The post simple explanation of spherical coordinate systems appeared first on EE-Vibes.

]]>What is the spherical coordinate system? A point P in space in spherical coordinates is uniquely represented in terms of r,θ,φ, where r is the radial distance from O to P, θ in the angle that r makes with the positive z-axis, and φ is the angle between the positive xz and OMPN planes and it varies from 0 to 2π.

The surface *r *= constant is a sphere,

The surface *θ *= constant is a cone,

**(****two surfaces, cone and sphere, are everywhere perpendicular along their ****intersection, which ****is a circle of radius ***r ***sin ***θ***)**

The surface *φ *= constant is a plane passing through the *θ *= 0 line (or the *z *axis).

The projection of r on x-plane is OM=r sinθ.”The positive direction of ” θ” is from the positive z-axis, where its value is 0, towards the negative z-axis, where its value is π. So ” θ” varies from 0 to π. “However, 0≤r≤∞.

These three surfaces are mutually perpendicular to each other as it can be seen in the figure below.

We consider any point as the intersection of three mutually perpendicular surfaces—a sphere, a cone, and a plane—each oriented in the manner just described.

The unit vector **a***r* is directed radially outward, normal to the sphere *r *= constant, and lies in the cone *θ *= constant and the plane *φ *= constant.

The unit vector **a***θ* is normal to the conical surface, lies in the plane, and is tangent to the sphere.

The third unit vector **a***φ* is the same as in cylindrical coordinates, being normal to the plane and tangent to both the cone and the sphere.

The relation between spherical and rectangular coordinate systems can be found using the following formulas.

A differential volume element may be constructed in spherical coordinates by increasing *r *, *θ*, and *φ *by *dr*, *dθ*, and *dφ*, as shown in figure below:

The surfaces have areas of :

An oscillating charge produces the radiations of spherical pattern.

The follow lecture describes the complete detail of spherical coordinate systems.

https://www.youtube.com/watch?v=zcoOkG7ppV0

Also read here

https://eevibes.com/electromagnetic-field-theory/what-is-the-cylindrical-coordinate-system/

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]]>The post what is the cylindrical coordinate system? appeared first on EE-Vibes.

]]>what is the cylindrical coordinate system? This system helps to analyze the geometry of a coaxial cable and we can analyze its electric field intensity or electric field more easily using them. Three coordinates that are used for representing a point in cylindrical are P(р,Ф,Ζ). р is the radial distance of a cylindrical surface, Ф is the angle that a point makes with the x-axis in counter clockwise direction, z is the height of a cylinder. These three coordinates give us constant planes. At top surface of the cylinder, z is constant while р andФ are variables. Similarly at outer cylindrical surface, р is constant while Ф and Ζ are variable. The cross sectional plane obtained by cutting the cylinder vertically gives us a plane on which Ф is constant while р and Ζ vary.

The following picture shows how these cylindrical coordinate systems with their unit vectors are represented.

1. **ρ (rho)**: As in the standard cylindrical system, ρ represents the radial distance from the origin to a point in space. In electrical engineering, this coordinate is used to specify distances from the central axis of a cylindrical structure, such as a wire or a coaxial cable.

2. **θ (theta)**: Theta still represents the angle measured counterclockwise from the positive x-axis to the projection of the point in the xy-plane. In electrical engineering, this angle helps determine the orientation of electric or magnetic fields around cylindrical structures.

3. **z (z)**: The z-coordinate remains the same, representing the height or distance along the central axis of the cylindrical structure.

Cylindrical coordinates in electrical engineering are particularly useful for solving problems related to:

– **Electrostatics**: Analyzing the electric fields around charged cylindrical objects.

– **Magnetostatics**: Studying the magnetic fields generated by current-carrying cylindrical conductors.

– Antennas: Modeling the radiation patterns of cylindrical antennas, like dipole antennas or slot antennas.

https://www.youtube.com/watch?v=h-oy0lklXHk

Also read here

https://eevibes.com/electromagnetic-field-theory/what-is-the-cartesian-coordinate-system/

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]]>The post Why Circuit Theory Fails at High Frequency? appeared first on EE-Vibes.

]]>Why Circuit Theory Fails at High Frequency? Circuit theory, which is based on the principles of Kirchhoff’s laws and Ohm’s law, is a powerful and widely used tool for analysing electrical circuits at low to moderate frequencies. However, it does have limitations when applied to high-frequency circuits, and these limitations arise from several factors:

At high frequencies, the parasitic elements of components, such as capacitance, inductance, and resistance, become significant. These parasitic elements can no longer be ignored and must be included in circuit models. For example, the parasitic capacitance of a wire or a component can cause unwanted capacitance coupling between different parts of the circuit, affecting its behaviour.

