Two-Port Matching and Power Gains in Networks

Two-port Matching and Power Gains 1. Scattering Parameters 2. Two-port Matching and Power Gains 3. Coding Example 4. Using Online Python 1

1. Scattering Parameters Scattering parameter, S-parameter - Scattering parameters: S11, S12, S21, S22 - Scatterin matrix = + b 11 1 S a 12 2 S a 1 = + b 21 1 S a 22 2 S a 2 :normalized incident voltage at port 1 a 1 :normalized reflected voltage at port 1 b 1 b b S S S S a a 1 11 12 1 = 2 21 22 2 S S S S 11 12 = [ ] S 21 22 2

- Incident and reflected waves + + 1 2 V V Z I Z = = = 0 i i i i ( 1,2) a i i |Re( )| |Re( )| Z 0 0 i i * 0 1 2 V V Z I Z = = i i i i a i |Re( )| |Re( )| Z 0 0 i i + , : incident and reflected voltages at port V V i i i , : total voltage and current at port V I i i i : reference or characte ristic impedance of port i Z 0 i 3

- Obtaining the scatterin parameters b a = 1 (port 2 terminated with ) S Z 11 0 1 = 0 a 2 b a = 2 (port 2 terminated with ) S Z 21 0 1 = 0 a 2 b a = 1 (port 1 terminated with ) S Z 12 0 2 = 0 a 1 b a = 2 (port 1 terminated with ) S Z 22 0 2 = 0 a 1 4

- Port power 2 | | (incident power upon port ) a + = i P i i 2 2 | | (reflected power from port ) b = i P i i 2 2 2 | | | | (power delivered to port ) a b + = = i i P P P i i i i 2 5

- Properties of the scattering parameters = (reciprocal network) S S mn nm 2 2 * t = ] [ ] [ ] (lossless network; unitary condition) S I = | | | | , [ a b S i i 2 2 ... | + + = | | | 1 (lossless network) S S 1 i in 2 2 | | | | (lossy network) a b i i = | | = (complex linear gain), | | (s calar linear gain) G S G S 21 21 = 20log | | (dB) (logarithmic gain) g S 10 21 = 20log | | (dB) (insertion loss) IL S 10 21 = 20log | | (dB) (input return loss) RL S in 10 11 = 20log | | (dB) (output return loss) RL S out 10 22 = = , 22 (reflection coefficient) S S in 11 out 6

Examples - 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 7

2. Two-port Matching and Power Gains Two-port network with input and output matching network Zs Two port network [S] Output matching network Input matching network ZL Vs Z0 Z0Z0 Z0 Z0 Z0 in,Zin out,Zout L,ZL L Zs s s, Input and output reflection coefficients - Two-port network without input/output matching = + = + Z Z Z Z 12 21 S S S in 0 L S in 11 1 22 in 0 L + 12 21 S S Z Z Z Z = + = out 0 S S out 22 1 S 11 out 0 S + + Z Z Z Z Z Z Z Z = = 0 0 L S , L S 0 0 L S 8

Input voltage and current + + + = + = + V 11 1 S V 12 2 S V 11 1 S V S V 1 12 2 L + + + = + = + V 21 1 S V 22 2 S V 21 1 S V S V 2 22 2 L Eliminate to obtain V 2 + Z Z Z Z V V 12 21 S S S = = + = in 0 1 L S in 11 + 1 22 in 0 L 1 Z + + + = = + = + in (1 ) V V V V V 1 1 1 1 in S Z Z in S + 1 1 = in Z Z in 0 in 1 (1 ) V V + V = = in S S S S , V 1 1 2 1 2 1 9 in in S S

Input voltage and current + (1 )(1 ) V = in S S V 1 2 1 in S + + (1 )(1 ) (1 )(1 ) 1 1 Z V V V Z = = = in in 1 in S S S S I 1 1 2 1 2 1 Z in 0 in in 0 in S S Power supplied by the source to the network 1Re( 2 * 1 = ) P V I S S Power input to the two-port network + 2 2 2 | | |1 | 1 2 | | V V * 2 2 = = (1 | = (1 | 1 Z S Z S Re( ) | ) | ) P 1 1 V I in in in 2 2 8 |1 | 0 0 in S 10

Output votage and current + + + = + = + V 21 1 S V 22 2 S V 21 1 S V S V 2 22 2 L (1 ) V S S S + = = 21 21 S S V V 2 1 1 2 (1 )(1 ) S 22 22 in L L S + = V V 2 2 L + (1 S )(1 ) V S = 21 S S L V 2 2 (1 )(1 ) 22 in L S + + (1 S )(1 ) (1 S )( 1 ) 1 1 Z V S V S V Z = = = 21 21 2 S S L S S )(1 L L I L 1 2 (1 )(1 ) 2 (1 ) Z 0 22 in 0 22 in L L L S L S 11

