Note that [V.sup.[+ or -].sub.L] = [V.sup.[- or +].sub.2] and [I.sub.L] = -[I.sub.2] by assuming zero length of the CCITL ([l.sub.2] = 0) in Figure 2.
Using (1), (2), and (10), the voltage reflection coefficient at the source [[GAMMA].sub.S] in the CCITL system can be expressed as
The difference between (6) and (11), which represent the input and source reflection coefficients respectively, is a result of the derivation following the definition of the CCITL traveling wave stated earlier.
The classic bilinear transformation or linear fraction transformation for a complex plane analysis  is applied here to map circles of interest in the [[GAMMA].sup.0]-plane (in the [Z.sub.0] system) onto those in the [GAMMA}-plane (in the CCITL system).
Then, the load impedance in a two-port network is rewritten in terms of [[GAMMA].sup.0.sub.L] and [Z.sub.0] and subsequently substituted in (9), yielding the bilinear transformation of the load reflection coefficient in the [Z.sub.0] system to that in the CCITL system in the form of
where [[GAMMA].sub.l] represents the load reflection coefficient in the CCITL system.
The circle in the [Z.sub.0] system that is located in the [[GAMMA].sup.0]-plane can be mapped into that in the CCITL system that is located in the [GAMMA]-plane using the following equations :
This bilinear transformation offers a simpler and faster approach to systematically derive associated circle equations for microwave transistor amplifiers in the CCITL system compared to rigorous parameter conversions reported previously [26,28].
APPLICATION OF THE BILINEAR TRANSFORMATION APPROACH TO MICROWAVE TRANSISTOR AMPLIFIER DESIGN IN THE CCITL SYSTEM
Figure 5 shows a schematic diagram of the microwave transistor amplifier using reciprocal open-circuited single-stub shunt tuners as the IMN and OMN in the CCITL system.
At a given frequency in the CCITL system, oscillations of the microwave transistor amplifier are possible when the input and/or output ports present a negative resistance resulting in [absolute value of [[GAMMA].sub.IN]] > 1 and/or [absolute value of [[GAMMA].sub.OUT]] > 1.
By using the bilinear transformation, [C.sup.0.sub.s] and [r.sup.0.sub.s] in (17) and (18) are substituted as [C.sub.0] and [r.sub.0] respectively into (15) and (16) resulting in the center [C.sub.s] and the radius [r.sub.s] in the CCITL system, respectively.