ECE 6130 Impedance and Admittance Matrices and S-Parameters Text Sections: 4.2, 4.3 Describe Z and S matrices, how to compute them, and how to convert between them. See for example Chapter 4, Problems 7,9 Impedance 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 ≠ j 1) Open all other ports except j 2) Drive port j with current Ij 3) Read Vi 4) Compute Zij OR: V = Z I Admittance Matrix: I = Y V Y = Z –1 (matrices are inverses of eachother) Reciprocal Network: Zij = Zji Examples of reciprocal networks: any R,L,C network Examples of non-reciprocal networks: transistors, amplifiers, attenuators Lossless 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 + Rc Find Z12 : I1=0; V2 = I2 (Zb + Zc); V1 = V2 Zc / (Zb+Zc); Z12 = V1 / I2 = Zc Find Z21: I2 = 0; V1 = I1 (Za + Zc); V2 = V1 Zc / (Za+Zc) ; Z21 = V2/ I1 = Zc = Z12 Find Z22 : I1 =0; V2 = I2 (Zb+Zc); Z22 = V2 / I2 = Zb + Zc Scattering Matrix (S-parameters) Where Sij = Vi- / Vj+ when Vk+ = 0 for k≠j 1) Terminate all ports except j with matched load. 2) Drive port j with Vj+ 3) Measure reflected voltage Vi- on port i. EXAMPLE: 3dB attenuator Find S11: Z2 = 50 ohms; Zin(port 1) = 8.56 + (141.8 || (8.56 + 50) ) = 50 ohms V1- = 0 (no reflection) S11 = V1- / V1+ =0 ⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡+++−−−NNNNNNVVVSSSSSSVVV211211121121Find S22: Circuit is symmetric. S22 = S11 Find
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