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# Maximum Gain Conjugate Match according to Pozar

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#### StoppTidigare

##### Full Member level 2
According to Pozar's book conjugate match is accomplished when

ΓS=Γ∗inΓS=Γin∗​
and
ΓL=ΓoutΓL=Γout.​

The expression are

Γin=S11+S12S21ΓL1−S22ΓL=Γ∗SΓin=S11+S12S21ΓL1−S22ΓL=ΓS∗​

and

Γout=S22+S12S21ΓS1−S11ΓS=Γ∗LΓout=S22+S12S21ΓS1−S11ΓS=ΓL∗​

. This is solved and he obtains

ΓS=B1±√B21−4|C1|22C1ΓL=B2±√B22−4|C2|22C2

ΓS=B1±B12−4|C1|22C1ΓL=B2±B22−4|C2|22C2​

where
B1=1+|S11|2−|S22|2−|Δ|2B2=1+|S22|2−|S11|2−|Δ|2C1=S11−Δ⋅S∗22C2=S22−Δ⋅S∗11Δ=S11S22−S21S12

B1=1+|S11|2−|S22|2−|Δ|2B2=1+|S22|2−|S11|2−|Δ|2C1=S11−Δ⋅S22∗C2=S22−Δ⋅S11∗Δ=S11S22−S21S12​

Why do I get reflection coefficients ΓSΓS and ΓLΓL greater than one though Rollet stability test says that I am OK?

Code:
i=sqrt(-1);
%S-Params at 2.4GHz
disp("S-Params at 2.4GHz BFG520 Common Emitter 6V10mA")
S11_r = 0.535
disp("")

S21_r=2.798
disp("")
S12_r=0.097
disp("")
S22_r=.280
disp("")

%Maximum gain calculations
disp("Maximum gain calculations")
Delta = S11*S22-S12*S21
B1=1+abs(S11)^2-abs(S22)^2 -abs(Delta)^2
B2=1+abs(S22)^2-abs(S11)^2 -abs(Delta)^2
C1=S11 + Delta*conj(S22)
C2=S22 + Delta*conj(S11)
disp("")

Gamma_S_plus_2400MHz= (B1 + sqrt(B1^2 -4*abs(C1)^2) )/(2*C1)
Gamma_S_minus_2400MHz=(B1 - sqrt(B1^2 -4*abs(C1)^2) )/(2*C1)
Gamma_L_plus_2400MHz= (B2 + sqrt(B2^2 -4*abs(C2)^2) )/(2*C2)
Gamma_L_minus_2400MHz=(B2 - sqrt(B2^2 -4*abs(C2)^2) )/(2*C2)
disp("")
%%Absolute values of Gammas must be less than 1 but are not
disp("Absolute values of Gammas must be less than 1 but are not")
AbsGamma_S_plus=abs(Gamma_S_plus)
AbsGamma_S_minus=abs(Gamma_S_minus)
AbsGamma_L_plus=abs(Gamma_L_plus)
AbsGamma_L_minus=abs(Gamma_L_minus)
disp("")

%Gamma to impedances. Real part are negative!?
disp("Gamma to impedances. Real part are negative!?")
z_S_plus2400MHz =(1+Gamma_S_plus_2400MHz)/(1-Gamma_S_plus_2400MHz)
z_S_minus2400MHz =(1+Gamma_S_minus_2400MHz)/(1-Gamma_S_minus_2400MHz)
z_L_plus2400MHz =(1+Gamma_L_plus_2400MHz)/(1-Gamma_L_plus_2400MHz)
z_L_minus2400MHz =(1+Gamma_L_minus_2400MHz)/(1-Gamma_L_minus_2400MHz)
disp("")
%Stability check
disp('Stability check')
K_Stab=(1-abs(S11)^2-abs(S22)^2+abs(Delta)^2)/(2*abs(S12*S21))
Abs_Delta = abs(Delta)
if(K_Stab>1 & Abs_Delta<1)
disp("Stability OK")
else
disp("Stability not OK")
endif
disp("")
%S21 gain
disp("S21 gain")
G_0=abs(S21)^2
G_0_dB=10*log10(G_0)
disp("")
%Gains of matching networks
disp("Gains of matching networks")
G_S=1/(1-AbsGamma_S_minus^2)
G_S_dB=10*log10(G_S)
disp("")
G_L=(1-AbsGamma_L_minus^2)/(abs(1-S22*Gamma_L_minus)^2)
G_L_dB=10*log10(G_L)

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