Problem 35-2    Source:    Problem 2 of the Quadripoles section of the list of Electrical Circuits I, of the School of Engineering of UFRGS - 2011 - Prof. Dr. Valner Brusamarello.

Determine the values of A , B , C and D for the circuit shown in the Figure 35-02.1, which is described by the following equations:

Solution of the Problem 35-2

By the equations of the circuit we can determine the value of A by doing I₁ = 0. At the output port, a voltage source with a convenient value is placed, for example, V₂ = 4 volts. Thus, the circuit with the modifications and indication of some current values is shown in the Figure 35-02.2.

Looking at the circuit we conclude that:

Applying LKC to the input node, we obtain:

Substituting eq. 35-02.1 in the above equation and remembering that V₂ = 4, we obtain:

By rearranging the equation and performing the calculation, we find the value of I₂, or:

Thus, we can calculate the value of A.

We know that for I₁ = 0, we obtain  I₂ = 9/5. Then we conclude that V₁ = 8/5. As V₂ = 4, we can find a relation between C and D replacing these values in the second equation of the circuit. Like this:

Rearranging the terms:

By making I₂ = 0 and placing a voltage source of 8 volts in the input port, C and D are determined. Therefore, the source of current V₁ / 8 = 1 A. See the circuit shown in the Figure 35-02.3 as the modifications were.

We can find the value of I₁ making the mesh indicated by the green arrow. So:

Working algebraically this equation and effecting the calculation, we find:

It was easy to calculate the V₂, or:

Knowing that I₂ = 0 , V₂ = 2 volts and I₁ = 4.5 A, we can find a relationship between A and B using the first equation of the circuit shown in the statement. After the substitutions and making an algebraic arrangement, we obtain:

As we already know that A = 9/20 , then the value of B is:

As I₁ = 4.5, I₂= 0 and V₁ = 2, using the second equation we will find the value of C.

Previously, we already found the relationship between C e D, or D = -(8/9) C. So:

And so, all the variables requested in the problem were determined. To conclude, the equations of the quadripole are: