105:- BASIC ELECTRICAL & ELECTRONICS ENGINEERING NOTES FOR UNIT-I

Unit 1

Syllabus : Basic Concepts of Electrical Engineering: Electric Current, Electromotive force, Electric Power, Ohm’s Law, Basic Circuit Components, Faraday’s Law of Electromagnetic Induction, Lenz’ Law, Kirchhoff’s laws, Network Sources, Resistive Networks, Series-Parallel Circuits, Node Voltage Method, Mesh Current Method, Superposition, Thevenin’s, Norton’s and Maximum Power Transfer Theorems.

Electric Current:-An electric current is a flow of electric charge. In electric circuits this charge is often carried by moving electrons in a wire. It can also be carried by ions in an electrolyte. The SI unit for measuring an electric current is the ampere, which is the flow of electric charge across a surface at the rate of one coulomb per second. Electric current is measured using a device called an ammeter.

The conventional symbol for current is I, which originates from the French phrase intensité de courant, or in English current intensity.

Electromotive force: - Electromotive force, also called emf (denoted e and measured in volt), is the voltage developed by any source of electrical energy such as a battery or dynamo. It is generally defined as the electrical potential for a source in a circuit. A device that supplies electrical energy is called a seat of electromotive force or emf. Emfs convert chemical, mechanical, and other forms of energy into electrical energy. The product of such a device is also known as emf.

Electric Power: - Electric power is the rate at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt, one joule per second.

Ohm's law:- states that the current through a conductor between two points is directly proportional to the potential difference across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship.

Where ‘I’ is the current through the conductor in units of amperes, V is the potential difference measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms. More specifically, Ohm's law states that the R in this relation is constant, independent of the current.

Basic Circuit Components:-

Resistance: - The electrical resistance of an electrical conductor is the opposition to the passage of an electric current through that conductor. The inverse quantity is electrical conductance, the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical conductance is measured in siemens (S).

An object of uniform cross section has a resistance proportional to its resistivity and length and inversely proportional to its cross-sectional area. All materials show some resistance, except for superconductors, which have a resistance of zero. The resistance (R) of an object is defined as the ratio of voltage across it (V) to current through it (I), while the conductance (G) is the inverse:

   

where l is the length of the conductor, measured in metres [m], A is the cross-sectional area of the conductor measured in square metres[m²], σ (sigma) is the electrical conductivity measured in siemens per meter (S·m⁻¹), and ρ (rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm-metres (Ω·m). The resistivity and conductivity are proportionality constants, and therefore depend only on the material the wire is made of, not the geometry of the wire. Resistivity and conductivity are reciprocals:

Resistivity is a measure of the material's ability to oppose electric current.

Inductance:- In electromagnetism and electronics, inductance is the property of an electrical conductor by which a change in current flowing through it induces an electromotive force in both the conductor itself and in any nearby conductors by mutual inductance. The relationship between the self-inductance L of an electrical circuit, the voltage v(t), and the current i(t) through the circuit is

Capacitance: - Capacitance is the ability of a body to store an electrical charge. Any object that can be electrically charged exhibits capacitance. A common form of energy storage device is a parallel-plate capacitor. In a parallel plate capacitor, capacitance is directly proportional to the surface area of the conductor plates and inversely proportional to the separation distance between the plates. If the charges on the plates are +q and −q respectively, and V gives the voltage between the plates, then the capacitance C is given by

Which gives the voltage/current relationship

The energy stored in a capacitor is found by integrating this equation. Starting with an uncharged capacitance (q = 0) and moving charge from one plate to the other until the plates have charge +Q and −Q requires the work W:

Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon called electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.

Faraday's law states that the EMF is also given by the rate of change of the magnetic flux:

The negative sign due to Lenz’ Law.

Where e is the electromotive force (EMF) and ΦB is the magnetic flux. The direction of the electromotive force is given by Lenz's law.

For a tightly wound coil of wire, composed of N identical turns, each with the same ΦB, Faraday's law of induction states that

Kirchhoff's circuit laws:- are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhoff. This generalized the work of Georg Ohm and preceded the work of Maxwell. Widely used in electrical engineering, they are also called Kirchhoff's rules or simply Kirchhoff's laws.

  

Kirchhoff's current law (KCL):-

The current entering any junction is equal to the current leaving that junction. i₂ + i₃ = i₁ + i₄

This law is also called Kirchhoff's first law, Kirchhoff's point rule, or Kirchhoff's junction rule (or nodal rule).

