Question 19.7: Find the g parameters as functions of s for the circuit in F...

In the s domain,

1 H \quad \Rightarrow \quad sL=s, \quad 1 F \quad \Rightarrow \quad \frac{1}{s C}=\frac{1}{s}

To get \pmb{g_{11}} \text{ and } \pmb{g_{21}}, we open-circuit the output port and connect a voltage source \pmb{V_1} to the input port as in Fig. 19.29(a). From the figure,

\pmb{I _1}=\frac{ \pmb{V _1}}{s+1}

or

\pmb{g _{11}}=\pmb{\frac{ I _1}{ V _1}}=\frac{1}{s+1}

By voltage division,

\pmb{V _2}=\frac{1}{s+1} \pmb{V _1}

or

\pmb{g _{21}}=\pmb{\frac{ V _2}{ V _1}}=\frac{1}{s+1}

To obtain \pmb{g_{12}} \text{ and } \pmb{g_{22}}, we short-circuit the input port and connect a current source \pmb{I_2} to the output port as in Fig. 19.29(b). By current division,

\pmb{I _1}=-\frac{1}{s+1} \pmb{I _2}

or

\pmb{g _{12}}=\pmb{\frac{ I _1}{ I _2}}=-\frac{1}{s+1}

Also,

\pmb{V _2}= \pmb{I _2}\left(\frac{1}{s}+s \| 1\right)

or

\pmb{g _{22}}=\pmb{\frac{ V _2}{ I _2}}=\frac{1}{s}+\frac{s}{s+1}=\frac{s^2+s+1}{s(s+1)}

Thus,

[ \pmb{g} ]=\left[\begin{array}{cc} \frac{1}{s+1} & -\frac{1}{s+1} \\ \space \\  \frac{1}{s+1} & \frac{s^2+s+1}{s(s+1)} \end{array}\right]