Current Division Rule:

Current Divider Rule is a renowned method of solving parallel circuits. You can solve parallel circuits using Ohm's law. But Ohm's law has a limitation. You have to know the voltage across parallel elements. This method allows you to find out the current flowing through parallel elements without knowing the voltage across it.  This procedure is derived from two very popular circuit solving laws, is Ohm's law and KCL. In this article, I am going to derive the expression of the current divider rule as well. 

Explanation:

It is only acceptable to parallel circuits where the voltage remains the same throughout the circuit. Consider a parallel circuit given below.

Figure 1: A parallel circuit

Let,

i1 = Current across R1

i2 = Current across R2

i3 = Current across R3

IT = total current across the circuit

V1 = voltage across each element

Apply KCL,

IT = i1 + i2 + i3  ….. Equation 1

From Ohm's law, i = v/R, replace "i" in Equation 1.

IT = V1/ R1 + V1/R2 +V1/R3 

IT = V1 (1/R1 + 1/R2 + 1/R3)

As you know formula for parallel resistances is

1/RT = 1/R1+1/R2+1/R3

RT = R1 || R2 || R3

IT = V1 /RT 

IT = V1/RT..... Equation 2

Now, if you want to find the current across R2, then using Ohm's law:

i2 = V1/R2 …... Equation 3

Now, solve Equation 2 and Equation 3 to get the value of i2 which is independent of V1.

IT*RT = i2*R2

Or

i2 = IT*RT/R2

Similarly, 

i1 = IT*RT/R1

i2 = IT*RT/R2

i3 = IT*RT/R3

These are the equations of current in parallel circuits which are independent of voltage. It is only applicable to parallel circuits only. These types of circuits are also called Current Divider Circuits because of the current divide among all the resistances. As you know the current adopts the least resistive path. So, the lower the resistance the higher the current flows through it.

Solved Example:

In the previous article (parallel resistance formula), I analysed and solved a parallel circuit using Ohm's law. In that case I use the following circuit.

Figure 2: Parallel circuit with a voltage source

In this tutorial, I am going to solve the same examples with the help of the current divider rule. In this example I am going to use the circuit in figure 3. Both circuits (Circuits in figure 2 and figure 3) are the same irrespective of the current and voltage sources. In the later circuit a voltage source is replaced by the equivalent current source. 

Figure 3: Parallel circuit with an equivalent current source

Example 1:

Determine total or equivalent resistance and the current flows through each resistor with the help of the current divider rule.

I solved this problem in my previous article (parallel resistance formula).

This is another way to solve this problem.

Find total resistance RT

RT = 1/R1 + 1/R2 + 1/R3

RT = 545.5 Ω

Apply CDR on each resistor.

Current through R1 is i1,

i1 = IT*RT/R1

i1 = 18*545/1000

i1 = 9.8 mA

Current through R2 is i2,

i2 = IT*RT/R2

i2 = 18*545/2000

i2 = 4.9 mA

Current through R3 is i3,

i3 = IT*RT/R3

i3 = 18*545/3000

i3 = 3.2 mA

Conclusion:

Subsequently reviewing the current divider rule, its derivation and a solved example, it is deduced that:

• This technique is useful in finding the current flows through the resistors without knowing the voltage

• The lesser the resistance the larger the current 

• It is only acceptable to parallel circuits only

Resistors & Resistances - Definitions, Composition, Construction

Resistors - Introduction, Effect of Temperature on conductors and Semiconductors

In this article I am going to introduce the most important and basic passive element, that is a resistor. Resistor is a tiny two terminal element, that is used in electrical and electronic circuits for variety of purposes. Like divide voltages, limit currents, adjust signal levels, bias active elements etc. It is mainly used to limit the current through each component. The larger the resistance, the smaller the current.

Mostly students think this is a redundant element or the resistance is responsible for signal degradation. The article will help you to understand the importance and practicality of resistors in different types of applications. This is a detailed article, I tried to cover every aspect and every question on this topic.

Outline:

• What is resistor and resistance?
• Resistance of metals and semiconductors
• Temperature dependence of resistance
• 2 practical ways to measure resistance
• What is inside the resistor (construction, advantages, disadvantages, applications of different types of resistors)

• Types of resistors

• Linear resistors
• Variable resistors
• Semi Fixed resistors
• Fixed resistors
• Non linear resistors
• Thermistors
• LDR (Photoresistor)
• Varistors
• Special type of resistors
• Zero ohm
• Fusible resistor

• Series connected resistors
• Voltage division rule for series connected resistors
• Parallel connected resistors
• Current division rule for parallel connected resistors

Resistor, Resistance & Resistivity :

What do you know about resistors? It decreases the rate of flow of charges through the circuit.

