The post Top 3 Generators of Harmonic Currents | Electrical Engineering appeared first on Engineering Notes India.

]]>ASDs are generators of large harmonic currents. Fluorescent lights use less electrical energy for the same light output as incandescent lighting but produce substantial harmonic currents in the process. The explosion of personal computer use has resulted in harmonic current proliferation in commercial buildings. This section is devoted to describing, in no particular order, a few of the more common nonlinear loads that surround us in our everyday life.

Figure 4.10 shows a current waveform at a distribution panel supplying exclusively fluorescent lights. The waveform is primarily comprised of the third and the fifth harmonic frequencies. The waveform also contains slight traces of even harmonics, especially of the higher frequency order. The current waveform is flat topped due to initiation of arc within the gas tube, which causes the voltage across the tube and the current to become essentially unchanged for a portion of each half of a cycle.

While several technologies exist for creating a variable voltage and variable frequency power source for the speed control of AC motors, the pulse-width modulation (PWM) drive technology is currently the most widely used.

Figure 4.11 shows current graphs at the ASD input lines with a motor operating at 60 Hz. The characteristic double hump for each half cycle of the AC waveform is due to conduction of the input rectifier modules for a duration of two 60° periods for each half cycle. As the operating frequency is reduced, the humps become pronounced with a large increase in the total harmonic distortion.

The THD of 74.2% for 45 Hz operation is excessive and can produce many deleterious effects. Figure 4.12 is the waveform of the voltage at the ASD input power lines. Large current distortions can produce significant voltage distortions.

In this particular case, the voltage THD is 8.3%, which is higher than levels typically found in most industrial installations. High levels of voltage THD also produce unwanted results.

This drive contains line side inductors which, along with the higher inductance of the motor, produce a current waveform with less distortion.

Figures 4.13 and 4.14 show the nonlinear current characteristics of a personal computer and a computer monitor, respectively. The predominance of the third and fifth harmonics is evident. The current THD for both devices exceeds 100%, as the result of high levels of individual distortions introduced by the third and fifth harmonics.

The total current drawn by a personal computer and its monitor is less than 2 A, but a typical high-rise building can contain several hundred computers and monitors. The net effect of this on the total current harmonic distortion of a facility is not difficult to visualize. So far we have examined some of the more common harmonic current generators.

The examples illustrate that a wide spectrum of harmonic currents is generated. Depending on the size of the power source and the harmonic current makeup, the composite harmonic picture will be different from facility to facility.

The post Top 3 Generators of Harmonic Currents | Electrical Engineering appeared first on Engineering Notes India.

]]>The post Resistance: Meaning and Laws | Current | Electrical Engineering appeared first on Engineering Notes India.

]]>Resistance may be defined as that property of a substance which opposes (or restricts) the flow of an electric current (or electrons) through it.

The practical as well as mks (or SI) unit of resistance is ohm (Ω), which is defined as that resistance between two points of a conductor when a potential difference of one volt, applied between these points, produces in this conductor a current of one ampere, the conductor not being a source of any emf.

For insulators having high resistance, much bigger unit’s kilo-ohm or kΩ (10^{3} ohm) and mega-ohm or MΩ (10^{6} ohm) are used. In case of very small resistances smaller units like milli-ohm (10^{-3} ohm) or micro- ohm (10^{-6} ohm) are employed.

Each resistor has two main characteristics i.e., its resistance value in ohms and its power dissipating capacity in watts. Resistors are employed for many purposes such as electric heaters, telephone equipment, electric and electronic circuit elements, and current limiting devices. As such their resistance values and tolerances and their power rating vary widely. Resistors of 0.1 Ω to many mega-ohms are manufactured.

Acceptable tolerances may range from ± 20% (resistors serving as heating elements) to ± 0.001 percent (precision resistors in sensitive measuring instruments). The power rating may be as low as 1/10 W and as high as several hundred watts.

Since no single resistor material or type can be made to encompass all the required ranges and tolerances, many different designs are available. Most common commercially available resistors with their properties are given in Table 1.1.

The value of R is selected to have a desired current I or permissible voltage drop I R. At the same time wattage of the resistor is selected so that it can dissipate the heat losses without getting itself overheated. Too much heat may burn the resistor.

From the operating conditions point of view the resistors can be broadly classified into two categories viz., fixed and variable (or adjustable) resistors.

**(a) Fixed Resistors:**

The symbols for fixed resistors used in circuit diagrams are given in Fig. 1.8 (a).

**(b) Variable or Adjustable Resistors:**

For circuits requiring a resistance that can be adjusted while it remains connected in the circuit (such as the volume control on a radio), variable resistors are required. They usually have three leads, two fixed and one movable.

If contacts are made to only two leads of the resistor (stationary lead and moving lead), the variable resistor is being used as a rheostat. The symbols for a rheostat are given in Fig. 1.8 (b). Rheostats are usually employed to limit current flowing in circuit branches.

If all the three contacts are employed in a circuit, it is termed a potentiometer or “pot”. Pots are often used as voltage dividers to control or vary voltage across a circuit branch. The symbols for “pots” are given in Fig. 1.8 (c). Thus the potentiometer (or the pot) is a three terminal resistor with an adjustable sliding contact that functions as an adjustable voltage divider and makes it possible to mechanically vary the resistance.