Power consumed in the load 2 2 2 | | | (1 S )| (1 | )(1 | ) )| 1Re( 2 V S * L = = 21 S Z S L ) P V I 2 L 2 8 |(1 0 22 in L S Power available from the source: input power with input conjugate matching + 2 2 2 | | |1 | 1 2 | | V V * 2 2 = = (1 | = (1 | 1 Z S Z S Re( ) | ) | ) P 1 1 V I in in in 2 2 8 |1 | 0 0 in S * in * in = = (input conjugate matching for maximum power tranfer) Z Z S S * in + Transform from ( )/( ) to using an input match ing network Z Z Z Z 0 0 S S S 2 2 | | |1 | | V = = S Z S avs P P * in in = 2 1 | 8 S 0 S 12

Power available at the output of the two-port network * out * out = = (output conjugate matching for maximum power tranfer) Z Z L L * out + Transform from ( )/( ) to using an output matching network Z Z Z Z 0 0 L L L 2 2 2 | | | (1 S )| (1 | )(1 | ) 1Re( 2 V S * L = = = 21 S Z S L ) avn P P V I * out 2 L = 2 8 |(1 )| L * out 0 22 in L S = L 12 21 S S 12 21 S S S = + = + S L , S S in 11 out 22 1 1 S 22 11 L S = ( )(1 ) 12 21 S S S S out 22 11 S S 13

Power available at the output of the two-port network * out * out ( )(1 S ) S S S S = = 12 21 1 out 22 11 S S 1 1 1 S S in 11 11 S S S * out * out 1 S 22 22 2 1 (1 )(1 | S | ) S S * out * out = = 11 11 1 out S S 1 ( ) S S 22 out 22 * out * out 1 S 22 22 * out 2 = (1 )(1 ) (1 )(1 | | ) S S in 22 11 out S S 2 2 | | | (1 )| V S = 21 S Z S avn P 2 2 8 |1 | (1 | | ) S 0 11 out S 14

Operating power gain 2 2 | | (1 | | (1 | L | ) P P S S = = 21 L L G 2 2 |1 | ) in 22 in Available power gain: input conjugate matchng, output conjugate matching 2 2 | | (1 | | (1 | S | ) avn P P S S = = 21 S G A 2 2 |1 | ) avs 11 out Transducer power gain: input conjugate matching 2 2 2 | | (1 | | )(1 | )(1 | ) S P = = 21 S L L G T 2 avs P |(1 )| S in 22 S L 15

Rollet's stability factor 2 2 2 1 | + | | | | | S S = 11 22 K 2| | 12 21 S S = 11 22 S S 12 21 S S | 1 (unconditionally stable) 1 and | K Edwards-Sinsky stability factor 2 1 | S | S = 11 | | + * 11 | | S 12 21 S S 22 1 (unconditionally stable) 16

Input and output conjugate matching for maximum power transfer * S * L = = and (simultaneous conjugate matching) in out 2 1 2 2 4| | B B C = 1 1 ( 1) K S C 1 2 2 2 2 4| | B B C = 2 2 ( 1) K L C 2 2 2 2 1 | = + | | | | | B S S 1 11 22 2 2 2 1 | = + | | | | | B S S 2 22 11 * 22 = C S S 1 11 * 11 = C S S 2 22 = 11 22 S S 12 21 S S 2 2 | | (1 | | (1 | L | ) S S = = = 21 L G G G A T 2 2 |1 | ) 22 S 17

Input and output impedances for conjugate matching S + + 1 1 1 1 S = Z Z 0 S L = L Z Z 0 L Input matching network: ZS Z'S Output matching network: ZL Z'L 18

Example 19

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Example - With Z0 = ZS = ZL = 50 and scattering parameters give by - After input and output conjugate matching 21

3. Coding Example Two-port network input & output matching for maximum power transfer - Input data: vs: source voltage (Vpeak) (real) z0: system reference impedance (ohm) (real) zs: source impedance in real/imaginary form (ohm) zL: load impedance in real/imaginary form (ohm) s11, s12, s21, s22: two-port scattering parameters in polar or real/imaginary form - Output results: 1) Before input/output conjugate matching zin : input impedance at port 1 (ohm) gs : source reflection coefficient (polar) gin : input reflection coefficient at port 1 (polar) v1 : voltage at port 1 (Vpeak) i1 : current at port 1 (Apeak) 22

