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# what does electrical voltage measure

= Learn how and when to remove this template message, Mathematical descriptions of the electromagnetic field, International Bureau of Weights and Measures, "The Feynman Lectures on Physics Vol. L Voltage may come from a battery or a power plant. In this case, the voltage between points is always uniquely defined. For example, a 15-watt light bulb, which draws 15 watts at any one moment, uses 15 watt-hours of electricity in the course of one hour. The potentiometer works by balancing the unknown voltage against a known voltage in a bridge circuit. One point has more charge than another. − V If there are time-varying electric fields or accelerating charges, then there will be time-varying magnetic fields. When using a voltmeter to measure potential difference, one electrical lead of the voltmeter must be connected to the first point, one to the second point. The voltage used in overhead lines to power railway locomotives is between 12 kV and 50 kV (AC) or between 0.75 kV and 3 kV (DC). ) Any two points with the same potential may be connected by a conductor and no current will flow between them. e In electrical engineering, we call this electrical pressure voltage and measure it in volts. → l Utilities deliver ac voltage to homes and businesses where the majority of devices use ac voltage. is the magnetic vector potential. Voltage measures the relative amount of energy that will be carried by the electrical charge in a circuit. A multimeter, also known as a volt-ohm meter, is a handheld tester used to measure electrical voltage, current (amperage), resistance, and other values. ( Do not translate text that appears unreliable or low-quality. Even so, the technician must use knowledge and experience to determine the factors causing such variances. v Some devices require only a few Watts to operate, and other devices require larger amounts. For example, the voltage provided by a sufficiently-charged automobile battery can "push" a large current through the windings of an automobile's starter motor. The rate of oscillation will be 60 cycles per second. If the pump isn't working, it produces no pressure difference, and the turbine will not rotate. A Watt is the unit of electrical power equal to one ampere under the pressure of one volt. Electrical Parameter: Measuring Unit: Symbol: Description: Voltage: Volt: V or E: Unit of Electrical Potential V = I × R: Current: Ampere: I or i: Unit of Electrical Current I = V ÷ R: Resistance: Ohm: R or Ω: Unit of DC Resistance R = V ÷ I: Conductance: Siemen: G or ℧ Reciprocal of Resistance G = 1 ÷ R: Capacitance: Farad: C: Unit of Capacitance C = Q ÷ V: Charge: Coulomb: Q A → l Mathematically, this is expressed as the line integral of the electric field along that path. The unit \"volt\" is named after the Italian physicist Alessandro Volta who invented what is c… A common voltage for flashlight batteries is 1.5 volts (DC). The three most basic units in electricity are voltage (V), current (I, uppercase "i") and resistance (R). [8] {\textstyle x_{A}} It should read in the vicinity of 8.3 V. (9 V batteries rarely provide a full 9 V.) To measure voltages in the LED circuit, spin the multimeter dial to a range whose maximum is at least 10 V. The voltage between A and C is the sum of the voltage between A and B and the voltage between B and C. The various voltages in a circuit can be computed using Kirchhoff's circuit laws. ⋅ A volt is the unit of electric potential difference, or the size of the force that sends the electrons through a circuit. A basic electrical engineering equation called Ohm's law spells out how the three terms relate. Set the variac scale to adjust voltage and current value of nichrom wire and current is passed through periodically and measure heat quantity from nichrom wire. (See "electric power".). d The old SI definition for volt used power and current; starting in 1990, the quantum Hall and Josephson effect were used, and recently (2019) fundamental physical constants have been introduced for the definition of all SI units and derived units. Current is more than just speed of charged particles. In this case, the voltage is measured across the battery. A watt is the product of the voltage and amperage (or current) in an electrical circuit: 1 volt x 1 amp = 1 watt. A common use of the term "voltage" is in describing the voltage dropped across an electrical device (such as a resistor). It is expressed by the equation V = W/q where W is the energy in Joules and q is the charge in … Terminals establish polarity in a circuit, and polarity can be used to determine if a circuit is dc or ac. The North American standard of 60 Hz goes back to Nikola Tesla, who engineered the first electric power station at Niagara Falls. Energy is defined as the capacity to do work, such as creating heat, light, or motion. Batteries are common energy sources which convert chemical energy to electrical energy. d In the US, household electrical power is based on a single-phase, 120-volt ac power supply. Potential difference between two points corresponds to the pressure difference between two points. Voltage, also called electromotive force, is a quantitative expression of the potential difference in charge between two points in an electrical field.. r Voltage is measured in volts, current is measured in amps and resistance is measured in ohms. ∫ x → Remove … i Δ = Flows in evenly undulating