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Impedance Mapping 241
r-- Z,
s"
I, I c 3 I,
"~
I
b
~"
+ Port 2
+
+
'" E, V, V, I',
/'V
~b, Z2 '" jwL b,----+
i
I,
Figure 7,4.
Grounded-emitter transistor two-port network.
where normalized z,=Z,/R" Substituting this into (7.32) yields
, a\z2+ a2
SI I = -a-'-z-"-+---!'-I
3 ,
where the bilinear coefficients are
SIl+A
a,=!+S,
"
I-S"
a3=~S'
"
(7.35)
(7.36)
Thus the bilinear function (7,32) of the bilinear function (7.34) is shown to be
bilinear, as in (7.35) with coefficients (7.36),
Consider the transistor in Figure 7.4. The emitter inductor is unavoidahle at
high frequencies, The reverse transducer gain s\3 was given in (7.31) when
emitter impedance Z,=200+jO; it was S,)=0.3847 /63,03°, For perfect
neutralization, S\3 = 0, Setting SIJ = 0 in the (7,29) expression yields the required
port-2 termination:
(7.37)
Example 7.3. Use the three-port scattering parameters in (7,30) for the
transistor in Figure 7.4, Assume a 50-ohm port normalization, Evaluating
(7.37), perfect neutralization occurs when [,= 1.3055 /164,94°, Since the
required magnitude is greater than unity, Re(Z,) would be negative, Therefore,
consider setting [, equal to 1.0 (164,94°; using the RTN,R/S feature in
Program A7-2 converts [, to an inductor with a reactance of +j6,6093 ohms,
Then keys A and 0 show that S\3 = 0,0450 /98.46°, Thus a 6,6-ohm inductive
emitter reactance produces much better neutralization than does a 200-ohm
resistance (ISn! = 0,3847) or a 50-ohm resistance (ISn! = 0,236),
7.2.3. Mapping. Mapping is the most important single concept and tool in
Chapter Seven, The classical analysis of bilinear functions according to
Churchill (1960, p, 74) is to express (7,1) in the form of (2.2) in order to show
that the bilinear functions amount to linear transformations and inversions.
Linear transformations such as in the denominator of (7,1) do not change the
shape of curves in the Z plane, Churchill (1960, p. 69) shows that inversions