ELECTRIC CIRCUITS UNIT II

                                                  UNIT – II

Basic Terms used in a Circuit

1.  Circuit.   A circuit is a closed conducting path through which an electric current either flows

                    or is intended flow.

2. Network.   A combination of various electric elements, connected in any manner.

3. Linear Circuit. A linear circuit is one whose parameters are constant i.e. they do not change

                       with voltage or current.

4.Non-linear Circuit. It is that circuit whose parameters change with voltage or current.

5. Bilateral Circuit. A bilateral circuit is one whose properties or characteristics are the same

                            in either direction. The usual transmission line is bilateral, because it can be                

                            made to perform   its function equally well in either direction.

6. Unilateral Circuit. It is that circuit whose properties or characteristics change with the

                              direction of its operation.

                              A diode rectifier is a unilateral circuit, because it cannot perform

                              rectification in both directions.

7. Parameters.  The various elements of an electric circuit are called its parameters like resistance,

                        inductance and capacitance. These parameters may be lumped or distributed.

8. Passive Network is one which contains no source of e.m.f. in it.

9. Active Network is one which contains one or more than one source of e.m.f.

10. Node    I t is a junction in a circuit where two or more circuit elements are connected together.

11. Branch   It is that part of a network which lies between two junctions.

12. Loop.   It is a close path in a circuit in which no element or node is encountered more than once.

13. Mesh.   It is a loop that contains no other loop within it.

 Kirchhoff`s Laws

  Kirchhoff’s laws are more comprehensive than Ohm's law and are used for solving electrical networks which may not be readily solved by the latter.

Kirchhoff`s laws, two in number, are particularly useful in determining the equivalent resistance of a complicated network of conductors and for calculating the currents flowing in the various conductors.

I.            Kirchhoff`s Point Law or Current Law (KCL)

In any electrical network, the algebraic sum of the currents meeting at a point (or junction) is

Zero.

That is the total current entering a junction is equal to the total current leaving that junction.  

Consider  I_1,I_{4 are incoming and} I_2,I_3,I_{5 are outgoing then}

I₁+(-I₂)+(I₃)+(+I₄)+(-I₅) = 0

I₁+I₄-I₂-I₃-I₅ = 0

Or

 I₁+I₄ = I₂+I₃+I₅

Or

Incoming currents =Outgoing currents

 II   Kirchhoff's Mesh Law or Voltage Law (KVL)

In any electrical network, the algebraic sum of the products of currents and resistances in each of the conductors in any closed path (or mesh) in a network plus the algebraic sum of the e.m.f.’s. in that path is zero.

That is, ∑IR +  ∑e.m.f = 0 round a mesh

It should be noted that algebraic sum is the sum which takes into account the polarities of the

voltage drops.

That is, if we start from a particular junction and go round the mesh till we come back to the starting point, then we must be at the same potential with which we started.

 Hence, it means that all the sources of emf met on the way must necessarily be equal to the voltage drops in the resistances, every voltage being given its proper sign, plus or minus.

 Determination of Voltage Sign

In applying Kirchhoff's laws to specific problems, particular attention should be paid to the

algebraic signs of voltage drops and e.m.fs.

(a)  Sign of Battery E.M.F.

A rise in voltage should be given a + ve sign and a fall in voltage a -ve sign. That is, if we go from the -ve terminal of a battery to its +ve terminal there is a rise in potential, hence this voltage should be given a + ve sign.

And on the other hand, we go from +ve terminal to -ve terminal, then there is a fall in potential, hence this voltage should be preceded by a -ve sign.

(b) Sign of IR Drop

If we go through a resistor in the same direction as the

current, then there is a fall in potential because current flows from a higher to a lower potential..

Hence, this voltage fall should be taken -ve. However, if we go in a direction opposite to that of the

current, then there is a rise in voltage. Hence, this voltage rise should be given a positive sign.

As we travel around the mesh in the clockwise direction, different voltage drops will have the following signs :

I₁R₁ is  - ve     (fall in potential)

I₂R₂ is  - ve     (fall in potential)

I₃R₃ is + ve     (rise in potential)

I₄R₄ is  - ve     (fall in potential)

E₂ is  - ve        (fall in potential)

E₁ is  + ve       (rise in potential)

Using Kirchhoff's voltage law, we get

-I₁R₁ – I₂R₂ – I₃R₃ – I₄ R₄ – E₂ + E₁ = 0

         Or       I₁R₁ + I₂R₂ – I₃R₃ + I₄R₄ = E₁ –E₂

Assumed Direction of Current:

In applying Kirchhoff's laws to electrical networks, the direction of current flow may be assumed either clockwise or anticlockwise. If the assumed direction of current is not the actual direction, then on solving the question, the current will be found to have a minus sign.

If the answer is positive, then assumed direction is the same as actual direction. However, the important point is that once a particular direction has been assumed, the same should be used throughout the solution of the question.

 Kirchhoff's laws are applicable both to d.c. and a.c. voltages and currents. However, in the case of alternating currents and voltages, any e.m.f. of self-inductance or that existing across a capacitor should be also taken into account.

Resistance in series:
If three conductors having resistances R1, R2 and R3 are joined end on end as shown in fig below, then they are said to be connected in series. It can be proved that the equivalent resistance between points A & D is equal to the sum of the three individual resistances.

For a series circuit, the current is same through all the three conductors but voltage drop across each is different due to its different values of resistances and is given by ohm`s Law and the sum of the three voltage drops is equal to the voltage supplied across the three conductors.
 ∴ V= V1+V2+V3 = IR1+IR2+IR3
 But V= IR
where R is the equivalent resistance of the series combination.
IR = IR1+IR2+IR3
 or R = R1 + R2+ R3
 The main characteristics of a series circuit are
 1. Same current flows through all parts of the circuit.
 2. Different resistors have their individual voltage drops.
3. Voltage drops are additive.
4. Applied voltage equals the sum of different voltage drops.
 5. Resistances are additive.
6. Powers are additive
 Voltage Divider Rule
In a series circuit, same current flows through each of the given resistors and the voltage drop varies directly with its resistance. Consider a circuit in which, a 24- V battery is connected across a series combination of three resistors of 2,4,6Ω. Determine the voltage drops across each resistor? Total resistance R = R 1 + R2 + R3= 12 Ω According to Voltage Divider Rule, voltages divide in the ratio of their resistances and hence the various voltage drops are

Resistances in Parallel:
(1) Potential difference across all resistances is the same
(2) Current in each resistor is different and is given by Ohm's Law and
(3) The total current is the sum of the three separate currents.

 The main characteristics of a parallel circuit are:
1. Same voltage acts across all parts of the circuit
2. Different resistors have their individual current.
3. Branch currents are additive.
4. Conductances are additive.
5. Powers are additive