8.4 Amplifiers

(a transistor), the S -parameters of which are determined in reference to the reference impedance Z 0 (characteristic impedance of the transmission line). The small-signal S -parameters of a transistor depend on frequency and operating point, that is, on bias voltages and currents. The manufacturer usually reports typical values of the S -parameters (in reference to 50V) versus frequency in a few operating points. The load impedance of the two-port is Z L , and the impedance of the feeding generator is Z S . The reflection coefficients of the terminations in the input and output are

The input reflection coefficient seen toward the two-port can be presented using the normalized voltage waves as

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A transistor amplifier provides the maximum available power gain when both ports are conjugately matched. This condition can be

written as

By substituting these into (8.18) and (8.20) we obtain generator and load reflection coefficients (in reference to 50V) r SM and rLM , respectively, that both the input and output are simultaneously matched:

where

The use of (8.28) requires that K ≥ 1; otherwise Ga , max becomes complex. Generally, the matching circuit provides the maximum available power gain only at the design frequency. When the frequency moves away from

the design frequency, gain decreases rapidly. The bandwidth can be made wider if we are satisfied with a gain lower than Ga , max . In that case a small mismatch is purposely designed in both the input and output. Design is aided by using constant gain circles drawn on the Smith chart.

A transistor amplifier is not necessarily stable. Oscillation is possible if the real part of the input or output impedance is negative. This means

that | r in | > 1 or | rout | > 1. If | r in | < 1 and | r out | < 1 with all generator and load impedance values, the amplifier is unconditionally stable. In other

cases the amplifier is potentially unstable. From conditions | rin | = 1 or | r out | = 1 we can calculate boundaries for the stable regions of input and output impedances. On the Smith chart these boundaries are circles, as shown in Figure 8.14. The output impedance stability circle is defined by the center point cL and radius r L as

Figure 8.14 Output stability circles on the Smith chart. Shaded areas are stable when | S11 | < 1. The center point of the Smith chart is (a) outside the stability circle, and (b) inside the circle.

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Either the inside or outside of the stability circle presents the stable region. If | S11 | < 1, the center point of the Smith chart is in the stable region. If, however, | S11 | > 1, the Smith chart center point is in the unstable region. The center point cS and radius r S of the input stability circle are obtained by replacing S22 with S11 and vice versa in (8.30) and (8.31). It can be proven that the necessary and sufficient conditions for the unconditional stability are K > 1 and | D | < 1. It may be worth checking that r S and r L are in the stable region not only at the operating frequency but also at other frequencies.

In addition to gain and stability, a third important characteristic of an LNA is its noise factor F (for more detail see Section 11.2). A transistor has four noise parameters: minimum noise factor Fmin , magnitude and phase of optimum input reflection coefficient ropt (F = Fmin , when rS = r opt ), and equivalent noise resistance R n [11]. The noise factor can be presented as a function of the generator admittance YS = 1/Z S as

where GS is the real part of YS and Yopt is the admittance corresponding to ropt . The load impedance Z L affects the gain but not the noise factor. In general, it is not possible to obtain the minimum noise factor and maximum gain simultaneously, but one must make a compromise. This can be helped by using constant noise circles calculated from (8.32) on the Smith chart, together with constant gain circles as shown Figure 8.15.

In addition to the design of matching circuits according to proper rS and r L , the completion of the design work requires also design of circuits for bias voltages. Figure 8.16 shows an example of a practical realization: a 22-GHz HEMT amplifier. The low-impedance microstrip line sections in the input and output provide r S for the minimum noise factor and rL for the conjugate match of output (in order to optimize gain). Bias voltages to the gate and drain are supplied through high-impedance lines. At a distance of l/4 from the feed point there is a radial stub short circuit in each of