Radio Frequency Integrated Circuit Design - Webs

Impedance Matching
4.10 Quality Factor of an LC Resonator
The Q (quality factor) of an LC resonator is another figure of merit used. It
is defined as
Q = 2��E stored/cycle
E �
(4.32)
lost/cycle
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
E lost/cycle =� T
0
V 2
osc sin 2 (�osct)
R
dt = 1 2 T
Vosc 2 R
85
(4.33)
Energy is also stored each cycle in the capacitor and the Q is therefore
given by
E stored/cycle = 1 2
CVosc ⇒ Q = 2�
2 CR
T = CR� C
osc = R√ L
Another definition of Q that is particularly useful is [5]
Q = �o
2 | d�
d� |
(4.34)
(4.35)
where � is the phase of the resonator and d� /d� 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 (4.32).
The Q of a resonator can also be related to its center frequency and
bandwidth, noting that
Q = R √ C
L
= RC
√ LC = � o
BW
(4.36)
Example 4.6 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
200-� resistor. Use a transformer with a coupling factor of 0.8 to match it to