Formula For Total Resistance In A Series Parallel Circuit

Formula For Total Resistance In A Series Parallel Circuit

A series-parallel circuit is a circuit that contains both series and parallel connections. The total resistance of a series-parallel circuit can be calculated using the following formula: ``` Rtotal = R1 + R2 + ... + Rn + 1/(1/R1 + 1/R2 + ... + 1/Rn) ``` where: *

Rtotal

is the total resistance of the circuit *

R1

,

R2

, ...,

Rn

are the resistances of the individual resistors in the circuit To understand how this formula works, let's consider a simple series-parallel circuit consisting of three resistors:

In this circuit, the resistors

R1

and

R2

are connected in series, and the resulting series combination is connected in parallel with resistor

R3

. The total resistance of this circuit can be calculated as follows: ``` Rtotal = R1 + R2 + R3 + 1/(1/R1 + 1/R2) ``` This formula can be derived by first considering the equivalent resistance of the series combination of resistors

R1

and

R2

. The equivalent resistance of a series circuit is equal to the sum of the individual resistances: ``` R12 = R1 + R2 ``` Now, we can consider the equivalent resistance of the parallel combination of resistors

R12

and

R3

. The equivalent resistance of a parallel circuit is equal to the reciprocal of the sum of the reciprocals of the individual resistances: ``` Rtotal = 1/(1/R12 + 1/R3) ``` Substituting the expression for

R12

into this equation, we get the following formula for the total resistance of the circuit: ``` Rtotal = R1 + R2 + R3 + 1/(1/R1 + 1/R2) ``` This formula can be used to calculate the total resistance of any series-parallel circuit. ## Example Let's use the formula for total resistance in a series-parallel circuit to calculate the total resistance of the following circuit:

In this circuit, the resistors

R1

,

R2

, and

R3

have resistances of 10 ohms, 20 ohms, and 30 ohms, respectively. The total resistance of this circuit can be calculated using the following formula: ``` Rtotal = R1 + R2 + R3 + 1/(1/R1 + 1/R2) ``` ``` Rtotal = 10 ohms + 20 ohms + 30 ohms + 1/(1/10 ohms + 1/20 ohms) ``` ``` Rtotal = 60 ohms + 1/(5/20 ohms) ``` ``` Rtotal = 60 ohms + 4 ohms ``` ``` Rtotal = 64 ohms ``` Therefore, the total resistance of this circuit is 64 ohms. ## Conclusion The formula for total resistance in a series-parallel circuit can be used to calculate the total resistance of any series-parallel circuit. This formula can be derived by first considering the equivalent resistance of the series combination of resistors and then the equivalent resistance of the parallel combination of resistors. The total resistance of the circuit is equal to the sum of the individual resistances and the equivalent resistance of the parallel combination of resistors.


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