Resistance in Series and Parallel | Current Divider Rule and Voltage Divider Rule

Need of Combining of Resistances

Combining resistances is a fundamental concept in electronics and circuit design. Resistors are used to limit current flow and voltage, and they can be combined in different ways to achieve different results. Here are some of the reasons why combining resistances is needed:

1. To increase or decrease resistance: By combining resistors in series, you can increase the total resistance, while combining resistors in parallel can decrease the total resistance.
2. To achieve a specific resistance value: Sometimes, you need a specific resistance value that cannot be obtained with a single resistor. In such cases, you can combine resistors in series or parallel to obtain the desired resistance value.
3. To distribute power: When resistors are combined in parallel, they distribute the power evenly among themselves. This helps to prevent overheating and damage to individual resistors.
4. To achieve different voltage levels: By combining resistors in series, you can divide the voltage across them in different ratios. This can be useful in voltage dividers and other applications where you need to obtain specific voltage levels.

Series Combination of Resistances

The resistances are said to be in series combination when the current flowing in the resistance are same as shown in figure (1) below

In the figure above three Resistances R₁ , R₂ and R₃ are connected in series. The Total Resistance of all the three resistance are called equivalent Resistance.

Let “V” is voltage across the resistances and “I” is the current flowing in the Resistance. \\\;\\Applying Kirchhoff’s voltage Law\\\;\\V – I R₁ – I R₂ – I R₃ = 0\\\;\\V – I( R₁ + R₂ + R₃ )= 0\\\;\\V = I( R₁ + R₂ + R₃ )\;\;\;\;\;\;\;\;\;\;(1)

Consider the equivalent circuit of the above circuit as shown in figure (2) below.

The equivalent circuit has same parameters (Current, Voltage and Total Resistance) as the original circuit. “R_eqv ” is the equivalent resistance that is equal to total resistance of the original circuit.

Applying Kirchhoff’s voltage law to the equivalent circuit. \\\;\\V – IR_eqv = 0\\\;\\V = IR_eqv \;\;\;\;\;\;\;\;(2)

Comapring equation (1) and equation (2) \\\;\\R_eqv = R₁ + R₂ + R₃ \\\;\\ Thus the equivalent resistance (Net resistance/ Total Resistance) of resistances in series is equal to sum of resistance in series.

Voltage Divider Rule

In a series circuit, the total supply voltage is divided into smaller voltages across each element of the circuit. This is because the series circuit acts like a voltage divider. Consider the figure below in which total supply voltage “V” is divided into voltages “V₁” , “V₂” and “V₃” across the Series Resistances “R₁” , “R₂“, and “R₃“

Let Current Through the resistances is “I”. Then According to Kirchhoff’s Voltage law

V – I (R₁ + R₂ + R₃ ) = 0

V = I (R₁ + R₂ + R₃ ) = I R_eqv \;\;\;\;\;\;\;\;\;\left( \because R_{1} + R_{2} + R_{3} = R_{eqv} \right) \\\;\\I = \frac {V}{R_{eqv}}\;\;\;\;\;\;\;\;\;\;(3)

in the above figure \\\;\\V_{1} = I R_{1}\\\;\\\textrm{Putting The value of "I" from equation (3)}\\\;\\V_{1} = \frac{V R_{1}}{R_{eqv}}\\\;\\\textrm{Similarly}\\\;\\V_{2} = \frac{VR_{2}}{R_{eqv}}\\\;\\V_{3}=\frac{VR_{3}}{R_{eqv}}

Parallel Combination of Resistances

The resistances are said to be in series combination when the voltage across the terminals of the resistances are same as shown in figure (3) below

The resistance “R₁ ” , “R₂” and “R₃” are connected in parallel. \\\;\\Let “V” is Volatge across the resistances “R₁ ” , “R₂” and “R₃“. \\\;\\I, I₁ , I₂ and I₃ are the total current in the circuit and currents through the resistances “R₁ ” , “R₂” and “R₃” respectively.

Applying Kirchhoff’s current law at node “a” \\\;\\I = I₁ + I₂ + I₃\\\;\\I = \frac{V}{R_{1}}+\frac{V}{R_{2}}+\frac{V}{R_{3}}\;\;\;\;\;\;\;\;\left ( \because V = I_1R_1=I_2R_2=I_3R_3\right )\\\;\\I = V\left(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\right)\;\;\;\;\;\;(4)

Let the figure below is the equivalent circuit of the above figure.

From the circuit above \\\;\\I = \frac{V}{R_{eqv}}\;\;\;\;\;\;\;(5)\\\;\\Putting the value of “I” from equation (5) into equation (4) \\\;\\ \frac{V}{R_{eqv}}=V \left(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\right)\\\;\\\frac{1}{R_{eqv}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}

Current Divider Rule

A parallel circuit acts as a current divider as it divides the total circuit current in its all branches. From figure (3)\\\;\\V = I₁ R₁ \;\;\;\;\;\; (6)\\\;\\ From equation (5) and Equation (6)\\\;\\IR_{eqv} = I_1R_1\\\;\\I_1=\frac{IR_{eqv}}{R_1}

Similarly \\\;\\I_2= \frac{IR_{eqv}}{R_2}\\\;\\I_3=\frac{IR_{eqv}}{R_3}

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