The resistance of a wire depends upon its length, area of x-section, type of material, purity and hardness of material of which it is made of and the operating temperature.

**Resistance of a wire is: **

(a) Directly proportional to its length, I i.e., R a I

(b) Inversely proportional to its area of x-section, a

i.e., R α I/a

Combining above two facts we have R α l/a

Or R = ρ l/a

Where ρ (rho) is a constant depending upon the nature of the material and is known as the specific resistance or resistivity of the material of the wire.

To determine the nature of the constant ρ let us imagine a conductor of unit length and unit cross-sectional area, for example a cube whose edges are each of length one unit, and let the current flow into the cube at right angles to one face and out at the other face. Then putting l = 1 and a = 1 in Eq. (1.8) we have R = ρ. Hence resistance of a material of unit length having unit cross-sectional area is defined as the resistivity or specific resistance of the material.

Specific resistance or resistivity of a material is also defined as the resistance between opposite faces of a unit cube of that material.

Resistivity is measured in ohm-metres (Ω-m) or ohms per metre cube in mks (or SI) system and ohm- cm (Ω-cm) or ohm per cm cube in cgs system.

1 Ω-m = 100 Ω-cm

The reciprocal of resistance i.e. 1/R is called the conductance and is denoted by English letter G. It is defined as the inducement offered by the conductor to the flow of current and is measured in Siemens (S). Earlier, the unit of conductance was mho (ʊ).

1 Siemen = 1 mho

From Eq. (1.8)

Where σ = 1/ρ and is known as specific conductance or conductivity of the material. Hence conductivity ρ is the reciprocal of the resistivity and is defined as the conductance between the two opposite faces of a unit cube. The unit of conductivity is Siemens/metre (S/m).

**Example 1:**

**A coil consists of 2,000 turns of copper wire having a cross-sectional area of 0.8 mm ^{2}. The mean length per turn is 80 cm, and the resistivity of copper is 0.02 µ Ω-m. Find the resistance of the coil.**

**Solution: **

Length of the coil, I – Number of turns × mean length per turn

= 2,000 × 0.8 = 1,600 m

Cross-sectional area of wire, a = 0.8 mm^{2} = 0.8 × 10^{-6} m^{2}

Resistivity of copper, ρ = 0.02 µΩ-m = 2 × 10^{-2} × 10^{-6} Ω-m = 2 × 10^{-8}Ω-m

Resistance of the coil, R = l/a = 2 × 10^{-8} × 1,600 / 0.8 × 10^{-6 }= 40 Ω Ans.

**Example 2:**

A heater element is made of nichrome wire having resistivity equal to 100 × 10^{-8} Ω- m. The diameter of the wire is 0.4 mm. Calculate the length of the wire required to get a resistance of 40 Ω.

**Solution: **

Resistance of nichrome wire, R = 40 Ω

Resistivity of nichrome wire, ρ = 100 × 10^{-8}Ω-m

The post Resistance: Meaning and Laws | Current | Electrical Engineering appeared first on Engineering Notes India.

]]>The post Distribution of Currents in a Network: 2 Laws | Electrical Engineering appeared first on Engineering Notes India.

]]>According to this law in any network of wires carrying currents, the algebraic sum of all currents meeting at a point (or junction) is zero or the sum of incoming currents towards any point is equal to the sum of outgoing currents away from that point. If I_{1}, I_{2}, I_{3}, I_{4}, I_{5}, and I_{6} are the currents meeting at junction O, flowing in the directions of arrowheads marked on them (Fig. 2.28); taking incoming currents as positive and outgoing currents as negative, according to Kirchhoff’s first law (KCL).

According to this law in any closed circuit or mesh the algebraic sum of emfs acting in that circuit or mesh is equal to the algebraic sum of the products of the currents and resistances of each part of the circuit.

If the circuit shown in Fig. 2.29 is considered, then according to Kirchhoff’s second law (or KVL).

**Application of Kirchhoff’s Laws on Circuits: **

First of all the current distribution in various branches of the circuit is made with directions of their flow complying with first law of Kirchhoff. Then Kirchhoff’s second law is applied to each mesh (one by one) separately and algebraic equations are obtained by equating the algebraic sum of emfs acting in a mesh equal to the algebraic sum of respective drops in the same mesh. By solving the equations so obtained unknown quantities can be determined. While applying Kirchhoff’s second law, the question of algebraic signs may be troublesome and is a frequent source of error. If, however, the following rules are kept in mind, no difficulty should occur.

The resistive drops in a mesh due to current flowing in clockwise direction must be taken positive drops.

The resistive drops in a mesh due to current flowing in counter-clockwise direction must be taken as negative drops.

Similarly the battery emf causing current to flow in clockwise direction in a mesh must be taken as positive emf and the battery emf causing current to flow in counter-clockwise direction in a mesh must be taken as negative emf.

For example, for the circuit shown in Fig. 2.29 let the current distribution be made as shown, which satisfy Kirchhoff’s first law fully.

