega }}^2\mathrm{L}}, \\ \mathrm{\omega }=\text{2}\mathrm{\pi f}, \end{gathered}$$

Where, f is frequency, $\mathrm{\omega }$is angular frequency, R is resistance,C is capacitance.

(a) Determining the Resistance of the RLC circuit

The phase angle is given by,

$$\begin{gathered} \text{tan}\mathrm{\varphi }=\left(\frac{{\mathrm{\omega }}_{\mathrm{d}}\mathrm{L}-\frac{1}{{\mathrm{\omega }}_{\mathrm{d}}\mathrm{C}}}{\mathrm{R}}\right) \\ \text{tan}{45}^0=\left(\frac{{\mathrm{\omega }}_{\mathrm{d}}\mathrm{L}-\frac{1}{{\mathrm{\omega }}_{\mathrm{d}}\mathrm{C}}}{\mathrm{R}}\right) \end{gathered}$$

Therefore,

$\mathrm{R}={\mathrm{\omega }}_{\mathrm{d}}\mathrm{L}-\frac{\text{1}}{{\mathrm{\omega }}_{\mathrm{d}}\mathrm{C}}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \left(1\right)$

It is also known that,

$\mathrm{z}=\sqrt{{\mathrm{R}}^2+{\left({\mathrm{\omega }}_{\mathrm{d}}\mathrm{L}-\frac{\text{1}}{{\mathrm{\omega }}_{\mathrm{d}}\mathrm{C}}\right)}^2}$

Substitute,${\mathrm{\omega }}_{\mathrm{d}}\mathrm{L}-\frac{\text{1}}{{\mathrm{\omega }}_{\mathrm{d}}\mathrm{C}}=\mathrm{R}$

$$\begin{gathered} \mathrm{z}=\sqrt{{\mathrm{R}}^2+{\mathrm{R}}^2} \\ \mathrm{z}=\sqrt{2{\mathrm{R}}^2} \\ \mathrm{z}=\sqrt{2}\mathrm{R} \\ \mathrm{R}=\frac{\mathrm{z}}{\sqrt{2}} \\ \mathrm{R}=\frac{1000}{\sqrt{2}} \\ =707\mathrm{\Omega } \end{gathered}$$

Hence, the Resistance of the RLC circuit is $707\mathrm{\Omega }$.

(b) Determining the Inductance of the RLC circuit

It is known that,

$\mathrm{C}=\frac{\text{1}}{{\mathrm{\omega }}^2\mathrm{L}}$

Substitute the value of c in the equation (1),

$$\begin{gathered} \mathrm{R}={\mathrm{\omega }}_{\mathrm{d}}\mathrm{L}-\frac{\text{1}}{{\mathrm{\omega }}_{\mathrm{d}}\left(\frac{\text{1}}{{\mathrm{\omega }}^2\mathrm{L}}\right)} \\ \mathrm{R}={\mathrm{\omega }}_{\mathrm{d}}\mathrm{L}-\frac{{\mathrm{\omega }}^2\mathrm{L}}{{\mathrm{\omega }}_{\mathrm{d}}} \\ \mathrm{R}=\mathrm{L}\left({\mathrm{\omega }}_{\mathrm{d}}-\frac{{\mathrm{\omega }}^2}{{\mathrm{\omega }}_{\mathrm{d}}}\right) \end{gathered}$$

Rearranging the terms,

$$\begin{gathered} \mathrm{L}=\frac{\mathrm{R}}{\left({\mathrm{\omega }}_{\mathrm{d}}-\frac{{\mathrm{\omega }}^2}{{\mathrm{\omega }}_{\mathrm{d}}}\right)} \\ \mathrm{\omega }=\text{2}\mathrm{\pi f} \\ \mathrm{L}=\frac{\mathrm{R}}{\text{2}\mathrm{\pi }\left({\mathrm{f}}_{\mathrm{d}}-\frac{{\mathrm{f}}^2}{{\mathrm{f}}_{\mathrm{d}}}\right)} \end{gathered}$$

Substituting the given quantities,

$$\begin{gathered} \mathrm{L}=\frac{707}{\left(2\mathrm{\pi }\right)(8000-{\displaystyle \frac{{6000}^2}{8000}})} \\ \mathrm{L}=32\times {10}^3\mathrm{H}=32\mathrm{mH} \end{gathered}$$

Hence, the Inductance of the RLC circuit is $32\mathrm{mH}$.

(c) Determining the Capacitance of the RLC circuit

It is known that,

$$\begin{gathered} \mathrm{C}=\frac{\text{1}}{{\mathrm{\omega }}^2\mathrm{L}} \\ \mathrm{c}=\frac{1}{{\left(2\mathrm{\pi f}\right)}^2\mathrm{L}} \\ \mathrm{c}=\frac{1}{{\left(2\mathrm{\pi }6000\right)}^{2}32\times {10}^3} \\ \mathrm{c}=21.9\times {10}^{-9}\mathrm{F}=21.9\mathrm{nF} \end{gathered}$$

Hence, the capacitance of the RLC circuit is $21.9\mathrm{nF}$.

Using the formula for phase angle and impedance, found the resistance. Using the relation between inductance and capacitance, found inductance and capacitance.