DC Circuits

Ohm’s Law 

Ohm's law state that the current through any two points of the conductor is directly proportional to the potential difference applied across the conductor, provided physical conditions i.e. temperature, etc. do not change. It is measured in (Ω) ohm. Mathematically it is expressed as 

This constant is also called the resistance (R) of the conductor (or circuit) R= 𝑉 𝐼 In a circuit, when current flows through a resistor, the potential difference across the resistor is known as voltage drops across it, i.e., V = IR.

Limitations of Ohm’s Law 

• Ohm's law is not applicable in unilateral networks. Unilateral networks allow the current to flow in one direction. Such types of network consist of elements like a diode, transistor, etc. 
• It is not applicable for the non-linear network (network containing non-linear elements such as electric arc etc). In the nonlinear network, the parameter of the network is varied with the voltage and current. Their parameters like resistance, inductance, capacitance and frequency, etc., won't remain constant with the times. So ohms law is not applicable to the nonlinear network. Ohm's law is used for finding the resistance of the circuit and also for knowing the voltage and current of the circuit. 

Resistance: The opposition offered to flow of current is called resistance. It is represented by R. The unit of resistance is ohms (Ω)

Law of Resistance: 

The resistance of a wire depends upon 

1. It is directly proportional to its length. 
R α L ………………. 1 

2. It is inversely proportional to its area of cross-section.                                                R α 1/𝐴 ………………..2 

3. It depends upon the nature of material of which the wire is made.

4. It also depends upon the temperature of the wire. 
Combining 1 and 2 

R α 𝐿/𝐴 

R = ρL /A

The proportionality constant  ρ is called the specific resistance or resistivity. Its value depends on the material of the conductor. The inverse of resistance is called conductance and inverse of resistivity is called conductivity

Electric Circuit: The close path for flow of electric current is called electric circuit. The electric circuit is an arrangement of electrical energy sources and various circuit elements such as R, L and C are connected in series, parallel or series parallel combinations. 

Voltage and Current equations of-

Resistance (R) = Voltage (V)/Current (I) 

Inductance (L) -

Capacitance (C) -

Circuit Elements: The circuit elements can be categorized as: 

1. Active and passive elements 

2. Unilateral and bilateral elements 

3. Linear and non-linear elements 

4. Lumped and distributed elements 

1. Active and passive elements: Active elements are those who supply energy or power in the form of a voltage or current to the circuit or network. Examples of the active components are batteries or generators etc. Passive elements are those who receive energy in the form of voltage or current. Examples of the passive components are resistor, capacitor and inductor. 

2. Unilateral and bilateral elements: 

Unilateral elements: The elements which conduct the current in one direction only are called unilateral elements such as diodes, transistors, vacuum tubes, rectifiers etc.

Bilateral elements: The elements which conduct the current in both the directions are called bilateral elements such as resistors. 

3. Linear and non-linear elements 

Linear Elements: The elements which follow the linear relation between current and voltage. e.g. resistors. 

Non Linear Elements: The elements which don't follow the linear relation between current and voltage. e.g. Diode and transistors 

4. Lumped and distributed elements 

Lumped elements: The elements in which action takes place simultaneously are lumped elements such as resistor, capacitor and inductor. These elements are smaller in size. 

Distributed elements: The elements in which for a given cause is not occurring simultaneously at the same instant but it is distributed are called distributed elements such as transmission lines.

Voltage and Current Source: 

To deliver electrical energy to the electrical circuits, a source is required and a load is connected to source as shown in fig. 

The source may be DC source or AC source. 

DC source: Any source that produces direct voltage continuously and has ability to deliver direct current is called DC source such as batteries and generators etc. 

AC source: Any source that produces alternating voltage continuously and has ability to deliver the alternating current is called AC source such as alternators, oscillators or signal generators. 

Independent and dependent sources: There are two types of sources- Voltage source and current source. Sources can be either independent or dependent upon some other quantities. 

Independent voltage/ current source: The voltage ( AC or DC) does not dependent on other voltages or current in the circuit. Symbol for independent voltage and current source respectively.

Examples of independent voltage source: batteries and generators. 

Examples of independent current source: semiconductor devices such as Diode and transistors.

Dependent voltage/ current source: 

The voltage does dependent on another voltage or current in the circuit. Symbol for dependent voltage and current source

 

Ideal and practical voltage sources: 

Ideal voltage sources: 

An imaginary voltage source, which can provide a constant voltage to load ranging from zero to infinity. Such voltage source is having zero internal resistance, Rs and is called Ideal Voltage Source. Practically it is not possible to build a voltage source with no internal resistance and constant voltage for that long range of the load.

