Design of the Frequency-Tunable CMOS RF Power Amplifie - Linear CMOS RF Power Amplifiers for Wireless Applications - page 79

Table 6.1 π -matching
network for a target
R opt = 18 . High- Q 0 case:
C 1 = 14 . 34 pF and
C 2 = 8 . 68 pF

freq (GHz) L 1 (nH) Q 0 R opt () X opt () Γ opt (dB)

3.7 0.336 8 . 04 17.95 0.14 − 47 . 43

3.8 0.319 8 . 25 17.90 0.32 − 40 . 50

3.9 0.303 8 . 47 17.92 0.33 − 40 . 39

4.0 0.288 8 . 69 17.93 0.34 − 40 . 29

4.1 0.275 8 . 91 17.89 0.51 − 36 . 66

4.2 0.262 9 . 13 17.90 0.52 − 36 . 67

4.3 0.250 9 . 35 17.92 0.52 − 36 . 70

4.4 0.239 9 . 57 17.93 0.52 − 36 . 75

4.5 0.228 9 . 78 17.89 0.68 − 34 . 28

4.6 0.219 10 . 00 17.90 0.68 − 34 . 36

4.7 0.210 10 . 22 17.92 0.67 − 34 . 46

4.8 0.201 10 . 44 17.87 0.84 − 32 . 55

4.9 0.193 10 . 66 17.89 0.83 − 32 . 66

5.0 0.185 10 . 88 17.90 0.82 − 32 . 78

5.1 0.178 11 . 09 17.86 0.98 − 31 . 20

− 31 . 33

5.2 0.171 11 . 31 17.87 0.97

The loaded quality factor 1 is associated with the bandwidth of the network. Dif-
ferent expressions for Q 0 can be found in the literature [ 27 , Appendix]. We chose
( 6.9 ) because it is in conformity with the definition of quality factor as the ratio
between the maximum instantaneous energy stored and the energy dissipated per
cycle.
The importance of Q 0 is twofold. First, we can verify if the network is designable
according to the following criteria taken from [ 27 ]:
⎧
⎨

R opt
R L

1
2

Q 0 ≥ − 1 or R opt ≥ R L ,

2 R L

(6.10)

⎩

1

Q 0 ≥

R opt −

1 or R opt ≤ R L .

Second, an interesting property of the π -matching network of Fig. 6.2 reveals
that, for high Q 0 , capacitors C 1 and C 2 can be kept constant while L 1 is changed
to achieve the same transformation factor at different frequencies. This property can
be noticed through the following example.
Consider that we would like to transform an antenna impedance of 50 into an
optimum resistance of 18 in a frequency range from 3.7 to 5.2 GHz. If we choose
a high- Q 0 π -matching network with C 1 = 8 . 68 pF, Table 6.1

reveals that the value of R opt can be kept very close to 18 throughout the whole
frequency range from 3.7 to 5.2 GHz. This is achieved by varying just L 1 whereas

14 . 34 pF and C 2 =

1 The term loaded describes the quality factor of a circuit element under loaded conditions
[ 4 , Chap. 2].