What is Norton’s theorem?

Norton’s Theorem is a fundamental principle in circuit analysis that simplifies complex linear electrical circuits. It states that any linear electrical network with voltage and current sources and resistances can be simplified into an equivalent circuit with:

  1. A single current source (I_N) in parallel with a single resistor (R_N).

This equivalent circuit will produce the same current and voltage at the terminals of the load resistor as the original circuit.

To apply Norton’s Theorem:

  • Step 1: Find the Norton current (I_N), which is the current that would flow through a short circuit placed at the load terminals.
  • Step 2: Find the Norton resistance (R_N), which is the equivalent resistance seen from the load terminals with all independent sources turned off (voltage sources replaced by short circuits and current sources replaced by open circuits).

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For obtaining the current 𝑰𝑵, short the load terminals AB as shown in the below figure. Then find the current 𝐼𝑁 using any of the network simplification techniques. This is called “Norton’s current”.

To calculate 𝑹𝒆𝒒 use the same procedure as discussed earlier for Thevenin’s theorem.

 

Kirchhoff’s laws  

There are two more important laws governing the performance of a circuit known as,

1. Kirchhoff’s Voltage Law ( KVL )

2. Kirchhoff’s Current Law ( KCL)

Kirchoff’s Current Law ( KCL or KIL )

➢ Kirchhoff's current law may be stated as follows, The sum of the currents entering a node is equal to the sum of the currents leaving that node. This means that the algebraic sum of the currents meeting at a node is equal to zero.

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Kirchoff’s Voltage Law ( KVL )

➢ Kirchoff’s Voltage Law states that, in a closed electric circuit the algebraic sum of E.M.F.s and Voltage drops is zero.

➢ In the closed circuit ABCDA given in Figure 03, applying Kirchoff’s Voltage Law, we have,

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What is Nodal Analysis?

Nodal analysis is a method that provides a general procedure for analyzing circuits using node voltages as the circuit variables. Also called Node-Voltage Method. Types of Nodes in Nodal Analysis:

Ø   Non-Reference Node: Node which has a definite Node-Voltage.

Ø   Reference Node: It is the node which acts as reference point to all the other node. Also called Datum Node/Ground.

Steps to find Node Voltages:

The steps you mentioned are correct for solving for node voltages in a circuit using Node Voltage Method (NVM), which is a systematic way of applying Kirchhoff’s Current Law (KCL) to find the voltages at the nodes of a circuit. Here’s a detailed explanation of each step:

1. Select a reference (ground) node and assign voltages to other nodes:

  • Reference node (ground): Choose one node in the circuit to be the reference node (ground), usually the one with the most connections or a point with a known potential.
  • Assign node voltages: Label the remaining nodes with variables V1,V2,V3,…,Vn−1, ​. These are the unknown voltages that need to be solved.

2. Apply Kirchhoff’s Current Law (KCL) to each non-reference node:

  • KCL equation: For each node, the sum of currents leaving the node must equal zero. Express the currents leaving or entering each node in terms of node voltages.
  • KCL equation for node i (assuming vi is the voltage at that node) would look like: the resistance between nodes i and j.

3. Use Ohm's Law to express currents:

  • Ohm’s Law: The current between two nodes connected by a resistor is given by

where V is the voltage difference between the nodes, and R is the resistance between them.

  • Substitute this expression for current into the KCL equations to represent the currents in terms of node voltages.

4. Solve the resulting simultaneous equations:

  • After applying KCL to all non-reference nodes, you will obtain a system of simultaneous equations with the node voltages as unknowns.
  • Solve these equations (typically using matrix methods or substitution) to find the values of the node voltages V1, V2,…, Vn−1.

Final Output:

Once the node voltages are found, you can use them to calculate the currents and voltages in the other parts of the circuit.

This method is particularly useful for complex circuits with many components, as it can be systematically solved using matrix algebra or computational tools.

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What is Mesh Analysis?

Mesh Analysis (or Mesh Current Analysis) is a method used to analyze electrical circuits with multiple loops or meshes, simplifying the process of finding the currents in the circuit. Mesh current analysis involves applying Kirchhoff's Voltage Law (KVL) to the loops of the circuit and solving for the mesh currents.

Steps to Find Current Flowing in a Circuit Using Mesh Analysis:

1. Identify Loops or Meshes in the Circuit and Label Mesh Currents:

  • Identify Meshes: A mesh is a loop in a circuit that does not contain any other loops inside it. In a planar circuit (one that can be drawn on a flat plane without crossing elements), identify all the independent loops (meshes).
  • Label Mesh Currents: Assign a mesh current I1,I2,I3,…,In ​ to each mesh. These mesh currents circulate in the respective loops in a clockwise or counterclockwise direction. The direction of the current is usually assumed, but if the solution gives a negative value, the current flows in the opposite direction.