Practical voltage sources:                                                            Practical voltage sources always have some resistance value in series with an ideal voltage source and because of that series resistance, voltage drops when current passes through it. So, Practical Voltage Source has internal resistance and slightly variable voltage.

Ideal Current Source:

An ideal current source is a two terminal device which supplies constant current irrespective of load resistance. The value of current will be constant with respect to time and load resistance. This means that the power delivering capability is infinite for this source. An ideal current source has infinite parallel resistance connected to it. Therefore, the output current is independent of voltage of the source terminals. No such current source exists in the world, this is just a concept. However, every current source is designed to approach closer to the ideal one.

Practical Current Source:

A practical current source is a two terminal device having some resistance connected across its terminals. Unlike ideal current source, the output current of practical source depends on the voltage of the source. The more this voltage, the lesser will be the current.

Source Transformation:

 

D. C. circuit: 

The closed path for flow of direct current is called D.C. circuit.

D.C Circuit is of two types: 

1. Series Circuit 

2. Parallel Circuit 

Current in Parallel Circuit:

                        

Kirchhoff’s Current Law or KCL

Kirchhoff's Current Law or KCL, states that the “total current or charge entering a junction or node is exactly equal to the charge leaving the node”. In other words the algebraic sum of all the currents entering and leaving a node must be equal to zero, I(exiting) + I(entering) = 0. 

Kirchhoff's Current Law or KCL  

Here, the three currents entering the node, I1, I2, I3 are all positive in value and the two currents leaving the node, I4 and I5 are negative in value. Then this means we can also rewrite the equation as; 

I1 + I2 + I3 – I4 – I5 = 0 

The term Node in an electrical circuit generally refers to a connection or junction of two or more current carrying paths or elements such as cables and components. Also for current to flow either in or out of a node a closed circuit path must exist. We can use Kirchhoff's current law when analysing parallel circuits.

Kirchhoff's Voltage Law or KVL

Kirchhoff’s Voltage Law, states that “in any closed loop network, the total voltage around the loop is equal to the sum of all the voltage drops within the same loop” which is also equal to zero. In other words the algebraic sum of all voltages within the loop must be equal to zero.

Kirchhoff's Voltage Law or KVL

 

Starting at any point in the loop continue in the same direction noting the direction of all the voltage drops, either positive or negative, and returning back to the same starting point. It is important to maintain the same direction either clockwise or anti-clockwise or the final voltage sum will not be equal to zero. We can use Kirchhoff's voltage law when analysing series circuits.

Analysis of simple circuits with Kirchhoff’s law 

Q: Calculate current in given circuit using Kirchhoff's law.

Superposition Theorem 

Superposition theorem states that in any linear, active, bilateral network having more than one source, the response across any element is the sum of the responses obtained from each source considered separately and all other sources are replaced by their internal resistance. The superposition theorem is used to solve the network where two or more sources are present and connected. Resulting current in any branch is the algebraic sum of all the currents that would be produced in it. 

Procedure for using superposition theorem 

Step-1: Retain one source at a time in the circuit and replace all other sources with their internal resistances. 

Step-2: Determine the output (current or voltage) due to the single source acting alone.

Step-3: Repeat steps 1 and 2 for each of the other independent sources. 

Step-4: Find the total contribution by adding algebraically all the contributions due to the independent sources.

Limitations of Superposition Theorem

1. It is not applicable when the circuit contains only dependent sources.
2.We cannot apply superposition theorem when a circuit contains nonlinear elements like diodes, transistors etc.
3. As we know that superposition theorem is applicable only for Linear networks, so it cannot be used for power calculations, since the power is proportional to the square (nonlinear) of current or voltage.
4. Superposition theorem is of no use if the circuit contains less than two independent sources.

Numerical 

Calculate the total voltage in given fig using Superposition theorem 

Norton’s Theorem Statement: 

Norton's Theorem states that – A linear active network consisting of the independent or dependent voltage source and current sources and the various circuit elements can be substituted by an equivalent circuit consisting of a current source in parallel with a resistance. The current source being the short-circuited current across the load terminal and the resistance being the internal resistance of the source network. The Norton's theorems reduce the networks equivalent to the circuit having one current source, parallel resistance and load. Norton's theorem is the converse of Thevenin's Theorem. It consists of the equivalent current source instead of an equivalent voltage source as in Thevenin's theorem. 

To understand Norton's Theorem in detail, let us consider a circuit diagram given below

 

 

Steps for Solving a Network Utilizing Norton’s Theorem 

Step 1 – Norton's equivalent circuit is drawn by keeping the equivalent resistance Req in parallel with the short circuit current ISC. 

