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# Circuit Connections in Resistors

A Resistor when connected in a circuit, that connection can be either series or parallel. Let us now know what will happen to the total current, voltage and resistance values if they are connected in series as well, when connected in parallel.

## Resistors in Series

Let us observe what happens, when few resistors are connected in Series. Let us consider three resistors with different values, as shown in the figure below.

### Resistance

The total resistance of a circuit having series resistors is equal to the sum of the individual resistances. That means, in the above figure there are three resistors having the values 1KΩ, 5KΩ and 9KΩ respectively.

Total resistance value of the resistor network is −

$$R\:\:=\:\:R_{1}\:+\:R_{2}\:+\:R_{3}$$

Which means 1 + 5 + 9 = 15KΩ is the total resistance.

Where R_{1} is the resistance of 1^{st} resistor, R_{2} is the resistance of 2^{nd} resistor and R_{3} is the resistance of 3^{rd} resistor in the above resistor network.

### Voltage

The total voltage that appears across a series resistors network is the addition of voltage drops at each individual resistances. In the above figure we have three different resistors which have three different values of voltage drops at each stage.

Total voltage that appears across the circuit −

$$V\:\:=\:\:V_{1}\:+\:V_{2}\:+\:V_{3}$$

Which means 1v + 5v + 9v = 15v is the total voltage.

Where V_{1} is the voltage drop of 1^{st} resistor, V_{2} is the voltage drop of 2^{nd} resistor and V_{3} is the voltage drop of 3^{rd} resistor in the above resistor network.

### Current

The total amount of Current that flows through a set of resistors connected in series is the same at all the points throughout the resistor network. Hence the current is same 5A when measured at the input or at any point between the resistors or even at the output.

Current through the network −

$$I\:\:=\:\:I_{1}\:=\:I_{2}\:=\:I_{3}$$

Which means that current at all points is 5A.

Where I_{1} is the current through the 1^{st} resistor, I_{2} is the current through the 2^{nd} resistor and I_{3} is the current through the 3^{rd} resistor in the above resistor network.

## Resistors in Parallel

Let us observe what happens, when few resistors are connected in Parallel. Let us consider three resistors with different values, as shown in the figure below.

### Resistance

The total resistance of a circuit having Parallel resistors is calculated differently from the series resistor network method. Here, the reciprocal (1/R) value of individual resistances are added with the inverse of algebraic sum to get the total resistance value.

Total resistance value of the resistor network is −

$$\frac{1}{R}\:\:=\:\:\frac{1}{R_{1}}\:\:+\:\:\frac{1}{R_{2}}\:\:+\frac{1}{R_{3}}$$

Where R_{1} is the resistance of 1^{st} resistor, R_{2} is the resistance of 2^{nd} resistor and R_{3} is the resistance of 3^{rd} resistor in the above resistor network.

For example, if the resistance values of previous example are considered, which means R_{1} = 1KΩ, R_{2} = 5KΩ and R_{3} = 9KΩ. The total resistance of parallel resistor network will be −

$$\frac{1}{R}\:\:=\:\:\frac{1}{1}\:\:+\:\:\frac{1}{5}\:\:+\frac{1}{9}$$

$$=\:\:\frac{45\:\:+\:\:9\:\:+\:\:5}{45}\:\:=\:\:\frac{59}{45}$$

$$R\:\:=\:\:\frac{45}{59}\:\:=\:\:0.762K\Omega\:\:=\:\:76.2\Omega$$

From the method we have for calculating parallel resistance, we can derive a simple equation for two-resistor parallel network. It is −

$$R\:\:=\:\:\frac{R_{1}\:\:\times\:\:R_{2}}{R_{1}\:\:+\:\:R_{2}}\:$$

### Voltage

The total voltage that appears across a Parallel resistors network is same as the voltage drops at each individual resistance.

The Voltage that appears across the circuit −

$$V\:\:=\:\:V_{1}\:=\:V_{2}\:=\:V_{3}$$

Where V_{1} is the voltage drop of 1^{st} resistor, V_{2} is the voltage drop of 2^{nd} resistor and V_{3} is the voltage drop of 3^{rd} resistor in the above resistor network. Hence the voltage is same at all the points of a parallel resistor network.

### Current

The total amount of current entering a Parallel resistive network is the sum of all individual currents flowing in all the Parallel branches. The resistance value of each branch determines the value of current that flows through it. The total current through the network is

$$I\:\:=\:\:I_{1}\:+\:I_{2}\:+\:I_{3}$$

Where I_{1} is the current through the 1^{st} resistor, I_{2} is the current through the 2^{nd} resistor and I_{3} is the current through the 3^{rd} resistor in the above resistor network. Hence the sum of individual currents in different branches obtain the total current in a parallel resistive network.

A Resistor is particularly used as a load in the output of many circuits. If at all the resistive load is not used, a resistor is placed before a load. Resistor is usually a basic component in any circuit.