Q-circle measurement error

High-Q lumped and distributed networks near resonance are generally modeled as elementary three element RLC circuits. The widely used Q-circle measurement technique is based on this assumption. It is shown that this assumption can lead to errors when measuring the Q-factor of more complex resonators, particularly when heavily loaded by the external source. In the Q-circle technique, the resonator is assumed to behave as a pure series (or parallel) RLC circuit and the intercept frequencies are found experimentally at which the components of impedance satisfy the absolute value of Im(Z) = Re(Z) (unloaded Q) and absolute value of Im(Z) = Ro+Re(Z) (loaded Q). The Q-factor is then determined as the ratio of the resonant frequency to the intercept bandwidth. This relationship is exact for simple series or parallel RLC circuits, regardless of the Q-factor, but not for more complex circuits. This is shown to be due to the fact that the impedance components of the circuit vary with frequency differently from those in a pure series RLC circuit, causing the Q-factor as determined above to be in error.