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

In the s domain,

1 H            ⇒            sL  =  s,            1 F        ⇒          \frac{1}{sC}  =  \frac{1}{s}

To get g_{11}  and  g_{21}   , we open-circuit the output port and connect a voltage source V_{1}   to the input port as in Fig. 19.29(a). From the figure,

I_{1}  = \frac{ V_{1}}{s  +  1}

or

g_{11}  = \frac{ I_{1}}{V_{1}} = \frac{1}{s  +  1}

By voltage division ,

V_{2}  = \frac{1}{s  +  1} V_{1}

or

g_{21}  =  \frac{V_{2}}{V_{1}} = \frac{1}{s  +  1}

To obtain  g_{12}  and  g_{22}   , we short-circuit the input port and connect a current source I_{2}   to the output port as in Fig. 19.29(b). By current division,

I_{1}  = –  \frac{1}{s  +  1} I_{2}

or

g_{12}  =  \frac{I_{1}}{I_{2}} = – \frac{ 1}{s  +   1}

Also,

V_{2}  =  I_{2} (\frac{1}{s}  + s  \parallel  1)

or

g_{22}  =  \frac{V_{2}}{I_{2}} =  \frac{1}{s}  + \frac{s}{s  +   1} =   \frac{s^{2}  +  s  +  1}{s(s  +  1)}

Thus,

[g] = \begin{bmatrix} \frac{1}{s  +  1}  &  – \frac{1}{s  +  1}  \\ \frac{1}{s  +  1}  &  \frac{s^{2}  +  s  +  1}{s(s  +  1)} \end{bmatrix}