INTRODUCTION:-

In 1845, German physicist Gustav Kirchhoff described two laws. In electromagnetic fields

these laws are generalizations of Ohm's law. These Kirchhoff's circuit law are very useful in

solving circuit problems. Kirchhoff's circuit laws are two equalities that deal with

the conservation of charge and energy in electrical circuits, and it widely used in electrical

engineering, they are also called Kirchhoff's rules or simply Kirchhoff's laws. The two

Kirchhoff’s Laws tell us about the relationships between voltages and currents in circuits.

Both circuit rules can be directly derived from Maxwell's equations, but Kirchhoff

preceded Maxwell and instead generalized work by Georg Ohm.

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

The first law is Kirchhoff's current law. Kirchhoff's current law is also known as Kirchhoff's

first law and Kirchhoff's junction law. This law states that 'The sum of current into a junction

equals the sum of current out of the junction'. This is the same as Kirchhoff's junction law. In

a junction, theelectric charge's sum preservation law is applied. If the entering value of the

current is i2 and i3, this current splits into the current of i1 and i4. Then the equation (i1 + i4

= i2 + i3) is satisfied. The right picture gives an example. This Kirchhoff's first law is that

charge is not destroyed or created in a junction point. This is based by an electric charge

preservation law.

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

Another is Kirchhoff's voltage law. Kirchhoff's voltage law is also known as Kirchhoff's

second law, a closed circuit law, and Kirchhoff's loop law. The algebraic sum of the voltage

(potential) differences in any loop must equal zero.(This circuit is a closed circuit) Any

complex circuit can be divided into many closed circuits. This law means that in the circuit

there is an electric cell and electric resistance. The electric cell gives the charge a

electromotive force, and then the electric resistance dissipates this force. But in electric

resistance if the direction is opposite of the current's direction, this electric resistance adds to

the electromotive force. This Kirchhoff's second law is based on potential energy

preservation law.

Contents:-  

SOME BASIC POINTS FOR KCL LAW

Kirchhoff's current law (KCL)

o (a)- Changing charge density

o   Uses

Kirchhoff's voltage law (KVL)

o Positive and Negative Signs in Kirchhoff's Voltage Law

o (b)   Electric field and electric potential

o Application

Application of Kirchoff's Circuit Laws

Limitations

Conclusion

4- References

SOME BASIC POINTS FOR KCL LAW:-

When analysing either DC circuits or AC circuits using Kirchoff's Circuit Laws a number of definitions and terminologies are used to describe the parts of the circuit being analysed such as: node, paths, branches, loops and meshes. These terms are used frequently in circuit analysis so it is important to understand them.

Circuit - a circuit is a closed loop conducting path in which an electrical current flows.

Path - a line of connecting elements or sources with no elements or sources included more than once.

Node - a node is a junction, connection or terminal within a circuit were two or more circuit elements are connected or joined together giving a connection point between two or more branches. A node is indicated by a dot.

Branch - a branch is a single or group of components such as resistors or a source which are connected between two nodes.

Loop - a loop is a simple closed path in a circuit in which no circuit element or node is encountered more than once.

Mesh - a mesh is a single open loop that does not have a closed path. No components are inside a mesh.

Components are connected in series if they carry the same current.

Components are connected in parallel if the same voltage is across them.

(1)-Kirchhoff's current law (KCL)

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

and Kirchhoff's first 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

The algebraic sum of currents in a network of conductors meeting at a point is zero.

(Assuming that current entering the junction is taken as positive and current leaving the

junction is taken as negative).

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 also 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 (which is measured in seconds).

Kirchhoff’s Current Law states that: ‘the algebraic sum of currents at a node is zero’ . Two points might need further explanation:

(a)-a ‘node’ is the technical term for a junction in a circuit, where two or more branches are joinedtogether. Fig. 2.1 shows a node with four branches connected.

(b)- the phrase ‘algebraic sum’ reminds us that we have to take account of the current direction, as well as magnitude, when applying Kirchhoff’s Current Law.