Taking first, mesh AFCBA for the application of Kirchhoff s second law, we see that there is only one emf acting in the mesh (E_{1}) and since it tries to send current in clockwise direction so E_{1} be taken as positive, similarly all the resistive drops i.e. R_{1} (I_{1} + I_{2}), R_{2} (I_{1} + I_{2}) and R_{5} I_{1} are clockwise, so these must be taken as positive.

.** ^{.}**. According to Kirchhoff s second law in mesh AFCBA

E_{1} = R_{1 }(I_{1} + I_{2}) + R_{2 }(I_{1} + I_{2}) + R_{5}I_{1}

In mesh FEDCF, there is only one emf acting in the mesh (E_{2}) and since it tries to send current in counter-clockwise direction through the mesh under consideration, it may be taken as negative. Since all of the resistive drops R_{3 }I_{2}, R_{4} I_{2}, R_{2 }(I_{1} + I_{2}) and R_{1} (I_{1} +1_{2}) are counter-clockwise, these may be taken as negative.

**Hence according to Kirchhoff s second law in mesh FEDCF we get:**

– E_{2} = – R_{3 }I_{2} – R_{4 }I_{2}– R_{4} I_{2} – R_{2} (I_{1} + I_{2}) – R_{1} (I_{1} + I_{2})

or E_{2} = (R_{3} + R_{4}) I_{2} + (R_{1} + R_{2}) (I_{1} + I_{2})

In mesh AFEDCBA, emf E_{1} tries to cause current in clockwise direction so be taken as positive and emf E_{2} tries to cause current in counter-clockwise direction so be taken as negative. Similarly resistive drop R_{5}I_{1} being clockwise be taken as positive and resistive drops R_{3 }I_{2} and R_{4 }I_{2} being counter – clockwise be taken as negative.

Hence E_{1} – E_{2} = – R_{3 }I_{2} – R_{4 }I_{2} + R_{5} I_{1} = R_{5 }I_{1}, – (R_{3} + R_{4}) I_{2}.

The post Distribution of Currents in a Network: 2 Laws | Electrical Engineering appeared first on Engineering Notes India.

]]>The post Fundamentals of Alternating Current | Electrical Engineering appeared first on Engineering Notes India.

]]>**Contents: **

- Meaning of Alternating Current
- Generation of Alternating Emf
- Sinusoidal Quantities (Emf, Voltage or Current)
- Average and Effective (RMS) Values of Alternating Voltage and Current
- Average and Effective (RMS) Values of Sinusoidal Current and Voltage
- Form Factor and Peak Factor of Sinusoidal Wave
- RMS Value, Average Value, Peak Factor and Form Factor of Half Wave Rectified Alternating Current
- RMS And Average Values of a Triangular Waveform Alternating Current

** 1. Meaning of Alternating Current****: **

A current (or voltage) is called alternating if it reverses periodically in direction, and its magnitude undergoes a definite cycle of changes in definite intervals of time. Each cycle of alternating current (or voltage) consists of two half cycles, during one of which the current (or voltage) acts in one direction; while during the other in opposite direction. In more restricted sense, alternating current is a periodically varying current, the average value of which, over a period, is zero.

The direct current always flows in one direction, and its magnitude remains unaltered. In order to produce an alternating current through an electric circuit, a source capable of reversing the emf periodically (ac generator) is required while for generating dc in an electric circuit, a source capable of developing a constant emf is required such as a battery or dc generator. The graphical representations of alternating current and direct current are given in Figs. 3.1(a) and (b) respectively.

At present a large percentage of the electrical energy (nearly all) being used for domestic and commercial purposes is generated as alternating current. In fact, almost the whole of the vast amount of electrical energy used throughout the world for every imaginable purpose is generated by alternating current generators. This is not due to any superiority of alternating current over direct current in the sphere of applicability to industrial and domestic use.

In fact there are certain types of works for which alternating current is unsuitable and, therefore, direct current is absolutely necessary such as for electroplating, charging of storage batteries, refining of copper, refining of aluminium, electrotyping, production of industrial gases by electrolysis, municipal traction etc.

In some power applications, the ac motor is unsatisfactory such as for metal rolling mills, paper making machines, high speed gearless elevators, automatic machine tools and high-speed printing presses. Direct current required for these applications is nowadays derived from an ac supply by the use of suitable convertors or rectifiers. For lighting and heating dc and ac are equally useful.

**The reasons for generation of electrical energy in the form of alternating current are given below: **

1. AC generators have no commutator and can, therefore, be built in very large units to run at high speeds producing high voltages (as high as 11,000 volts), so that the construction and operating cost per kW is low, whereas dc generator capacities and voltages are limited to comparatively low values.

2. Alternating current can be generated at comparatively high voltages and can be raised and lowered readily by a static machine called the transformer which makes the transmission and distribution of electrical energy economical. In direct current use of transformers is not possible.

3. AC induction motor is cheaper in initial cost and in maintenance since it has got no commutator and is more efficient than dc motor for constant speed work, so it is desirable to generate power as alternating current.

4. The high transmission efficiency in ac makes the generation of electrical energy economical by generating it in large quantities in a single station and distributing over a large territory.

