The S-parameters of a two port network is given as

$[S] = [\begin{matrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{matrix}]$

with reference to 𝑍₀. Two lossless transmission line sections of electrical lengths 𝜃₁ = 𝛽𝑙₁ and 𝜃₂ = 𝛽𝑙₂ are added to the input and output ports for measurement purposes, respectively. The S-parameters [𝑆′] of the resultant two port network is

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The S-parameters of a two port network is given as

$[S] = [\begin{matrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{matrix}]$

with reference to 𝑍₀. Two lossless transmission line sections of electrical lengths 𝜃₁ = 𝛽𝑙₁ and 𝜃₂ = 𝛽𝑙₂ are added to the input and output ports for measurement purposes, respectively. The S-parameters [𝑆′] of the resultant two port network is

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$[\begin{matrix} S_{11} e^{- j 2 θ_{1}} & S_{12} e^{- j (θ_{1} + θ_{2})} \\ S_{21} e^{- j (θ_{1} + θ_{2})} & S_{22} e^{- j 2 θ_{2}} \end{matrix}]$
$[\begin{matrix} S_{11} e^{j 2 θ_{1}} & S_{12} e^{- j (θ_{1} + θ_{2})} \\ S_{21} e^{- j (θ_{1} + θ_{2})} & S_{22} e^{j 2 θ_{2}} \end{matrix}]$
$[\begin{matrix} S_{11} e^{j 2 θ_{1}} & S_{12} e^{j (θ_{1} + θ_{2})} \\ S_{21} e^{j (θ_{1} + θ_{2})} & S_{22} e^{j 2 θ_{2}} \end{matrix}]$
$[\begin{matrix} S_{11} e^{- j 2 θ_{1}} & S_{12} e^{j (θ_{1} + θ_{2})} \\ S_{21} e^{j (θ_{1} + θ_{2})} & S_{22} e^{- j 2 θ_{2}} \end{matrix}]$

Answer 1

F1 Engineering Arbaz 27-12-23 D12

$[\begin{array}{l} V_{1}^{-} \\ V_{2}^{-} \end{array}] = [\begin{array}{ll} S_{11} & S_{12} \\ S_{21} & S_{22} \end{array}] [\begin{array}{l} V_{1}^{+} \\ V_{2}^{+} \end{array}]$ ....(1)

To be calculated,

$\begin{aligned} [\begin{matrix} V_{1 N}^{-} \\ V_{2 N}^{-} \end{matrix}] = [\begin{array}{ll} S_{11 N} & S_{12 N} \\ S_{21 N} & S_{22 N} \end{array}] [\begin{matrix} V_{1 N}^{+} \\ V_{2 N}^{+} \end{matrix}] \end{aligned}$

$V_{1}^{+} = V_{1 N}^{+} e^{- j θ_{1}}$

$V_{1 N}^{-} = V_{1}^{-} e^{- j θ_{1}} = V_{1}^{-} = V_{1 N}^{-} e^{j θ_{1}}$

$V_{2}^{+} = V_{2 N}^{+} e^{- j θ_{2}}$

$V_{2 N}^{-} = V_{2}^{-} e^{- j θ_{2}} = V_{2}^{-} = V_{2 N}^{-} e^{j θ_{2}}$

Feeding the data in equation (i),

$[\begin{matrix} V_{1 N}^{-} \\ V_{2 N}^{-} \end{matrix}] = [\begin{array}{cc} S_{11 N} e^{- j (2 θ_{1})} & S_{12 N} e^{- j (θ_{1} + θ_{2})} \\ S_{21 N} e^{- j (θ_{1} + θ_{2})} & S_{22 N} e^{- j 2 (θ_{2})} \end{array}]$

Hence, the correct option is (A).

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