**Example**

Parasitic capacitance refers to the inadvertent or undesired capacitance that arises within an electronic or electrical circuit due to the proximity and inherent characteristics of components, conductors, or traces, often resulting in unintended capacitive effects.

Parasitic capacitance at high frequencies is a critical consideration in the design and performance of electronic circuits and systems. This phenomenon refers to the unintended capacitance that arises due to the inherent properties and geometries of components, traces, and conductors in a circuit when it is subjected to high-frequency signals. Understanding and managing parasitic capacitance is essential because it can have a substantial impact on signal integrity, circuit stability, and overall system functionality.

Key points to consider about parasitic capacitance at high frequencies include:

**Proximity Effects:** At high frequencies, even small gaps between conductors can act as capacitors, as electric fields are more concentrated. These unintended capacitors can introduce impedance and affect signal propagation.

**Signal Integrity:** Parasitic capacitance can lead to signal distortion and loss, especially in high-speed digital circuits and RF (Radio Frequency) systems. It can cause signal reflections, phase shifts, and attenuation, degrading the quality of transmitted or received signals.

**Power Consumption:** Parasitic capacitance can also increase power consumption in integrated circuits. Charging and discharging these capacitances require energy, which can be a significant concern in battery-powered devices.

**Frequency Dependency:** The impact of parasitic capacitance becomes more pronounced as the operating frequency of a circuit increases. Engineers must account for this capacitance when designing high-frequency circuits, as it can affect the intended circuit behavior.

**Mitigation Techniques:** Engineers employ various techniques to mitigate the effects of parasitic capacitance. These include careful layout and routing of traces, the use of shielded cables, and the selection of components with lower parasitic capacitance characteristics.

**Simulation and Modeling:** Advanced simulation tools and modeling software are invaluable for analyzing and predicting the effects of parasitic capacitance at high frequencies. They enable engineers to optimize circuit designs and minimize undesirable capacitive effects.

**Balancing Act:** Managing parasitic capacitance involves a trade-off between factors like component size, trace length, and signal speed. Engineers must strike a balance to achieve the desired circuit performance while minimizing parasitic capacitance.

At high frequencies, the physical dimensions of conductors and traces on a circuit board become comparable to the wavelength of the signals being carried. This leads to transmission line effects, where the behaviour of the circuit is more accurately described by transmission line theory, which accounts for impedance mismatches, reflections, and wave propagation effects. Circuit theory assumes ideal connections without transmission line effects, which can lead to incorrect predictions at high frequencies.

Real-world components are not ideal at high frequencies. For instance, an inductor may exhibit parasitic capacitance, and a capacitor may have parasitic inductance. These non-ideal characteristics can significantly affect the performance of the circuit at high frequencies and are not accounted for in basic circuit theory.

At high frequencies, the skin effect becomes more pronounced, causing the current to concentrate near the surface of conductors. This leads to increased resistance and can affect the performance of components like resistors and inductors. Dielectric materials in capacitors can also exhibit losses at high frequencies, which are not considered in simple circuit models.

In the field of electrical engineering, the skin effect is a phenomenon observed in high frequency alternating currents. This is manifested in the tendency of these currents to concentrate near the outer surface of a conductive material. This occurrence causes the current flow to be limited to only a fraction of the total cross-sectional area of the material, and consequently increases the resistance of the conductor.

Due to the presence of the skin effect, the induction heating can be limited exactly to the surface of the material. This localization of heat can be managed by choosing an appropriate inductor coil configuration. The degree of heating can be precisely controlled (as demonstrated in the practice of induction heating). It should be noted that the intensity of the skin effect becomes more pronounced with magnification of frequency.

At high frequencies, circuits can behave as antennas and radiate electromagnetic energy. This is particularly important in RF (radio frequency) and microwave circuits, where energy loss through radiation can be significant. Circuit theory does not account for radiation effects.

At high frequencies, the wave nature of signals becomes more pronounced, and concepts like phase, wavelength, and impedance matching become critical. Circuit theory, which is based on lumped-element models, is not well-suited for describing these wave phenomena accurately.

To analyse circuits at high frequencies, more advanced techniques, such as RF circuit analysis, microwave engineering, and electromagnetic field simulation, are often employed. These methods take into account the complex behaviours and interactions that arise at high frequencies and are necessary for designing high-frequency and high-speed electronic systems accurately.

