Gandhinagar Institute of

Technology(012)Sub:- Elements of Electrical Engineering

(2110005)

Topic:- Ohm’s law, Kirchhoff's law, nodal & mesh analysis Prepared by :- Kathan Patel

(150120119064) Kundariya Ankit

(150120119073) Makwana Mitesh

(150120119075)

Guided by :- Prof. Abhishek Harit

Georg Simon Ohm • “At Constant temperature and pressure, current

flowing through the conductor is directly proportional to the potential difference across the conductor and inversely proportional to the resistance experienced by the conductor”.

• German physicist who experimentally determined that the if the voltage across a resistor is increased, the current through the resistor will increase.

• Ohm's work long preceded Maxwell's equations and any understanding of frequency-dependent effects in AC circuits.

• Modern developments in electromagnetic theory and circuit theory do not contradict Ohm's law when they are evaluated within the appropriate limits.

Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points.

Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship

where I is the current through the conductor in units

of amperes, V is the voltage measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms.

More specifically, Ohm's law states that the R in this relation is constant, independent of the current.

• V = I * R– For a constant resistance, if the current increases, the voltage

increases at the same rate• I = V / R

– For a constant resistance, if the voltage increases, the current will increase at the exact same rate

• R = V / I– For a constant resistance, if the voltage increases, the current

must increase at the exact same rate.

Is ohm’s law applicable to only individual components?

A question arise in our mind that is this law is applicable to only individual components or we can apply it to a complete circuit?

Ans:- ohm’s law is applicable to individual components as well as the complete circuit.

If it is to be applied to the complete circuit, then voltage across the complete circuit and effective resistance of the complete circuit should be used.

If the law is to be applied to only a part of the circuit, then the voltage and resistance corresponding to that part should be taken into consideration.

Is ohm’s law applicable to nonlinear devices?

The answer is No. The ohm’s law is not applicable to any nonlinear device such as diode, transistor, zener diodes etc. it is applicable only to the linear devices.

A linear device exhibits a linear relation between voltage across it and the current flowing through it (e.g. resistor). This is the biggest limitation of ohm’s law.

Examples:-A Resistance of 15Ω carries a current of 5Amp. Calculate the voltage developed across the resistor.Soln:- According to ohm’s law : I=V/R = 50/5 = 10Amp

Calculate the resistance of an iron filament if it operates on 230V supply and draws a current equal to 2Amp.Soln:- Note that the iron filament is a linear device. Hence the ohm’s law is applicable here even though it is an AC circuit. Resistance of the iron filament = R = 230/2 = 115 Ω

The total resistance of a circuit is dependant on the number of resistors in the circuit and their configuration

1 2

1 2

...

1 1 1 1 ...

total

total

R R R R

R R R R

Series Circuit

Parallel Circuit

Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits.

They were first described in 1845 by German physicist Gustav Kirchhoff.

This generalized the work of Georg Ohm and preceded the work of Maxwell. Widely used in electrical engineering, they are also called Kirchhoff's rules or simply Kirchhoff's laws.

Kirchhoff's circuit laws

Gustav Kirchhoff

Both of Kirchhoff's laws can be understood as corollaries of the Maxwell equations in the low-frequency limit.

They are accurate for DC circuits, and for AC circuits at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuits.

Kirchhoff's current law (KCL) This law is also called Kirchhoff's first law, Kirchhoff's point rule, or Kirchhoff's junction rule (or nodal rule).

The principle of conservation of electric charge implies that:

At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node or equivalentlyThe algebraic sum of currents in a network of conductors meeting at a point is zero.

The current entering any junction is equal to the current leaving that junction. i2 + i3 = i1 + i4

Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be stated as:

n is the total number of branches with currents flowing towards or away from the node.

This formula is valid for complex currents:

The law is based on the conservation of charge whereby the charge (measured in coulombs) is the product of the current (in amperes) and the time (in seconds).

