# Resistance In Series and Parallel Formula

Table of Contents

## Series Circuit or Resistance in Series

(1). A circuit wherein two or more than two resistances are connected parallel to a voltage source in such a way that there is just one path available for the flow of current through these resistances, then such circuits are called a series circuit.

(2). When some resistances (R_{1}, R_{2}, R_{3}, etc.) are connected to each other in such a fashion that the 2nd end of the 1st resistance and 2nd end of the 2nd resistance is linked with 3rd resistance, then such a circuit is called a series circuit. If such a circuit is connected to a supply, only one passage for the flow of current remains available in this circuit. This type of circuit has been illustrated in the below figure.

(3). If the negative end of one resistor is connected to the positive end of the other resistor, and the negative end of the 2nd resistor is associated with the positive end of the 3rd resistance, then such resistance is called to be in a series. This type of circuit is known as a series circuit because only one path is available for the flow of current in such circuit after touching with supply.

(4). If different ends of various resistances are connected together in such a way that only one path or way is available for the flow of current through them, then the joining of resistances in such a manner is called series connection.

(5). Joining the different resistances end to end is called resistance in series. If these resistances are connected across a source in such a way that there is only one way for the current to flow is called a series circuit or circuit elements are said to be connected in series when they are joined together end to end.

## Characteristics of Series Circuit

A series circuit consists of the following characteristics;

(i). The quantity of current flowing through resistances in a series tends to be the same. In other words, the same current flows through different parts of a series circuit owing to having just one way available for the current, i.e., I = I_{1} = I_{2} = I_{3}

(ii). Total resistance of a circuit equals to the sum of all resistances installed on the circuit, i.e., R_{T} = R_{1} + R_{2 }+ R_{3} …

(iii). If the resistances of a circuit are increased continuously, then total resistance will keep on increasing.

(iv). An isolated voltage drop occurs parallel to every resistance according to its own value

(v). Sum of isolated voltage drops taking place parallel to every resistance equals to the applied voltage drop provided on the circuit, i.e.,

V = V_{1} + V_{2} + V_{3 }= IR_{1 }+ IR_{2 }+ IR_{3 }

(vi). If resistance or equipment becomes damaged or fused, then all circuit renders OFF.

(vii). Voltage drops occur after every resistor (i.e., from point A towards point D). See figure 1.29 (b). If the value of resistance is high, a voltage drop will also be high, i.e.,

V = IR … (Ohm’s Law)

(viii). The individual dissipated power (P = I_{2} R) of every element (resistor) on a series circuit equals to the sum of total power provided on the circuit, i.e.,

Power are additives i.e., P_{T} = P_{1 }+ P_{2} + P_{3} + … + Pn

(ix). The value of converted energy (W = I_{2} R_{t}) of every element of a circuit up to a specified time equals to the total energy of the circuit, i.e.,

Energies are additive i.e., W_{T} = W_{1} + W_{2} + W_{3} + … + W_{n}

(x). As only one way is available for the flow of current, that’s why the application of this type of circuit is rare.

## Parallel Circuit or Resistances in Parallel

(1). A circuit, in which two or more than two resistances are connected parallel to a voltage source in such a way that several substitute paths are available (according to the number of connected resistances) for the flow of current through these resistances, then such a circuit is called parallel circuit or resistances in parallel.

(2). If connections of some of the resistances (e.g., R_{1}, R_{2}. R_{3}, etc.) are done in such a manner that one end of each of all resistances installed on the circuit are interconnected, and other ends of resistances are also joined on one point, then joint having first ends are connected to the positive wire of the supply, whereas joint with 2nd ends connected to negative ends of the supply, such connections are called parallel connection or parallel circuit.

(3). A circuit in which two or more than two resistances are connected parallel to the source in such a way that voltage across every resistance is equal to the applied voltage, and more than one way is available for the flow of current from this circuit, then such a circuit is called a parallel circuit.

Figure 1.30 – Resistances in parallel

Or when all the elements (or resistors) of a circuit are connected between two common points, they are said to be connected in parallel.

When resistors are arranged so that each forms a separate path for a part of that total current, they are said to be connected in parallel.

## Characteristics of Parallel Circuit

A parallel circuit consists of the following characteristics;

(i). There are equal voltages at every point of the circuit, i.e., V_{1} = V_{2} = V_{3}

(ii). The current flowing through each resistance (or element) depends on its own value. That’s high current flows through a low resistance and low current passes through a high resistance.

