Series-Parallel Circuits

In practice we come across combined circuits, known as series- parallel circuits, which are very complicated in solution. The circuit of this type is shown in. To solve such circuits, the following method illustrating the solution of method illustrating the solution of circuit shown in must be adopted.

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

Equivalent resistances of parallel branch AB,

R_AB =   1     =   R₁R₂
          1 + 1     R₁+R₂
          R₁+R₂

and equivalent resistance of parallel branch CD

R_CD =    1  =    R4 R5 R6
         1+1+1  R₅R₆+ R₄R₆+R₄R₅  
        R₄ R₅ R₆

Now the circuit shown in gets reduced to a simple series circuit shown in consisting of three resistance,

R_AB  =   R₁ R₂  , Rac = R₂ and R_CD =    R₁ R₂ R₃
            R₁+R₂                                    R₅R₆+ R₄R₆+R₄R₅

Total resistance of the circuit, R_T = R_AB + R_BC + R_CD
Now circuit current may be determined from the relation, I = V
                                                                                      R_T
After knowing I, potential differences across branches AB, BC and CD are determined form the relations.

P.D. across AB. V_AB = I. R_AB = V . R_AB
                                               R_T

P.D. across CD, Vac = I. Rac = V .Rac
                                              R_T

and P.D. across CD, V_CD = I. R_CD = V . R_CD
                                                      R_T

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

Current in resistance    R₁ = I₁ =V_AB
                                                R₁

Current in resistance     R₂ = I₂ V_AB
                                                R₂

Current in resistance      R₃ = I₃ = I

Current in resistance     R₄ = I₄  =V_CD
                                                  R₄

Current in resistance      R₅ = I₅ = V_CD
                                                    R₅                  

and current in resistance  R₆ = I₆ = V_CD
                                                     R₆

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

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