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]]>The post What is the Vector Algebra? addition, subtraction and multiplication of vectors appeared first on EE-Vibes.

]]>what is the vector algebra? vector algebra is the first part of vector analysis where we perform basic algebraic operations on vectors that are :

- addition
- subtraction and
- multiplication.

In order to add two or more vectors, we follow the head to tail rule that is implemented using parallelogram rule. If we want to add any two vectors A and B then it is necessary to join the head of one vector with the tail of another vector and after joining them together, the tail of one vector is joined to the head of another vector as shown below.

The vectors subtraction can be performed just like the vectors addition. Because if you want to perform the operation A-B then it can also be written as A+(-B)=A-B. While performing the vectors subtraction you just need to invert the sign of the vector that is being subtracted and then add it into the first vector. Consider the following example.

Here it can been that two vectors A and B are subtracted. Fist step is to draw the negative of B that will be in opposite direction to B. Then draw a parallel vector of A on -B. Finally they are added using head to tail rule.

There are three types of vector multiplication that are

- simple multiplication
- scalar or dot product
- vector or cross product

In simple product, a vector is multiplied with a constant K which extends its magnitude and results in a new vector B. So, **B**=k**A. **If k is positive then the direction of B is same as that of A and if it is negative, then the direction of B is opposite to that of A.

If two vectors are multiplied such that the resultant is a scalar quantity or a number then it is called scalar or dot product. The scalar product of two vectors A and B is written as

**A.B**=ABCOSʘ

scalar product helps to determine the projection of one vector in the direction of another vector. So the scalar projection of vector A on B is given as

ACOSʘ=A.Ḃ

where Ḃ is the unit vector of vector B.

In order to convert this scalar projection into the vector projection you need to multiply it with unit vector of B again as shown below

ACOSʘ=A.Ḃ.Ḃ

If the multiplication of two vectors results in a vectors result in a vector quantity then it is called vector or cross product. The formula for cross multiplication of A and B is given as

**AxB**=ABSinʘ.an

where an is the unit vector perpendicular to the plane containing A and B.

**EMT BOOK**

Electromagnetic Field Theory Fundamentals by Bhag Singh Guru

**EMT notes**

https://www.youtube.com/watch?v=gbsz9-Jn9So

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]]>The post A train travels at 225 km in 3.5 hours and 370 km in 5 hours. Find the average speed of the train. appeared first on EE-Vibes.

]]>To find the average speed of the train, we can use the formula:

Average Speed = Total Distance / Total Time

Given that the train travels 225 km in 3.5 hours and 370 km in 5 hours, we can calculate the total distance and total time:

Total Distance = 225 km + 370 km = 595 km

Total Time = 3.5 hours + 5 hours = 8.5 hours

Now, plug these values into the formula to find the average speed:

Average Speed = Total Distance / Total Time

Average Speed = 595 km / 8.5 hours ≈ 70 km/h

Therefore, the average speed of the train is approximately 70 kilometers per hour.

Five persons can be seated at a round table in 4! (4 factorial) ways, which is equal to 4 × 3 × 2 × 1 = 24 ways.

To calculate the time it takes for an amount of money to produce a certain interest, you can use the formula for simple interest:

Simple Interest (SI) = (Principal) × (Rate) × (Time)

Given:

Principal (P) = Rs. 6000

Interest (I) = Rs. 600

Rate (R) = 5% = 0.05 (decimal)

We want to find the Time (T).

Using the formula for simple interest:

Interest (I) = (Principal) × (Rate) × (Time)

600 = 6000 × 0.05 × Time

Now, solve for Time:

Time = 600 / (6000 × 0.05)

Time = 600 / 300

Time = 2 years

Therefore, it will take 2 years for Rs. 6000 to produce Rs. 600 in interest at a rate of 5%.

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]]>The post A train 100 meters long passes a bridge at the rate of 72 kilometers per hour in 25 seconds. What is the length of the bridge? appeared first on EE-Vibes.

]]>Let’s break down the information given:

The train is 100 meters long.

The train passes a bridge.

The train’s speed is 72 kilometers per hour (km/h).

The time taken to pass the bridge is 25 seconds.

First, let’s convert the train’s speed from kilometers per hour to meters per second, since the length of the train is given in meters and the time is given in seconds.

1 kilometer = 1000 meters

1 hour = 3600 seconds

So, the train’s speed in meters per second is:

(72 km/h) * (1000 m/km) / (3600 s/h) = 20 m/s

Now, we can calculate the distance the train travels in 25 seconds at a speed of 20 m/s:

Distance = Speed × Time

Distance = 20 m/s × 25 s = 500 meters

The train travels a distance of 500 meters during the 25 seconds it takes to pass the bridge. Since the train is 100 meters long, the remaining distance must be the length of the bridge.

