Problem 33-2    Source:    Question 2 - Test of the School of Engineering of UFRGS - 1976.

In the circuit below determine the parameters "h"

Solution of Problem 33-2

The parameters "h" are given by the following equations:

To calculate h₁₁ and h₂₁ we must put the output in short circuit, that is, E₂ = 0 volt. Ans in the input, we put a source current I₁, with a convenient value. In this case, we choose I₁ = 1 A. Now, note that if E₂ = 0 volt, then the current source that is in parallel with 4 ohms resistance assumes ZERO value, and therefore can be suppressed from the circuit. As I₁ = 1 A, then the voltage source 6 I₁ = 6 volts. See the Figure 33-02.2, as was the redesigned circuit.

A transformation can be made with the 6 volts voltage source and the 3 ohms series resistor. A 2 A current source with 3 ohms resistor results in parallel. So, we have the circuit of the Figure 33-02.3.

The equivalent resistance of the three resistors that are in parallel can be calculated. Calculating the parallel of 2, 3 and 6 ohms results in 1 ohm. Since to node a reaches a total current of 3 A, it is obvious that this current must pass through the equivalent resistance of 1 ohm. And this generates a voltage on the node a of 3 volts. Knowing the tension in the node a, we can calculate the current I₂, ou:

Knowing I₁, we can calculate E₁, because we know the value of V_a and the voltage drop in the resistance of 5 ohms, that is:

With these data, we can calculate h₁₁ and h₂₁. Logo:

To calculate h₁₂ and h₂₂ we need to leave the port open, that is, by making I₁ = 0. In port 2, or output port, a current source of 3 A is connected. Soon I₂ = 3 A. See the Figure 33-02.4 as was the transformation of the circuit.

As I₁ = 0 , then the current source current 5 E₂ it will be confined to it and the 4 ohms resistor. Therefore, the current I₂ will be divided into two parts: one by the resistor of 2 ohms and another by the resistor of 3 ohms , as shown in the figure above, already with its values indicated. It was used a current divider to calculate them. Based on the circuit of the figure above, we can calculate E₁ and E₂.

Note that the current source 5 E₂ transforms into a current source of 108 A , and this current passes through the 4 ohms resistor, then:

With the calculated values, you can calculate the values of h₁₂ and h₂₂. Then: