GaAsFET amplifier: gains, Γin, Γout, Zin, Zout

Design flowchart

 +------------------------------+
 | Specs: f, gain, Z0, BW, NF   |
 +------------------------------+
               |
 +------------------------------+
 | Choose transistor, bias;     |
 | get [S] at design frequency  |
 +------------------------------+
               |
 +------------------------------+
 | Compute Δ, K (and μ)         |
 +------------------------------+
               |
      K>1 and |Δ|<1 ?
     yes |          | no
         v          v
 Uncond. stable   Draw stability circles;
         |        pick Γs, ΓL in stable
         |        region (or stabilise)
         +----+-----+
              |
   Unilateral? (U small, S12≈0)
   yes: Γs=S11*, ΓL=S22*
   no : Γs=ΓMS, ΓL=ΓML (bilateral)
              |
 +------------------------------+
 | Compute GS, G0, GL -> GT     |
 +------------------------------+
              |
 +------------------------------+
 | Design input/output matching |
 | (stubs or L-C) on Smith chart|
 +------------------------------+
              |
 | Bias network, layout, simulate
              |
   Specs met? -- no --> revise
              | yes
            Build & test

Stability

Δ = S11S22 − S12S21
  = (0.0981 + j0.0557) − (−0.0039 + j0.2806)
  = 0.1020 − j0.2248 = 0.2469∠−65.59°
K = (1 − 0.4303 − 0.0296 + 0.0610)/(2×0.2806)
  = 0.6010/0.5612 = 1.071

K = 1.071 > 1, |Δ| = 0.247 < 1 → unconditionally stable.

Maximum gain: bilateral model

GT,max = (2.30/0.122)(1.071 − √(1.071² − 1))
       = 18.85 × 0.6876 = 12.96 → 11.13 dB
B1 = 1.3398, C1 = S11 − ΔS22* = 0.6612∠150.37°
ΓMS = 0.8504∠−150.37°
B2 = 0.5383, C2 = S22 − ΔS11* = 0.2467∠−76.27°
ΓML = 0.6548∠76.27°
GS = 3.613 (5.58 dB), G0 = 5.29 (7.23 dB),
GL = 0.678 (−1.69 dB)  → GT = 11.13 dB

Maximum gain: unilateral model

GS,max = 1/(1 − 0.656²) = 1.755 → 2.44 dB
G0     = 2.30² = 5.29         → 7.23 dB
GL,max = 1/(1 − 0.172²) = 1.030 → 0.13 dB
GTU,max = 9.57 → 9.81 dB
U = 0.057 → error −0.48 dB to +0.51 dB

The bilateral maximum (11.13 dB) is 1.32 dB higher, because |S12| = 0.122 is not small.

Γin, Γout and Zin, Zout

With the bilateral match, Γin = ΓMS* and Γout = ΓML*:

Γin  = 0.8504∠150.37°
zin  = (1+Γin)/(1−Γin) = 0.0864 + j0.2627
Zin  = 50·zin = 4.32 + j13.13 Ω
Γout = 0.6548∠−76.27°
zout = 0.511 − j1.138
Zout = 25.55 − j56.90 Ω

(Unilateral model: Γin = S11 → Zin = 11.27 + j14.25 Ω; Γout = S22 → Zout = 40.90 − j12.91 Ω.)

Tracing on the Smith chart

1. Draw a circle of radius 0.850 (of the chart radius) about the centre and mark the angle 150.37° on the outer angle scale. The point lies near r = 0.086, x = +0.263: upper half, inductive, very close to the short-circuit end. Read Zin = 50(0.086 + j0.263).
2. Draw a circle of radius 0.655 and mark −76.27°. The point lies near r = 0.51, x = −1.14 in the lower (capacitive) half. Read Zout = 50(0.51 − j1.14).
3. The source must present Zs = Zin* = 4.32 − j13.13 Ω and the load ZL = Zout* = 25.55 + j56.90 Ω; matching networks are designed from these points to the chart centre.

Answer: GT,max = 12.96 (11.13 dB) bilateral; GTU,max = 9.57 (9.81 dB) unilateral; Zin = 4.32 + j13.13 Ω, Zout = 25.55 − j56.90 Ω.