Length of the bridge = Total distance traveled – Length of the train

Length of the bridge = 500 m – 100 m = 400 meters

Therefore, the length of the bridge is 400 meters.

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]]>The post What are synchronous sequential circuits? Discuss with examples. appeared first on EE-Vibes.

]]>What are synchronous sequential circuits? Discuss with examples. A sequential circuit is a logical circuit where the output of depends on the present value of the input as well as the sequence of past inputs and output.

The circuit whose behavior can be defined from the knowledge of its signal at discrete instant of time is known as synchronous sequential circuit. Synchronous sequential circuits use level inputs and clock signals as circuit inputs with circuit constraints producing time and beat width to produce performance. A sequential circuit is a logic circuit in which the output depends on the current value of the input and the order of the past inputs and outputs.

There are two types of synchronous circuit:

- Clocked sequential circuit
- Un-clocked sequential circuit

** **Clocked circuits utilize use flip-flop and gated laches as the memory components. The activity of the circuit is worked by the periodic clock pulse where these are associated with the clock contributions to synchronize every one of the inner changes of the state.

This circuit does not require a clock. The internal state change is based on the pulse transition between 0 pulse or 1 pulse. It is designed to act in response to a specific time impulse; periodic signal between the zero signal and the spacer.

- These circuit are easy to design as compared to other types of sequential circuit,
- A clocked flip flop acts as a memory elements.
- They are slower.
- The status of memory elements is affected only at the active edge of clock, if input is changed.
- They use clock signal.

Synchronous sequential circuits are applicable in these:

- Counter
- Shift registers
- Storage units

**Examples:**

Flip flop:

This is an example of a sequential circuit that generates an output from Inputs and Outputs at different time intervals, but not periodically. A flip-flop is an edge-sensitive device.

SR flip-flop is formed from the combination of NAND gates and the enable input. The output occurs when the input is active.

**Block diagram:**

This is the combination of two SR flip-flops that connect the received output of the second FF to the first FF. Master is considered the positively activated level. However, if there is a clock line inverter, the slave will only respond at the negative level. If the clock is in state ‘1’, the master is on and the slave is in sleep mode. And when the clock is in state ‘0’, the master becomes inactive and the slave is in active mode.

Its function is correct if its connection is identical to that shown in the following diagram:

**Truth table:**

The output is clearly described in the following truth table:

Flip -delay flop is the combination of a simple SR flip-flop with a NAND inverter, which is connected between the S and R inputs. The delay flip-flop has an input and the input information appears at the output after a certain delay. This delay is what this flip-flop calls a delay flip-flop. The input condition will not appear here under the conditions S = R = 0 or S = R = 1

**Block diagram:**

This is the actual diagram of the delay flip flop:

The Toggle Flip Flop is generally a JK FF with inputs J and K connected. There is only one input here and this flip flop is triggered with positive edges.

**Diagram:**

This the actual diagram of this:

This is truth table of toggle flip flop:

Related topics:

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]]>The post Light Emitting Diode and its Radian Pattern appeared first on EE-Vibes.

]]>LEDs have revolutionized the lighting industry with their energy efficiency, longevity, and versatility. In this article, we will explore the light emitting diode and its radian patterns, its wide range of applications, and the recent advancements that have propelled its adoption in various fields. So, let’s delve into the fascinating world of LEDs.

An LED is a semiconductor device that emits light when an electric current passes through it. It operates on the principle of electroluminescence, where the movement of electrons within the semiconductor material produces light.

An LED typically consists of a semiconductor chip (die) that is encapsulated in a transparent or colored epoxy resin.

The chip is connected to electrical leads, known as anode and cathode, which provide the necessary current for light emission.

- Principles of Operation:
- Electroluminescence:

LEDs are based on the phenomenon of electroluminescence, where electrons recombine with electron holes within the semiconductor material, releasing energy in the form of photons.

LEDs are predominantly made using compound semiconductor materials, such as gallium arsenide (GaAs), gallium nitride (GaN), and indium gallium nitride (InGaN).

The choice of materials determines the LED’s wavelength and color of light emitted.

The energy band gap of the semiconductor material determines the energy and, hence, the color of light emitted.

By controlling the composition of the semiconductor material, LEDs can be designed to emit various colors, including red, green, blue, and beyond.

III. Applications of LEDs:

LEDs have transformed the lighting industry, offering energy-efficient and long-lasting lighting solutions for residential, commercial, and industrial purposes.

