Radio Frequency Integrated Circuit Design - Webs

80 Radio Frequency Integrated Circuit Design
Example 4.5 Equivalent Impedance of Transformer Networks
Referring to the diagram of Figure 4.20, find the equivalent impedance of each
structure, noting the placement of the dots.
Solution
For each structure we apply a test voltage and see what current flows. In the
first case, the current flows into the side with the dot of each inductor. In this
case, the flux from each structure is added. If we apply a voltage V to the circuit
on the left in Figure 4.20, then V /2 appears across each inductor. Therefore,
for each inductor,
V
2
= j�LI + j�MI
We can solve for the impedance by
Z = V
I
= 2j�(L + M )
Thus, since Z = j�Leq, we can solve for L eq:
L eq = 2L + 2M
In the second case for the circuit on the right in Figure 4.20, the dots
are placed in such a way that the flux is reduced. We repeat the analysis:
V
2
= j�LI − j�MI
Z = V
= 2j�(L − M )
I
L eq = 2L − 2M
Figure 4.20 Circuits to find the equivalent impedance.