The principle of conservation of electric charge implies that:

At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node or equivalently

The algebraic sum of currents in a network of conductors meeting at a point is zero.

Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node; this principle can be stated as:

‘n’ is the total number of branches with currents flowing towards or away from the node.

  

Kirchhoff's voltage law (KVL):-

The sum of all the voltages around a loop is equal to zero.
v₁ + v₂ + v₃ - v₄ = 0

This law is also called Kirchhoff's second law, Kirchhoff's loop (or mesh) rule, and Kirchhoff's second rule.

The principle of conservation of energy implies that

The directed sum of the electrical potential differences (voltage) around any closed network is zero, or:

More simply, the sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop, or:

The algebraic sum of the products of the resistances of the conductors and the currents in them in a closed loop is equal to the total emf available in that loop.

Similarly to KCL, it can be stated as:

Network Sources:-

Voltage and Current sources

(a) An ideal voltage source is a circuit element that maintains a prescribed voltage across its terminals regardless of the current flowing in those terminals.

(b) An ideal current source is a circuit element that maintains a prescribed current through its terminals regardless of the voltage across those terminals.

Independent Voltage Source

Output of an independent source does not depend upon the voltage or current  of any other part of the network. When terminal voltage of a voltage source is not affected by the current or voltage of any other part of the network, then the source is said to be an independent voltage source. This type of sources may be referred as constant source or time variant source. When terminal voltage of an independent source remains constant throughout its operation, it is referred as time–invariant or constant independent voltage source.

Independent Current Source

Similarly, output current of independent current source does not depend upon the voltage or current of any other part of the network. It is also categorized as independent time-invariant and time-variant current source.

Symbolic representations of independent time-invariant and time-variant voltage and current source s are shown below.

Dependent Voltage Source & Dependent Current Source

There are four possible dependent sources as are represented below,

1.     Voltage dependent voltage source.

2.     Current dependent voltage source.

3.     Voltage dependent current source.

4.     Current dependent current source.

Dependent voltage source s and dependent current sources can also be time variant or time invariant. That means, when the output voltage or current of a dependent source is varied with time, referred as time invariant dependent current or voltage source and if not varied with time, it is referred as time variant.

Current Source to Voltage Source Conversion

All sources of electrical energy give both current as well as voltage. This is not practically possible to distinguish between voltage source and current source.  Any electrical source can be represented as voltage source as well as current source. It merely depends upon the operating condition. If the load impedance is much higher than internal impedance of the source, then it is preferable to consider the source as a voltage source on the other hand if the load impedance is much lower than internal impedance of the source; it is preferable to consider the source as a current source. Current source to voltage source conversion or voltage source to current source conversion is always possible.

Now we will discuss how to convert a current source into voltage source and vice-versa.

Let us consider a voltage source which has no load terminal voltage or source voltage V and internal resistance r. Now we have to convert this to an equivalent current source. For that, first we have to calculate the current which might be flowing through the source if the terminal A and B of the voltage source were short circuited. That would be nothing but I = V / r. This current will be supplied by the equivalent current source and that source will have the same resistance connected across it.

 Resistive Networks:-

Series and parallel circuits

Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur frequently. Components connected in series are connected along a single path, so the same current flows through all of the components. Components connected in parallel are connected so the same voltage is applied to each component.

A circuit composed solely of components connected in series is known as a series circuit; likewise, one connected completely in parallel is known as a parallel circuit

Series circuits

Series circuits are sometimes called current-coupled or daisy chain-coupled. The current in a series circuit goes through every component in the circuit. Therefore, all of the components in a series connection carry the same current. There is only one path in a series circuit in which the current can flow. The formula to search resistance in series is R_s = R₁ + R₂ + R₃ .... R_n

A series circuit's main disadvantage or advantage, depending on its intended role in a product's overall design, is that because there is only one path in which its current can flow, opening or breaking a series circuit at any point causes the entire circuit to "open" or stop operating. For example, if even one of the light bulbs in an older-style string of Christmas burns out or is removed, the entire string becomes inoperable until the bulb is replaced.

Current

In a series circuit the current is the same for all of elements.

Resistors

The total resistance of resistors in series is equal to the sum of their individual resistances:

Electrical conductance presents a reciprocal quantity to resistance. Total conductance of series circuits of pure resistors, therefore, can be calculated from the following expression:

.