They are used in general lighting, architectural lighting, automotive lighting, and even in specialized applications like horticulture lighting.

LEDs are extensively used in displays and signage, such as LED billboards, scoreboards, digital signage, and alphanumeric displays.

Their ability to emit vibrant colors, high brightness, and sharp contrasts makes them ideal for these applications.

LEDs are ubiquitous in consumer electronics, including smartphones, televisions, laptops, and gaming consoles, providing backlighting for displays and indicating device status.

LEDs are increasingly being adopted in automotive lighting systems, including headlights, taillights, brake lights, and interior lighting, offering improved energy efficiency, durability, and design flexibility.

- Advancements and Future Trends:
- Efficiency Improvements:

Advances in LED technology have led to significant improvements in efficiency, resulting in higher luminous efficacy and reduced energy consumption.

Researchers have made strides in developing LEDs with tunable colors, allowing dynamic adjustments to meet specific lighting requirements, such as color temperature and circadian lighting.

Miniature LEDs and flexible LEDs are gaining popularity, enabling new form factors and applications, such as wearable technology, flexible displays, and smart textiles.

Quantum dot LEDs utilize nanocrystals called quantum dots to enhance color purity, widen the color gamut, and improve efficiency, holding promise for future display and lighting applications.

Light Emitting Diode and its Radian Pattern. Compared to a silicon diode, an LED has a forward voltage across it that is significantly higher. Depending on the material, the maximum VF for LEDs is typically between 1.2 V and 3.2 V. Compared to a silicon rectifier diode, an LED experiences far less reverse breakdown (3 V to 10 V is normal).

Following Figure illustrates how an adequate forward current causes the LED to emit light. As shown in Figure below, the quantity of power output converted into light is exactly proportional to the forward current. The amount of light produced increases proportionally to an increase in IF. Temperature has an impact on both the color and intensity of the light emission. As shown in the graph, light intensity decreases as temperature rises.

The spectral output curves are shown that an LED emits light over a specific range of wavelengths. The curve in part (b) is for a typical infrared LED, and the curve in part (a) is the light output vs wavelength for common visible LEDs.

LED. The unit of wavelength measurement is nanometers (nm). The visible red LED’s normalized output peaks at 660 nm, the yellow LED at 590 nm, the green LED at 540 nm, and the blue LED at 460 nm. The infrared LED’s output reaches its maximum at 940 nm.

The graphs in the following figure illustrate typical LED radiation patterns for tiny devices. Unlike filament or fluorescent bulbs, LEDs are directional light sources. The radiation pattern is typically perpendicular to the emitting surface, although it can be influenced to favor one direction by the form of the emitter surface, lenses, and diffusion layers.

For some applications, like traffic lights, where the light is only meant to be seen by specific cars, directional patterns can be advantageous. The pattern for a forward-directed LED, like those found in tiny panel indicators, is depicted in figure part (a). The pattern with a broader viewing angle, like those found in many super-bright LEDs, is depicted in figure part(b).

There are several manufacturers of LEDs (Light-Emitting Diodes) in the market. Here are some well-known LED manufacturers:

Nichia Corporation: Nichia is a Japanese company and one of the world’s largest LED manufacturers. They are known for their high-quality LEDs and have a wide range of products for various applications.

Cree, Inc.: Cree is an American company that specializes in LED lighting and semiconductor products. They are known for their innovative LED technology and have a strong presence in the market.

Lumileds: Lumileds is a global leader in LED technology and a subsidiary of the Dutch company Royal Philips. They offer a broad portfolio of LED products, including high-power LEDs for lighting applications.

Osram Opto Semiconductors: Osram is a German company that manufactures optoelectronic semiconductors, including LEDs. They offer a wide range of LED products for general lighting, automotive lighting, and specialty lighting applications.

Seoul Semiconductor: Seoul Semiconductor is a South Korean company that produces LED products for various applications. They are known for their high-performance LEDs and have developed innovative technologies like Wicop and Acrich.

Everlight Electronics Co., Ltd.: Everlight is a Taiwanese company that manufactures LEDs and optoelectronic components. They offer a diverse range of LED products, including through-hole, surface-mount, and high-power LEDs.

Epistar Corporation: Epistar is a Taiwanese LED manufacturer and one of the largest LED chip producers globally. They focus on LED chip design and production, supplying chips to various LED packaging companies.

Samsung LED: Samsung is a well-known Korean conglomerate that produces a wide range of products, including LEDs. They offer LED products for various applications, such as general lighting, displays, and automotive lighting.

