What Does Stability Mean?• Stability circles determine what load or source

impedances should be avoided for stable or non-oscillatory 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 2 211 22

12 21

11

2S S

kS S

− − + ∆= >

with |S11| < 1 or |S22| < 1and

11 22 12 21 1S S S S∆ = − <

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

2 22

21 02 211 22

1 1

1 1S L

TU S LS L

G S G G GS S

− Γ − Γ= =

− Γ − Γ

• Individual blocks are: 2 2

20 212 2

11 22

1 1

1 1S L

S LS L

G ; G S ; GS S

− Γ − Γ= = =

− Γ − Γ

• GTU (dB) = GS(dB) + G0(dB) +GL(dB)

Unilateral Gain Circles

max max2 211 22

1 11 1

S LG ; GS S

= =− −

• If |S11| < 1 and |S22 |< 1 maximum unilateral power gain GTUmax when ΓS = S11* and ΓL = S22*

• Normalized GS w.r.t. maximum:

( )2

2112

max 11

11

1SS

SS S

Gg SG S

− Γ= = −

− Γ

Unilateral Gain Circles

• Results in circles with center and radii:

( )2

2222

max 22

11

1LL

LL L

Gg SG S

− Γ= = −

− Γ

• Normalized GL w.r.t. maximums:

( )( )( )

2

2 2

1 1

1 1 1 1i i

i iii iig g

ii i ii i

g Sg Sd ; rS g S g

− −= =

− − − −

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

12 21 1111

22 221 1* L LS

L L

S S SSS S

Γ − Γ ∆Γ = + =− Γ − Γ

12 21 2222

11 111 1* S SL

S S

S S SSS S

Γ − Γ ∆Γ = + =− Γ − Γ

Bilateral Conjugate Match• Matched source reflection coefficient

21 1 1

1 1 1

1 42 2

*

MSB B CC C C

Γ = − −

2 2 2

1 11 22 1 22 111*C S S ; B S S= − ∆ = − − ∆ +

• Matched load reflection coefficient2

2 2 2

2 2 2

1 42 2

*

MLB B CC C C

Γ = − −

2 2 22 22 11 2 11 221*C S S ; B S S= − ∆ = − − ∆ +

Optimum Bilateral Matching

12 2111

221MS

* ML

ML

S SSS

ΓΓ = +− Γ

12 2122

111ML

* MS

MS

S SSS

ΓΓ = +− Γ

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

( )1 0 211R Z S= − 0

2

21

1

m

ZR

R g= −

Cm

T

IgV

= S21 calculated from desired gain G

Distortion: 1 dB Compression

Distortion: 3rd Order IntermodulationDistortion

Distortion: 3rd Order IMD[ ] ( )[ ] [ ]2 2 13 dB dBm (2 ) dBmout outIMD P f P f f= − −

[ ] [ ] [ ] [ ]( )0 ,2dB dBm dB dBm3f in mdsd IP G P= − −

Spurious Free Dynamic Range