Frequency-Tunable Capability - Linear CMOS RF Power Amplifiers for Wireless Applications - page 69

Fig. 5.3 Coupled inductors
used as a tunable inductance

Fig. 5.4 Coupled inductors
used as a tunable inductance
(parasitic series resistances
included)

To make the inductors of Fig. 5.3 more realistic, a parasitic series resistance is
included as shown in Fig. 5.4 . As a result, when we inject a current I ctrl through
L 2 , besides the change in the inductance, a resistive part appears in series with the
impedance seen by the RF circuit connected to L 1 . By varying the amplitude and
phase of I ctrl , one can find adequate values for which the resistive part added by
the mutual effect is negative and cancels (or at least decreases) the original parasitic
series resistance ( R Ls1 )of L 1 .
The relationship between the currents through L 1 and L 2 can be expressed, in
rectangular notation, by

I ctrl
I RF =

α + i β, (5.6)

where α and β denote the real and imaginary parts of the current ratio. In polar
notation, ( 5.6 ) becomes

I ctrl
I RF =

r( cos φ + isin φ), (5.7)

where r is the magnitude and φ is the phase of the ratio between I ctrl and I RF .Al-
ternatively, r can be seen as the attenuation in the magnitude of I ctrl when compared
to I RF and φ as the phase shift between these two currents.
If we consider that the inductors in Fig. 5.4 have the same inductance L 1 = L 2 =
L and the same series parasitic resistance R Ls1 = R Ls2 = R Ls , the impedance seen

by the RF circuit is

Z eq =− ωkβL + R Ls + i [ ωL( 1 + αk) ] , (5.8)

which can be split into a resistance and a reactance:

R eq =− ωkβL + R Ls , (5.9)

X eq = ωL( 1 + αk). (5.10)