2. Apply Kirchhoff’s Voltage Law (KVL) to Each Mesh:

  • KVL Equation: For each mesh, apply Kirchhoff's Voltage Law, which states that the sum of all voltages around a closed loop must be zero.
  • Write KVL for each mesh by considering all the resistances and voltage sources in the mesh. When writing the KVL equation for a mesh, take into account:
    • The voltage drops across resistors, which are calculated using Ohm’s Law:

V = IR

    • The voltage sources, which may cause either a rise or a drop in voltage, depending on the assumed direction of the mesh current relative to the source.
  • If there are shared resistors between adjacent meshes, the voltage drop across those resistors will depend on the difference between the mesh currents.

3. Solve the Resulting Simultaneous Linear Equations:

  • After applying KVL to each mesh, you will get a set of simultaneous linear equations for the mesh currents.
  • Solve these equations using algebraic techniques (such as substitution, elimination, or matrix methods) to find the unknown mesh currents.

Example of Mesh Analysis in a Simple Circuit:

Consider a circuit with two meshes, and label the mesh currents I1 and I2​. Apply KVL to each mesh, incorporating the resistances and any voltage sources present. You'll get two equations, and you can solve them simultaneously to find I1​ and I2.

Final Output:

Once the mesh currents are found, you can easily calculate the currents through all components in the circuit (by considering the mesh current contributions to each resistor or source) and the voltage drops across components using Ohm’s Law.

Mesh analysis is especially useful for circuits with many loops and is typically easier to apply when the circuit is planar and the components are arranged in a way that makes it clear how to define the loops.

What is Thevenin’s theorem?

Ø  Thevenin theorem tells us that we can replace the entire network, exclusive of the load resistor, by an equivalent circuit that contains only an independent voltage source in series with a resistor in such a way that the current-voltage (i-v) relationship at the load resistor is unchanged.

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Steps to Find the Thevenin Equivalent Circuit:

Step 1: Remove the Branch Resistance

  • Remove the load resistor (or the branch through which the current is to be calculated) from the circuit. This allows you to focus on the remaining part of the circuit to find the Thevenin equivalent.

Step 2: Calculate the Open-Circuit Voltage (V_Th)

  • Open-circuit voltage: This is the voltage across the terminals where the load resistor was removed. You can calculate this voltage using any network simplification technique (e.g., nodal analysis, mesh analysis, or voltage divider).
  • The voltage you calculate across these open terminals is the Thevenin voltage (V_Th).

Step 3: Calculate the Thevenin Resistance (R_Th)

  • Find the equivalent resistance seen from the open terminals where the load resistor was removed.
    • For voltage sources: Replace all independent voltage sources with short circuits.
    • For current sources: Replace all independent current sources with open circuits.
  • After replacing sources, calculate the resistance between the open terminals. This is the Thevenin resistance (R_Th).

Step 4: Draw the Thevenin Equivalent Circuit

  • After finding VTh​ and RTh, draw the Thevenin equivalent circuit:
    • Thevenin equivalent circuit consists of the Thevenin voltage source (V_Th) in series with the Thevenin resistance (R_Th).
    • The load resistor (RLR_LRL​) is then connected across the Thevenin equivalent circuit.

Step 5: Reconnect the Load Resistor

·         Reconnect the load resistor (RLR_LRL​) to the Thevenin equivalent circuit.

·         Now, the current through the load resistor RLR_LRL​ can be calculated using Ohm’s Law:

·         This gives the required current through the load resistor.

Final Thevenin Equivalent Circuit:

The simplified Thevenin equivalent circuit will consist of:

·      A Thevenin voltage source ​ in series with

·      A Thevenin resistance ​, and the load resistor ​ will be connected across this series combination.

What is Norton’s theorem?

Norton theorem tells us that we can replace the entire network, exclusive of the load resistor, by an equivalent circuit that contains only an independent current source in parallel with a resistor in such a way that the current-voltage (i-v) relationship at the load resistor is unchanged.

Steps to Find the Norton’s theorem Circuit:

Step 01: Short the branch through which the current is to be calculated.

Step 02: Obtain the current through this short-circuited branch, using any of the network simplification techniques. This current is Norton’s current 𝑰𝑵.

Step 03: Calculate the equivalent resistance between the terminals. (Voltage sources should be short circuit and current source should be open circuit)

Step 04: Draw Norton’s equivalent circuit showing current source 𝑰𝑵 with the 𝑹𝒆𝒒 series resistance.

Step 05: Reconnect the branch resistance. Let it be 𝑹𝑳, then using the current division rule

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