Resistance is the measure of flow of charges through the circuit. While the opposite quantity is conductance, which is defined as the ease with which charges flow through the circuit. Every component in a circuit offers some internal resistance, it could be an ohmic resistance or non ohmic resistance. 

The electrical resistance of a component can also be defined as the ratio of voltage applied to the current flows through it.

R = V/I

The electrical resistance of a wire is dependent on size and geometry. Larger (L) wires offer more resistance. For larger cross sectional area (A) of wires offer less resistance.

R = ρ * L/A

Where, ρ is the coefficient called resistivity of the material.

Resistivity can be defined as resistance offered by per unit length and per unit cross sectional area. It is measured in (Ω⋅m). It is the ability of material to conduct or resist the flow of charges through the material. The higher the resistivity, the lower the flow of charges through the conductor.

Resistance and resistivity are temperature dependent. So it's time to understand temperature dependence of resistance.

Resistance and temperature dependence:

While, learning basic electronics we come across two types of conducting materials that are metals and semiconductors. Connecting wires, contacts are made from metals whereas electronic components like diodes, transistors are made from semiconductors. We have to visualize the behaviour of both materials at higher temperatures.

Effect of temperature on resistance of a conductor:

In pure metals, the resistance increases with increase in temperature. They have positive temperature coefficient (α)

                    Rt = Ro (1+ αot)

In metals or good conductors, there are large number of free electrons available. As the temperature increases, molecular vibrations increases violently. Due to these vibrations, the mean free path of molecules decreases. The molecular vibrations disturbs the motion of free electrons, and the electrons can't not move freely. They experience more frequent collisions from vibrating molecules. This is the basic cause of resistance or hindrance in the flow of free electrons at higher temperature. Or in other words increased temperature decreases conductivity of the metals.

Effect of temperature on resistance of a semiconductor:

In semiconductors, the resistance decreases with increase in temperature. They have negative temperature coefficient.

In semiconductors, there are a few free electrons at room temperature. An increase in temperature, the loosely bound valence band electrons get enough energy and escape from their parent atom. It produce greater number of free electrons. Resistance of semiconductors decreases with an increase in temperature. That's why it has negative temperature coefficient. You can further read about temperature dependence of semiconductors here.

What is inside the resistor?

Have you ever thought about what is inside the resistor? A 10 ohm resistor and a 10k resistor both have same size, two terminals but different values.

Actually resistors are made up of different types of materials having different resistivity values. R = p*L/A where p is the resistivity. Resistance can vary by varying these three quantities.

• Resistivity (p) varies different for each material
• Length (L)
• Cross sectional area (A)

Frequently Asked Questions:

Difference between ohmic and non ohmic resistance:

Ohmic resistance, which obeys Ohm's law or voltage and current have linear relation. Or the IV characteristics graph is a straight line. Current and resistance are temperature dependent. It is static resistance. Example, a metal.

Ohmic and non-ohmic resistances (linear and non-linear resistance)

Non ohmic resistance, which doesn't obey Ohm's law or voltage and current doesn't have linear relation. Or the IV characteristics graph is not a straight line.Current and resistance are not temperature dependent. It is dynamic resistance.  Example, semiconductors.

Difference between resistance and resistivity:

Resistivity: Each material has different ability to resist the flow of current. Some materials allow more charges to flow, while other offers more resistance. It measures how strongly a material can oppose charges to flow.  This is known as resistivity or specific electrical resistance. It is the physical property of the material of the conductor, it doesn't depends on shape or geometry of the conductor. Resistivity depends on nature of material and temperature. A good conductor offers low resistance at lower temperatures.

Resistance: It is the measure of flow of charges. It is the property of the conductor. The resistance of a conductor depends upon its geometry, nature of material and temperature as well. For example, thicker wires have less resistance.

How does resistivity change with temperature? Temperature coefficient of resistance | resistivity

As you have seen, resistance and resistivity both are temperature dependent. Increase in temperature will increase in resistivity.

Temperature coefficient of resistivity can be defined as rate of change of resistivity per degree change in temperature. It is denoted by“alpha” (α).

The resistivity of metallic conductors linearly changes with temperature. As temperature increases resistivity increases. So they have positive temperature coefficient.

Similarly for semiconductors, resistivity decreases with an increase in temperature. So they have negative temperature coefficient.

Figure: Resistivity depends on temperature

Circuit Analysis - Methods, Laws & Theorems

This tutorial is intended to provide detailed information about the circuit or network analysis, methods of analysis and network theorems. It is just an introductory article, in which I gather all the laws, methods and theorems. This topic is usually a part of the ECE basic course on circuits. The term solving a circuit means determining all the voltage across each element and currents through each element present in a circuit.

Key Concepts:

• What are fundamental laws for solving circuits?
• Methods used for network/circuit analysis
• Network theorems
• What is the purpose of network theorems or why do we study network theorems

Fundamental Circuit Laws:

To determine current and voltage in an electric circuit, we need to gain knowledge of fundamental laws that govern electronic circuits.

