Radio Frequency Integrated Circuit Design - Webs

116 Radio Frequency Integrated Circuit Design
V1 = Z 11I1 + Z 12I2
0 = Z 21I 1 + Z 22I 2
The second equation can be solved for I2:
I2 =− Z 21I1
Z 22
(5.15)
(5.16)
Thus, I2 can now be removed from the first equation, and solving for
Z in = V1 /I1:
Z port1 = Z 11 − Z 12Z21
Z 22
(5.17)
Equivalently, if we look from port 2 to ground, the impedance becomes
Z port2 = Z 22 − Z 12Z21
Z 11
(5.18)
Note that, referring to Figure 5.12, this effectively grounds out both C 1
and R 1 or C 2 and R 2. Thus, the Q will not necessarily be the same looking
from both ports. In fact, the Q will be marginally higher in the case of a regular
structure looking from the side with no underpass, as there will be less loss.
Also note that the side with no underpass will have a higher self-resonance
frequency.
Often designers want to use inductors in a differential configuration. This
means that both ends of the inductor are connected to active points in the
circuit and neither side is connected to ground. In this case, we can define the
impedance seen between the two ports:
Starting again with the Z parameters, the voltage difference applied across
the structure is now
V1 − V2 = Z 11I1 + Z 12I2 − Z 21I1 − Z 22I2
(5.19)
V1 − V2 = I1(Z 11 − Z 21) − I2(Z 22 − Z 12) (5.20)
Because the structure is symmetric, we make the assumption that I1 =
−I2. Thus,
Z diff = V1 − V2
I1
= Z 11 + Z 22 − Z 12 − Z 21
(5.21)