(iii). The sum of current flowing through every branch of the circuit, equals to the total circuit current, i.e.,

I = I_{1 }+ I_{2} + I_{3} = V/ R_{1 }+ V/ R_{2} + V/ R_{3} = V / R_{t}

(v). The total resistance of a circuit equals to the inverse of total conductance (i.e., G = 1 / R) installed on the circuit, i.e.,

1 / R_{t} = 1 / R_{1} + 1/ R_{2} + 1/ R_{3} or G = G_{1} + G_{2} + G_{3}

(v). If further resistance is added to the circuit, then total resistance will reduce.

(vi). As more than one path are available for the flow of current, therefore, if any equipment or fuse gets damaged, rest of the circuit continues to operate normally.

(vii). The quantity of total resistance tends to be even less than the minimum resistance existing on the circuit.

(viii). The number of current flowing paths tends to be equal to the number of parallel resistances existing on the circuit.

(ix). Powers and energies are also additive similar to a current.

(x). This circuit is commonly used.

## Series and Parallel Circuit

(1). If on any one circuit, some of the resistances are mounted in a series, and some resistances fitted in parallel, then such a circuit is called series parallel or a compound circuit.

(2). Joining various resistances in such a way that circuit becomes a sum total of series as well as parallel resistances, is known as a series parallel circuit.

## Division of Voltage in Series Circuits

As same current flows through every resistor installed on a series circuit, therefore, voltage drop changes directly with each and every resistance of this circuit. In figure 1.31, a 24-volt battery installed in a series parallel to three resistors, has been shown.

Figure 1.31 – Division of voltage in series circuit

The total resistance of this circuit will be as follows;

Total Resistance, R_{T }= R_{1 }+ R_{2 }+ R_{3}

= 2 + 4 + 6 = 12 ohms

According to voltage divider rule, different voltage drops occurring within the circuit, are as follows;

V_{1} = V x R_{1}/R = 24 x 2/ 12 = 4 V

V_{2} = V x R_{2} / R = 24 x 4 /12 = 8 V

V_{3} = V x R_{3} /R = 24 x 6 /12 = 12 V

We know that total voltages found parallel to resistors fitted on a series circuit, tend to be equal to the total of separate voltage drops occurring parallel to every resistor, i.e.,

V = V_{1} + V_{2} + V_{3 }= 4 + 8 + 12 = 24 V

## Division of Current in Parallel Circuits

We know that voltages found parallel to each branch of any parallel circuit always tend to be equal. Whereas, current flowing through each branch, tends to be different. (This current depends on the value of resistance). Moreover, the total of individual currents passing through every branch, tends to be equal to the overall circuit current.

Figure 1.32 – Division of current in parallel circuit

In figure 1.32, two resistances joined in parallel have been shown connected to a same voltage source (V). The quantity of current passing through each branch can be determined according to ohm’s law, as follows;

I_{1} = V/ R and I_{2} = V / R_{2}

So, I_{1} / I_{2} = R_{2 }/ R_{1}

As 1 / R_{1 }=G_{1}, and 1 / R_{2} = G_{2}

So, I_{1} / I_{2 }= G_{1} / G_{2}

The division of current in various branches of any parallel circuit, tends to be directly proportional to the branch’s conductance, or inversely proportional to its resistances.

G_{1} = 1 / R_{1}, and G_{2} = 1/ R_{2}

Thus, G_{1 }/ G_{2} = I_{1} / I_{2}

Therefore, the division of current in any parallel circuit can be determined as follows;

I_{1} + I_{2 }= I

I_{2} = I – I_{1}

Put value of I_{2} in equation I_{1} / I_{2 }= R_{2 }/ R_{1}

So, I_{1} / I -I_{1} = R_{2}/ R_{1} or I_{1} R_{1 }= R_{2 }(I – I_{1})

Thus, I_{1} = I [R_{2}/ R_{1} + R_{2}]

I_{2} = I [R_{1 }/ R_{1 }+ R_{2}]

Similarly, if three resistances are set on a circuit in parallel, then the quantity of current flowing through all the three branches, can be determined with the help of the following formulae;

I_{1} = I [R_{2} R_{3}/ R_{1} R_{2} + R_{2} R_{3 }+ R_{3} R_{1}

I_{2 }= I [R_{1} R_{3} / R_{1} R_{2} + R_{2} R_{3} + R_{3} R_{1}

I_{3} = I [R_{1} R_{2} / R_{1} R_{2} + R_{2 }R_{3 }+ R_{3} R_{1}]

Remember that in AC circuits, impedance (Z) is used instead of resistance. Therefore, different quantities of current can be determined in any AC circuit by placing “Z”instead of “R”.

