Problem 31-3    Source:    problem 19-6 - page 808 -    SADIKU, Matthew N. O. , ALEXANDER, Charles K. - Book: Fundamentos de Circuitos Elétricos - Mc Graw Hill - 5ª edição - 2013.

In the circuit shown in Figure 31-03.1, determine the parameters "Z"

Solution of the Problem 31-3

The Z parameters are given by the following equations:

Because this circuit presents a dependent voltage source of the input current, we can not use the methodology used in the previous problems. In this way, we will introduce a current source I₁ on port 1, and we will do I₂ = 0, as shown in the Figure 31-03.2.

From the circuit, as we adopted I₂ = 0, then there will be no voltage drop on the resistor of 10 ohms. Also it is possible to realize that in the resistor of 20 ohms will pass the current I₁. Then, by making the mesh in the direction indicated by the arrow red, in the figure above, V₂ is:

On the other hand, by making the mesh in the direction of the violet arrow, in the figure above, the current I₁ will circulate through the 5 and 20 ohms resistors. Therefore, the input voltage V₁ will be equal to:

With this data, it is already possible to calculate Z₁₁ and Z₂₁.

In the same way, to calculate Z₂₂ and Z₁₂ we should have I₁ = 0. Se I₁ = 0, then there will be no voltage drop over the 5 ohm resistor and the voltage source 4 I₁, will become null. With this we can eliminate these two components from the circuit without prejudice to our calculation. Thus, we can easily calculate V₁ and V₂. See in the Figure 31-03.3, as was the modification in the circuit.

Making the mesh equations for V₁ and V₂, we obtain:

With this data, it is already possible to calculate Z₂₂ and Z₁₂.

As can be seen, Z₁₂ ≠ Z₂₁and with this we conclude that the circuit NOT is PASSIVE or RECIPROCAL.