120 Impedance Matchingfrequency at which the gain of the transfer function is down by 3 dB relative to the gain at the center frequency. 5.10 Quality Factor of an LC ResonatorThe quality factor, Q, of an LC resonator is another figure of merit used. It is de-fined as Stored / CycleLost / Cycle2EQEπ= (5.34)This can be used as a starting point to define Q in terms of circuit parameters.We first note that all the loss must occur in the resistor, because it is the only element present capable of dissipating any energy and the energy dissipated per cycle is: ω==ò22oscosc2Lost / Cycleosc0sin ()12T VtTEdtVRR (5.35)Energy is also stored each cycle in the capacitor and the Q is therefore given by: 2Stored / Cycleoscosc122CRCECVQCRRTLπω=Þ=== (5.36)Another definition of Q that is particularly useful is : 2o dQdωφω= (5.37)where f is the phase of the resonator and df/dw is the rate of change of the phase transfer function with respect to frequency. This can be shown to give the same value in terms of circuit parameters as (5.34).The Q of a resonator can also be related to its center frequency and bandwidth, noting that: BWoCRCQ RLLCω=== (5.38)Example 5.6: Matching a Transistor Input with a Transformer A circuit has an input that is made up of a 1-pF capacitor in parallel with a 200W resistor. Use a transformer with a coupling factor of 0.8 to match it to a source re-sistance of 50W. The matching circuit must have a bandwidth of 200 MHz and the circuit is to operate at 2 GHz.