This Law is used in circuit analysis to define relationships between currents flowing in branches of the circuit. For example, in Fig. 2.1 the currents flowing in the four branches connected to the node have been defined as I1, I2, I3, I4 and Kirchhoff’s Current Law allows us to write down an equation relating these currents. Looking closely at Fig. 2.1, we see that two of the currents (I1, I2) are flowing towards the node, while the other two currents (I3, I4) are flowing outwards. The ‘algebraic sum’ needs to take account of this difference in relative direction.

To apply Kirchhoff’s Current Law rigorously, we must first make an arbitrary choice of positive current direction. Suppose currents flowing in to the node (I1, I2) are treated as positive contributions to the algebraic sum (and conversely currents flowing from the node are treated as negative contributions), then the algebraic sum of currents would be written: + I1 + I2 - I3 - I4 , and according to Kirchhoff’s Current Law this algebraic sum is equal to zero:

+ I1 + I2 - I3 - I4 =0 ----------------------------------------------------------- (2.1)

The same result could be obtained with the opposite choice of positive current direction. If currents flowing from the node (I3, I4) are treated as positive contributions to the algebraic sum, then the algebraic sum of currents would be written: - I1 - I2 + I3 + I4 , and equating this algebraic sum to zero:

- I1 - I2 + I3 + I4 = 0 ----------------------------------------------------------- (2.2)

which is the same relationship as Eqn. 2.1 with all terms multiplied by –1.It must be emphasised that the choice of sign convention when using Kirchhoff’s Current Law is entirely arbitrary and, of course, makes no difference to the result obtained. However, it is good practice to be consistent in your choice, because this minimises the chance of making a mistake when writing down the algebraic sum. Eqns. 2.1 and 2.2 can be re-arranged to show that:

I1+ I2 = I3 + I4 -------------------------------------------------------------------(2.3)

and referring back to Fig. 2.1 we see that this equation is showing that current flowing into the node is equal to the current flowing out. This formulation arises naturally from physical considerations of current as the flow of charge. Charge does not accumulate at a node and

therefore any charge flowing into the node through one or more branches must flow out from the node through other branches. Therefore, current flowing into is equal to current flowing out of the node.

Worked example 2.1

Calculate the current I flowing into the node.Solution:-

Choosing currents flowing into the node as positive and applying Kirchhoff’sCurrent Law: +3 –2 +I = 0, so I = -1 AThe current flowing into the node is –1IA, which is the same as +1IA flowing out of the node.

Worked example 2.2

Calculate the current I defined in the diagram.

Solution:-

In this problem, there are two nodes, each with three branches connected. Begin by defining the current I’ flowing in the branch between the two nodes. The direction of I’ has been chosen randomly: it may turn out to have a positive or negative value. Choosing currents flowing out of the nodes aspositive and applying Kirchhoff’s Current Law at each node:

-(-4) + 2 + I’ = 0, so I’ = -6 A and: -I’ – 6 + I = 0, so I = I’ + 6 = 0 A

but is there any easier way? Yes! We can merge the two separate nodes into a single ‘supernode’, shown in red in the lower diagram. The supernode can’t accumulate charge, so Kirchhoff’s Current Law can be applied to currents in branches connected to it. Making the same choice of current direction:

-(-4) + 2 + I – 6 = 0, so I = 0 A________________________________________________________________________

(a)-Changing charge density

Physically speaking, the restriction regarding the "capacitor plate" means that Kirchhoff's

current law is only valid if the charge density remains constant in the point that it is applied

to. This is normally not a problem because of the strength of electrostatic forces: the charge

buildup would cause repulsive forces to disperse the charges.

