Electrical resistance is the property of a material by virtue of which it opposes the flow of electrons through the material. Thus the opposition offered by a substance to the flow of electric current is called resistance. Any complex circuit of resistances can be easily solved, if circuit is divided into small parts and then every part is replaced by its equivalent resistance.
Resistances can be connected in series or in parallel in any complex circuit. If, we can calculate the equivalent resistance of the series connected resistances and parallel connected resistances we can replace it and solve the complex circuit very easily.
Resistances in Series
The circuit in which resistances are connected end to end so that the same current flows through all the resistances is called a series circuit as shown in the figure
By ohm’s law, voltage across various resistances is:`V_1 = IR_1` ; `V_2 = IR_2` ; `V_3 = IR_3`Now, `V = V_1 + V_2 + V_3 = IR_1 + IR_2 + IR_3``V = I (R_1 + R_2 + R_3)`Or `V/I = R_1 + R_2 + R_3`
But V/I is the total resistance RT between points A and B. RT is called the total or equivalent resistance (Equivalent resistance is the single resistance, which, if substituted for the series resistances would provide the same current in the circuit) of the three resistances.
Therefore, `R_T = R_1 + R_2 + R_3`
Hence when a number of resistances are connected in series, the total resistance is equal to the sum of individual resistances.
Resistances in Parallel
When one end of each resistance is joined to a common point and the other end of each resistance is joined to another common point so that there are as many paths for current flow as the number of resistances, it is called parallel circuit as shown in the figure. Note that voltage across each resistance is same (i.e. V volts in this case).
Now, current through respective resistance is `I_1 = V/R_1``I_2 = V/R_2``I_3 = V/R3`Now, `I = I_1 + I_2 + I_3 = V/R_1 + V/R_2 + V/R_3``I = V (1/R_1 + 1/R_2 + 1/R_3)`Or `I/V = 1/R_1 + 1/R_2 + 1/R_3`But V/I is the total resistance RT of the parallel resistances (see figure 2) so that` I/V = 1/R_T`Therefore, `1/R_T = 1/R_1 + 1/R_2 + 1/R_3`Hence when a number of resistances are connected in parallel, the reciprocal of total resistance is equal to the sum of reciprocals of individual resistances.
Example for Finding Equivalent ResistanceExample: A resistive network is shown in the figure find the equivalent resistance looking from the terminal x-y.
Solution: - Let R be the equivalent resistance of the network looking from terminal x-y to the network.First we see that 2Ω and 2 Ωresistances are in parallel, there equivalent resistance is 1Ω.This 1Ω resistance is in series with 3Ω resistance,Therefore there equivalent resistance is 4Ω.Again this 4Ω resistance and 4Ω resistance are in parallel combination there equivalent resistance is2Ω`1/R = 1/4 +1/4` , R = 2ΩThus the equivalent resistance of the network looking from terminal x-y is R = 2Ω.