1. Ohm's law
2. Kirchhoff's Voltage Law (KVL)
3. Kirchhoff's Current Law (KCL)

Ohm's Law defines a linear relationship between voltage and current in an ideal conductor. This is one of the most important and fundamental laws of electric circuits.

Kirchhoff’s Current Law is used when analysing a parallel circuit. It is based on the idea of conservation of charge. According to KCL 'current entering the node is equal to current leaving the node’.

Kirchhoff's Voltage Law is used when analysing a series circuit. It is based on the idea of conservation of potential. According to KVL 'the sum of voltage drop or potential difference across a closed loop is zero’.

Methods of Analysis:

Analysis of a circuit is the determination of output response. These methods of analysis are based on basic circuit laws. These methods are restricted to linear circuits only. The two most common methods are

1. Mesh analysis
2. Nodal analysis

Nodal analysis is nothing but an application of KCL. In the nodal analysis, we apply KCL to each node. If we have 'n’ number of nodes, then there are (n-1) linearly independent node equations. These equations are in terms of node potential.

Mesh | loop analysis is nothing but an application of KVL. In mesh analysis, we apply KVL around each loop in a circuit. If we have 'b’ branches and 'n’ nodes in a circuit then we have b-(n-1) linearly independent equations. These equations are in terms of mesh currents.

Why do we study network theorems?

I suppose you are familiar with basic network analysis methods. Like node analysis and mesh analysis. When dealing with KCL and KVL we will have a fairly large number of independent equations. As I discussed in methods of analysis, we get (n-1) node equations and b-(n-1) mesh current equations, so total ‘b’ number of equations. Did you find them time-consuming, you have to deal with so many equations (the number of node and mesh equations are equal to the number of elements/branches present in a network). For example, you have a network/circuit having 4 passive elements, then there are 4 current or voltage equations. And if there are 5 passive elements you have to solve 5 equations and so on. Analysing more complex circuits by using these methods seem to be ridiculous. What do you think?? So instead of applying tedious mesh and nodal analysis to complex networks, we develop network theorems.

Network theorems

1. Superposition theorem
2. Reciprocating theorem
3. Thevenin’s theorem
4. Norton's theorem
5. Compensation theorem
6. Millman's theorem
7. Substitution theorem
8. Maximum power transfer theorem
9. Tellegen's theorem

Applications

• Some of these apply to linear as well as non-linear circuits
• With the help of these theorems, we can solve DC as well as AC Circuits
• Easy comprehension of complex circuits

Duality In Electric Circuits

It is interesting to know how systems relate to one another. How a mechanical system can be modelled as an electrical system and observed. The concept of duality in electrical circuits is of great importance. Two phenomena are said to be dual if they can be expressed by same form of mathematical equations. This topic is usually covered under the network topology or graph theory.

Key Questions:

• What is principle of duality in electric circuits?
• List of dual pairs and their explanation
• Formation of dual networks

Principle of Duality:

Principle of duality in context of electrical networks states that

• A dual of a relationship is one in which current and voltage are interchangeable
• Two networks are dual to each other if one has mesh equation numerically identical to others node equation

List of Dual Pairs:

For evaluating a dual network, you should follow these points

1. The number of meshes in a network is equal to number of nodes in its dual network
2. The impedance of a branch common to two meshes must be equal to admittance between two nodes in the dual network
3. Voltage source common to both loops must be replaced by a current source between two nodes
4. Open switch in a network is replaced by a closed switch in its dual network or vice versa

	Elements	Dual Elements
1	Voltage (v) v = iR	Current (i) i = vG
2	Short Circuit	Open Circuit
3	Series	Parallel
4	Norton	Thevenin
5	Resistance (R)	Conductance (G)
6	Impedance	Admittance
7	KVL	KCL
8	Capacitance	Inductance

Formation of Dual Networks:

The principle of duality is applicable to planar circuits only. Carefully read the points stated below, follow each step and draw the dual circuit

1. Place a dot within each loop, these dots will become nodes of the dual network
2. Place a dot outside of the network, this dot will be the ground/datum node of the dual network
3. Carefully draw lines between nodes such that each line cuts only one element
4. If an element exclusively present in a loop, then connect the dual element in between node and ground/datum node
5. If an element is common in between two loops, then dual element is placed in between two nodes
6. Branch containing active source, consider as a separate branch
7. Now to determine polarity of voltage source and direction of current sources, consider voltage source producing clockwise current in a loop. Its dual current source will have a current direction from ground to non-reference node

Example#01: Draw dual of the given circuit.

Graphical method of drawing dual network

Example#02: Draw dual of the given circuit 

Reference:

1. Fundamentals of Electric Circuits by Alexander
2. Circuit Theory by A.V. Bakshi and U.A. Bakshi