5. The switchgear (e.g., switches, circuit breakers etc.) for ac system is simpler than that required in a dc system.

6. The maintenance cost of ac equipment is less.

** 2. Generation of Alternating Emf****: **

We know that an alternating emf can be generated either by rotating a coil within a stationary magnetic field, as illustrated in Fig. 3.2 (a) or by rotating a magnetic field within a stationary coil, as illustrated in Fig. 3.2 (b). The emf generated, in either case, will be of sinusoidal waveform.

The magnitude of emf generated in the coil depends upon the number of turns on the coil, the strength of magnetic field and the speed at which the coil or magnetic field rotates. The former method is employed in case of small ac generators while the later one is employed for large sized ac generators.

Now consider a rectangular coil of N turns rotating in counter-clockwise direction with angular velocity of co radians per second in a uniform magnetic field, as illustrated in Fig. 3.3.

Let the time be measured from the instant of coincidence of the plane of the coil with the X-axis. At this instant maximum flux, ɸ_{max} links with the coil. Let the coil assume the position, as shown in Fig. 3.3, after moving in counter-clockwise direction for t seconds.

The angle θ through which the coil has rotated in t seconds = ωt

In this position, the component of flux along perpendicular to the plane of coil = ɸ_{max} cos ωt.

Hence flux linkages of the coil at this instant = Number of turns on coil × linking flux i.e., instantaneous flux linkages = N ɸ_{max} cos ωt

**Since emf induced in a coil is equal to the rate of change of flux linkages with minus sign: **

When ωt = 0, sin ωt = 0, therefore, induced emf is zero, when ωt = π/2, sin π/2 =1, therefore, induced emf is maximum, which is denoted by E_{max} and is equal to ɸ_{max} N ω sin ωt

**Substituting ɸ _{max} Nω = E_{max} in Eq. (3.1) we have: **

Instantaneous emf, e = E_{max} sin ω t … (3.2)

So the emf induced varies as the sine function of the time angle ωt, and if emf induced is plotted against time, a curve of sine wave shape is obtained as illustrated in Fig. 3.4. Such an emf is called the sinusoidal emf. The sine curve is completed when the coil rotates through an angle of 2π radians.

The induced emf e will have maximum value, represented by E_{max}, when the coil has turned through π/2 radians (or 90°) in counter-clockwise direction from the reference axis (i.e., OX axis).

3. Sinusoidal **Quantities (Emf, Voltage or Current)****: **

It is not an accident that the bulk of electric power generated in electric power stations throughout the world and distributed to the consumers appears in the form of sinusoidal variations of voltage and current.

There are many technical and economic advantages associated with the use of sinusoidal voltages and currents. For example, it will be learned that the use of sinusoidal voltages applied to appropriately designed coils results in a revolving magnetic field which has the capacity to do work.

As a matter of fact it is this principle which underlies the operation of almost all the electric motors found in home appliances and about 90% of all electric motors found in commercial and industrial applications. Although other waveforms can be used in such devices, none leads to an operation which is as efficient and economical as that achieved through the use of sinusoidal quantities.

**The other advantages of using sinusoidal voltages and currents are: **

1. The waveform from generation to utilization remains the same if a sinusoidal waveform is generated.

2. Electromagnetic torque developed in three phase machines (generators and motors) with balanced three-phase currents is uniform (constant), and therefore, there are no oscillations in developed torque and absence of noise in operation.

3. Non-sinusoidal voltages which contain harmonic frequencies, according to Fourier analysis, are harmful to the system on account of-

(i) increased losses in generators, motors, transformers, and transmission and distribution systems,

(ii) More interference (noise) to nearby communication circuits,

(iii) Resonance may result in over-voltages or over-currents at many pockets on the way from generating station to consumer’s premises which may damage the equipment and increase losses, and

(iv) Increased current through power factor improvement capacitors.

In practical electrical engineering it is assumed that the alternating voltages and currents are sinusoidal, though they may slightly deviate from it. The advantage of this assumption is that calculations become simple. It may be noted that alternating voltage and current mean sinusoidal voltage and current unless stated otherwise.

**Alternating emf following sine law (i.e., sinusoidal emf) is illustrated in Fig. 3.4 and is expressed in the form:**

e = E _{max} sin^{ }ωt^{ }… (3.3)

Where e is the instantaneous value of alternating emf (or voltage), E_{max} is the maximum value of the alternating emf (or voltage) and ω is angular velocity of the coil.

The rotating coil moves through an angle of 2π radians in one cycle, so angular velocity ω = 2πf where f is the number of cycles completed per second.

**Substituting ω = 2π f in Eq. (3.3) we have: **

e = E_{max} Sin 2πft … (3.4)

If the alternating emf (or voltage) given by Eq. (3.3) is applied across a load, alternating current flows through the circuit which would also vary sinusoidal i.e., following a sine law.

**The expression for alternating current is given as:**

i = I_{max} sin ωt = I_{max} sin^{ }2πft … (3.5)

Provided the load is pure resistive.

**Plotting of Sine Waveform****: **

Sine curve may be graphically drawn, as illustrated in Fig. 3.8. Draw a circle of radius equal to the maximum value of sinusoidal quantity. Divide the circumference of the circle drawn so into any number of equal parts, say 12, and draw a horizontal line AB (the base on which the sine wave is to be drawn) passing through the centre of the circle.

