Problem 15-14    Source:    Problem elaborated by the author of the site.

Based on the previous problem, considering the circuit shown in Figure 15-14.1, determine the Thévenin equivalent of the circuit when analyzed from the point of view of R.

Solution of the Problem 15-14

Since we have dependent and independent sources in the circuit, one of the techniques that can be used is to remove the resistance R from the circuit, as shown in Figure 15-14.2, and calculate the open circuit voltage between the points r - s . In this way the Thévenin Voltage is calculated, V_th. In a second moment, apply a short circuit between the points r - s and calculate the current of short circuit, I_cc . The ratio between these two quantities allows the calculation of Thévenin Resistance.

Note that we represent only the part of the circuit that matters for the calculation of V_th . This voltage will be given as a function of current I . Remembering that V_a = 2 I and therefore the current source 2 V_a = 4 I. And the current I_b = 4 I, because the resistor R has been removed from the circuit. So the voltage source 5 I_b = 5 x 4 I = 20 I and by the resistor of 8 ohms circulates a current equal to 5 I and over this resistor we will have a voltage drop equal to 8 x 5 I = 40 I.

From this data, we can make the mesh indicated by the orange arrow on the Figure 15-14.2. Then:

After calculating Thévenin Voltage we will calculate the short circuit current when we short the r - s terminals. For that we will be based on Figure 15-14.3

Here we also represent only the part of the circuit that matters for the calculation of I_cc. Remembering that V _a = 2 I and therefore the current source 2 V_a = 4 I. Note that resistor R has been replaced by short circuit. Thus, by the resistor of 8 ohms circulates a current equal to I + I_b. On the other hand, by analyzing the point r in the circuit we find that the short circuit current is I_cc = 4 I - I_b. Then making the mesh indicated by the green arrow, we have:

Performing the algebraic calculation, we arrive at a relationship between I e I_b, or:

By substituting this value in I_cc we can calculate its value as a function of I, or:

So, we have the values needed to calculate the Thévenin resistance. Then:

Performing the calculation, we find the value of Thévenin resistance. So:

In the Figure 15-14.4, it is clear that the value of Thévenin resistance confirms the value found in the previous problem of the value of R resistor required to dissipate maximum power, where we find the value R = 13 ohms.

The maximum power transfer theorem, studied in chapter 7, explicitly states that for maximum power transfer the load must have the same value as the internal resistance of the source, represented by R_th in this problem.

Finally, we will express the values of I and I_R in R function. We know that the voltage on R is given by V_R = R (4 I - I_b). Developing this equation, after some algebraic work, we arrive at:

On the other hand, we know that:

Considering that I_R = 4 I - I_b, we find:

And so, we explored all that was possible of the problem 15-12 using the methods learned on this site. Have fun !!!!!!