Radio Frequency Integrated Circuit Design - Webs

Impedance Matching
Z 2 = Z in + j�L
where Z in = R in − jX in. Solving for Y3 and equating it to the reference admittance
Yo ,
Y3 = Y 2 + j�C = 1
Z o = Y o
Using the above two equations, to eliminate Y2 leaving only L and C as
unknowns,
1
Z in + j�L = Y o − j�C
Solving the real and imaginary parts of this equation, values for C and L
can be found. With some manipulation,
the real part of this equation gives
�L = X in + √ R in − Y o R 2 in
Yo
R in − j (�L − X in)
R 2 in + (�L − X in) 2 = Yo − j�C
= 30 + √ 40 − (0.02)(40) 2
0.02
Now using the imaginary half part of the equation,
�C =
�L − X in
R 2 2 =
in + (�L − X in)
50 − 30
40 2 2 = 0.01
+ (50 − 30)
= 50
At 2 GHz it is straightforward to determine that L is equal to 3.98 nH
and C is equal to 796 fF. We note that the impedance is matched exactly only
at 2 GHz. We also note that this matching network cannot be used to transform
all impedances to 50�. Other matching circuits will be discussed later.
Although the preceding analysis is very useful for entertaining undergraduates
during final exams, in practice there is a more general method for determining
a matching network and finding the values. However, first we must review
the Smith chart.
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