- Output results: zout : output impedance at port 2 (ohm) gL : load reflection coefficient (polar) gout : output reflection coefficient at port 2 v2 : voltage at port 2 (Vpeak) i2 : current at port 2 (Apeak) ps : power supplied by the source (W) pin : input power to the two-port network (W) pavs : power available from the source (W) pL : power consumed in the load (W) pavn : power available from the two-port network output g : operating power gain in linear and dB ga : available power gain in linear and dB gt : transducer power gain in linear and dB k : Rollet's stability factor d_mag : magnitude of the determinant of the scattering matrix of the two-port mu: Edwards-Sinsky stability factor 23

2) After input/output conjugate matching If k 1, simultaneous conjugate matching not possible If k > 1, then zsp, zLp : transformed source and load impedances (ohm) g : operating power gain in linear and dB ga : available power gain in linear and dB gt : transducer power gain in linear and dB 24

# Two-port network input & output matching for maximum power transfer # Input data: # vs : source volatage (Vpeak) (real) # z0 : system reference impedance (ohm) (real) # zs : source impedance in real/imaginary format (ohm) # zL : load impedace in real/imaginary format (ohm) # s11, s12, s21, s22 : scattering parameters of the two-port network # in polar or rectangular form # Output results: # 1) Before conjugate matching # v1 : voltage at port 1 (Vpeak) # i1 : current at port 1 (Apeak) # zin : input impedance at port 1 (ohm) # v2 : voltage at port 2 (Vpeak) # i2 : current at port 2 (Apeak) # zout : output impedance at port 2 (Vpeak) # ps : power supplied by the source (W) # pin : input poer to the two-port network (W) # pavs : power available from the source (W) 25

# g : operating power gain (linear/dB) # ga: avalable power gain (linear/dB) # gt : transducer power gain (linear/dB) # k : Rollet's stability factor # d_mag : absolute value of scattering-matrix determinant # mu : Edwards-Sinsky stability factor # 2) After conjugate matching # If k <= 1: # print 'Input/output conjugate matching not possible...' # If k > 1: # zsp: transformed source impedance (ohm) # zLp: transformed load impedance (ohm) # g : operating power gain (linear/dB) # ga: avalable power gain (linear/dB) # gt : transducer power gain (linear/dB) 26

import cmath import math rad=3.141593/180 while True: ipolar=int(input('S-parameter form (1:polar, 2:real/imaginary) = ')) vs=float(input('Source voltage (Vpeak), real = ')) 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) 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) 27

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) # Quantities at input side gs=(zs-z0)/(zs+z0); gsm=abs(gs); gsp=cmath.phase(gs)/rad gL=(zL-z0)/(zL+z0); gLm=abs(gL); gLp=cmath.phase(gL)/rad gin=s11+s12*s21*gL/(1-s22*gL); ginm=abs(gin); ginp=cmath.phase(gin)/rad zin=z0*(1+gin)/(1-gin) v1=vs*zin/(zs+zin); v1m=abs(v1); v1p=cmath.phase(v1)/rad i1=v1/zin; i1m=abs(i1); i1p=cmath.phase(i1)/rad 28

print(' ') print(' Source voltage (Vpeak) = ',vs) print(' Input impedance at port 1 (ohm) = ', zin) print(' Source reflection coefficient (polar) = ',gsm,gsp) print(' Input reflection coeffcient at port 1 (polar) = ',ginm,ginp) print(' Input voltage (Vpeak) at port 1 in polar = ',v1m,v1p) print(' Input current (Apeak) at port 1 in polar = ',i1m,i1p) # Quantities at output side gout=s22+s12*s21*gs/(1-s11*gs) zout=z0*(1+gout)/(1-gout) goutm=abs(gout); goutp=cmath.phase(gout)/rad v2=(vs/2)*s21*(1-gs)*(1+gL)/((1-s22*gL)*(1-gs*gin)) v2m=abs(v2); v2p=cmath.phase(v2)/rad i2=v2/zout; i2m=abs(i2); i2p=cmath.phase(i2)/rad print(' ') print(' Output impedance at port 2 (ohm) = ',zout) print(' Load reflection coefficient (polar) = ',gLm,gLp) print(' Output reflection coeffcient (polar) = ',goutm,goutp) print(' Output voltage (Vpeak) in polar = ',v2m,v2p) print(' Output current (Apeak) in polar = ',i2m,i2p) 29

