Week 11: Thevenin's Theorem with Dependent Sources

Thevenin's Theorem with Dependent Sources

Cases in Thevenin's Theorem:

■ CASE 1: If the network has no dependent sources, we turn off all independent sources. R_TH is the input resistance of the network looking between terminals A and B. 

■ CASE 2: If the network has dependent sources, we turn off all independent sources. As with superposition, dependent sources are not to be turned off because they are controlled by circuit variables. We apply a voltage source at terminals A and B and determine the resulting current I_o. Then, R_TH = V_o/I_o . Alternatively, we may insert a current source I_o at terminals A-B and find the terminal voltage V_o. Again R_TH = V_o/I_o. Either of the two approaches will give the same result. In either approach we may assume any value of and V_o and I_o. For example, we may use V_o = 1V or I_o = 1A or even use unspecified values of V_o or I_o.

It often occurs that R_TH takes a negative value. In this case, the negative resistance (V = -iR) implies that the circuit is supplying power. This is possible in a circuit with dependent sources.

Thevenin’s theorem is very important in circuit analysis. It helps simplify a circuit. A large circuit may be replaced by a single independent voltage source and a single resistor. This replacement technique is a powerful tool in circuit design.

As mentioned earlier, a linear circuit with a variable load can be replaced by the Thevenin equivalent, exclusive of the load. The equivalent network behaves the same way externally as the original circuit. Consider a linear circuit terminated by a load R_L, . The current I_L through the load and the voltage V_L across the load are easily determined once the Thevenin equivalent of the circuit at the load’s terminals is obtained.

Suppose that R₁ =5Ω , R₂ =3Ω and I_S =2A .

Solution

The circuit has both independent and dependent sources. In these cases, we need to find the open circuit voltage to determine the Thevenin Equivalent Circuit

Open Circuit Voltage

Open circuit voltage means the voltage across the terminals of the network without connecting any extra element or connection:

Since there is no connection, the current of R₂ is zero. To solve the circuit lets write KVL for the left hand side loop assuming I_R1 defined from left to right:

+V_x+R₁×I_R1+V_x=0
→2V_x=−R₁×I_R1
→V_x=−R₁×I_R12
But what is I_R1? R₁ is in series with the current source; they have only one node shared and there is no other element connected there. This means that all current of I_S must pass through R₁ . Therefore, I_R1=I_S=2A .
If we apply this to the equation above, we have

V_x=−R₁×I_R12→V_x=−5V.
Since no current passing through R₂ we can easily see that V_OC=V_x . If it is not clear, you could find this by applying KVL to the right hand side loop:
−V_x+R₂×I_R2+V_OC=0
→−V_x+R₂×0+V_OC=0
→V_OC=V_x=−5V

Calculating R_TH

The only thing left is to calculate R_Th which can be easily found by
R­_TH=V_OC / I_SC=−5−53=3Ω

Thevenin's Equivalent Network

V_Th=V_OC=−5V
R_Th=3Ω 

Reflection/Learnings:

This week, We discussed about Thevenin's Theorem with Dependent Sources. I didnt include this in the previous section because I wanted to explain it in more detail. Actually, In Thevenin's Theorem, There were two cases. Case 1 are for circuits that doesn't have any dependent sources while Case 2 are for circuits that contains at least one dependent source. To solve for problems with dependent sources, We first turn off all independent sources by replacing all voltage sources as a closed branch and replacing all current sources with an open branch. Dependent sources cannot be turned off because they are controlled by circuit variables and then we simply remove one load in a circuit and instead of replacing it with an open circuit and measuring the open circuit voltage, We add an additional voltage or current source. The values of the added sources completely depends on you. You can assume any value for it as long as it is not 0. 

Video(s):

Thevenin's Theorem with Dependent Source

Thevenin's Theorem with one Dependent Source 

Electricity is often called wonderful, beautiful; but it is so only in common with the other forces of nature. The beauty of electricity or of any other force is not that the power is mysterious, and unexpected, touching every sense at unawares in turn, but that it is under law, and that the taught intellect can even govern it largely. The human mind is placed above, and not beneath it, and it is in such a point of view that the mental education afforded by science is rendered super-eminent in dignity, in practical application and utility; for by enabling the mind to apply the natural power through law, it conveys the gifts of God to man.

--Michael Faraday