Problem 33-1    Source:    Problem 31 - List of Electrical Circuits I, of the School of Engineering of UFRGS - 2011 - Prof. Dr. Valner Brusamarello.

In the circuit below, determine the parameters "h" and find the Thévenin equivalent of the output when the S switch is closed.

Solution of Problem 33-1

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

We calculate h₁₁ and h₂₁ when E₂ = 0. Therefore, if E₂ = 0 the source of controlled current E₂/5 is canceled. In this way, we calculate h₁₁:

In the Figure 33-01.2 we see the circuit with the necessary modifications for the calculation of h₁₁ and h₂₁. To calculate h₂₁, we must calculate the current I₂. Then:

Now, we can calculate h₂₁, or:

Using the same reasoning, we can calculate h₁₂ e h₂₂ making I₁ = 0.

Following the circuit in the Figure 33-01.3 it is easy to calculate E₁, because the entire current of the source passes through the resistor of 50 ohms. Like this:

Therefore, the value of h₁₂ is:

On the other hand, if I₁ = 0 the voltage source on the right side of the circuit is annulled and therefore:

Thus, the equations of the quadripole are as follows:

Equivalente de Thévenin

In order to determine the Thévenin equivalent of the circuit, we close the key S and write the following equation for the input circuit:

Reorganizing the equation, we have:

Replacing the found value of E₁ in the first equation of the quadripole, we obtain

Reorganizing the equation we find the value of I₁ in function of E₂, or:

To find Thévenin's voltage, we need to calculate the open circuit voltage at the output. For an open circuit on the output we have to I₂ = 0. Then, using the second equation of the quadripole and replacing the value of I₂ by zero, we obtain:

By performing the calculation, you will find the value of E₂. And this is the Thévenin voltage. Soon:

On the other hand, to calculate Thévenin's resistance we need to cancel the independent sources. If we cancel the source of 50 volts in the input circuit of the quadripole, we find the relation between E₁ and I₁ given by the equation:

Substituting this relation into the first equation of the quadripole, we get:

Reorganizing the equation, we find the relation between I₁ and E₂. Then:

Following the same reasoning, this equation is substituted in the second equation of the quadriplo. Thus, we get:

As we know the Thévenin resistance is given by the relation between E₂ and I₂, that is:

In the Figure 33-01.4 we can see the Thévenin equivalent for the problem circuit.