1.a) Thévenin Theorem

The Thévenin Theorem says that any electric circuit, it can always be reduced to a voltage source in series with a resistor. This voltage source is called Thévenin Voltage and symbolized by V_th. The equivalent resistance is called Thévenin Resistance and symbolized by R_th.

In the Figure 15-01 the voltage source represented by V_th, is in series with the resistance of Thevenin, R_th. At the terminals a - b will be connected the circuit that we want to determine its characteristics. Pay attention to the polarity of the voltage source. The positive pole points to the terminal a.

1.b) Norton Theorem

Like the Thévenin Theorem, the Norton Theorem says that any electric circuit, for more complex that is, can always be reduced to a current source in parallel with a resistor. This current source is called Norton Current and symbolized by I_n. The equivalent resistance is called Norton resistance and is symbolized by R_n.

In the Figure 15-02 the current source represented by I_n, is in parallel with the Norton resistor, R_N. At the terminals a - b will be connected the circuit that we want to determine its characteristics.

Notice the fact that the current source arrow points to terminal a.

In the Figure 15-03 we see how we can transform a circuit Thévenin on a Norton circuit. If we short-circuit the terminals a- b and compute the current I_n between these points, then this will be the value of the source of current on a Norton circuit. And the Norton resistance will have the same value as the Thévenin resistance, however, it will be in parallel with the current source, as can be seen in the figure below. Note that the arrow of the current source points in the same direction as the positive pole of the voltage source. This rule must be followed whenever we do this kind of transformation.

Let's see how we can transform a circuit Norton on a Thévenin circuit. If we calculate the potential difference between the terminals a - b, when there is no load connected to these terminals, this will be the voltage of Thévenin, ou V_th. And the resistance of Thévenin will have the same value of the resistance of Norton, however, will be in series with the voltage source, as can be seen in the Figure 15-04. Again, we pay attention to the fact that the positive pole of the voltage source points in the same direction as the arrow of the current source.

To calculate the equivalent of Thévenin, we have to analyze the circuit and in this case we have three options. Let's look at each one of them.

3.1   1st Option or Method 1

All sources are INDEPENDENT

We can use this option if all the sources that are part of the circuit are independent sources. In this case, we must analyze in which part of the circuit we want to calculate the Thévenin equivalent. If it is in some component of the circuit, we must remove it from the circuit and in its place, two points emerge that we will call terminals a-b. Therefore, these terminals characterize an open circuit. Calculating the voltage V_ab, that is, open circuit voltage, we will call this voltage Thévenin voltage, represented by V_th.

To calculate the Thévenin resistance, in this case, we must eliminate all sources of voltage and sources of current. To eliminate a source of voltage, we must put it in short circuit. In the case of the source of current, we must open the circuit. Then, we must calculate the equivalent resistance seen by the terminal a-b, using equivalent series, parallel, or star-triangle associations. This equivalent resistance is the call Thévenin resistance.

There are cases where we want to compute the Thévenin equivalent, but the circuit already presents the terminals a-b as an open circuit. In this case, there is no need to remove any component of the circuit. However, to calculate the Thévenin equivalent, we use the same technique described above.

To see an example  click here!

3.2   2nd Option or Method 2

To use this option or method, we must have at least one independent source (either voltage or current) in the circuit. There may be one or several dependent fonts, either voltage or current. Let's see how we determine the Thévenin equivalent.

Calculation of the Voltage of Thévenin

Initially, we must determine in which part of the circuit we want to calculate the Thévenin equivalent. Then, we remove the element from the circuit and have two points, a and b. If there are already two points a and b, then there is no need to remove any components from the circuit. Now we must calculate the voltage at the open circuit. This voltage is the Thévenin voltage, V_th. We can use any circuit analysis technique to calculate this voltage.

Calculation of Thévenin resistance

The next step is to calculate the Thévenin resistance. To do this, we must return to the original circuit, with all dependent and independent sources, and apply a short circuit between the terminals a - b. From this moment we must calculate what is the short-circuit current at points a and b. With this data we are able to calculate the Thévenin resistance. For this, we will use the following equation:

3.3   3rd Option or Method 3

For any type of circuit

This option or method is applicable if there are only dependent sources in the circuit, or only independent sources or both together in the circuit.

To find the value of R_th, the technique used is to determine the points a and b. Depending on the characteristics of the circuit, we connect a known voltage or current source to the points a-b. To do so, we must eliminate all independent voltage and current sources. We can not eliminate dependent fonts. Those are in the circuit. The next step is to insert a voltage source (which we will call V_s ) or current (which we will call I_s ), in terminals a - b of the circuit. From that moment we must find a relationship between V_s and I_s, as stated in the equation eq. 15-02 below.

If we choose a voltage source, then we already know the value of V_s. Therefore, you must compute I_s. And vice versa.

In order to determine the value of V_th, we must return to the original circuit and calculate the open circuit voltage at terminals a - b. This voltage will be the Thévenin Voltage.

To see an example  click here!

To calculate the Norton equivalent follow the same calculation steps for the Thévenin equivalent, except that to calculate the Norton current, place the points a - b in short-circuit and calculate the electric current passing through the short-circuit. This will be the Norton current, or I_N. The Norton resistance is exactly equal to Thévenin resistance and can be calculated by the method described above for the Thévenin resistance.