Divide the line AB into the same number of equal parts i.e., 12 and number the points correspondingly. Draw perpendicular ordinates from each point. Project the points on the circle horizontally to meet the perpendicular ordinates having corresponding numbers. Draw smooth curve through these points. Curve so drawn will be of sine waveform.

** 4. Average and Effective (RMS) Values of Alternating Voltage and Current****: **

In a dc system, the voltage and current are constant and, therefore, there is no problem in specifying their magnitude. But in case of ac system, an alternating voltage or current varies from instant to instant and so poses a problem how to specify the magnitude of an alternating voltage or current. An alternating voltage or current may possibly be expressed in terms of peak (maximum) value, average (mean) value or effective (rms) value.

In specifying an alternating voltage or current, its peak or maximum value is rarely used because it has that value only twice each cycle. Furthermore, the average or mean value cannot be used because it is positive as much as it is negative, so the average value is zero.

Although the average value over half cycle might be used, it would not be as logical a choice as what we shall find effective (virtual or rms) value which is related to the power developed in a resistance by the alternating voltage or current.

**Average Value of Alternating Current:**

The average (or mean) value of an alternating current is equal to the value of direct current which transfers across any circuit the same charge as is transferred by that alternating current during a given time.

Since in the case of a symmetrical alternating current (i.e., one whose two half cycles are exactly similar, whether sinusoidal or non-sinusoidal) the average or mean value over a complete cycle is zero hence for such alternating quantities average or mean value means the value determined by taking the average of instantaneous values during half cycle or one alternation only. However, for unsymmetrical alternating current, as half wave rectified current, the average value means the value determined by taking the mean of instantaneous values over the whole cycle.

The average value is determined by measuring the lengths of a number of equidistant ordinates and then taking their mean i.e. of i_{1}, i_{2}, i_{3 }… i_{n} etc. which are mid-ordinates.

**Using the integral calculus the average (or mean) value of a function f (t) over a specific interval of time between t _{1} and t_{2} is given by:**

**Any function, whose cycle is repeated continuously, irrespective of its wave shape, is termed as periodic function, such as sinusoidal function, and its average value is given by:**

Where T is time period of periodic function.

In case of a symmetrical alternating current, whether sinusoidal or non-sinusoidal the average value is determined by taking average of one half cycle or one alternation only.

**RMS Value or Effective Value of Alternating Current:**

The rms or effective value of an alternating current or voltage is given by that steady current or voltage which when flows or applied to a given resistance for a given time produces the same amount of heat as when the alternating current or voltage is flowing or applied to the same-resistance for the same time.

Consider an alternating current of waveform shown in Fig. 3.10 flowing through a resistor of R ohms. Divide the base of one alternation into n equal parts and let the mid-ordinates be i_{1}, i_{2}, i_{3} …i_{n}. etc.

Now if I_{eff} is the effective current, then heat produced by this current in time T = I^{2}_{eff} RT joules. By definition these two expressions are equal-

Hence the effective or virtual value of alternating current or voltage is equal to the square root of the mean of the squares of successive ordinates and that is why it is known as root-mean-square (rms) value.

**Using the integral calculus the root mean square (rms) or effective value of an alternating quantity over a time period is given by:**

** 5. Average and Effective (RMS) Values of Sinusoidal Current and Voltage****: **

**i. Average Value for Sinusoidal Current or Voltage:**

The average value of a sine wave over a complete cycle is zero. Therefore, the half cycle average value is intended.

**ii. Effective (RMS) Value for Sinusoidal Current or Voltage:**

**A sinusoidal alternating current is represented by:**

** 6. Form Factor and Peak Factor of Sinusoidal Wave: **

**Form Factor:**

In certain cases it is convenient to have calculations at first upon the mean value of the emf over half a period, therefore, it becomes essential to have some means of connecting this mean value with the effective or rms value. The knowledge of form factor, which is defined as the ratio of effective value to the average or mean value of periodic wave is, therefore, necessary.

**Mathematically form factor is given by the relation:**

**Peak Factor:**

Knowledge of peak factor of an alternating voltage is very essential in connection with determining the dielectric strength since the dielectric stress developed in any insulating material is proportional to the maximum value of the voltage applied to it.

**Peak or crest or amplitude factor of a periodic wave is defined as the ratio of maximum or peak to the effective or rms value of the wave:**

** 7. RMS Value, Average Value, Peak Factor and Form Factor of Half Wave Rectified Alternating Current: **

Half wave rectified alternating current is one whose one-half cycle has been suppressed i.e. one which flows for half the time during one cycle. It is illustrated in Fig. 3.16 where suppressed half cycle is shown dotted.

For determining rms and average values of such an alternating current summation would be carried over the period for which current actually flows i.e., 0 to π but would be averaged for the whole cycle i.e., from 0 to 2π.

** 8. RMS And Average Values of a Triangular Waveform Alternating Current: **

Let the maximum value of the current be I_{max} amperes.

Since I = I_{max} when θ = π.

**Hence, expression for the instantaneous current can be written as:**

The post Fundamentals of Alternating Current | Electrical Engineering appeared first on Engineering Notes India.