Bridgelux, Inc.: Bridgelux is an American company that specializes in LED lighting solutions. They offer a range of LED products, including chips, arrays, and modules, with a focus on energy efficiency and high performance.

LG Innotek: LG Innotek is a subsidiary of the LG Group and manufactures a variety of electronic components, including LEDs. They provide LEDs for general lighting, automotive lighting, and display applications.

Please note that this list is not exhaustive, and there are many other manufacturers of LEDs in the market. The LED industry is dynamic, and new players continue to emerge as the technology advances. Light Emitting Diode and its Radian Pattern

LED (Light-Emitting Diode) technology has seen significant growth and has become increasingly popular in various industries. Here are some market trends related to LEDs:

LED lighting is highly energy-efficient compared to traditional lighting technologies such as incandescent bulbs and fluorescent lamps. LEDs consume less power while providing the same or better illumination, making them a preferred choice for energy-conscious consumers and businesses. The demand for energy-efficient lighting solutions has been a driving force behind the LED market’s growth.

Over the years, the cost of LED technology has significantly decreased, making it more affordable for a wide range of applications. This cost reduction has resulted from advancements in manufacturing processes, economies of scale, and improvements in LED efficacy. As LED prices continue to drop, they become increasingly competitive with other lighting options.

LEDs have a significantly longer lifespan compared to traditional lighting sources. LED bulbs can last up to 25 times longer than incandescent bulbs and several times longer than fluorescent lamps. The longer lifespan reduces maintenance and replacement costs, particularly in commercial and industrial settings where lighting is often used for extended periods.

LED lighting is considered more environmentally friendly due to its energy efficiency and lower carbon footprint. LEDs consume less electricity, leading to reduced greenhouse gas emissions. Additionally, LEDs do not contain hazardous materials such as mercury, which is commonly found in fluorescent lamps. The growing focus on sustainability and environmental regulations has fueled the adoption of LED lighting.

LEDs can be easily integrated into smart lighting systems and connected to the Internet of Things (IoT). This allows for advanced features such as remote control, dimming, color tuning, and automated lighting control. The integration of LED lighting with IoT technology has opened up new opportunities for energy management, customization, and enhanced user experiences.

LEDs have become a popular choice for automotive lighting applications. They are used in headlights, taillights, interior lighting, and other automotive lighting functions. LED technology offers improved visibility, energy efficiency, and design flexibility for automotive manufacturers.

LEDs are widely used in display applications, including televisions, computer monitors, and digital signage. The demand for high-resolution displays with better color accuracy and energy efficiency has driven the adoption of LED backlighting and direct LED display technologies.

LEDs are increasingly used in horticulture and indoor farming to provide specific light spectra for plant growth. LED grow lights can be customized to deliver the ideal wavelengths required for different stages of plant development, resulting in improved crop yields, energy efficiency, and reduced dependence on natural sunlight.

It’s important to note that the LED market is dynamic, and these trends may continue to evolve as technology advances and new applications emerge.

In conclusion, light-emitting diodes (LEDs) have revolutionized the lighting and display industries with their energy efficiency, longevity, and versatility. Their applications span across lighting, displays, consumer electronics, and automotive lighting. With ongoing advancements in efficiency, color tunability, miniaturization, and quantum dot technology, LEDs continue to evolve and open up new possibilities for lighting and beyond. As we move forward, we can expect further breakthroughs in LED technology, shaping the future of illumination and visual communication. Thank you.

Also read here:

** **

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]]>The post Representation of Floats in Single Precision appeared first on EE-Vibes.

]]>In this article I have discussed about how to represent floats in computer system? Real numbers are stored as floats in computer system. The more bit size will provide us larger range of numbers to be represented. We have two approaches for storing floats that are:

- single precision method
- double precision method

In single precision, we have reserved 32 bits for representing a single number. The distribution of 32 bits is as follows:

The 31-bit is reserved for representing the sign of the number. If we are representing a positive number then sign bit stores ‘0’ and if the negative number is being stored, then sign bit=1.

Here exponent has 8-bits. The exponent is always stored as a biased component. By biasing we mean that it is always stored by adding a constant ‘127’ in it. The purpose of biasing is to store the more small number in single precision. Since the more smaller number will have negative value of exponent. The value of the biased component is determined from the range of 8-bits signed numbers. (-128 to 127).

23-bits has been reserved for storing the mantissa: the fractional part of the number. There is a hidden bit ‘1’ that needs to be amended every time you read the stored number. The hidden ‘1’ indicates that the number is actually normalized.