However, a charge build-up can occur in a capacitor, where the charge is typically spread

over wide parallel plates, with a physical break in the circuit that prevents the positive and

negative charge accumulations over the two plates from coming together and cancelling. In

this case, the sum of the currents flowing into one plate of the capacitor is not zero, but rather

is equal to the rate of charge accumulation. However, if the displacement current dD/dt is

included, Kirchhoff's current law once again holds. (This is really only required if one wants

to apply the current law to a point on a capacitor plate. In circuit analyses, however, the

capacitor as a whole is typically treated as a unit, in which case the ordinary current law holds

since exactly the current that enters the capacitor on the one side leaves it on the other side.)

More technically, Kirchhoff's current law can be found by taking the divergence of Ampère's

law with Maxwell's correction and combining with Gauss's law, yielding:

This is simply the charge conservation equation (in integral form, it says that the current

flowing out of a closed surface is equal to the rate of loss of charge within the enclosed

volume (Divergence theorem)). Kirchhoff's current law is equivalent to the statement

that the divergence of the current is zero, true for time-invariant ρ, or always true if the

displacement current is included with J.

Uses:-

A matrix version of Kirchhoff's current law is the basis of most circuit simulation software, such as SPICE. This law can be used, along with Kichoff's voltage law, Ohm's law, Norton and Thevanin equivalents, and other transformations to analyze various circuits. 

A specific example of Kirchoff's current law is the analysis of a transistor. If you know the current through the collector or emitter resistor, you can assume (within reasonable limits) that the current through the other resistor is the same. (The "limits" have to do with the contribution from the base current, but that is generally negligible)

(2)-Kirchhoff's voltage law (KVL):-

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

This law is also called Kirchhoff's second law, Kirchhoff's loop (or mesh) rule,

and Kirchhoff's second rule.

The principle of conservation of energy implies that

The directed sum of the electrical potential differences (voltage) around any closed circuit is

zero.

or

More simply, the sum of the emfs in any closed loop is equivalent to the sum of the potential

drops in that loop.

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 emfavailable in that loop.

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 given/taken by potential field" (not

including energy taken by dissipation). Given a voltage potential, a charge which has

completed a closed loop doesn't gain or lose energy as it has gone back to initial potential

level.

This law holds true even when resistance (which causes dissipation of energy) is present in a

circuit. The validity of this law in this case can be understood if one realizes that a charge in

fact doesn't go back to its starting point, due to dissipation of energy. A charge will just

terminate at the negative terminal, instead of positive terminal. This means all the energy given by the

potential difference has been fully consumed by resistance which in turn loses the energy as heat dissipation.

To summarize, Kirchhoff's voltage law has nothing to do with gain or loss of energy by electronic components

(resistors, capacitors, etc.). It is a law referring to the potential field generated by voltage sources. In this

potential field, regardless of what electronic components are present, the gain or loss in "energy given by the

potential field" must be zero when a charge completes a closed loop.

Positive and Negative Signs in Kirchhoff's Voltage Law:-

Using the Voltage Rule requires some sign conventions, which aren't necessarily as clear as those in the Current Rule. You choose a direction (clockwise or counter-clockwise) to go along the loop.

When travelling from positive to negative (+ to -) in an emf (power source) the voltage drops, so the value is negative. When going from negative to positive (- to +) the voltage goes up, so the value is positive.

When crossing a resistor, the voltage change is determined by the formula I*R, where I is the value of the current and R is the resistance of the resistor. Crossing in the same direction as the current means the voltage goes down, so its value is negative. When crossing a resistor in the direction opposite the current, the voltage value is positive (the voltage is increasing).

(b)-Electric field and electric potential

Kirchhoff's voltage law could be viewed as a consequence of the principle of conservation of

energy. Otherwise, it would be possible to build a perpetual motion machine that passed a

current in a circle around the circuit.

Considering that electric potential is defined as a line integral over an electric field,

Kirchhoff's voltage law can be expressed equivalently as

which states that the line integral of the electric field around closed loop C is zero.

In order to return to the more special form, this integral can be "cut in pieces" in order to get

the voltage at specific components.

