91Ó°ÊÓ

Resonance angular frequency is given by $\mathit{Ó\neg }\mathbf{=}\frac{\mathbf{1}}{\mathbf{L}\mathbf{C}}$.The reactance are given by ${\mathbf{X}}_{\mathbf{L}}\mathbf{=}\mathbf{Ó\neg L}$for inductor and ${\mathbf{X}}_{\mathbf{C}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{Ó\neg C}}$for capacitor.

$\mathbf{Z}\mathbf{=}\sqrt{{\mathbf{R}}^{\mathbf{2}}\mathbf{+}{\left(X_L-X_{C)}\right)}^{\mathbf{2}}}$, where Z is the impedance

${\mathrm{Ó\neg }}_1=\frac{1}{\sqrt{\left(\mathrm{L}\right)\left(\mathrm{C}\right)}}$

Now, all the parameters are doubled

$$\begin{gathered} {\mathrm{R}}_2=2{\mathrm{R}}_1 \\ {\mathrm{C}}_2=2{\mathrm{C}}_1 \\ {\mathrm{L}}_2=2{\mathrm{L}}_1 \end{gathered}$$

Find the new resonance frequency

$$\begin{gathered} \mathrm{Ó\neg }=\frac{1}{\sqrt{\left(2\mathrm{L}\right)\left(2\mathrm{C}\right)}} \\ =\frac{1}{\sqrt{4\mathrm{LC}}} \\ =\frac{1}{2\sqrt{\mathrm{LC}}} \\ =\frac{{\mathrm{Ó\neg }}_1}{2} \end{gathered}$$

Therefore, resonance angular frequency is halved.

First calculate the inductive reactance;

$$\begin{gathered} {\left({\mathrm{X}}_{\mathrm{L}}\right)}_2={\mathrm{Ó\neg L}}_2 \\ =\mathrm{Ó\neg }\left(2{\mathrm{L}}_1\right) \\ =2{\left({\mathrm{X}}_{\mathrm{L}}\right)}_1 \end{gathered}$$

And then the capacitive reactance;

$$\begin{gathered} {\left({\mathrm{X}}_{\mathrm{C}}\right)}_2=\frac{1}{{\mathrm{Ó\neg C}}_2} \\ =\frac{1}{\mathrm{Ó\neg }\left(2{\mathrm{C}}_1\right)} \\ =\frac{{\left({\mathrm{X}}_{\mathrm{C}}\right)}_1}{2} \end{gathered}$$

Then calculate the impedance;

$$\begin{gathered} \mathrm{Z}=\sqrt{{\mathrm{R}}^2+{\left({\mathrm{X}}_{\mathrm{L}}-{\mathrm{X}}_{\mathrm{C}}\right)}^2} \\ \mathrm{Z}=\sqrt{{\left(2\mathrm{R}\right)}^2+{\left(2{\mathrm{X}}_{\mathrm{L}}-\frac{{\mathrm{X}}_{\mathrm{C})}}{2}\right)}^2} \\ \mathrm{Z}=2\sqrt{{\left(\mathrm{R}\right)}^2+{\left({\mathrm{X}}_{\mathrm{L}}-\frac{{\mathrm{X}}_{\mathrm{C})}}{4}\right)}^2} \end{gathered}$$

Therefore, a) resonance angular frequency is halved, b) the inductive reactance doubles, c) capacitive reactance halves and d) Z increases without doubling.