Radio Frequency Integrated Circuit Design - Webs

82 Radio Frequency Integrated Circuit Design
4.8 Tuning a Transformer
Unlike the previous case where the transformer was assumed to be ideal, in a
real transformer there are losses. Since there is inductance in the primary and
secondary, this must be resonated out if the circuit is to be matched to a real
impedance. To do a more accurate analysis, we start with the equivalent model
for the transformer loaded on the secondary with resistance R L , as shown in
Figure 4.21.
Next, we find the equivalent admittance looking into the primary. Through
circuit analysis, it can be shown that
Yin = −R L� 2 (L s L p − M 2 ) − j� 3 (L s L p − M 2 ) − j�R 2
L L p + � 2 L s L p R L
� 4 (L s L p − M 2 ) 2 + � 2 R 2 s L 2 p
(4.24)
Taking the imaginary part of this expression, the inductance seen looking
into the primary L eff-p can be found, making use of (4.19) to express the results
in terms of the coupling coefficient k:
L eff-p = � 2 L 2 s L p (1 − k 2 ) 2 + R 2
L L p
� 2 L 2 s (1 − k 2 ) + R 2
L
(4.25)
When k = 1 or 0, then the inductance is simply L p . When k has a value
between these two limits, then the inductance will be reduced slightly from
this value, depending on circuit values. Thus, a transformer can be made to
resonate and have a zero reactive component at a particular frequency using a
capacitor on either the primary C p or secondary C s :
�o =
1
√ L eff-p C p
=
1
√ L eff-s C s
where L eff-s is the inductance seen looking into its secondary.
Figure 4.21 Real transformer used to transform one resistance into another.
(4.26)