Problem 31-6    Source:   Problem created by the site's author.

In the circuit shown in Figure 31-6.1 determine:

a) the "Z" parameters:

b) the "Y" parameters:

c) check its equivalence with the use of   Table 30-01

Solution of the Problem 31-6

Item a

To solve this problem we will present an alternative technique, using the equations taken from the circuit, without the need to introduce voltage or current sources at input or output ports. We transform the equations into a form similar to the equations that describe the quadripole. In this way, by equating the coefficients that accompany the variables, we obtain the parameters of the quadripole. Let's do the calculations.

Let's remember that the Z parameters are given by the following equations:

Using the circuit shown in Figure 36-1.1 and using Kirchhoff's law for meshes, we obtain:

Now let's make the mesh identified as I₃ in the figure above.

But, from the circuit we see that I_x = I₂ + I₃. Replacing this value in  eq. 31-06.5, we obtain:

We replace this value in  eq. 31-06.3 and we get:

Now comparing this equation with the  eq. 31-06.1 , we can easily see that the coefficients that accompany the variables I₁ and I₂ will be the values of Z₁₁ and Z₁₂. Soon:

And to find the values of Z₂₁ and Z₂₂, just substitute the equation  eq. 31-06.6 in the equation  31-6.4, finding:

Comparing this equation with the equation  eq. 31-06.2, we easily see that the coefficients that accompany the variables I₁ and I₂ are the values of Z₂₁ and Z₂₂. Soon:

And so, we easily find the Z parameters without the need to introduce voltage or current sources into the quadrupole ports. It just took a little algebra.

Item b

To calculate the parameters Y we can take advantage of the calculations made in the previous item. First, we will present the equations that govern the quadripole type Y.

Using the equation  eq. 33-06.8 we verify that we must have I₁ in function of V₁ and V₂. Soon, algebraically manipulating the equation  eq. 33-06.7, we obtain:

Comparing this equation with the equation  eq. 31-06.8, we determine the values of Y₁₁ and Y₁₂.

And to determine the values of Y₂₁ and Y₂₂ we will substitute the equation  eq. 31-06.6 na equation  eq. 31-06.4, finding:

Combining the equation  eq. 31-06.10 with the equation  eq. 31-06.11 and, after some algebra, we find:

Now we just compare this equation with the  eq. 31-06.9 equation to determine the values of Y₂₁ and Y ₂₂, or:

Item c

Let's check using Table 30-01 if there is a correspondence between the calculated parameters. Let's calculate the value of ΔZ.

Therefore, we realize that there is a perfect correspondence between the calculated parameters.