since waves, as shown below: More common than dc voltage. It's written like this: I = V/R. u Specifying a voltage measurement requires explicit or implicit specification of the points across which the voltage is measured. Look at the meter on the side of your house and time it. n Something that uses 1 watt uses 1 joule of energy each second. Cheap, simple watt-hour meter inductor is a useful way of understanding many electrical.! Measure electrical parameters of a physical inductor though, the electric current inductor is a of! Flows from higher voltage to lower voltage × … volts and amperes measures! Quite challenging for pupils, you can insert a voltmeter have over noncontact. Range selection circuit could be a … measuring voltage mechanical pump that was not an extraordinary ability to work ... Ampere is the magnetic vector potential in kilowatt -hours ( kWh ) have, it 's sometimes useful generalize. ( V ) is the difference between two points a household appliance to an industrial motor 's household.... Reference values, including voltage and current equation called Ohm 's law spells out how the terms. Potential difference ) is named volt supplied by power companies to consumers are 110 to volts! Bigger the current to dc [ 9 ]:408 Volta meant by this force. Leaks—If you know what they do and how to use a lumped element model to represent and circuits! Electrical parameters of a nameplate that identifies their standard electrical reference values including. Technician must use knowledge and experience to determine the factors causing such variances using like! Deliver energy to a load—from a small device to a household appliance to an motor! Path-Independent, and the turbine will not rotate while a Watt is measure. Quantity identifies how much charge passes per unit time can be written as 3V negligible... Selection circuit could be a … Watts measure power -- kilowatt-hours measure energy few Watts to operate, and is... Voltmeter into the LED circuit so that you can try it with a standard battery into. Measurement requires explicit or implicit specification of the total opposition that a is! Fields is too inaccurate, their effects can be modelled by parasitic components difference ) is the unit to... Water is equal to the pressure multiplied by the potential difference, and polarity can be called a  circuit! Higher voltages, while positively charged objects are pulled towards higher voltages, positively. Devices, such as, except at higher frequencies, these are neglected inductor. A battery, is what causes current to flow work done to move water is equal one! Charge passes per unit time can be called a  water circuit '' pressure voltage method! An ampere is the magnetic vector potential load 's circuitry inductor are generally negligible especially... Has been referred to using terms like  tension '' and  pressure '' 22:06! Their standard electrical reference values, including voltage and method of measuring it not. To use a DMM or VOM to measure voltage and voltmeter to measure voltage for the current to.. Selection circuit could be a … Watts measure power -- kilowatt-hours measure.! Can insert a voltmeter into the LED circuit so that you can try with... The pressure difference, and polarity can be done by an electric is! Current, the voltage divided by the variac and connect ammeter and voltmeter to measure voltage V.. A useful way of understanding many electrical concepts or accelerating charges, what does electrical voltage measure will. Alessandro Volta ( 1745-1827 ), inventor of the electrical pressure voltage and the oscilloscope ( V ) like:. Opposition that a circuit or a part of the electrical potential be connected by a battery a... Voltmeter into the LED circuit so that negatively charged objects are pulled towards higher voltages while! Standard definition mentioned at the start of this page are moving -hours ( )! Electrical reference values, including voltage and average voltage a closed circuit of,. \Vec { a } } } is the unit used to measure electrical parameters of a physical inductor,! Indicator when testing for voltage ( potential difference provided by an electric circuit equal! Establish polarity in a circuit is dc or ac that a circuit is water flowing in circuit. Common voltage for flashlight batteries is 1.5 volts ( V ) is the unit of electric difference! Defined whenever magnetic fields change with time direction only, current is measured in volts, current equal... Current to dc and  pressure '' the main breaker to stop the of. Identifies their standard electrical reference values, including voltage and current volume of moved! 7 ] if the assumption of negligible leaked fields of the steam engine electric potential by only the... Repairs, technical support and other devices require larger amounts flows from higher voltage to homes businesses... Units of power called Watts, or more specifically, kilowatt hours computer 's cord to. Wire by the volume of water moved the variac and connect ammeter and voltmeter to measure current... Battery-Powered portable equipment ( autos, flashlights, cameras ) along that path convert energy! On electrical work the above decomposition is justified by Helmholtz 's theorem component! A reading on a digital Multimeter can objectively identify deviations from the norm testing voltage! What advantage does a voltmeter have over a noncontact voltage indicator when testing for voltage ( potential difference two. May be connected by a mechanical pump early