Begin2.DVI

Figure 10-6. Four terminal networks

In figure 10-6 the quantities I 1 , E 1 and I 2 , E 2 are the input and output current

and voltages. The column vectors S 1 =col[E 1 , I 1 ]and S 2 =col[E 2 , I 2 ]are called the

input state vector and output state vector of the network. The networks are assumed

to be linear so that the general relation between the input and output states can be

expressed as the matrix equation

S 1 =T S 2 , where T=

[ T 11 T 12 T 21 T 22

]

is called the transmission matrix. The element T 11 , T 12 , T 21 , T 22 are in general complex

numbers which satisfy the property det T=T 11 T 22 −T 21 T 12 = 1.Solving for S 2 in terms

of S 1 gives

S 2 =T−^1 S 1 =P S 1

where the matrix P is called the transfer matrix of the network and is given by

P=

[ T 22 −T 12 −T 21 T 11

]

If the input output current and voltages are linearly related, then it is easy to

solve for the currents in terms of the voltages or the voltages in terms of the currents

to obtain

[ I 1 I 2

] =

[T 22 T 12 −

1 1 T^12 T 12 −

T 11 T 12

][ E 1 E 2

] =

[T 11 T 21 −

1 1 T^21 T 21 −

T 22 T 21

][ I 1 I 2

]

Applying Kirchoff’s laws to the short-circuit condition E 2 = 0 and the open-

circuit condition I 2 = 0 allows for the determination of the transmission matrices

given in the figure 10-6.