Kirchhoff's voltage law (KVL) This law is also called Kirchhoff's second law, Kirchhoff's loop (or

mesh) rule, and Kirchhoff's second rule.

The directed sum of the electrical potential differences (voltage) around any closed network is zero, or:

The algebraic sum of the products of the resistances of the conductors and the currents in them in a

closed loop is equal to the total emf available in that loop.

The sum of all the voltages around a loop is equal to zero.v1 + v2 + v3 - v4 = 0

Similarly to KCL, it can be stated as:

Here, n is the total number of voltages measured. The voltages may also be complex:

This law is based on the conservation of energy whereby voltage is defined as the energy per unit charge. The total amount of energy gained per unit charge must be equal to the amount of energy lost per unit charge, as energy and charge are both conserved.

Kirchhoff’s Current LawKCLConservation of charge

The algebraic sum of all the currents at any node in a circuit equals zero.

Kirchhoff’s Voltage LawKVLConservation of energy

The algebraic sum of all the voltages around any closed path in a circuit equals zero.

Definition of Nodal AnalysisNodal analysis is a method that provides a general procedure for analysing circuits using node voltages as the circuit variables. Nodal Analysis is also called the Node –Voltage Method.

In analysing a circuit using Kirchhoff's circuit laws, one can either do nodal analysis using Kirchhoff's current law (KCL) or mesh analysis using Kirchhoff's voltage law (KVL)

Nodal analysis writes an equation at each Electrical node, requiring that the branch currents incident at a node must sum to zero.

NODAL ANALYSIS

Types of Nodes in Nodal Analysis

• Non Reference Node - It is a node which has a definite Node Voltage. e.g. Here Node 1 and Node 2 are the Non Reference nodes

• Reference Node - It is a node which acts a reference point to all the other node. It is also called the Datum Node.

Steps Taken While Solving Problem By Nodal Analysis:

1) Mark all nodes. Normally there is a return path which is datum node D. All voltages are to be determined w.r.t . node D.

2) Certain nodes are super nodes whose potential are already known.3) At each node, find the currents through various branches and equate the algebraic

sum to 04) If a branch consists only one resistance, then normally current is assumed to flow

away from node, such as VA/R4 or VB/R5. 5) For a common branch between two nodes, one of the node voltage is assumed to

be of higher value and other of lower value. then difference of voltages will make the current to flow from higher node voltage to lower node voltage. For example

current through branch AB is (VA - VB)/R2.

But the same current can be written as (VB - VA)/R1. Flowing from B to A. 6) Write down all such equation on node basis.

For node A it is , (VA – E1)/R1 + (VA/R4) + (VA –VB)/R2 = 0For node B it is , (VB –VA )/R2 + (VB/R5) + (VB + E2 )/R3 = 0

7) Solve these equations & also current in different branches can also calculated.

Examples of nodal analysis

1) Calculate the value of branch for network shown in fig.

2) Calculate the node voltage VB using the nodal analysis

3)Using nodal voltage method find current in the 3 resistance

Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other.

A more general technique, called loop analysis (with the corresponding network variables called loop currents) can be applied to any circuit, planar or not.

Mesh analysis and loop analysis both make use of Kirchhoff’s voltage law to arrive at a set of equations guaranteed to be solvable if the circuit has a solution.

Mesh Analysis

The following is the same circuit from above with the equations needed to solve for all the currents in the circuit.

Potential rise and potential drop for a Resistor:

Potential rise : If we trace the path along a closed loop from negatively marked terminal of a resistor, then the associated potential change is called as the potential rise.

Potential drop : If we trace the path along the closed loop from the positive marked terminal to negative marked terminal, then the associated potential change is called as the potential drop.

1) Determine the current and power consumed in the 3 resistance of the circuit shown in fig

2) Using Kirchhoff's law calculate the current delivered by the battery

3) In fig (a) determine V2 which results in zero current in the branch containing v1 , using mesh analysis.