S-Parameters

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S-parameters

MUSE Material developed by Prof. L. Dunleavy, USF

1 © University of South Florida 22--PortPort ParametersParameters

z Recall Z-Parameters:

VZIZI=+ ⎡V ⎤ ⎡ZZ⎤⎡I ⎤ 1111122 ==> 1 = 11 12 1 VZIZI=+ ⎢V ⎥ ⎢ZZ⎥⎢I ⎥ 2211222 ⎣ 2 ⎦ ⎣ 21 22 ⎦⎣ 2 ⎦

I1 I2 V12-port V2

2 © University of South Florida 22--PortPort ParametersParameters

z Recall Z-Parameters:

VZIZI=+ ⎡V ⎤ ⎡ZZ⎤⎡I ⎤ 1111122 ==> 1 = 11 12 1 VZIZI=+ ⎢V ⎥ ⎢ZZ⎥⎢I ⎥ 2211222 ⎣ 2 ⎦ ⎣ 21 22 ⎦⎣ 2 ⎦

I1 I2 V12-port V2 network

3 © University of South Florida 22--PORTPORT ParametersParameters (cont(cont’’d)d)

z Y-Parameters:

IYVYV=+ 1111122 ⎡I1 ⎤ ⎡YY11 12 ⎤⎡ V1 ⎤ IYVYV=+ ==> ⎢ ⎥ = ⎢ ⎥⎢ ⎥ 2211222 ⎣I 2 ⎦ ⎣YY21 22 ⎦⎣V2 ⎦

z h-Parameters:

VhIhV= + ⎡V ⎤ ⎡hh⎤⎡ I ⎤ 1111122==> 1 = 11 12 1 ⎢I ⎥ ⎢hh⎥⎢V ⎥ IhIhV2211222=+ ⎣ 2 ⎦ ⎣ 21 22 ⎦⎣ 2 ⎦

4 © University of South Florida 22--PortPort ParametersParameters (cont(cont’’d)d)

z 2-port Parameter Determination:

v1 h11 = |v = 0 I1 2 (Put a Short Circuit at Port #2) I2 h21 = |V =0 I1 2 V h = 1| 12 V I =0 (Put an Open Circuit at Port # 1) 2 1 I 2 h22 = |I =0 V 1 2 5 © University of South Florida S-Parameters At high RF and Microwave frequencies direct measurement of Y- , Z-, or H- parameters is difficult due to: • Unavailability of equipment to measure RF/MW total current and voltage. • Difficulty of obtaining perfect opens/shorts • Active devices may be unstable under open/short conditions.

6 © University of South Florida SS--ParametersParameters z For a two-port device there are four S-

parameters S11, S21, S12, and S22

z S11, and S22 are simply the forward and reverse reflection coefficients, with the

opposite port terminated in Z0 (usually 50 ohms.)

z S21 and S12 are simply the forward and reverse gains assuming a Z0 source and load (again usually 50 ohms).

7 © University of South Florida SS--ParametersParameters (cont(cont’’d)d)

z S-Parameters: bSasa= + 1111122 ⎡b1 ⎤ ⎡SS11 12 ⎤⎡a1 ⎤ bSaSa=+ ==> ⎢ ⎥ = ⎢ ⎥⎢ ⎥ 2211222 ⎣b2 ⎦ ⎣SS21 22 ⎦⎣a 2 ⎦

a1b1 b2 a2

2-port

8 © University of South Florida SS--ParametersParameters (cont(cont’’d)d)

z Q. So what’s the deal with the a’s and b’s? a1b1 b2 a2 Zs 2-port Vg V1 V2 ZL

z A. a1 and a2 are incident waves; b1 and b2 are reflected waves

9 © University of South Florida IncidentIncident && ReflectedReflected Waves:Waves: SimplifiedSimplified Case:Case: Z1=Zs=Z2=ZL=ZoZ1=Zs=Z2=ZL=Zo (real)(real)

VZI101+ Incident port 1 voltage E i1 a1 = == 2 ZZ00Z0

VZI202+ Incident port 2 voltage E i2 a 2 = == 2 ZZ00Z0

VZI101− reflected port 1 voltage E r1 b1 = == 2 ZZ00Z0

VZI202− reflected port 2 voltage E r2 b 2 = == 2 ZZ00Z 0

10 © University of South Florida S-Parameter Determination b s = 1 | = Input reflection coefficient Γin 11 a = 0 a 1 2 for case of ZL=Z0 b s = 2 | 21 a =0 = Forward transmission (insertion) gain a1 2 for case of ZL=Z0 b s = 1 | 12 a =0 = Reverse transmission (insertion) gain a2 1 for case of Zs=Z0 b s = 2 | 22 a =0 = Output reflection coefficient Γout a2 1 for case of Zs=Z0 11 © University of South Florida Desired Measurement Conditions a1 b1 b2 a2 = 0 Zs = Z0

2-port ZL = Vg Z0 Z0 V2 Z0

Note…the input and output are terminated in Z0 a1 b1

Zs = Z0 1-port Vg Z0

Γ S11=Γ

Reverse Gain Insertion Loss, S11, Transmission Phase Input S22, Refl. Output Device Refl. Coeff. Γin, Under Test Return Coeff. Γout, Loss, Return VSWR Loss, S21, Forward Gain VSWR Insertion Loss, Transmission Phase

dB Meaning or interpretation

Corresponds to the algebraic negative of the input S11 20 log10 |S11| return loss of a 2-port with an R0 termination on the opposite port. Reverse isolation (active device or amplifier), or S12 20 log10 |S12| algebraic negative of the insertion loss (I.L.) for a

passive device, with R0 at ports 1 and 2. Power gain (active device or amplifier), or algebraic S21 20 log10 |S21| negative of the insertion loss (I.L.) for a passive device, under matched R0 at ports 1 and 2. Corresponds to the algebraic negative of the S22 20 log10 |S22| output return loss of a 2-port with an R0 termination on the opposite port.

Ideal “X” dB Attenuator S11=S22=0 S21=S12=xe-jθpad PAD S21DB= X=20log(|S21|) x =10-X/20 Ideal “G” dB Gain Amp S11 = S22 = 0 = S12 -jθamp S21 = gve S21DB= G=20log(S21)

Ideal Band Pass Ideal Low Pass Ideal High Pass

S21 S21 S21 0dB

fc1 fc2 fc fc

GENERAL “IN-BAND” GENERAL “OUT-OF-BAND”

S11=S22=0 |S11|=|S22|=1 -jθ(f) S21=S12=1e S21=S12=0 S21(dB)=S12(dB)=0dB S11(dB)=S22(dB)=0dB