The Quality Factor (Q)

The quality factor Q of a series resonant circuit is defined as the ratio of the reactive power of either the inductor or the capacitor to the average power of the resistor at resonance; that is,
$$ \bbox[10px,border:1px solid grey]{Q_s = { \text{reactive power} \over \text{average power}}} \tag{1}$$
The quality factor is also an indication of how much energy is placed in storage (continual transfer from one reactive element to the other) compared to that dissipated. The lower the level of dissipation for the same reactive power, the larger the $Q_s$ factor and the more concentrated and intense the region of resonance.
Substituting for an inductive reactance in Eq. (1) at resonance gives us
$$ Q_s = { I^2 X_L \over I^2 R}$$
$$ \bbox[10px,border:1px solid grey]{Q_s = { X_L \over R} = { w_s L \over R}} \tag{2}$$
If the resistance R is just the resistance of the coil ($R_l$), we can speak of the Q of the coil, where
$$ \bbox[10px,border:1px solid grey]{Q_{coil} = Q_l = { X_L \over R_l}} \tag{3}$$
Since the quality factor of a coil is typically the information provided by manufacturers of inductors, it is often given the symbol Q without an associated subscript. It would appear from Eq. (3) that $Q_l$ will increase linearly with frequency since $X_L = 2\pi fL$. That is, if the frequency doubles, then $Q_l$ will also increase by a factor of 2. This is approximately true for the low range to the midrange of frequencies such as shown for the coils of Fig. 1.
Fig. 1: $Q_l$ versus frequency for a series of inductors of similar construction.
Unfortunately, however, as the frequency increases, the effective resistance of the coil will also increase, due primarily to skin effect phenomena, and the resulting $Q_l$ will decrease. In addition, the capacitive effects between the windings will increase, further reducing the $Q_l$ of the coil.
For this reason, $Q_l$ must be specified for a particular frequency or frequency range. For wide frequency applications, a plot of $Q_l$ versus frequency is often provided. The maximum $Q_l$ for most commercially available coils is less than 200, with most having a maximum near 100. Note in Fig. 1 that for coils of the same type, $Q_l$ drops off more quickly for higher levels of inductance.
If we substitute
$$ w_s = 2 \pi f_s$$
and then
$$ f_s = { 1 \over 2 \pi \sqrt{LC}}$$
into Eq. 2, we have
$$ \begin{split} Q_s &= { w_s L \over R} = { 2 \pi f_s L \over R} \\ & = { 2 \pi\over R } ({ 1 \over 2 \pi \sqrt{LC}}) L\\ &= {L \over R } ({ 1 \over \sqrt{LC}})\\ & = ({\sqrt{L}\over \sqrt{L} }) { L \over R \sqrt{LC}} \end{split}$$
$$ \bbox[10px,border:1px solid grey]{Q_s = { 1 \over R} \sqrt{{L \over C}}} \tag{4}$$
providing Qs in terms of the circuit parameters.
For series resonant circuits used in communication systems, $Q_s$ is usually greater than 1. By applying the voltage divider rule to the circuit of Fig. 2,
Fig. 2: Series resonant circuit.
we obtain
$$ V_L = {X_L E \over Z_T} = {X_L E \over R} \, \text{at resonance}$$
$$\bbox[10px,border:1px solid grey]{V_{L_{s}} = Q_sE} \tag{5}$$
$$ V_C = {X_C E \over Z_T}={X_C E \over R}$$
$$\bbox[10px,border:1px solid grey]{V_{C_{s}} = Q_sE} \tag{6}$$
Since $Q_s$ is usually greater than 1, the voltage across the capacitor or inductor of a series resonant circuit can be significantly greater than the input voltage. In fact, in many cases the $Q_s$ is so high that careful design and handling (including adequate insulation) are mandatory with respect to the voltage across the capacitor and inductor.
High-Q series resonant circuit.
Fig. 3: High-Q series resonant circuit.
In the circuit of Fig. 3, for example, which is in the state of resonance,
$$ Q_s = { X_L \over R} = { 480 Ω \over 6Ω} = 80$$
$$ V_L = Q_C = Q_s E = (80)(10V) = 800V$$
which is certainly a potential of significant magnitude.

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