According to IEEE standards, some of the bit patterns have been reserved for representing the specific numbers. These are shown below

From here we conclude that two combinations of exponent i.e., 1111 1111 and 0000 0000 can not be used for storing the numbers as they are used for representing the special numbers (infinity and zero). The rest of the combinations that are available are from (1111 1110 and 0000 0001).

In order to store a decimal number 10.5 in single precision, follow these steps.

- First find the binary equivalent of the number which is 1010.10 in this case.
- Convert that binary equivalent into the scientific notation of binary representation: 1.01010 X 2
^{3} - From here you will get exponent=3.
- Add 127 into exponent (3+127=130)
- Find binary equivalent of 124 which is 1000 0010
- Store it in place of exponent
- Also drop 1 from mantissa and you will have remaining part that is 0101000000000…..
- Always start storing mantissa from left hand side i.e., from bit number 23. Since point is placed there.
- Since it is a positive number, that is why sign bit=0.
- The number is stored as follows

In order to read which number is stored in the bit pattern follow these steps:

- First note down the sign bit, if it is zero, then the number is positive otherwise it is a negative number.
- read 8 bits of exponents, find its decimal equivalent. Whatever that decimal equivalent is subtract 127 from it.
- Then read bits from mantissa and place 1 before it.
- Lets do an example of reading a stored number.

sign=0 so its a positive number

exponent= 1000 0010 = 130

subtract 127 from it i.e., 130-127=3

mantissa=1.01010

So it can be written as

1.01010 X 2^{3}

convert it into decimal form

where 2^{3 is equal to 8}

so this becomes 1.3125*8=10.5 which was the actual number we stored.

Also watch here

https://www.youtube.com/watch?v=0SNjrlBC1C4

https://www.youtube.com/watch?v=WCxb8mxiSTw

https://www.youtube.com/watch?v=DU2ZenZSLfs

Also read here

https://eevibes.com/computing/introduction-to-computing/what-are-number-systems/

What are the Different Types of Number Systems in Computing?

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]]>The post Integral calculus: Definition, formulas, types, and examples appeared first on EE-Vibes.

]]>In calculus, integrals are used to find the volume and area of the region. This branch of calculus also provides graphs of the integral functions. Integral is used to solve the complex problems of calculus with respect to integrating variables.

In this post, we will learn the definition, formulas, types, and examples of integrals.

In mathematics, an integral assigns numbers to functions in a way that defines the area, displacement, volume, and other concepts that arise by joining the infinitesimal data.

The process of finding the integrals of the functions by applying the rules and formulas is known as integration. Integral is applied to the function by using an integral notation. The integral notation may or may not have upper and lower limits.

The formulas of the integral can be used to integrate the functions with or without taking the limits. The equation used for integral is given below.

**ʃ f(x) dx **

- The integral notation used to integrate the function is
**ʃ.** - f(x) is the function.
- dx is the integrating variable.

The types of the integral are used to integrate the function with or without using the limits. In integral calculus, there are two types of integral.

- Indefinite integral
- Definite integral

The definite integral is the other type of integral. It is used to integrate the functions by taking the upper and lower limits of the function. This type of integral integrates the function first and then applies the limits according to the fundamental theorem of calculus.

The equation used to integrate the function using a definite integral is given below.

**ʃ****f(x) dx = F(b) – F(a) = L**

- In
**a**and**b**are the lower and upper limits respectively and it is known as the integral notation of definite integral. **f(x)**is the given function.**dx**is the integrating variable.**F(b) – F(a)**is the fundamental theorem of calculus.**L**is the result of the definite integral.

the indefinite integral is used to integrate the integral without taking the upper and lower limits of the function. In this type of integral, integrate the function and write the integral with the constant of integration.

The equation of indefinite integral is given below.

**ʃ f(x) dx = F(x) + c**

- The integral notation used to integrate the function is
**ʃ.** - f(x) is the function.
- dx is the integrating variable.
- F(x) is the integral of the function.
- C is the constant of integration.

You can use an integral calculator to integrate the functions according to the above types of integral.

Following are a few examples of integral solved by using the types and formulas of integral.

**Example 1: For indefinite integral**

Calculate the indefinite integral of 6x^{3} + 7cos(x) – 12x^{3}y^{2} + xy – cot(x) having** x** as an integrating variable.

