Question

There is a circuit with an inductor and two capacitors in series. Each capacitor is adjacent to the inductor. Capacitor 1 has charge Q1 and Capacitor 2 has charge Q2. Current is moving in the clockwise direction check this figure

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We are asked to derive a formula for the resonant frequency.

Answer 1

Solution: In the figure, two capacitors are connected in series and in series with inductance. The impedance of the circuit with alternating current frequency $w$ can be written as.

(1) $Z(w)=iwL+\frac {1}{iwC_1}+\frac {1}{iwC_2}$

Find the magnitude of impedance (1)

(2) $|Z(w)|=|i(Lw-\frac{C_1 + C_2}{w C_1\cdot C_2})|=|Lw-\frac{C_1 + C_2}{w C_1\cdot C_2}|$ .

Resonance in a sequential circuit is achieved at a frequency at which $|Z(w)|\to min$. In our case of the absence of resistive elements the exact condition must be met $|Z(w)|=0$, that is

(3) $Lw_r-\frac{C_1 + C_2}{w_r C_1\cdot C_2}=0$

From (3) we determine the resonant frequency $w_r$

(4) $w_r=\sqrt{\frac{C_1+C_2}{L\cdot C_1\cdot C_2}}$

The initial charges of the capacitors do not affect the resonant frequency of the circuit. Expression (4) corresponds to the well-known formula of the oscillatory contour $w_r=\frac{1}{\sqrt{LC}}$ if we take into account (5) $C=\frac{C_1\cdot C_2}{C_1+C_2}$ for the capacity of serial capacitors.

This formula can be obtained from the following consideration. When connected in series, electrical voltages of the components are added to each other. $U=U_1+U_2$ . According to the capacity definition we have $U_1=\frac {Q_1}{C_1}$, $U_2=\frac {Q_2}{C_2}$ and If they change over time $U(t)=\frac{Q_1(t)}{C_1}+\frac{Q_2(t)}{C_2}$ . The current or what is the same amount of charge flowing per unit of time through the capacitors in a serial circuit is the same i.e. $I(t)=\frac{dQ_1(t)}{dt}=\frac{dQ_2(t)}{dt}$ . This means that

(6) $\dot{U}(t)=I(t)\cdot (\frac{1}{C_1}+\frac{1}{C_2})$ . The voltage in the AC circuit is written as $U(t)=U(w)\cdot e^{iwt}$ and has a time derivative $\dot{U}(t)=iw\cdot U(w)\cdot e^{iwt}$ substitute to (6) we get

(7) $U(w)=I(w)\cdot (\frac{1}{iwC_1}+\frac{1}{iwC_2})$ when $I(t)=I(w)\cdot e^{iwt}$ , and the multiplier $e^{iwt}$ has decreased.

Using the definition of impedance $Z(w)=\frac{U(w)}{I(w)}$ we get (1) and (5).

Answer:  The formula for the resonant frequency is $w_r=\sqrt{\frac{C_1+C_2}{L\cdot C_1\cdot C_2}}$