In the area of Electrical Engineering, we have two extremely important theorems in the resolution of electrical circuits.
One of them is Thévenin Method or Theorem and the other is
Norton Method or Theorem.
These theorems are fundamental in the analysis of electrical circuits, allowing to simplify
the complexity of electrical networks and facilitate the calculation of currents and voltages in different parts of the circuit.
Both theorems are used to analyze a specific point in the circuit,
considering only the voltage, current and resistance values that arrive at that point, regardless of
the complexity of the original circuit. They are particularly valuable in electrical engineering and
electronics, as they reduce the need for extensive calculations and allow a clearer understanding of the
circuit operation, helping to identify potential problems or optimize circuit performance.
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 Vth. The equivalent resistance is called Thévenin Resistance and symbolized by Rth.
Figura 15-01
In the Figure 15-01 the voltage source represented by Vth, is in series with the resistance of Thevenin, Rth. 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.
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 In. The equivalent resistance
is called Norton resistance and is symbolized by Rn.
Figura 15-02
In the Figure 15-02 the current source represented by In, is in parallel with the Norton resistor, RN. 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.
4. Transformation of Thévenin Circuit in Nortonand Vice versa
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
In 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.
Figura 15-03
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 Vth. 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.
5.1 Method 1 -
All sources areINDEPENDENT
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 Vab, that is, open circuit voltage, we will call this voltage Thévenin voltage, represented by Vth.
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 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, Vth. 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:
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 Rth, 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 Vs ) or current
(which we will call Is ), in
terminals a - b of the circuit. From that moment we must find a relationship
between Vs and Is, as stated in the equation eq. 15-02 below.
eq. 15-02
If we choose a voltage source, then we already know the value of Vs.
Therefore, you must compute Is. And vice versa.
In order to determine the value of Vth, 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 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 IN.
The Norton resistance is exactly equal to Thévenin resistance and can be calculated by the method
described above for the Thévenin resistance.
