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

It may be seen form that resistances R₄ and R₅ are connected in series and the same current (Ia) 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 = R₂+R₄+R₅

Thus a simpler equivalent circuit shown in is obtained. Form circuit diagram shown in 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 R₁ can be combined into a single equivalent resistance given by the expression

R = R₂R’ =  R₂(R₃+R₅)
     R₂+R₁‘  R₂(R₃+R₅)

Thus the circuit is further simplified. Resistances R₁, R" and R₆ in the circuit of are seen to be connected in series. Replacing all the three resistances by a single equivalent resistance connected a cross the battery (supply source), a simple series circuit 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, can be readily determined by a plying 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 form circuit diagram shown in

I₁+I₂+I₃

Further more, the voltage across points B and E will be I, R” knowing the voltage across points B and E, currents I₂ and I₃ can be easily determined from the expressions

I₂ = I₁R" and I₃ = I₁R"  =      I₁R"      
        R₃                R"      R₃+R₄+R₅

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