Electrical Circuits

Figure 4.1:

Electrical Circuits are the most common technological application where we see resonances. The LCR circuit shown in Figure 4.1 characterizes the typical situation. The circuits includes a signal generator which produces an AC signal of voltage amplitude $V$ at frequency $\omega$. Applying Kirchoff's Law to this circuit we have,

which may be written solely in terms of the charge $q$ as

$$L \ddot{Q} + R \dot{Q} + \frac{Q}{C}=V e^{i \omega t} ,.$$ (4.2)

We see that this is a damped oscillator with an external sinusoidal force. The equation governing this is

$$\ddot{Q} + 2 \beta \dot{Q} + \omega_0^2 Q =\tilde{v}e^{i \omega t}$$ (4.3)

where $\omega_0^2=1/LC$, $\beta=R/2L$ and $v=(V/L)$.

We next consider the power dissipated in this circuit. The resistance is the only circuit element which draws power. We proceed to calculate this by calculating the impedence

$$\tilde{Z}(\omega)=i \omega L - \frac{i}{\omega C} + R$$ (4.4)

which varies with frequency. The voltage and current are related as , which gives the current

$$\tilde{I}=\frac{\tilde{V}}{i(\omega L-1/\omega C)+R}\,.$$ (4.5)

The average power dissipated may be calculated as $\langle P(\omega) \rangle =R \tilde{I}\tilde{I}^*/2$ which is

$$\langle P(\omega)\rangle = \frac{\omega^2}{(\omega_0^2-\ome... ...4 \beta^2 \omega^2} \left( \frac{R V^2}{2 L^2} \right)\,.$$ (4.6)

Problem For an Electrical Oscillator with $L= 10 mH$ and $C=1 \mu F$,

a.

what is the natural (angular) frequency $\omega_0$? ($10 KHz$)

b.

Choose $R$ so that the oscillator is critically damped. ($200 \, \Omega$)

c.

For $R= 2 \, \Omega$, what is the maximum power that can be drawn from a $10 \, V$ source? ($25 W$)

d.

What is the FWHM of the peak? ($200 \, Hz$)

e.

At what frequency is half the maximum power drawn? ($10.1 KHz$ and $9.9 KHz$)

f.

What is the value of the quality factor $Q$? ( $Q=\omega_0/2 \beta=50$ )

g.

What is the time period of the oscillator? ( $T=2 \pi/\omega$ where $\omega=\omega \sqrt{1-\beta^2/\omega_0^2} \approx 10 KHz$ and $T=2 \pi 10^{-4} sec$.)

h.

What is the value of the log decrement $\lambda$? ( $\tilde{x}(t)=[\tilde{A} e^{- \beta t}] e^{i \omega t}$ , $\lambda=\ln(x_n/x_{n+1})=\beta T=2 \pi 10^{-2}$)