Series Circuit | Parallel Circuit |

1. I_{1} = I_{2 }+ I_{3} = I_{3} = …
2. V 3. R Or 1/ G 4. I = V 5. Voltage Divider Rule: V1 = V |
V I = I 1/ R Or G V= I Current Divider Rule I |

## Complex Networks

A circuit consisting of such active and passive components e.g., resistors, capacitors, inductors, and voltage sources, etc., which instead of being solved through general and simple methods (e.g., series, parallel, ohm law, and Kirchhoff law, because if such a circuit is solved through ohm’ law or Kirchhoff’ s law, too ⁰many equations build up, and resultantly, the solution of the circuit becomes complicated instead of becoming easy), is easily solved through a specific theorem, is known as complex network.

In other words, a circuit, which comprises multiple resistors, capacitors, and voltage source, is called a complex network and numerous resistors and voltage source etc., are connected in the circuit in such an arrangement that circuit assumes the form of a complex network. The voltage source provided to the circuit can either be AC or DC. Knowledge about the following points is essential for understanding and easily solving such circuits;

**(1). Circuit**

The circuit is a closed path wherein current flows to complete its circle. In other words, a closed conducting path, through which electric current can transmit, is called circuit.

**(2). Parameters or Constants**

Different elements used in any electric circuit, are known as parameters of this circuit, e.g., resistance, inductance, and capacitance, etc. These parameters can either be lumped or distributed.

**(3). Linear Circuit**

A linear circuit is one, having constant parameters. That’s a circuit, whose parameters do not change according to current or voltage, is called a linear circuit.

**(4). Non–linear Circuit**

A circuit whose parameters change according to voltage and current, that’s changing the values of voltage and current, the values of these parameters also change.

**(5). Bilateral Circuit**

A circuit, having same characteristics both ways, is called bilateral circuit, e.g., transmission line, is a bilateral circuit, because its characteristics do not change supplying from any direction.

**(6). Unilateral Circuit**

A circuit, characteristics or properties of which changes as a result of a change in the direction of supply. Diode rectifier is a unilateral rectifier, because it cannot perform the function of rectification in both directions.

**(7). Electric Network**

A combination of different elements, in which elements are connected in any form, is called electric network.

**(8). Active Network**

A circuit consisting of one ore more than one EMF sources, is known as an active network circuit.

**(9). Passive Network**

A circuit, having no EMF source, is called a passive network circuit.

**(10). Node**

A junction (or any point) existing in any circuit where two or more than two elements (e.g., resistors, capacitors, and inductors, etc.) combine together, is called node.

**(11). Branch**

That part of any network, which is located between two junctions, is called branch. In branch, one or more than one element may be connected in a series and it has two terminals.

**(12). Loop**

A closed path found in a circuit, having more than two mesh, is known as a loop. That’s a loop may consist of multiple mesh, however a mesh does not consist of a loop.

**(13). Mesh**

It is that type of a loop, wherein no other loop exists, or any path which does not comprise any other paths, is known as a mesh.

For example, the circuit shown in figure 1.33 (a), consists of 7 branches, 6 nodes, 3 loops, and 2 meshes. Whereas the circuit shown in figure 1.33 (b) consists of 4 branches, 2 nodes, 6 loops, and 3 meshes.

## Method for Solution of Complex Networks

Different theorems are used in order to solve a complex network. These will be discussed in the coming chapters. A complex network can generally be solved with the help of the following two methods;

**(i). Direct Method**

According to this method, a network is allowed to be retained in its genuine form, and its values are determined directly according to Ohm’s law and Kirchhoff’s’ law (that’s its different voltages and its different currents are determined). These types of methods are generally used for the solution of simple circuits and Kirchhoff’s laws, loop analysis, nodal analysis, super position theorem, and reciprocity etc., questions.

**(ii). Equivalent Direct Method**

This method is also known as the network reduction method. In this method, the original network is changed into a very simple equivalent circuit, so that different quantities can be calculated easily. A simple circuit can be solved out through this method, and a complex network can also be solved. Star/delta and delta/star conversion, theonym theorem, and Norton Theorem etc., are examples of this method.

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