Question 404: AC fundamentals

Question

A circuit containing resistance R$$_1$$, inductance L$$_1$$ and capacitance C$$_1$$ connected in series resonates at the same frequency 'f$$_0$$' as another circuit containing R$$_2$$, L$$_2$$ and C$$_2$$ in series. If two circuits are connected in series then the new frequency at resonance is

Accepted Answer

When R$$_1$$, L$$_1$$ and C$$_1$$ are connected in series the resonant (angular) frequency is given by
$$\omega_{\mathrm{r}}=\frac{1}{\sqrt{\mathrm{L}_1 \cdot \mathrm{C}_1}} \quad 	ext{.... (1)}$$
Also, when $$\mathrm{R}_2, \mathrm{~L}_2, \mathrm{C}_2$$ are connected in series
then $$\omega_r=\frac{1}{\sqrt{L_2 \cdot C_2}}\quad 	ext{..... (2)}$$
By (1) and (2): $$\mathrm{L}_1 \mathrm{C}_1=\mathrm{L}_2 \mathrm{C}_2$$
If the two circuits are connected in series then the equivalent inductance is given by $$\mathrm{L}=\mathrm{L}_1+\mathrm{L}_2$$ and the equivalent capacitance is given by
$$\mathrm{C}=\frac{\mathrm{C}_1 \mathrm{C}_2}{\mathrm{C}_1+\mathrm{C}_2}$$
$$	herefore$$ Resonant (angular) frequency $$\omega=\frac{1}{\sqrt{	ext { LC }}}$$
$$\begin{aligned}
& \mathrm{LC}=\left(\mathrm{L}_1+\mathrm{L}_2ight) \cdot \frac{\mathrm{C}_1 \mathrm{C}_2}{\mathrm{C}_1+\mathrm{C}_2} \
&=\frac{\mathrm{L}_1 \mathrm{C}_1 \mathrm{C}_2+\mathrm{L}_2 \mathrm{C}_1 \mathrm{C}_2}{\mathrm{C}_1+\mathrm{C}_2} \quad\left[\because \mathrm{L}_1 \mathrm{C}_1=\mathrm{L}_2 \mathrm{C}_2ight] \
&=\frac{\mathrm{L}_2 \mathrm{C}_2\left(\mathrm{C}_2+\mathrm{C}_1ight)}{\mathrm{C}_1+\mathrm{C}_2}=\mathrm{L}_2 \mathrm{C}_2 \
& 	herefore \quad \omega=\frac{1}{\sqrt{\mathrm{L}_2 \mathrm{C}_2}}=\omega_{\mathrm{r}}
\end{aligned}$$

Answer

$$\mathrm{\frac{3}{4} f_r}$$

Answer

$$\frac{3}{2} \mathrm{f}_{\mathrm{r}}$$

Answer

$$\mathrm{2 f_r}$$

Answer

$$\mathrm{f}_{\mathrm{r}}$$

AC fundamentals

A circuit containing resistance R$_1, inductance L_1 and capacitance C_1 connected in series resonates at the same frequency 'f_0' as another circuit containing R_2, L_2 and C_2$ in series. If two circuits are connected in series then the new frequency at resonance is

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