Understanding Two-Port Network Parameters and Conversions
Delve into the realm of two-port network parameters including impedance, admittance, a-parameters, b-parameters, h-parameters, g-parameters, and parameter conversions. Explore the definitions, complexities, and applications of these network parameters for low-frequency circuits, transmission, and cascading. Unravel the intricacies of network analysis through a diverse range of parameters and their significance in understanding network behavior and design.
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Two-port Network Parameters 1. Two-port network parameters 2. Scattering parameters 3. T-parameters 4. Multiport network parameters 1
1. Low-frequency Network Parameters Port voltage and current definition Two-port network parameters - Select 2 variables out of 4 variables (V1, I1, V2, I2) - 4C2 = 6 cases z (impedance) parameters y (admittance) parameters a parameters b parameters h parameters g parameters 2
a-parameters (1) - Other names: ABCD-parameters, chain (cascade, transmission) parameters 7
a-parameters (2) - ABCD-parameters of simple networks 8
b-parameters (1) - Inverse ABCD-parameters 9
b-parameters (2) - Inverse ABCD-parameters of simple networks 10
h-parameters (2) - Example: CB-amp 12
g-parameters = Inverse hybrid parameters (2) - Example: CB-amp 14
4. Multiport Network Parameters Multiport networks - 3-port: circulator, power divider, mixer - 4-port: hybrid coupler, directional coupler, magic T junction Network parameters for multiports y-parameters z-parameters s-parameters 23
Maximum power transfer for AC networks Z + R + 1 + * S 2 L L = = = Re( *) Re | | P V I V V V L L S S * 2 2 + + Z Z ( ) ( ) ( ) Z Z R R X X S L S L S L S L P R P X L L = = = = 0 and 0 and R R X X L S L S L L 2 V 1 2 1 2 S = = = = P P P V I ,max L L S S 4 R S = + P P P S R R S L 24
3. Scattering Parameter Scattering parameter, S-paramter - Lossless networ: unitary condition - Reciprocal network: = S S mn nm 25
Scattering parameters of simple networks - Transmission line l j 0 e = [ ] S l j 0 e - Series element on transmission line Z Z Z Z Z S Z Z Z Z Z + + + Z Z 0 + + 2 2 0 0 Z = [ ] 0 + 2 2 Z 0 0 - Shunt element on transmission line 2 Y Y + 0 + 2 2 Y Y Y Y 0 0 = [ ] S 2 Y Y + 0 + 2 2 Y Y Y Y 0 0 26
4. Conjugate Matching Maximum power transfer theorem for two-ports - S. D. Stearns, ARRL Pacificon Antenna Seminar, 2011 * (conjugate matching) s = Z Z L * s * L = = (simultaneous conjugate matching) Z Z Z Z out in 27
Two-port matching network design [www.ittc.ku.edu/~jstiles/723/ - Find the equivalent circuit at the output of the two-port network using the open-circuit and short-circuit method. - Design a two-port network for the conjugate matching at the load. * L = Z Z out 28
5. Impedance Matching of Two-port Networks Two-port network power gains [S] : scattering parameters are defined with Z0. 30
- Power gain G: source & load not matched - Available power gain GA: source & load matched - Transducer power gain GT: source matched, load not matched G G G T A - Gain in dB: G (dB) = 10log10G 32
Two-port network power gains with conjugate matching = max( ) P P in avs * S * S = = : maximum if or P Z Z in in in = max( ) P avn P L * out * out = = : maximum if or P Z Z L L L 33
Example 35
Input and output matching of two-port networks 2 1 2 2 4| | B B C 1 1 = s C 1 2 2 2 4| | B B C 2 2 = L 2 C 2 37
+ 1 1 S = S Z Z S S + 1 1 L = L Z Z L L Input matching network: ZS Z'S Output matching network: ZL Z'L 38
Example - With Z0 = ZS = ZL = 50 39
6. Coding Example Two-port network input & output matching - Input data: System reference impedance Z0 ( ) Source impedance ZS ( ) in real/imaginary form Load impedance Z2 ( ) in real/imaginary form Two-port scattering parameters S11, S21, S12, S22 in polar or rectangular form - Output results: Transformed source and load impedances: ZS', ZL' in ohms Power gain G in natural and dB units Available power gain GA in natural and dB units Transducer power gain GT in natural and dB units 40
