MicrowaveNetwork AnalysisMicrowave EngineeringEE 172Dr. Ray KwokMicrowave Network - Dr. Ray KwokMicrowave Matrices N – portnetworkV1+, I1+V1−, I1−V2+, I2+V2−, I2−VN+, IN+VN−, IN−• Use matrices to relate the voltages and currents of the portswithout knowing details of the network. • Z, Y, ABCD, S – matrix (+ others not covered here. e.g. T-matrix)• Useful to cascade structures, instead of working with Maxwell Eqns.Microwave Network - Dr. Ray KwokZ Matrix0I12210I111121222112112122IVZIVZIIZZZZVV==≡≡=2 – portnetworkV1 V2I1I20I22220I211211IVZIVZ==≡≡N x N matrixfor a N-port networkPort-2 openPort-1 openMicrowave Network - Dr. Ray KwokExample: T - networkCB0I222212C0I1221CA0I1111ZZIVZZZIVZZZIVZ122+=≡==≡+=≡===( )++=CBCCCAZZZZZZZV1 V2I1I2ZAZBZCZ11−Z12Z12Z22−Z12Likewise, knowing the Z-matrix, one can construct an equivalent circuit.Microwave Network - Dr. Ray KwokY Matrix0V12210V111121222112112122VIYVIYVVYYYYII==≡≡=2 – portnetworkV1 V2I1I20V22220V211211VIYVIY==≡≡N x N matrixfor a N-port networkPort-1 shortPort-2 shortMicrowave Network - Dr. Ray KwokExample: π π π π - networkCB0V222212C0V1221CA0V1111YYVIYYYVIYYYVIY122+=≡=−=≡+=≡===( )+−−+=CBCCCAYYYYYYYV1 V2I1I2YBYCYA−Y12Y11 +Y12Y22 +Y12Likewise, knowing the Y-matrix, one can construct an equivalent circuit.Microwave Network - Dr. Ray KwokReciprocal NetworkNo active elementNo ferriteZij= ZjiYij= YjiLossless NetworkAll Zij& Yijare pure imaginary“symmetric” matrixinterchangeable input – output portsFor low loss systems, this is the first approximation people useto simplify the problem significantly.Symmetrical Networkall Ziiare the same, e.g. Z11= Z22all Yiiare the sameMicrowave Network - Dr. Ray KwokABCD Matrix0I210I21221122VICVVAIVDCBAIV==≡≡=2 – portnetworkV1 V2I1I20V210V2122IIDIVB==≡≡2 x 2 matrixOnly for 2-port networkA & D are dimensionlessB has an unit of impedanceC has an unit of admittancePort-2openPort-2shortMicrowave Network - Dr. Ray KwokCascade NetworkBATOTAL33BA22A1122A11DCBADCBADCBAIVDCBADCBAIVDCBAIVIVDCBAIV====NetworkAV1 V2I1I2NetworkBV3I2I3ABCD matrix is extremely useful to cascade 2-port networks.Pay attention to the order of multiplication !!=33B22IVDCBAIVMicrowave Network - Dr. Ray KwokExample: series element0VIC1VVA0I210I2122=≡=≡==1IIDZIVB0V210V2122=≡=≡==?DCBA=ZV2= V1I2 = I1 = 0V1− I2Z = 0I 2 = I1short V2=0V1I1I2Z10Z1openV2I2=0ZV1I1Microwave Network - Dr. Ray KwokExample: shunt elementYVIC1VVA0I210I2122=≡=≡==1IID0IVB0V210V2122=≡=≡==?DCBA=YV2= V1= I1/YV2= V1 = 0I 2 = I11Y01Yopen V2V1I2=0I1Yshort V2=0V1I1I2Microwave Network - Dr. Ray KwokExample: low pass filterωωωωω=10Lj11Cj0110Lj11Cj0110Lj1DCBA32211L1L2L3C1C2Microwave Network - Dr. Ray KwokExample: T - network[ ]CCA2CA11C12ZZZVZZIVZIV+=+==CB0V21CBABA0V21ZZ1IIDZZZZZIVB22+=≡++=≡==C0I21CA0I21Z1VICZZ1VVA22=≡+=≡==V1 V2I1I2ZAZBZCOpen V2V1I2=0I1ZAZBZCshort V2=0V1I1I2ZAZBZC()[ ]++=+++=+=+=−=BACBA21CBCBACCB21CBA11CCB21C21B2ZZZZZIVZZZZZZZZIVZ//ZZIVZZZIIZIIZIMicrowave Network - Dr. Ray KwokT – network cascadeV1 V2I1I2ZAZBZC++++=+==CBCCBABACACBCBABCAZ/Z1Z/1Z/ZZZZZ/Z1DCBA1Z/ZZ/1Z110Z1DCBA10Z11Z/10110Z1DCBAsame answerMicrowave Network - Dr. Ray KwokSymmetrical & ReciprocalV1V2I1I2ZAZBZC1DCBAZZZZZZZZZZZZZZ1ZZZZZZ1ZZ1ZZ1DCBADAZ/Z1Z/1Z/ZZZZZ/Z1DCBA2CBACBCA2CBACBCACBABACCBCACBCCBABACA=−−−+++=++−++==++++=IF symmetrical networkIFreciprocal networkMicrowave Network - Dr. Ray KwokS Matrix0a12210a111121222112112122abSabSaaSSSSbb==≡≡=0a22220a211211abSabS==≡≡2 – portnetworkb1b2a1a2N x N matrixfor N-port networkOutnetworkInnetworko2ii*ii,outo2ii*ii,inoiioiiZV21bb21PZV21aa21PZVbZVa−+−+====≡≡ith-portPort – 2terminatedPort – 1terminatedZoZoMicrowave Network - Dr. Ray KwokS-parameters (matched load)1120a12211110a1111VVabSVVabS22τ==≡Γ==≡+−=+−=2 – portnetworkb1b2a1a2=0MatchedLoadReflection coefficientTransmission coefficientReturn Loss = -20 log|S11|Insertion Loss = -20 log|S21|(called rejection or isolation for out-of-band)Input a1, measure b1& b2to characterize the network.Microwave Network - Dr. Ray KwokUnmatched load2 – portnetworkb1b2a1a2= ΓLb2ZLThis is what we measure.How does that affect our matching process?L2221L1211in11L22121L121111L2212122L2212122212122L121112121111212221121121S1SSSabS1aSSaSbS1aSbbSaSaSaSbbSaSaSaSbaaSSSSbbΓ−Γ+=Γ≡Γ−Γ+=Γ−=⇒Γ+=+=Γ+=+==Microwave Network - Dr. Ray KwokMatrix Operations()()()PxNMxPMxNBAC=MCMji1ij=−( )( )bcadMacbdM1MdcbaM1−=−−=≡−ifdeterminantInverse Matrix (2 x 2)rank of matrices have to matchcomponent notation, summation impliedInverse Matrix (general)Cijis the cofactor of ji-th elementwhich is (-1)i+j* minorjiMinorjiis the deteminant withoutthe j-th row & i-th column.kjikkkjikijBABAC ≡=∑240832C21=−=( )≡087654321M35421C33−=+=e.g.( )−−−−−=−3636214232448271M1Microwave Network - Dr. Ray KwokLossless Network( )( )( )( )( ) ( )( )( ) ( ) ( )( )( )(