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?

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,$\mathrm{\varphi }={45}^{°}$.

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.

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 $707\mathrm{\Omega }$.

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 $32\mathrm{mH}$.

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.