**Solution **

**Step 1:** Use the integral notation to write the given function.

ʃ (6x^{3} + 7cos(x) – 12x^{3}y^{2} + xy – cot(x)) dx

**Step 2:** Now use the rules of integration to apply the integral notation separately.

ʃ (6x^{3} + 7cos(x) – 12x^{3}y^{2} + xy – cot(x)) dx = ʃ (6x^{3}) dx + ʃ 7cos(x) dx – ʃ 12x^{3}y^{2} dx + ʃ xy dx – ʃ cot(x) dx

**Step 3:** Use constant function rule and write the constant outside the integral notation.

ʃ (6x^{3} + 7cos(x) – 12x^{3}y^{2} + xy – cot(x)) dx = 6 ʃ (x^{3}) dx + 7 ʃ cos(x) dx – 12 y^{2 }ʃ x^{3} dx + y ʃ x dx – ʃ cot(x) dx

**Step 4:** Now use the power and trigonometric rules to integrate the above equation.

ʃ (6x^{3} + 7cos(x) – 12x^{3}y^{2} + xy – cot(x)) dx = 6 (x^{3+1} / 3 + 1) + 7 (sin(x)) – 12y^{2}(x^{3+1} / 3 + 1) + y (x^{1+1} / 1 + 1) – ln(sin(x)) + C

ʃ (6x^{3} + 7cos(x) – 12x^{3}y^{2} + xy – cot(x)) dx = 6 (x^{4} / 4) + 7 (sin(x)) – 12y^{2 }(x^{4} / 4) + y (x^{2} / 2) – ln(sin(x)) + C

ʃ (6x^{3} + 7cos(x) – 12x^{3}y^{2} + xy – cot(x)) dx = 6/4 (x^{4}) + 7 (sin(x)) – 12y^{2}/4 (x^{4}) + y (x^{2} / 2) – ln(sin(x)) + C

ʃ (6x^{3} + 7cos(x) – 12x^{3}y^{2} + xy – cot(x)) dx = 3/2 (x^{4}) + 7 (sin(x)) – 3y^{2 }(x^{4}) + y (x^{2} / 2) – ln(sin(x)) + C

ʃ (6x^{3} + 7cos(x) – 12x^{3}y^{2} + xy – cot(x)) dx = 3x^{4}/2+ 7sin(x) – 3 x^{4}y^{2} + x^{2}y/ 2 – ln(sin(x)) + C

**Example 2: For definite integral**

Calculate the definite integral of 15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2} having** x** as an integrating variable and limits of x are from 2 to 5.

**Solution **

**Step 1:** Use the integral notation to write the given function.

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx

**Step 2:** Now use the rules of integration to apply the integral notation separately.

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx = (15x^{4}) dx + (2cos(x)) dx – (5x^{4}y^{3} dx + (15x^{2}) dx

**Step 3:** Use the constant function rule and write the constant outside the integral notation.

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx = 15(x^{4}) dx + 22cos(x) dx – 5y^{3}(x^{4} dx + 15(x^{2}) dx

**Step 4:** Now use the power and trigonometric rules to integrate the above equation.

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx = 15 (x^{4+1} / 4 +1)^{5}_{2} + 2 (-sin(x) )^{5}_{2} – 5y^{3} (x^{4+1} / 4 +1)^{5}_{2} + 15 (x^{2+1} /2 + 1)^{5}_{2}

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx = 15 (x^{5} / 5)^{5}_{2} + 2 (-sin(x) )^{5}_{2} – 5y^{3} (x^{5} / 5)^{5}_{2} + 15 (x^{3} / 3)^{5}_{2}

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx = 15/5 (x^{5})^{5}_{2} + 2 (-sin(x) )^{5}_{2} – 5y^{3}/5 (x^{5})^{5}_{2} + 15/3 (x^{3})^{5}_{2}

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx = 3 (x^{5})^{5}_{2} – 2 (sin(x) )^{5}_{2} – y^{3} (x^{5})^{5}_{2} + 5 (x^{3})^{5}_{2}

**Step 5:** Use the fundamental theorem of calculus to get the result.

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx = 3 (5^{5} – 2^{5}) – 2 (sin (5)) – sin (2))) – y^{3} (5^{5} – 2^{5}) + 5 (5^{3} – 2^{3})

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx = 3 (3125 – 32) – 2 (-0.9589 – 0.9093) – y^{3} (3125 – 32) + 5 (125 – 8)

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx = 3(3093) – 2 (-1.8692) – y^{3} (3093) + 5 (117)

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx = 9279 + 3.7364 – 3093y^{3} + 585

(15x^{4} + 2cos(x) – 5x^{4}y^{3} + 15x^{2}) dx = 9867.7364– 3093y^{3}

In this article, we have learned about the definition, formulas, types, and examples of integral. Now you can solve any problem of antiderivatives by learning this post.

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