]]>The post Combined Action of Alternating Quantities | Electrical Engineering appeared first on Engineering Notes India.

]]>**Let it be required to add two currents given by the equations:**

i_{1} = I_{1} max sin ωt and i_{2 }= I_{2} _{max} sin (ω t – ɸ)

**The resultant sum may be expressed as:**

I_{r} = i_{1} + i_{2} = I_{1} _{max} sin ω t + I_{2} _{max} sin (ω t – ɸ) but it is too awkward and gives no idea of the peak, value and phase angle of the resultant current.

The currents may be added graphically by plotting their curves in the same system of coordinates and then adding the ordinates of i_{1 }and i_{2} point by point, according to the equation i_{r} = i_{1} + i_{2}. Evidently this method is also too cumbersome and unwieldy to be practical. This is particularly so when situation arise where more than two sinusoidal quantities are to be added.

A simpler and more direct method consists in adding the sinusoidal quantities as phasors.

Consider the phasors I_{1} _{max} and I_{2 max }that would generate the two curves i_{1} and i_{2} and let them be in a position, as shown in Fig. 3.36 at one particular instant of time. If we now add I_{1 max} and I_{2} _{max} by completing the parallelogram as shown, the diagonal I_{r} _{max }when rotated, generate a third sine curve. It remains to be shown that this third sine curve coincides with the waveform of i_{r} obtained by adding i_{1} and i_{2} point by point.

Now the vertical component of I_{r max }is the sum of the vertical components of I_{1} _{max} and I_{2} _{max}. Therefore, the waveform of i_{r} is the graph generated by rotating I_{r} _{max} in counter-clockwise direction.

It follows, therefore, that two or more alternating quantities_{ }may be added in the same way as forces are added, namely by constructing parallelograms or closed polygons and either measuring or calculating the lengths of the diagonals or closing sides and the magnitude of the phase angles.

**Illustrations: **

The way in which the two given currents i_{1 }and i_{2} can be added by the parallelogram rule of phasor addition is illustrated in Fig. 3.37 where the currents are shown as phasor drawn from the origin O of the system of coordinates. The resultant phasor is the diagonal of the parallelogram formed by the phasors I_{1} max and I_{2} max.

This method is more convenient when more than two phasors are to be added, as shown in Fig. 3.38. From the end point of I_{1} _{max} a phasor is constructed parallel to I_{2} _{max} of the same magnitude and direction as the latter; then from the end point of I_{2 max} a phasor is constructed parallel to I_{3} _{max} and so on. Phasor I_{r max} from the origin of the first phasor (I_{1} _{max}) to the end point of the last phasor (I_{5 max}) represents the sum of all the phasors.

Phasors may also be subtracted by the above method. For example if phasors I_{2 max} and I_{3} _{max} are to be subtracted from phasor I_{1} _{max} each of the two phasors be reversed in direction and then added as explained above [Fig. 3.39]. This time, the phasor drawn from the origin of the first phasor I_{1} _{max} to the terminal point of the last phasor I_{3 max} gives the difference of phasors I_{2 max} and I_{3 max} from phasor I_{1 max}.

The post Combined Action of Alternating Quantities | Electrical Engineering appeared first on Engineering Notes India.

]]>The post Phasor Representation of Alternating Quantities | Electrical Engineering appeared first on Engineering Notes India.

]]>Assumed that alternating voltages and currents follow sine law and generators are designed to give emfs having sine waveform. The above said assumption makes the calculations simple. The method of representing alternating quantities by waveform or by the equations giving instantaneous values is quite cumbersome.

For solution of ac problems it is advantageous to represent a sinusoidal quantity (voltage or current) by a line of definite length rotating in counterclockwise direction with the same angular velocity as that of the sinusoidal quantity. Such a rotating line is called the phasor.

Consider a line OA (or phasor as it is called) representing to scale the maximum value of an alternating quantity, say emf i.e., OA = E_{max} and rotating in counter-clockwise direction at an angular velocity ω radians/second about the point O, as shown in Fig. 3.30. An arrow head is put at the outer end of the phasor, partly to indicate which end is assumed to move and partly to indicate the precise length of the phasor when two or more phasors happen to coincide.

Figure 3.30 shows OA when it has rotated through an angle θ, being equal to ωt, from the position occupied when the emf was passing through its zero value. The projection of OA on Y-axis, OB = OA sin θ = E_{max} sin ωt = e, the value of the emf at that instant.

Thus the projection of OA on the vertical axis represents to scale the instantaneous value of emf.

**It will be seen that the phasor OA rotating in counter-clockwise direction will represent a sinusoidal quantity (voltage or current) if:**

(i) Its length is equal to the peak or maximum value of the sinusoidal voltage or current to a suitable scale.

(ii) It is in horizontal position at the instant the alternating quantity (voltage or current) is zero and increasing, and

(iii) Its angular velocity is such that it completes one revolution in the same time as taken by the alternating quantity (voltage or current) to complete one cycle.

Since there is definite relation between maximum value and rms value (E_{max} = n√2 E_{rms}), the length of phasor OA can be taken equal to rms value if desired. But it should be noted that in such cases, the projection of the rotating phasor on the vertical axis will not give the instantaneous value of that alternating quantity.