# Two-port network input & output matching # Input data: # System reference impedance z0 # Source impedance (zs), load impedace (zL) in real/imaginar format # Scattering parameters (s11, s21, s12, s22) in polar form (magnitude in natural unit, phase in degrees) # Output results: # Transformed source impedance zs' in ohms # Transformed load impedance zL' in ohms # Power gain g, available power gain ga, transducer power gatin gt in dB import cmath import math rad=3.141593/180 while True: imatch=int(input('Calculate transformed input and output impedances?(0,1)=')) ipolar=int(input('S-parameter form(1:polar, 2:real/imaginary)')) z0=float(input('System reference impedance Z0 (ohm) =')) rs,xs=map(float,input('Source impedance Zs=Rs+jXs (ohm): Rs, Xs =').split()) zs=complex(rs,xs) rL,xL=map(float,input('Load impedance ZL=RL+jXL (ohm): RL, XL =').split()) zL=complex(rL,xL) 41
if ipolar == 1: s11m,s11p=map(float,input('Mag(S11), phase(S11)(deg) =').split()) s12m,s12p=map(float,input('Mag(S12), phase(S12)(deg) =').split()) s21m,s21p=map(float,input('Mag(S21), phase(S21)(deg) =').split()) s22m,s22p=map(float,input('Mag(S22), phase(S22)(deg) =').split()) s11=cmath.rect(s11m,s11p*rad) s12=cmath.rect(s12m,s12p*rad) s21=cmath.rect(s21m,s21p*rad) s22=cmath.rect(s22m,s22p*rad) elif ipolar == 2: s11r,s11i=map(float,input('Re(S11), Im(S11) =').split()) s12r,s12i=map(float,input('Re(S12), Im(S12) =').split()) s21r,s21i=map(float,input('Re(S21), Im(S21) =').split()) s22r,s22i=map(float,input('Re(S22), Im(S22) =').split()) s11=complex(s11r,s11i) s12=complex(s12r,s12i) s21=complex(s21r,s21i) s22=complex(s22r,s22i) gs=(zs-z0)/(zs+z0) # Source reflection gL=(zL-z0)/(zL+z0) # Load reflection gin=s11+s12*s21*gL/(1-s22*gL) # Input reflection looking into port 1 of the two-port network gout=s22+s12*s21*gs/(1-s11*gs) # Output reflection looking into port 2 of the two-port network 42
g=abs(s21)**2*(1-abs(gL)**2)/(abs(1-s22*gL)**2*(1-abs(gin)**2))g=abs(s21)**2*(1-abs(gL)**2)/(abs(1-s22*gL)**2*(1-abs(gin)**2)) ga=abs(s21)**2*(1-abs(gs)**2)/ (abs(1-s11*gs)**2*(1-abs(gout)**2)) gt= abs(s21)**2*(1-abs(gs)**2)*(1-abs(gL)**2)/ (abs(1-s22*gL)**2*abs(1-gs*gin)**2) print(' Power gain G =',g,'/',10*math.log10(g),'(dB)') print(' Available power gain Ga =',ga,'/',10*math.log10(ga),'(dB)') print(' Transducer power gain Gt =',gt,'/',10*math.log10(gt),'(dB)') if imatch == 1: d=s11*s22-s12*s21 dm=abs(d) # abs() function returns absolute values for integer, real, and complex numbers k=(1-abs(s11)**2-abs(s22)**2+abs(d)**2)/(2*abs(s12)*abs(s21)) b1=1+abs(s11)**2-abs(s22)**2-abs(d)**2 b2=1+abs(s22)**2-abs(s11)**2-abs(d)**2 c1=s11-d*s22.conjugate() # complex class conjugate function c2=s22-d*s11.conjugate() gsp=(b1-math.sqrt(b1**2-4*abs(c1)**2))/(2*c1) # Transformed source reflection gLp=(b2-math.sqrt(b2**2-4*abs(c2)**2))/(2*c2) # Transformed load reflection zsp=(1+gsp)*zs/(1-gsp) zLp=(1+gLp)*zL/(1-gLp) print(' Transformed source impedance: Zg =',zsp) print(' Transformed load impedance: ZL =',zLp) 43
''' Calculate transformed input and output impedances?(0,1)= 1 S-parameter form(1:polar, 2:real/imaginary) 2 System reference impedance Z0 (ohm) = 50 Source impedance Zs=Rs+jXs (ohm): Rs, Xs = 50 0 Load impedance ZL=RL+jXL (ohm): RL, XL = 50 0 Re(S11), Im(S11) = 0.5998 -0.5399 Re(S12), Im(S12) = 0.0675 0.0373 Re(S21), Im(S21) = -0.2194 1.1418 Re(S22), Im(S22) = 0.1166 -0.4004 Power gain G = 3.876276835462403 / 5.8841478587625 (dB) Available power gain Ga = 1.6364475547216564 / 2.139020914894498 (dB) Transducer power gain Gt = 1.3518436 / 1.3092644930828499 (dB) Transformed source impedance: Zg = (32.59169632182722+112.80182946859793j) Transformed load impedance: ZL = (30.386211590971747+29.502492322691676j) 44
Calculate transformed input and output impedances?(0,1)= 0 S-parameter form(1:polar, 2:real/imaginary) 1 System reference impedance Z0 (ohm) = 50 Source impedance Zs=Rs+jXs (ohm): Rs, Xs = 25 0 Load impedance ZL=RL+jXL (ohm): RL, XL = 40 0 Mag(S11), phase(S11)(deg) = 0.38 -158 Mag(S12), phase(S12)(deg) = 0.11 54 Mag(S21), phase(S21)(deg) = 3.50 80 Mag(S22), phase(S22)(deg) = 0.40 -43 Power gain G = 13.085822887992618 / 11.168010379791545 (dB) Available power gain Ga = 19.83537106617267 / 12.974403293619135 (dB) Transducer power gain Gt = 12.603900024194017 / 11.005049496342165 (dB) Calculate transformed input and output impedances?(0,1)= ''' 45