For a special case of two resistors in series, the total conductance is equal to:

Inductors

Inductors follow the same law, in that the total inductance of non-coupled inductors in series is equal to the sum of their individual inductances:

Capacitors

Capacitors follow the same law using the reciprocals. The total capacitance of capacitors in series is equal to the reciprocal of the sum of the reciprocals of their individual capacitances:

Parallel circuits

If two or more components are connected in parallel they have the same potential difference (voltage) across their ends. The potential differences across the components are the same in magnitude, and they also have identical polarities. The same voltage is applicable to all circuit components connected in parallel. The total current is the sum of the currents through the individual components, in accordance with Kirchhoff’s current law.

Voltage

In a parallel circuit the voltage is the same for all elements.

Resistors

The current in each individual resistor is found by Ohm's law. Factoring out the voltage gives

To find the total resistance of all components, add the reciprocals of the resistances  of  R_i  each component and take the reciprocal of the sum. Total resistance will always be less than the value of the smallest resistance:

.

For only two resistors, the unreciprocated expression is reasonably simple:

This sometimes goes by the mnemonic "product over sum".

For N equal resistors in parallel, the reciprocal sum expression simplifies to:

and therefore to:

.

To find the current in a component with resistance, use Ohm's law again:

The components divide the current according to their reciprocal resistances, so, in the case of two resistors

An old term for devices connected in parallel is multiple, such as a multiple connection for arc lamps.

Since electrical conductance  G is reciprocal to resistance, the expression for total conductance of a parallel circuit of resistors reads:

The relations for total conductance and resistance stand in a complementary relationship: the expression for a series connection of resistances is the same as for parallel connection of conductance, and vice versa.

Inductors

Inductors follow the same law, in that the total inductance of non-coupled inductors in parallel is equal to the reciprocal of the sum of the reciprocals of their individual inductances.

 Capacitors

The total capacitance of capacitors in parallel is equal to the sum of their individual capacitances:

The working voltage of a parallel combination of capacitors is always limited by the smallest working voltage of an individual capacitor.

Nodal Voltage Analysis

As its name implies, Nodal Voltage Analysis uses the “Nodal” equations of Kirchoff’s first law to find the voltage potentials around the circuit.

So by adding together all these Nodal Voltages the net result will be equal to zero. Then, if there are “n” nodes in the circuit there will be “n-1” independent nodal equations and these alone are sufficient to describe and hence solve the circuit.

At each node point write down Kirchoff’s first law equation, that is: “the currents entering a node are exactly equal in value to the currents leaving the node” then express each current in terms of the voltage across the branch. For “n” nodes, one node will be used as the reference node and all the other voltages will be referenced or measured with respect to this common node.

For Example:-

In the above circuit, node D is chosen as the reference node and the other three nodes are assumed to have voltages, Va, Vb and  Vc with respect to node D. For example;

As Va = 10v and Vc = 20v , Vb can be easily found by:

Mesh current analysis:-

Figure 1: Essential meshes of the planar circuit labelled 1, 2, and 3. R₁, R₂, R₃, 1/sc, and Ls represent the impedance of the resistors, capacitor, and inductor values in the s-domain. Vs and is the values of the voltage and current source, respectively.

Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. A more general technique, called loop analysis (with the corresponding network variables called loop currents) can be applied to any circuit, planar or not. Mesh analysis and loop analysis both make use of Kirchhoff’s voltage law to arrive at a set of equations guaranteed to be solvable if the circuit has a solution. Mesh analysis is usually easier to use when the circuit is planar, compared to loop analysis.

Figure 2: Circuit with mesh currents labelled as i₁, i₂, and i₃. The arrows show the direction of the mesh current.

Mesh analysis works by arbitrarily assigning mesh currents in the essential meshes (also referred to as independent meshes). An essential mesh is a loop in the circuit that does not contain any other loop. Figure 1 labels the essential meshes with one, two, and three.

A mesh current is a current that loops around the essential mesh and the equations are set solved in terms of them. A mesh current may not correspond to any physically flowing current, but the physical currents are easily found from them. It is usual practice to have all the mesh currents loop in the same direction. This helps prevent errors when writing out the equations. The convention is to have all the mesh currents looping in a clockwise direction. Figure 2 shows the same circuit from Figure 1 with the mesh currents labelled.

Solving for mesh currents instead of directly applying Kirchhoff's current law and Kirchhoff's voltage law can greatly reduce the amount of calculation required. This is because there are fewer mesh currents than there are physical branch currents. In figure 2 for example, there are six branch currents but only three mesh currents.