APPLICATION OF KIRCHHOFF'S VOLTAGE LAW:-

Kirchhoff's voltage law can be written as an equation, as shown below: 

Ea + Eb + Ec + . . . En = 0

where Ea, Eb, etc., are the voltage drops or emf's around any closed circuit loop. To set up the equation for an actual circuit, the following procedure is used. 

Assume a direction of current through the circuit. (The correct direction is desirable but not necessary.)Using the assumed direction of current, assign polarities to all resistors through which the current flows.Place the correct polarities on any sources included in the circuit.Starting at any point in the circuit, trace around the circuit, writing down the amount and polarity of the voltage across each component in succession. The polarity used is the sign AFTER the assumed current has passed through the component. Stop when the point at which the trace was started is reached.Place these voltages, with their polarities, into the equation and solve for the desired quantity.

Example: Three resistors are connected across a 50-volt source. What is the voltage across the third resistor if the voltage drops across the first two resistors are 25 volts and 15 volts?

Solution: First, a diagram, such as the one shown in figure 3-23, is drawn. Next, a direction of current is assumed (as shown). Using this current, the polarity markings are placed at each end of each resistor and also on the terminals of the source. Starting at point A, trace around the circuit in the direction of current flow, recording the voltage and polarity of each component. Starting at point A and using the components from the circuit: 

Substituting values from the circuit:

Figure 3-23. - Determining unknown voltage in a series circuit. 

Application of Kirchoff's Circuit Laws

These two laws enable the Currents and Voltages in a circuit to be found, ie, the circuit is said to be "Analysed", and the basic procedure for using Kirchoff's Circuit Laws is as follows:

1. Assume all voltage sources and resistances are given. (If not label them V1, V2 ..., R1, R2etc)

2. Label each branch with a branch current. (I1, I2, I3 etc)

3. Find Kirchoff's first law equations for each node.

4. Find Kirchoff's second law equations for each of the independent loops of the circuit.

5. Use Linear simultaneous equations as required to find the unknown currents.

As well as using Kirchoff's Circuit Law to calculate the voltages and currents circulating around a linear circuit, we can also use loop analysis to calculate the currents in each independent loop helping to reduce the amount of mathematics required using just Kirchoff's laws. In the next tutorial about DC Theory we will look at Mesh Current Analysis to do just that.

Limitations:-

This is a simplification of Faraday's law of induction for the special case where there is no

fluctuating magnetic field linking the closed loop. Therefore, it practically suffices for

explaining circuits containing only resistors and capacitors.

In the presence of a changing magnetic field the electric field is not conservative and it

cannot therefore define a pure scalar potential—the line integral of the electric field around

the circuit is not zero. This is because energy is being transferred from the magnetic field to

the current (or vice versa). In order to "fix" Kirchhoff's voltage law for circuits containing

inductors, an effective potential drop, or electromotive force (emf), is associated with

each inductance of the circuit, exactly equal to the amount by which the line integral of the

electric field is not zero by Faraday's law of induction.

Conclusion for Kirchoff's Circuit Law:-

We saw in the Resistors  above that a single equivalent resistance, ( RT ) can be found when two or more resistors are connected together in either series, parallel or combinations of both, and that these circuits obey Ohm's Law.

However, sometimes in complex circuits such as bridge or T networks, we can not simply use Ohm's Law alone to find the voltages or currents circulating within the circuit. For these types of calculations we need certain rules which allow us to obtain the circuit equations and for this we can use Kirchoff's Circuit Law.

we see that at junction,  there may be many conductors. Current enters from the top, then goes out in the other directions and, from the Current Law, we know the algebraic sum of these must be zero. If we knew the resistance values of the resistors, and the current coming into the junction, we could use the Voltage Law equations to determine the currents in each of the lower paths, if the resistors were unequal.

This is why Kirchhoff's Laws are such powerful tools

References

Paul, Clayton R. (2001). Fundamentals of Electric Circuit Analysis. John Wiley &

Sons. ISBN 0-471-37195-5.

Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers

(6th ed.). Brooks/Cole. ISBN 0-534-40842-7.

Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism,

Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.