Use Thévenin's theorem to determine $V_O$.

Solution
To find the Thévenin equivalent, we break the circuit at the $4\Omega$ load as shown below.

So, our goal is to find an equivalent circuit that contains only an independent voltage source in series with a resistor, as shown in Fig. (1-26-3), in such a way that the current-voltage relationship at the load is not changed.

Now, we need to find $V_{Th}$ and $R_{Th}$. $V_{Th}$ is equal to the open circuit voltage $V_{OC}$ shown in Fig. (1-26-2). The current of $2\Omega$ resistor is zero because one of its terminals is not connected to any element; therefore, current cannot pass through it. Since the current of $2\Omega$ resistor is zero, the $9V$ voltage source, $3\Omega$ and $6\Omega$ resistors form a voltage divider circuit and the voltage across the $6\Omega$ resistor can be determined by the voltage devision rule. Please not that we are able to use the voltage devision rule here just because the current of the $2\Omega$ resistor is zero. You may ask that there is no reason to prove that the current of the $2\Omega$ resistor is zero in the original circuit shown in Fig. (1-26-1). That is correct. However, we are calculating $V_{OC}$ for the circuit shown in Fig. (1-26-1) and this is a different circuit. The Thévenin theorem guarantees that $V_{Th}=V_{OC}$, it is not saying that $V_{OC}$ is the voltage across the load in the original circuit.

$V_{6\Omega}=\frac{6\Omega}{3\Omega+6\Omega}\times 9 V= 6V$
Since the current of the $2\Omega$ resistor is zero:
$V_{OC}=V_{6\Omega}=6V$
$V_{Th}=V_{OC}=6V$
Now, we need to find $R_{Th}$. An easy way to find $R_{Th}$ for circuits without dependent sources is to turn off independent sources and find the equivalent resistance seen from the port. Recall that voltage sources should be replace with short circuits and current sources with open circuits. Here, there is only a voltage source that should be replaced by short circuit as shown in Fig. (1-26-4).

It is trivial to see that the $3 \Omega$ and $6 \Omega$ resistors are connected in parallel and then wired in series to the $2\Omega$ resistor. Therefore,
$R_{Th}=(3\Omega || 6\Omega)+2\Omega=\frac{3\Omega \times 6\Omega}{3\Omega + 6\Omega}+2\Omega=4\Omega$.
Now that $V_{Th}$ and $R_{Th}$ are found, we can use the Thévenin equivalent circuit depicted in Fig. (1-26-3) to calculate $V_O$ in the original circuit shown in Fig. (1-26-1). The voltage devision rule can be used here to find $V_O$. We have,
$V_{O}=\frac{4\Omega}{R_{Th}+4 \Omega}\times V_{Th}=\frac{4\Omega}{4\Omega+4 \Omega}\times 6V=3V$.

Hi! Yaz is here. I am passionate about learning and teaching. I try to explain every detail simultaneously with examples to ensure that students will remember them later too.

## Join the Conversation

It is good.But,it could be more better to give some more information.

2. It is good.But,it could be more better to give some more information and examples.

1. Sure

3. manasa says:

How to find the current through given(particular) resistor when a source voltage is given?

1. Dear Mansa,

Your question is very general. All methods explained here can be used and the best method is totally depended on the circuit topology. Please explain or give a link to image of the circuit that you are talking about.

4. manasa says:

What are the steps to be followed to find the current through given (particular) resistor when a source voltage is given using norton's theorem?

5. syed humayoon shah says:

sir i am the student of electronic engineering but i am not satisfied from my university teachers,as they are not so much attentive and hardworking..........so plzzzzzzz tell me how can i learn my basics of electronic circuits course concept wise.....i need your kind supports,becoz i wanna to become a great engineer of the furture.
regsrds

1. Dear syed humayoon shah,

If you are a hardworking student, you do not need university teachers. There are a handful of resources such as books, notes, websites,... that you can read and learn.

6. smita says:

its very gud.....

7. bakhtawer says:

sir i have a prolum in which at last there is no paralaell circuit with the load so how can i ind the voltage

1. Would you show me an example?

8. Prachiti says:

Sir pls tell me how to solve problems on equivalent resistances based on star-delta transformation effectively....

9. kishore says:

sir, nodal analysis using kcl can lead us quick ans

1. Yes, but it is asked to solve this problem by Thévenin’s Theorem.

10. Mahesh says:

sir, why capacitor acts as open circuit at steady state current ?

1. This is unrelated to this post.
Anyway, $latex i_c(t)=C\frac{V_c(t)}{dt}$. $latex V_c(t)$ is constant in steady state, therefore $latex \frac{V_c(t)}{dt}=0$ and $latex i_c(t)=0$. This shows that no current passes through a capacitor so it acts like an open circuit.

11. sir i dont understand how to solve thevinens thearom with dependent sources

12. mufu tee says:

very very nice blog. More grease to your elbow

13. likitha says:

how can we find vth and rth if another resistor of 5 ohms is conected parallel to 4 ohm resistor?

14. Shrabanee says:

Sir,if v hv d load resistance parallel wth source voltge nd current n some resistances with it how cn v find d thevenin's equivqlent voltage across that load resistance?

15. george says:

great work,l think more examples willn't be less appreciated

16. Amit says:

How to solve a problem when load is connected with a current or voltage source?
Will the voltage across the load(Vth) will be equal to the voltage of the battery itself?

Thank you

17. ambi says:

how can we solve the superposition theorem problems using thevenin theorem... also when two voltage sources on opposite sides have opposite polarity.....

18. this web realy help me in my study.
Thanks for this Great work.

19. Anusha says:

I think i have a silly doubt. But i dont understand how 3 ohm and 6 ohm resistors are connected in parallel in that problem 🙁 .. can u pls explain?? I'm new to these concepts.

1. Two elements are connected in parallel if they are connected at both sets of terminals.
3 ohm and 6 ohm are in parallel only when we make the 9V voltage zero to find Rth, as shown in Fig. (1-26-4).

20. sir i truly appreciate ur problem solving and i would love to see more examples i want to get registered in ur website but i am encountering problems please it would be good if u fix it

21. Christina says:

dear sir,
please include complicated circuit's solutions also.

22. Susmita says:

hi,plz give some examples of thevenin's and norton's theorem with dependent source.

23. allan says:

sir, may i get Io by dividing V0 by 4 Ohms? thanks.

24. Osama says:

I am student of Material Science and Engineering but i am studying basic circuits as a subject in 4rth semester. You explained the theorem very well . I forward this link to my classmates hope they'll find it helpful .Thanks

25. Do you always enter the electric current from the cathode or the anode

26. sourav mukherjee says:

this examples are to simple give some complicated ones.

Hi, I don't understand why you directly eliminated the 4ohm resistance in order to calculate the thevenin equivalent voltage, if for instance you also have another 6ohm resistance between the 4ohm and the 6ohm resistance, would you take both of them out to also calculate the thevenin equivalent voltage?
Thanks.

28. Shubhamoy Maity says:

Please,solve my problem....three resistance of 8 ohm,8 ohm and 4 ohm conected in paraller with two series resistance 5 & 3 ohm which are conected with a voltage source of 10 volt.then,calculate current through 4 ohm.by thevinin theorem.

1. Mokoena M.M says:

sir I was asking if we are given two resistors that are parallel&series with A on the bottom left then B top right how do we find equivalent resistance

29. Prasant Nair says:

But the solution vo is 3v?

30. dhanushka says:

dear sir ,
what if the batteries are having internal resistance.must it be considered when calculating open cct voltage(Vth).?????????