ECE 6130 Impedance and Admittance Matrices and S-Parameters Text Sections: 4.2, 4.3Describe Z and S matrices, how to compute them, and how to convert between them.See for example Chapter 4, Problems 7,9Impedance Matrix:DRAW an N-port network.Impedance matrix is used to model V and I relations for all ports.Zij = Vi / Ij with Ik = 0 for k j1) Open all other ports except j2) Drive port j with current Ij 3) Read Vi 4) Compute ZijOR: V = Z IAdmittance Matrix:I = Y VY = Z –1 (matrices are inverses of eachother)Reciprocal Network:Zij = ZjiExamples of reciprocal networks: any R,L,C networkExamples of non-reciprocal networks: transistors, amplifiers, attenuatorsLossless Network:Real (Zij ) = 0 << Zij is strictly imaginary (change of phase, but no attentuation)EXAMPLE: T-Network NNNNNNIIIZZZZZZVVV211211121121Find Z11 : I2 = 0; V1 = I1 (Ra + Rc); Z11 = V1 / I1 = Ra + RcFind Z12 :I1=0; V2 = I2 (Zb + Zc); V1 = V2 Zc / (Zb+Zc); Z12 = V1 / I2 = ZcFind Z21:I2 = 0; V1 = I1 (Za + Zc); V2 = V1 Zc / (Za+Zc) ; Z21 = V2/ I1 = Zc = Z12Find Z22 :I1 =0; V2 = I2 (Zb+Zc); Z22 = V2 / I2 = Zb + ZcScattering Matrix (S-parameters)WhereSij = Vi- / Vj+ when Vk+ = 0 for kj1) Terminate all ports except j with matched load.2) Drive port j with Vj+ 3) Measure reflected voltage Vi- on port i.EXAMPLE: 3dB attenuatorFind S11:Z2 = 50 ohms; Zin(port 1) = 8.56 + (141.8 || (8.56 + 50) ) = 50 ohmsV1- = 0 (no reflection)S11 = V1- / V1+ =0NNNNNNVVVSSSSSSVVV211211121121Find S22:Circuit is symmetric.S22 = S11 Find
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