034/100 Loaded Q and External Q

November 2019
Michael Benker

Example 4.2-3: Analyze the parallel resonator that is attached to a 50 Ohm source and load as shown.


This problem is specifically asking to define the Q factor related to this circuit. The Q factor is a ratio of energy stored (by an inductor or capacitor) to the power dissipated in a resistor. The Q factor varies with frequency since the effect of a capacitor or inductor also vary with frequency. For a series resonant circuit, the “unloaded” Q factor is defined by the following function: Qu = X / R = 1/(wRC) = wL/R

The unloaded Q factor of a parallel resonant circuit: Qu = R / X = R/(wL) = wRC

Overall, the Q factor is a measure of loss in the resonant circuit. A higher Q corresponds to lower loss, while a lower Q indicated higher loss. An “unloaded” Q factor means that the resonator is not connected to a source or load. The above circuit can no longer apply the “unloaded” Q factor formulas due to the presence of a source and a load. There are two further Q factor formulas that need to be considered: loaded Q factor and external Q factor. The loaded Q factor includes the source resistance and load resistance with the resistance of the circuit. The external Q factor refers to only the source resistance and load resistance together.

For the above circuit, the loaded Q factor for the parallel resonator is defined as:

Loaded Q = (Rs + R + Rl)/(wL) = (Source resistance + R + load resistance) / (wL)

The external Q factor for the source resistance and load resistance is:

External Q = (Rs + Rl)/(wL) = (Source resistance + load resistance)/(wL)

The relationship between the different types of Q factors are:

1/(Loaded Q) = 1/(External Q) + 1/(Unloaded Q)

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