Series Circuit

A series circuit is one in which several resistances are connected one after the other. Such connection is also called end to end connection or cascade connection. There is only one path for the flow of current.

1. Resistor in series

       Consider resistor shown in the Fig. 1. The resistance R₁, R₂ and R₃ are said to be in series. The combination is connected across a source of a voltage V volts. Naturally the current flowing through all of them is same indicated as I amperes. e.g. the small chain of small lights, used for the decoration purposes is good example of series combination.

Fig.  1  A series circuit

       Now let us study the voltage distribution 

       Let V₁, V₂ and V₃ be the voltages across the terminals of resistances R₁, R₂ and R₃ respectively,

Then                                               V = V₁ +V₂ + V₃
       Now according to Ohm's law   V₁ = I R₁ ,     V₂ = I R₂     ,  V₃ = I R₃
      Current through of them is same i.e. I
                                                       V = I R₁ + I R₂ + I R₃ = I  (R₁ +R₂ +R₃ )
      Applying Ohm's law to overall circuit 

                                                       V =  I R_eq

     Where R_eq  = Equivalent resistance of the circuit, by comparison of tow equations, 

                                                        R_eq = R₁  + R₂ + R₃

      i.e. total or equivalent resistance of the series circuit is arithmetic sum of the resistances connected in series 

        For n resistances in series,                 R = R₁  + R₂ + R₃ +.......+ R_n 

Characteristic of series circuits

1) The same current flows through each resistance.

2) The supply voltage V is the sum of the individual voltage drops across the resistances. 

                                                       V = V₁ +V₂ + ..... + V_n

3) The equivalent resistance is equal to the sum of the individual resistances

4) The equivalent resistance is the largest of all the individual resistances. 

i.e                                                  R >R₁  ,  R >R₂ , ..........R > Rn

2. Inductors in series 

       Consider in the Fig. 2(a). Tow inductors L₁  and L₂ are connected in series. The current flowing through L₁ and L₂ are i₁ and i₂  while voltages developed across L₁ and L₂ are V_L1 and V_L2  respectively. The equivalent circuit as shown in the Fig. 2(b). 

Fig.  2

       For series combination,

                i = i₁  = i₂ 

and          V_L  = V_L1  + V_L2 

.^..              L_eq  di/dt = L₁ di/dt + L₂ di/dt

.^..               L_eq  di/dt = (L₁ +L₂ ) di/dt

.^..                L_eq  = L₁ +L₂

       That means, equivalent inductances of the series combination of tow inductances is the sum of inductances connected in series

       The total equivalent inductances of the series circuit is the sum of the inductances connected in series.

       For n inducatances in series, 

                  L_eq   = L₁ + L₂ + L₃ +  ............+ L_n

3. Capacitors in series

       Consider the Fig. 3(a). Tow capacitors C₁ and C₂ are connected in series. The current flowing through and voltages developed across C₁  and C₂ are i₁, i₂   and V_C1  and V_C2  respectively. The equivalent circuit is shown in the Fig. 3(b). 

Fig.  3

       For series combination,

                                  i = i₁ = i₂   and

                                 V_C  = V_C1  + V_C2 

 But                            i = i₁ = i₂

      That means, reciprocal of equivalent capacitor of the series combination is the sum of  the reciprocal of individual capacitances. 

       The reciprocal of the total equivalent capacitors of the series combination is the sum of the reciprocols of the individual capacitors, connected in series,