# Top Analysis of DC Circuits | Electrical Network | Part-2

Hello everyone. I hope that you all are doing good in your lives. Today we are going to continue the last topic which we discussed i.e Analysis of DC circuits. So if you haven’t already checked it, then go check it out first through the link below and then move further with this one.

### Superposition Theorem for analysis of DC circuits

It states that “In a linear network comprising of more than one independent source and dependent source; The resultant current in any element is the algebraic sum of the current that would be produced by each independent source being represented meanwhile by their respective internal resistance“.

The independent voltage source is shown by their internal resistance if present or with zero resistance i.e. short circuits. The independent current sources are shown by infinite resistance, i.e. open circuits.

A linear network or circuit is one whose parameters are constant that is why values do not change with voltage and current.

#### Steps to follow during the analysis

- Find the current through the resistance when only one independent source is acting, replacing all other independent sources by respective internal resistance.
- Find current through the resistance for each of the independent sources.
- Lastly Find the resultant current through the resistance by the superposition theorem considering magnitude and direction of each current.

#### Example Problems

##### Q.1 Find the current through 4Ω resistor in the figure below

Solution:- Step 1 – Whenever the 5 A source is acting alone

Thus by series-parallel reduction method;

Secondly when 20 V source is acting alone similar to the above series-parallel reduction method

Lastly by the superposition theorem,

##### Q.2 Find the current I_{y} in the figure below

Solution:- Step 1- Whenever the 120 V source is acting alone. Applying KVL to the mesh.

Step 2 – Whenever 12 A source is acting alone I_{Y}” = I_{1}. Meshes 1 and 2 will form a super mesh. Thus writing the current equation for the super mesh,

Consequently applying KVL to the outer path of super mesh.

Certainly by using calculator we get

Step 3 – Whenever the 40 V source is acting alone. applying KVL to the mesh,

Lastly by the superposition theorem,

### Thevenin’s Theorem for Analysis of DC circuits

It states that ” Any two terminals of a network can be replaced by an equivalent voltage source and an equivalent series resistance. However, the voltage source is the voltage across the two terminals with load, if any, removed. The series resistance is the resistance of the network measure between two terminals with load remove and the constant voltage source is replaced by its internal resistance (or if it is not given with zero resistance i.e. short circuit). And the current source by infinite resistance i.e. opens circuit“.

#### Steps to be followed

- Firstly remove the load resistance R
_{L}. - find the open circuit voltage V
_{TH}across point A and B. - Find the resistance R
_{TH}as observe from A and B. - Subsequently replace the network by a voltage source V
_{TH}in series with resistance R_{TH}. - Lastly find the current through R
_{L}using ohm’s law.

#### Example Problems

##### Q.1 Find the current through the 20Ω resistor in the figure below

Solution:- Step 1- Calculation of V_{TH}. Firstly by applying KVL to mesh 1.

Secondly applying KVL to mesh 2

However, solving the above equations we get

Thus the V_{TH} equation is written as,

Step 2- Calculation of R_{TH}

Certainly converting the delta into equivalent star network;

Thus RTH = (16.25 || 2.5) +2.5 =4.67Ω

Step 3- Lastly calculating I_{L}

##### Q.2 Find VTH and RTH between terminals A and B of the network below

Solution:- Firstly Calculating Vth; I_{x} = 0

The dependent source 2I_{x} certainly depends on the controlling variable Ix. Whenever I_{x} = 0, the dependent source vanishes i.e. 2I_{x} = 0. writing the Vth equation we get,

Secondly calculating I_{N} from the figure I_{x} = V_{1} / 2. Applying KCL at node 1.

Lastly calculating R_{TH}

### Norton’s Theorem for analysis of DC circuits

It certainly states that “Any two terminals of a circuit can be replaced by an equivalent current source and an equivalent parallel resistance”. The constant current is equal to the current which would flow in a short circuit placed across the terminals. The parallel resistance is the resistance of the network when viewed from these open-circuited terminals after all voltage and current sources have been remove and replace by internal resistances.

#### Steps to follow during the analysis

- Start with Removing the load resistance RL and put a short circuit across the terminals.
- Find the short-circuit current I
_{SC}or I_{N}. - Find the resistance R
_{N}between the points A and B. - Subsequently replace the network by a current source I
_{N}in a parallel with resistance R_{N}.

#### Example Problems

##### Q.1 Obtain the current flowing through the 10 Ω resistor in the figure below

Solution:- Firstly calculating the value I_{N}. applying KVL to mesh 1,

As mesh 2 and 3 are forming a super mesh we will be writing the current equation for super mesh.

Consequently applying KVL to the super mesh

Thus by solving the above polynomial equations we get;

Secondly calculating R_{N}

Lastly calculating I_{L}

##### Q.2 Find Norton’s equivalent circuit for the figure below

Solution:- Firstly calculating V_{TH}. applying KVL to the mesh,

Certainly writing V_{TH} equation,

Secondly calculating I_{N}. Whenever a short circuit is placed across the 1Ω resistor, it gets shorted. Thus, I_{1} = 0.

However, the dependent source of 0.5I_{1} depends on the controlling variable I_{1}. Whenever I_{1} = 0, the dependent source vanishes, i.e. 0.5I_{1} = 0

Lastly calculating R_{N}

Thus the equivalent Norton’s network.

### Maximum Power Transfer analysis of DC circuits

It states that the “maximum power is transferred from a source to a load resistance is equal to the source resistance.”

#### Steps to follow

- Firstly Remove the variable load resistor R
_{L}. - Find the open circuit voltage V
_{TH}across point A and B. - Find the resistance R
_{TH}between the points A and B. - Lastly find the resistance R
_{L}for maximum power transfer R_{L}= R_{TH}. - And thus find the maximum power

#### Example Problems

##### Q.1 For the value of resistance R_{L} in the figure below for maximum power transfer and calculate thee maximum power

Solution:- Firstly calculating V_{TH}. applying KVL to mesh 1,

Certainly writing the current equation for mesh 2, I_{2} = 2

Thus by Solving the above equations we get, I_{1} = 3.43 A

However writing the V_{TH} equation we get,

Secondly calculating R_{TH}, R_{TH} =15 || 20 = 8.57 Ω

Thirdly calculating R_{L}_{ } for maximum power transfer; R_{L} = R_{TH} = 8.57 Ω

Lastly calculating the P_{max}

### Conclusion

However here we are at the last part of the blog. I hope that you liked it and got all your doubts cleared. Besides if you are having any doubts then feel free to comment down below and ask and also mention the part which you liked the most. Above all, I would be really happy to know the topic which you like the most.

Have a nice day 🙂

Regards.

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