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    Charateristics of {A,B ; C,D} Matrix for lossless circuit

    As well known, Scattering Matrix [S] is a unitary matrix for lossless circuit.

    Is there any simple relation in [F]={A,B ; C,D} Matrix for lossless circuit ?
    Reciprocity is reflected to determinant(F)=1.

    Here {A,B ; C,D} Matrix is called as "Fundamental Matrix", "Ketten Matrix", "Cascade Matrix" or "Chain Matrix".

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    Re: Charateristics of {A,B ; C,D} Matrix for lossless circuit

    Hmm, I like these sort of puzzles. I'll give it a shot.

    If A and D are real valued and B and C are imaginary valued, then it should be lossless.

    I base this on the fact that impedance parameters of a lossless network are all imaginary, and the known conversions from Z parameters to ABCD parameters.



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    Re: Charateristics of {A,B ; C,D} Matrix for lossless circuit

    Thanks for response.

    Quote Originally Posted by mtwieg View Post
    If A and D are real valued
    and B and C are imaginary valued,
    then it should be lossless.
    [F] Matrix of Lossless Transmission Line is a simple example.

    A=D=cos(beta*L)
    B=j*R0*sin(beta*L)
    C=j*(1/R0*)sin(beta*L)



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    Re: Charateristics of {A,B ; C,D} Matrix for lossless circuit

    It is assumed that ABCD-parameters are defined by the following equations:
    V1 = A*V2 - B*I2,
    I1 = C*V2 - D*I2.

    No-loss condition:
    0 = real(conj(V1)*I1 + conj(V2)*I2)

    = real(conj(A*V2 - B*I2)*(C*V2 - D*I2) + conj(V2)*I2)

    = real(conj(A)*C*conj(V2)*V2)
    + real(conj(B)*D*conj(I2)*I2)
    + real((1 - conj(A)*D)*conj(V2)*I2)
    - real(conj(B)*C*conj(I2)*V2)

    = real(conj(A)*C*conj(V2)*V2)
    + real(conj(B)*D*conj(I2)*I2)
    + real((1 - conj(A)*D - B*conj(C))*conj(V2)*I2).

    Consequently,
    real(conj(A)*C) = 0,
    real(conj(B)*D) = 0,
    conj(A)*D + B*conj(C) = 1.



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