Question

For equivalent $\mathbf{\pi }$circuits of lossless lines, the$\mathbf{A}$ and$\mathbf{D}$ parameters are pure_____, whereas $\mathbf{B}$and $\mathbf{C}$ parameters are pure______.

Short answer

Therefore, parameter $A$ and$D$ are pure real and parameters$B$ and$C$ are pure imaginary.

Step-by-step solution

Determine A, B, C and D parameter of the π network.

The parameter $\mathbf{A}$and $\mathbf{D}$ are equal for the lossless line and it is given by,

$\begin{array}{l}\mathbf{A}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{D}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\\ \mathbf{A}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{cos}\mathbf{\left(}\mathbf{\beta x}\mathbf{\right)}\mathbf{pu}\end{array}$…… (1)

Here, $\mathbf{\beta }$is the phase constant.

The parameter$\mathbf{B}$ is given by,

$\mathbf{B}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{j}\sqrt{\frac{\mathbf{L}}{\mathbf{C}}}\mathbf{sin}\mathbf{\left(}\mathbf{\beta x}\mathbf{\right)}\mathbf{\Omega }$ …… (2)

Here,$\mathbf{L}$ is the inductance and $\mathbf{C}$ is the capacitance.

The parameter$\mathbf{C}$ is given by,

$\mathbf{C}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{jsin}\mathbf{\left(}\mathbf{\beta x}\mathbf{\right)}}{\sqrt{\frac{\mathbf{L}}{\mathbf{C}}}}\mathbf{S}$ …… (3)

Determine the nature of A, B, C and D parameter of the transmission line for π network.

By observing the equation (1), It clear that the imaginary part is zero. Therefore, the parameter $A$and $D$are purely real.

By observing the equation (2) and (3), it is clear that the real part is zero both the equation. Therefore, the parameter $B$and$C$ are purely imaginary.

Hence, the parameter$A$ and$D$ are purely real and parameters$B$ and$C$ are purely imaginary.