The phasor diagram drawn in rms values of the alternating quantities helps in understanding the behaviour of the ac machines under different loading conditions.

**Phase and Phase Angle: **

By phase of an alternating current is meant the fraction of the time period of that alternating current that has elapsed since the current last passed through the zero position of reference. The phase angle of any quantity means the angle the phasor representing the quantity makes with the reference line (which is taken to be at zero degrees or radians). For example the phase angle of current I_{2}, in Fig. 3.31 is (-ɸ).

**Phase Difference of Current or Voltage: **

When two alternating quantities, say, two emfs, or two currents or one voltage and one current are considered simultaneously, the frequency being the same, they may not pass through a particular point at the same instant. One may pass through its maximum value at the instant when the other passes through the value other than its maximum one. These two quantities are said to have a phase difference. Phase difference is always given either in degrees or in radians.

The phase difference is measured by the angular distance between the points where, the two curves cross the base or reference line in the same direction.

The quantity ahead in phase is said to lead the other quantity while the second quantity is said to lag behind the first one. In Fig. 3.31 (b) current I_{1} represented by phasor OA leads the current I_{2}, represented by phasor OB by ɸ or current I_{2}, lags behind the current I_{1} by ɸ. The leading current I_{1} goes through its zero and maximum values first and the current I_{2} goes through its zero and maximum values after time angle ɸ. The two waves representing these two currents are shown in Fig. 3.31 (a).

**If I _{1} is taken as reference phasor, the two currents can be expressed as:**

i_{1} = I_{1} _{max} sin ωt

And i_{2} = I_{2} sin (ωt – ɸ)

The two quantities are said to be in phase with each other if they pass through zero values at the same instant and rise in the same direction, as shown in Fig. 3.32.

But the two quantities passing through zero values at the same instant but rising in opposite directions, as shown in Fig. 3.33 are said to be in phase opposition, i.e., phase difference is 180°. When the two alternating quantities have a phase difference of 90° or π/2 radians they are said to be in quadrature.

The alternating quantities (voltages and currents) in practice are represented by straight lines having definite direction and length. Such lines are called the phasors and the diagrams in which phasors represent currents, voltages and their phase difference are known as phasor diagrams.

Though phasor diagrams can be drawn to represent either maximum or effective values of voltages and currents but since effective values are of much more importance, phasor diagrams are mostly drawn to represent effective values.

In order to achieve consistent and accurate results it is essential to follow certain conventions.

**Some of the common conventions in this regard are enlisted below: **

1. Counter-clockwise direction of rotation of phasors is usually taken as positive direction of rotation of phasors i.e., a phasor rotated in a counter-clockwise direction from a given phasor is said to lead the given phasor while a phasor rotated in clockwise direction is said to lag the given phasor.

2. For series circuits, in which the current is common to all parts of the circuit, the current phasor is usually taken as reference phasor for other phasors in the same diagram and drawn on horizontal line.

3. In parallel circuits in which the voltage is common to all branches of the circuit, the voltage phasor is usually taken as reference phasor and drawn on the horizontal line. Other phasors are referred to the common voltage phasor.

4. It is not necessary that current and voltage phasors are drawn to the same scale; in fact it is often desirable to draw the current phasor to a larger scale than the voltage phasor when the values of currents being represented are small. However, if several voltage phasors are to be used in the same phasor diagram, they should all be drawn to the same scale. Likewise all current phasors in the same diagram should be drawn to the same scale.

The post Phasor Representation of Alternating Quantities | Electrical Engineering appeared first on Engineering Notes India.

]]>The post Calculating Power Factor in an A.C. Circuit | Electrical Engineering appeared first on Engineering Notes India.

]]>The power in watt absorbed by a d.c. circuit is known when the supply pressure in volt across the circuit is multiplied by the current in ampere flowing through the circuit. But in an a.c. circuit the product of supply pressure and the circuit current gives the Apparent Power of the circuit. It is expressed in volt-ampere and not in watt. If the pressure applied across the circuit be V volt and the current flowing through the circuit be I ampere, the apparent power required for the circuit is VI volt-ampere.

The True Power or Actual Power absorbed by an a.c. circuit is VI cos θ watt, where θ is the phase angle between V and I.

Power factor of an a.c. circuit is the ratio = True Power / Apparent Power

Thus, power factor = [(VI cosθ)/ VI] = cosθ

i.e. cosine of the phase single of an a.c. circuit is the power factor of that circuit. If the current lags behind the applied voltage, the power factor is called a lagging power factor; when the current leads the applied voltage, the power factor is called a leading power factor.

Sometimes power factor is expressed in per cent. In that case power factor indicates the true power expressed as a percentage of the apparent power. For example, if the power factor of an a.c. circuit is 80 per cent, it indicates that the true power absorbed by the circuit is 80 per cent of its apparent power, i.e.,

(True Power / Apparent Power) x 100 = 80%,

or, True Power / Apparent Power = 0.8

The post Calculating Power Factor in an A.C. Circuit | Electrical Engineering appeared first on Engineering Notes India.

]]>The post Value of Alternating Current and EMF | Wiring | Electrical Engineering appeared first on Engineering Notes India.

