#### Series Circuits:

When the resistors are connected end to end, so that they form only one path for the flow of current, then resistors are said to be connected in series and such circuits are known as series circuits.

Let resistors R1, R2, and R3 be connected in series as shown in Fig. 2.5, and the potential difference of V volts be applied between extreme ends A and D to cause flow of current of I amperes through all the resistors R1, R2, and R3.

Now according to Ohm’s law:

Voltage drop across resistor R1, V1 = I R1

Voltage drop across resistor R2, V2 = I R2

Voltage drop across resistor R3, V3 = I R3

Voltage drop across whole circuit,

V = Voltage drop across resistor R1 + voltage drop across resistor R2 + voltage drop across resistor R3

i.e. V = I R1 + I R2 + I R3 = I (R1 + R2 + R3)

Or V/I = R1 + R2 + R3 …(2.4)

And according to Ohm’s law V/I gives the whole circuit resistance, say R

... Effective resistance of the series circuit,

R = R1 + R2 + R3 …(2.5)

Thus when a number of resistors are connected in series, the equivalent resistance is given by the arithmetic sum of their individual resistances.

i.e., R = R1 + R2 + R3 + … Rn …(2.6)

From the above discussions for a series circuit we conclude that:

1. Same current flows through all parts of the circuit,

2. Applied voltage is equal to the sum of voltage drops across the different parts of the circuit,

3. Different resistors have their individual voltage drops,

4. Voltage drop across individual resistor is directly proportional to its resistance, current being the same in each resistor,

Series circuits are common in electrical equipment. The tube filaments in small radios are usually in series. Current controlling devices are wired in series with the controlled equipment. Fuses are in series with the equipment they protect. A thermostat switch is in series with the heating element in an electric iron. Automatic house-heating equipment has a thermostat, electro-magnet coils, and safety cut-outs in series with a voltage source. Rheostats are placed in series with the coils in large motors for motor current control.

#### Parallel Circuits:

When a number of resistors are connected in such a way that one end of each of them is joined to a common point and the other ends being joined to another common point, as shown in Fig. 2.9, then resistors are said to be connected in parallel and such circuits are known as parallel circuits. In these circuits current is divided into as many paths as the number of resistances.

Let the resistors R1, R2 and R3 be connected in parallel, as shown in Fig. 2.9, and the potential difference of V volts be applied across the circuit.

Since potential difference across each resistor is same and equal to potential difference applied to the circuit i.e. V

Thus when a number of resistors are connected in parallel, the reciprocal of the equivalent resistance is given by the arithmetic sum of the reciprocals of their individual resistances.

In general if n resistors of resistances R1, R2, R3 …Rn are connected in parallel, then equivalent resistance R of the circuit is given by the expression:

From the above discussions for a parallel circuit we conclude that:

1. Same voltage acts across all branches of the circuit,

2. Different resistors (or branches) have their individual currents,

3. Total circuit current is equal to the sum of individual currents through the various resistors (or branches),

7. The reciprocal of the equivalent or combined resistance is equal to the sum of the reciprocals of the resistances of the individual branches.

Parallel circuits are very common in use. Various lamps and appliances in a house are connected in parallel, so that each one can be operated independently. A series circuit is an “all or none” circuit, in which either everything operates or nothing operates. For individual control, devices are wired in parallel.

Current Distribution in Parallel Circuits:

Let two resistors of resistances R1 and R2 be connected in parallel across a pd of V volts, According to Ohm’s law:

Hence current flowing through each resistor, when connected in parallel, is inversely proportional to their respective resistances.

Since conductance is reciprocal of resistance and if G1 and G2 are the respective conductance’s of resistors R1 and R2 then:

Adding 1 on both sides of the above expression we have:

Hence current in any branch of a parallel circuit is directly proportional to its respective conductance and is equal to the total current flowing through the circuit multiplied by the ratio of the conductance of the branch to that of the circuit.

The same relation holds good for parallel circuit consisting of more than two resistors and is very useful for its solution.

Let us consider a circuit consisting of resistances R1, R2, R3, and R4 ohms respectively connected in parallel across a potential difference of V volts, as shown in Fig. 2.12.

#### Series-Parallel Circuits:

So far, only simple series and simple parallel circuits have been considered. Practical electric circuits very often consist of combinations of series and parallel resistances. Such circuits may be solved by the proper application of Ohm’s law and the rules for series and parallel circuits to the various parts of the complex circuit.

There is no definite procedure to be followed in solving complex circuits; the solution depends on the known facts concerning the circuit and the quantities which one desires to find. One simple rule may usually be followed, however — reduce the parallel branches to an equivalent series branch and then solve the circuit as a simple series circuit.

For example consider a series-parallel circuit shown in Fig. 2.14 for solution.

First of all equivalent resistances of all parallel branches are determined separately e.g. of branches AB and CD by the law of parallel circuits,

Now the circuit shown in Fig. 2.14 gets reduced to a simple series circuit shown in Fig. 2.15 consisting of three resistances:

After knowing I, potential differences across branches AB, BC and CD are determined from the relations:

After determination of potential difference across each parallel branch, the currents in the various resistances are determined from the relations:

Thus equivalent resistance of the whole circuit, voltage drop across each branch and currents in the various resistors may be determined.

Network Simplification (Or Reduction):

Sometimes we come across so complicated circuits that they cannot be solved simply by applying Ohm’s law. Hence for solution of such a circuit first of all the circuit is reduced to simple series or simple parallel or series-parallel circuit and then solved by applying Ohm’s law.

Network reduction or simplification is a process by which currents and voltages acting in a circuit composed of resistors having a series, parallel or series-parallel combinations can be determined.

It may be seen from Fig. 2.18 (a) that resistances R3, R4 and R5 are connected in series and the same current I3 flows through them because the circuit does not branch off at points C and D.

By virtue of this fact, the three resistances R3, R4 and R5 can be combined into a single equivalent resistance given by the expression:

R’ = R3 + R4 + R5

Thus a simpler equivalent circuit shown in Fig. 2.18 (b) is obtained. From circuit diagram shown in Fig. 2.18 (b) it can be seen that resistance R2 is connected in parallel with the series combination of resistances R3, R4 and R5 i.e., equivalent resistance R’.

Resistances R2, and R’ can be combined into a single equivalent resistance given by the expression:

Thus the circuit is further simplified [Fig. 2.18 (c)]. Resistances R1, R” and R0 in the circuit of Fig. 2.18 (c) are seen to be connected in series.

Replacing all the three resistances by a single equivalent resistance connected across the battery (supply source), a simple series circuit [Fig. 2.18 (d)] is obtained where in:

R” = R1 + R6 + R”

In a similar way most of the series-parallel circuits containing a single supply source can be reduced to a simple series circuit.

Now the current in the simple series circuit, shown in Fig. 2.18 (d), can be readily determined by applying Ohm’s law. Now the branch currents can be determined by restoring the circuit step by step to its original form in the reverse order.

It is evident from circuit diagram shown in Fig. 2.18 (c)

I1 = I2 + I3

Furthermore, the voltage across points B and E will be I1R”. Knowing the voltage across points B and E, currents I2, and I3 can be easily determined from the expressions: