What Does Stability Mean?
• Stability circles determine what load or source
impedances should be avoided for stable or nonoscillatory amplifier behavior
• Because reactive loads are being added to amp the
conditions for oscillation must be determined
• So the Output Stability Circle determine the ΓL or
load impedance (looking into matching network
from output of amp) that may cause oscillation
• Input Stability Circle determine the ΓS or
impedance (looking into matching network from
input of amp) that may cause oscillation
Criteria for Unconditional Stability
• Unconditional Stability when amplifier
remains stable throughout the entire domain
of the Smith Chart at the operating bias and
frequency. Applies to input and output
ports.
• For |S11| < 1 and |S22| < 1, the stability
circles reside completely outside the |ΓS| = 1
and |ΓL| = 1 circles.
Unconditional Stability: Rollett Factor
• |Cin| – rin | >1 and |Cout| – rout | >1
• Stability or Rollett factor k:
2
k=
2
1 − S11 − S22 + ∆
2 S12 S21
2
>1
with |S11| < 1 or |S22| < 1
and
∆ = S11S22 − S12 S21 < 1
Stabilization Methods
• Stabilization methods can be used to for
operation of BJT or FET found to be
unstable at operating bias and frequency
• One method is to add series or shunt
conductance to the input or output of the
active device in the RF signal path to
“move” the source or load impedances out
of the unstable regions as defined by the
Stability Circles
Stabilization Using Series Resistance or
Shunt Conductance
Stabilization Method: Smith Chart
Constant Gain: Unilateral Design (S12= 0)
• Need to obtain desired gain performance
• Basically we can “detune” the amp
matching networks for desired gain
• Unilateral power gain GTU implies S12 = 0
Unilateral Power Gain Equations
• Unilateral Power gain
GTU =
1 − ΓS
2
1 − S11Γ S
2
S21
2
1− ΓL
2
1 − S22 Γ L
2
= GS G0GL
• Individual blocks are:
GS =
1 − ΓS
2
1 − S11Γ S
2
2
; G0 = S21 ; GL =
1− ΓL
2
1 − S 22Γ L
• GTU (dB) = GS(dB) + G0(dB) +GL(dB)
2
Unilateral Gain Circles
• If |S11| < 1 and |S22 |< 1 maximum unilateral
power gain GTUmax when ΓS = S11* and
ΓL = S22*
GS max =
1
1 − S11
2
; GL max =
1
1 − S22
• Normalized GS w.r.t. maximum:
2
1 − ΓS
GS
gS =
=
GS max 1 − S11Γ S
(1 − S )
2
2
11
2
Unilateral Gain Circles
• Normalized GL w.r.t. maximums:
2
1− ΓL
GL
gL =
=
GL max 1 − S22 Γ L
(1 − S )
2
2
22
• Results in circles with center and radii:
d gi =
gi Sii
1 − Sii
2
(1 − gi )
; rgi =
(
1 − gi 1 − Sii
1 − Sii
2
2
(1 − gi )
ii = 11 or 22 depending on i = S or L
)
Gain Circle Observations
• Gi max when Γi = Sii* and dgi = Sii* of radius
rgi = 0
• Constant gain circles all have centers on
line connecting the origin to Sii*
• For the special case Γi = 0 the normalized
gain is:
gi = 1 - | Sii |2 and dgi = rgi = | Sii |/(1 + | Sii |2)
• This implies that Gi = 1 (0dB) circle always
passes through origin of Γi - plane
Input Matching Network Gain Circles
ΓS is detuned
implying the
matching
network is
detuned
Bilateral Amplifier Design (S12 included)
• Complete equations required taking into
account S12: Thus ΓS* ≠ S11 and ΓL* ≠ S22
Γ*S
S12 S 21Γ L S11 − Γ L ∆
= S11 +
=
1 − S22 Γ L 1 − S 22Γ L
*
ΓL
S12 S21Γ S S22 − Γ S ∆
= S 22 +
=
1 − S11Γ S 1 − S11Γ S
Bilateral Conjugate Match
• Matched source reflection coefficient
2
Γ MS
C1 =
B1 1  B1 
C1*
=
−
  −4
2C1 2  C1 
C1
*
S11 − S22
∆;
2
2
B1 = 1 − S22 − ∆ + S11
2
• Matched load reflection coefficient
2
Γ ML
B2 1  B2 
C2*
=
−
  −4
2C2 2  C2 
C2
2
2
*
C2 = S 22 − S11
∆ ; B2 = 1 − S11 − ∆ + S22
2
Optimum Bilateral Matching
Γ MS
S12 S21Γ ML
= S11 +
1 − S22Γ ML
Γ ML
S12 S21Γ MS
= S22 +
1 − S11Γ MS
*
*
Design Procedure for RF BJT Amps
• Bias the circuit as specified by data sheet
with available S-Parameters
• Determine S-Parameters at bias conditions
and operating frequency
• Calculate stability |k| > 1 and |∆| < 1?
• If unconditionally stable, design for gain
• If |k| ≤ 1 and |∆| ≥1 then draw Stability
Circles on Smith Chart by finding rout, Cout,
rin, and Cin radii and distances for the circles
Design Procedure for RF BJT Amps
• Determine if ΓL ( S22* for conjugate match)
lies in unstable region – do same for ΓS
• If stable, no worries.
• If unstable, add small shunt or series
resistance to move effective S22* into stable
region – use max outer edge real part of
circle as resistance or conductance (do same
for input side)
• Can adjust gain by detuning ΓL or ΓS
Design Procedure for RF BJT Amps
• To design for specified gain, must be less than
GTU max (max unilateral gain small S12)
• Recall that (know G0 = |S21|2)
GTU [dB] = GS [dB] + G0 [dB] + GL [dB]
• Detune either ΓS or ΓL
• Draw gain circles for GS (or GL) for detuned ΓS (or
ΓL) matching network
• Overall gain is reduced when designed for (a)
Stability and (b) detuned matching netw0rk
Design Procedure for RF BJT Amps
• Further circles on the Smith Chart include
noise circles and constant VSWR circles
• Broadband amps often are feedback amps
RF Shunt-Shunt Feedback Amp Design
R1 = Z 0 (1 − S 21 )
IC
gm =
VT
2
Z0
1
R2 =
−
R1 g m
S21 calculated from desired gain G
Distortion: 1 dB Compression
Distortion: 3rd Order Intermodulation
Distortion
Distortion: 3rd Order IMD
IMD3 [ dB] = Pout ( f 2 ) [ dBm ] − Pout (2 f 2 − f1 ) [ dBm ]
2
d f [ dB] = ( IP [ dBm ] − G0 [ dB] − Pin ,mds [ dBm ])
3
Spurious Free Dynamic Range