Step 2 – Find the internal resistance Req of the source network by deactivating the constant sources. 

Step 3 – Now short the load terminals and find the short circuit current ISC flowing through the shorted load terminals using conventional network analysis methods. 

Step 4 – Reconnect the load resistance RL of the circuit across the load terminals and find the current through it known as load current IL. This is all about Norton's Theorem. 

Limitations of Norton's Theorem

• It's not for such modules which are not linear like diodes, the transistor. 
• It does not operate for such circuitries which has magnetic locking. 
• It also does not work for such circuitries which has loaded in parallel with dependent supplies.

Numerical 

Applying Norton's theorem, find the maximum power dissipated by the resistor 6.2Ω under that situation

  

 

Nodal Analysis in Electric Circuits 

Nodal analysis is a method that provides a general procedure for analysing circuits using node voltages as the circuit variables. Nodal Analysis is also called the Node-Voltage Method. In Node-Voltage Method, we can solve for unknown voltages in a circuit using KCL. 

Some Features of Nodal Analysis:

• Nodal Analysis is based on the application of the Kirchhoff's Current Law (KCL). 
• Having "n" nodes there will be "n-1" simultaneous equations to solve. 
• Solving "n-1" equations all the nodes voltages can be obtained. 
• The number of non reference nodes is equal to the number of Nodal equations that can be obtained. 

Types of Nodes in Nodal Analysis:

• Non Reference Node – It is a node which has a definite Node Voltage. e.g. Here Node 1 and Node 2 are the Non Reference nodes.
• Reference Node – It is a node which acts a reference point to all the other node. It is also called the Datum Node. 

Solving of Circuit Using Nodal Analysis

Basic Steps Used in Nodal Analysis: 

1. Select a node as the reference node. Assign voltages V1, V2… Vn-1 to the remaining nodes. The voltages are referenced with respect to the reference node. 

2. Apply KCL to each of the non reference nodes. 

3. Use Ohm's law to express the branch currents in terms of node voltages.

4. After the application of Ohm's Law get the "n-1" node equations in terms of node voltages and resistances. 

5. Solve "n-1" node equations for the values of node voltages and get the required node Voltages as result. 

Mesh analysis

Mesh Current Analysis is a technique used to find the currents circulating around a loop or mesh with in any closed path of a circuit. Therefore, a mesh analysis can also be known as loop analysis or mesh-current method.

Procedure of Mesh Analysis

The following steps are to be followed while solving the given electrical network using mesh analysis:

Step 1:

To identify the meshes and label these mesh currents in either clockwise or counter clockwise direction.

Step 2:

To observe the amount of current that flows through each element in terms of mesh current.

Step 3:

Writing the mesh equations to all meshes using Kirchhoff’s voltage law and then Ohm’s law.

Step 4:

The mesh currents are obtained by following Step 3 in which the mesh equations are solved.

Hence, for a given electrical circuit the current flowing through any element and the voltage across any element can be determined using the node voltages

Thevenin’s Theorem 

Thevenin's Theorem states that “Any linear circuit containing several voltages and resistances can be replaced by just one single voltage in series with a single resistance connected across the load“. In other words, it is possible to simplify any electrical circuit, no matter how complex, to an equivalent two-terminal circuit with just a single constant voltage source in series with a resistance (or impedance) connected to a load as shown below. 

Thevenin’s Theorem is especially useful in the circuit analysis of power or battery systems and other interconnected resistive circuits where it will have an effect on the adjoining part of the circuit.

As far as the load resistor RL is concerned, any complex “one-port” network consisting of multiple resistive circuit elements and energy sources can be replaced by one single equivalent resistance Rs and one single equivalent voltage Vs. Rs is the source resistance value looking back into the circuit and Vs is the open circuit voltage at the terminals. 

Limitations of Thevenin's Theorem 

• If the circuit consists of non linear elements, this theorem is not applicable. 
• Also to the unilateral networks it is not applicable. 
• There should not be magnetic coupling between the load and circuit to be replaced with the Thevenin's equivalent.

Q. Find the load current in 40 Ω in given fig using Thevenin’s theorem

Firstly, to analyse the circuit we have to remove the centre 40Ω load resistor connected across the terminals A-B, and remove any internal resistance associated with the voltage source(s). This is done by shorting out all the voltage sources connected to the circuit, that is v = 0, or open circuit any connected current sources making i = 0. The reason for this is that we want to have an ideal voltage source or an ideal current source for the circuit analysis.

The value of the equivalent resistance, Rs is found by calculating the total resistance looking back from the terminals A and B with all the voltage sources shorted. We then get the following circuit.

 

    To find the Equivalent Resistance (Rs)