]]>In a graph plot the instantaneous values of current along the ordinate and the corresponding angular displacements along the abscissa for one complete cycle or 360°. A Sine curve will be obtained as shown in figure 21.

Now plot the square curve. This is done by plotting a number of points a such that ac = (bc)^{2} and drawing a smooth curve through them.

**The average height of this square curve gives the mean value if i ^{2} which is determined as follows: **

Let I_{m} be the maximum value of the current. Then maximum height of the square curve is I^{2}_{m}. Draw a horizontal line pq at a height of I^{2}_{m}/2 above the horizontal axis OS. The area of the rectangle opqs will be equal to the area bounded by the square curve which is shown by the shaded portions in fig. 21.

Thus, it is seen that if an alternating current is sinusoidal, its R.M.S. value is 0.707 times its maximum value.

**Similarly, R.M.S. value of a sinusoidal e.m.f.:**

where E_{m} is the maximum value of e.m.f.

The post Value of Alternating Current and EMF | Wiring | Electrical Engineering appeared first on Engineering Notes India.

]]>The post Leading Current from the Supply Line | Wiring | Electrical Engineering appeared first on Engineering Notes India.

]]>Static capacitor takes a current which leads the applied voltage by about 90°. Thus, it acts in direct opposition to inductance, neutralizes the inductive influence of the circuit and improves the power factor.

There are two methods of connecting capacitor in the circuit. In one method every individual inductive load is provided with a capacitor of suitable capacitance in order to improve the power factor of that particular load circuit. In another method a big delta-connected capacitor bank (three similar capacitors connected in delta) is installed, usually at the supply mains, to improve the power factor of the whole installation.

Although the former method involves greater cost than the latter, it is more popular system as it provides greater flexibility in operation and helps to maintain power factor always at the desired value.

The power loss in a capacitor is negligible and it is much less in first cost than any other device used for the improvement of power factor. Therefore among all the power factor correction devices, it has the largest application in different electrical circuits and installations.

When the direct current flowing through the field coil of a synchronous motor exceeds the normal rated value (for which the e.m.f. induced per phase in the armature is equal to voltage applied per phase at the motor terminals), the motor is said to be over excited. Under this condition the armature of the motor draws a leading current from the a.c. bus-bars, and as a result the power factor of the circuit is improved.

In a factory or in an electrical installation where large number of induction motors are used, it is economical to use one or two synchronous motors for constant-speed drive as well as for power factor correction. Since the over-excited synchronous motor takes a leading current from the supply line like a capacitor, it is also called Synchronous Condenser or Synchronous Capacitor.

Phase advancer is seldom used for the improvement of power factor in an a.c. circuit, as this method involves very high cost. It is used occasionally with a big induction motor where the phase advancer supplies the reactive component of current taken by the motor. As a result the motor draws only the active component of current from the supply line and thereby the power factor of the motor circuit is raised almost to unity.

The post Leading Current from the Supply Line | Wiring | Electrical Engineering appeared first on Engineering Notes India.

]]>The post Calculating Capacitance of the Capacitor | Current | Electrical Engineering appeared first on Engineering Notes India.

]]>Two metal plates separated by an insulator constitute a capacitor or condenser, namely an arrangement which has the capacity of storing electricity as an excess of electrons on one plate and a deficiency on the other. The insulator between the plates is called dielectric. It may be paper, mica, glass, air or any other insulating material.

The most common type of capacitor used in practice consists of two strips of metal foil separated by strips of waxed paper wound spirally, forming two large surfaces near to each other. A capacitor can be charged or discharged like a storage battery. If electric current is supplied to a capacitor from an external source, the current is stored in the capacitor as electrostatic charge, and the process is known as charging the capacitor.

The quantity of charge is generally denoted by Q and is measured in coulomb. A capacitor is said to have a charge of one coulomb when one ampere current is supplied to it for one second. Thus, if a current of I ampere is supplied to a capacitor for t second, the quantity of charge stored in it-

Q = I x t coulomb.

Current can be drawn from a charged capacitor similar to a charged storage battery supplying current to load circuits. This process is known as discharging the capacitor.

The potential difference between the plates of a capacitor is nil when the capacitor is – uncharged. As the quantity of charge accumulated in the capacitor increases, the p.d. between the plates rises and the charging current decreases. Finally when the p.d. between the plates becomes equal to that of supply, the flow of current is stopped and the capacitor is considered to be fully charged at this point.

The property of a capacitor to store an electric charge when its plates are at different potentials is referred to as its capacitance. It is generally denoted by ‘C’ and is expressed in farad. A capacitance of one farad may be defined as the capacitance of a capacitor between the plates of which there appears a difference of potential of 1 volt when it is charged by 1 coulomb of electricity.

**Thus, if the quantity of charge stored in a capacitor be Q coulomb when the p.d. applied across its plates is V volt, the capacitance of the capacitor: **

C = Q/V farad

or Q = CV coulomb.

Farad is considered to be very large unit, and in practice, capacitance is usually expressed in microfarad.

1 microfarad = 10^{-6} farad.

The post Calculating Capacitance of the Capacitor | Current | Electrical Engineering appeared first on Engineering Notes India.

]]>