Setting up the equations:-

Each mesh produces one equation. These equations are the sum of the voltage drops in a complete loop of the mesh current. For problems more general than those including current and voltage sources, the voltage drops will be the impedance of the electronic component multiplied by the mesh current in that loop.

If a voltage source is present within the mesh loop, the voltage at the source is either added or subtracted depending on if it is a voltage drop or a voltage rise in the direction of the mesh current. For a current source that is not contained between two meshes, the mesh current will take the positive or negative value of the current source depending on if the mesh current is in the same or opposite direction of the current source. The following is the same circuit from above with the equations needed to solve for all the currents in the circuit.

Once the equations are found, the system of linear equations can be solved by using any technique to solve linear equations.

Superposition theorem:-

The superposition theorem for electrical circuits states that for a linear system the response (voltage or current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by their internal impedances.

To ascertain the contribution of each individual source, all of the other sources first must be "turned off" (set to zero) by:

1.     Replacing all other independent voltage sources with a short circuit (thereby eliminating difference of potential i.e. V=0; internal impedance of ideal voltage source is zero (short circuit)).

2.     Replacing all other independent current sources with an open circuit (thereby eliminating current i.e. I=0; internal impedance of ideal current source is infinite (open circuit)).

This procedure is followed for each source in turn, and then the resultant responses are added to determine the true operation of the circuit. The resultant circuit operation is the superposition of the various voltage and current sources.

Thévenin's theorem:-

Any black box containing resistances only and voltage and current sources can be replaced to a Thévenin equivalent circuit consisting of an equivalent voltage source in series connection with an equivalent resistance.

1.     Any linear electrical network with voltage and current sources and only resistances can be replaced at terminals A-B by an equivalent voltage source V_th in series connection with an equivalent resistance R_th.

2.     This equivalent voltage V_th is the voltage obtained at terminals A-B of the network with terminals A-B open circuited.

3.     This equivalent resistance R_th is the resistance obtained at terminals A-B of the network with all its independent current sources open circuited and all its independent voltage sources short circuited.

Example

In the example, calculating the equivalent voltage:

(notice that R₁ is not taken into consideration, as above calculations are done in an open circuit condition between A and B, therefore no current flows through this part, which means there is no current through R₁ and therefore no voltage drop along this part)

Calculating equivalent resistance: 

Norton's theorem:-

Any linear electrical network with voltage and current sources and only resistances can be replaced at terminals A-B by an equivalent current source I_NO in parallel connection with an equivalent resistance R_NO.

This equivalent current I_NO is the current obtained at terminals A-B of the network with terminals A-B short circuited.

This equivalent resistance R_NO is the resistance obtained at terminals A-B of the network with all its voltage sources short circuited and all its current sources open circuited.

For AC systems the theorem can be applied to reactive impedances as well as resistances.

The Norton equivalent circuit is used to represent any network of linear sources and impedances at a given frequency.

Any black box containing resistances only and voltage and current sources can be replaced by an equivalent consisting of an equivalent current source in parallel connection with an equivalent resistance.

Norton's theorem and its dual, Thévenin's theorem, are widely used for circuit analysis simplification and to study circuit's initial-condition and steady-state response.

To find the equivalent,

Find the Norton current I_No. Calculate the output current, I_AB, with a short circuit as the load (meaning 0 resistance between A and B). This is I_No.

Find the Norton resistance R_No. When there are no dependent sources (all current and voltage sources are independent), there are two methods of determining the Norton impedance R_No.

Calculate the output voltage, V_AB, when in open circuit condition (i.e., no load resistor – meaning infinite load resistance). R_No equals this V_AB divided by I_No.

or

Replace independent voltage sources with short circuits and independent current sources with open circuits. The total resistance across the output port is the Norton impedance R_No.

Maximum power transfer theorem:-

In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals.

Suppose we have a voltage source or battery that's internal resistance is R_i and a load resistance R_L is connected across this battery. Maximum power transfer theorem determines the value of resistance R_L for which, the maximum power will be transferred from source to it. Actually the maximum power, drawn from the source, depends upon the value of the load resistance. There may be some confusion let us clear it.

Power delivered to the load resistance,

To find the maximum power, differentiate the above expression with respect to resistance R_L and equate it to zero. Thus,

Thus in this case, the maximum power will be transferred to the load when load resistance is just equal to internal resistance of the battery.