# Power calculation ps=vs*i1/2; psr=ps.real pin=abs(vs)**2/(8*z0)*abs(1-gs)**2*(1-abs(gin)**2)/abs(1-gs*gin)**2 pavs=abs(vs)**2/(8*z0)*abs(1-gs)**2/(1-abs(gs)**2) pL=abs(vs)**2/(8*z0)*abs(s21*(1-gs))**2*(1-abs(gL)**2)/abs((1-s22*gL)*(1-gs*gin))**2 pavn=abs(vs)**2/(8*z0)*abs(s21*(1-gs))**2/(abs(1-s11*gs)**2*(1-abs(gout)**2)) print(' ') print(' Sourece power (W) = ',psr) print(' Input power at port 1 (W) = ',pin) print(' Available power from the source (W) = ',pavs) print(' Load power (W) = ',pL) print(' Available power from the network (W) = ',pavn) # Power gain calculation 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)*(1-gs*gin))**2 print(' ') print(' Operating 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)') 30

# Stability calculation d=s11*s22-s12*s21 k=(1-abs(s11)**2-abs(s22)**2+abs(d)**2)/(2*abs(s12*s21)) mu=(1-abs(s11)**2)/(abs(s22-d*s11.conjugate())+abs(s12*s21)) print(' ') print(' Rollet\'s stability factor K (>1 if stable) = ',k) print(' Rollet\'s stability factor d_mag (< 1 if stable) = ',abs(d)) print(' Edwards stability factor mu (>1 if stable) = ',mu) # Input/output conjugate matching if k > 1: 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)*z0/(1-gsp) zLp=(1+gLp)*z0/(1-gLp) 31

print(' ') print(' Transformed source impedance: Zsp = ',zsp) print(' Transformed load impedance: ZLp = ',zLp) gL=gLp; gs=gsp gin=gsp.conjugate(); gout=gLp.conjugate() 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)') print(' ') else: print(' ') print(' The network cannot be conjugate-matched...') print(' ') 32

''' Sample run S-parameter form (1:polar, 2:real/imaginary) = 1 Source voltage (Vpeak), real = 1 System reference impedance Z0 (ohm) = 50 Source impedance Zs=Rs+jXs (ohm): Rs, Xs = 20 20 Load impedance ZL=RL+jXL (ohm): RL, XL = 40 0 Mag(S11), phase(S11)(deg) = 0.1 2 Mag(S12), phase(S12)(deg) = 0.8 -100 Mag(S21), phase(S21)(deg) = 0.8 -100 Mag(S22), phase(S22)(deg) = 0.1 100 33

Source voltage (Vpeak) = 1.0 Input impedance at port 1 (ohm) = (69.91203191917708-3.1093010629658027j) Source reflection coefficient (polar) = 0.49526055654364864 130.36452219834987 Input reflection coeffcient at port 1 (polar) = 0.1680111613227668 -7.389831770229386 Input voltage (Vpeak) at port 1 in polar = 0.7649482501639194 -13.185992261202722 Input current (Apeak) at port 1 in polar = 0.010930777203899952 -10.639470812065685 Output impedance at port 2 (ohm) = (56.54525858808134-21.879898520912395j) Load reflection coefficient (polar) = 0.1111111111111111 179.99998015215937 Output reflection coeffcient (polar) = 0.20996778395869814 -61.74091356141908 Output voltage (Vpeak) in polar = 0.46706774279660934 -112.75294363306014 Output current (Apeak) in polar = 0.007703470768125124 -91.59925510327503 Sourece power (W) = 0.005371429766367877 Input power at port 1 (W) = 0.00417661086355489 Available power from the source (W) = 0.006250000000000003 Load power (W) = 0.0027269034545139956 Available power from the network (W) = 0.0029537049807950265 34

Operating power gain G = 0.6528986165096103 / -1.8515425166338568 (dB) Available power gain Ga = 0.47259279692720413 / -3.2551290199820686 (dB) Transducer power gain Gt = 0.4363045527222393 / -3.602102552418523 (dB) Rollet's stability factor K (>1 if stable) = 1.0804039274288189 Rollet's stability factor d_mag (< 1 if stable) = 0.634757455339351 Edwards stability factor mu (>1 if stable) = 1.2634491974830742 Transformed source impedance: Zsp = (78.08792105218402-17.565644004445534j) Transformed load impedance: ZLp = (34.89372207163361-16.192980191048612j) Power gain G = 0.6714141397768333 / -1.7300951688247814 (dB) Available power gain Ga = 0.6714141397768332 / -1.7300951688247823 (dB) Transducer power gain Gt = 0.6714141397768334 / -1.730095168824781 (dB) S-parameter form (1:polar, 2:real/imaginary) = End of sample run ''' 35

4. Using www.online-python.com Getting stared with Online Python 1. Connect to www.online-python https://www.online-python.com/ 2. Copy your code or write your code on the upper window 3. Click [Run] 4. Enter input data in the lower window 5. Results will be printed in the lower window 36

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