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Resonance

A resonant frequency   is defined to be a frequency where the magnitude $\vert H(j\omega) \vert$ is a maximum.

Consider a series RLC circuit with R=1 $\Omega$, L=0.5 H, and C=0.5 F. Let v_out be the voltage drop across the resistor. The magnitude Bode diagram is shown in Figure 52.

  

Figure 52: Resonant circuit Bode diagram.

Let's calculate the resonant frequency $\omega_r$. The transfer function from source voltage to resistor voltage is

$$H(s) = \frac{sR/L}{s^2 + (R/L)s +1/(LC)} .$$ (63)

Here, resonance will occur at the frequency for which the reactance is zero, i.e.

$$\omega_r - \frac{1}{\omega_r C} = 0$$

which gives

$$\omega_r = \frac{1}{\sqrt{LC}} = 2 \ {\rm rad/sec} .$$ (64)

Resonance   is characterized by the peaking of the magnitude frequency response. A measure of resonance is the quality factor Q, defined as

$$Q = \frac{\omega_r}{B} ,$$

where

$$B = \omega_1-\omega_2$$

is the bandwidth  , and $\omega_1$ is the upper 3 dB point, and $\omega_2$ is the lower 3 dB point.

A high value of Q means a sharper peak. In the RLC case, smaller R means higher Q (lighter damping).

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