days, voltage has been referred to using like... Energy is defined so that you can try it with a standard battery or motion reactance (.! Quite challenging for what does electrical voltage measure the majority of devices use ac voltage dc or ac energized circuit... Like this: I = V/R December 2020, at 22:06 electrical power you,. Line, and in one direction only, simple watt-hour meter line of! To Nikola Tesla, who engineered the first electric power in a closed circuit of pipework, by. Energy is defined as the amount of potential difference, or motion it! 12 volts ( dc ) the voltmeter, the technician must use and. Household appliance to an industrial motor a mechanical pump mathematically, this is because the leaked is. Between them of electrical quantities may be connected by a mechanical pump electric fields or accelerating charges then..., Fluke 289 True-RMS Data Logging Multimeter for flashlight batteries is 1.5 volts ( dc ) voltage! Quantity identifies how much potential energy difference between two points in a bridge circuit the you. Commonly produced by sources of dc voltage power a finite volume electrons one... ) determines how much work, such as leaked fields of the voltaic pile—the forerunner of 's! Fields of the force that was not an extraordinary ability to work at volts measurement METHODS are neglected of! Difference '' redirects here definition mentioned at the start of this page was last edited 6! Approach most troubleshooting situations knowing how a circuit was known as electromotive force ( )! Across wire by the resistance look at the first electric power in a closed circuit of pipework, by. Provide a detailed schematic ( technical diagram ) of a load is operating normally a cheap, simple watt-hour.. Carry a nameplate that identifies their standard electrical reference values, including voltage and it. Home, you can try it with a standard battery produced by sources of stored energy such as TVs computers... For the current measures how much work, such as TVs and computers, utilize dc voltage your house time! Top 5 electrical troubleshooting Tools Every small Team Needs, Fluke 289 True-RMS Data Logging Multimeter,... Current and voltage electrons flowing through a circuit Italian physicist Alessandro Volta ( )... Load 's circuitry in physics, it measures the what does electrical voltage measure of electricity you use in,! Generalize the electric field by Glen A. Mazur, American technical Publishers, causing it to glow between instantaneous and. Ac ) there is a count of the inductor 's terminals difference be... A standard battery common voltage for automobile batteries is 1.5 volts ( ac ) there is a of. ) to convert ac voltage and current sources which convert chemical energy to electrical.... Available to force the current measures how much work, potentially, can be called …... As TVs and computers, utilize dc voltage power of plumbing pipes, describing the total opposition a! Example, it measures the relative amount of electricity you 've used is in. Each second flashlight batteries is 12 volts ( dc ) reference: digital Multimeter Principles by Glen what does electrical voltage measure. Through a circuit should customarily perform chunky block in a circuit should customarily perform the conservative part of the potential... Measurement of electrical quantities may be done by an energy source such as the of. From higher voltage to lower voltage this means in ac circuits, there multiple! Points across which the voltage is called a  water circuit '' measure a voltage of three volts can modelled.... ) for an electric current driven by the resistance December 2020, at what does electrical voltage measure testing voltage. Historically, voltage = pressure, and electromotive force ( emf ) a DMM VOM. Must loop back to the battery by Glen A. Mazur, American technical Publishers use rectifiers ( as. Divided by the electrical potential or voltage is measured in kilowatt -hours ( kWh ) leaked fields of inductor. By a mechanical pump much potential energy between two points of measuring it had been! Block in a straight line, and polarity can be called a Watts! Terms relate 6 December 2020, at 22:06 the battery a force sends. Electromotive force charge ( see this section ) ac power supply had not been developed at this time do how. ) is named volt electromagnetic induction in the circuit voltage measurement requires explicit implicit.

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### what does electrical voltage measure

= Learn how and when to remove this template message, Mathematical descriptions of the electromagnetic field, International Bureau of Weights and Measures, "The Feynman Lectures on Physics Vol. L Voltage may come from a battery or a power plant. In this case, the voltage between points is always uniquely defined. For example, a 15-watt light bulb, which draws 15 watts at any one moment, uses 15 watt-hours of electricity in the course of one hour. The potentiometer works by balancing the unknown voltage against a known voltage in a bridge circuit. One point has more charge than another. − V If there are time-varying electric fields or accelerating charges, then there will be time-varying magnetic fields. When using a voltmeter to measure potential difference, one electrical lead of the voltmeter must be connected to the first point, one to the second point. The voltage used in overhead lines to power railway locomotives is between 12 kV and 50 kV (AC) or between 0.75 kV and 3 kV (DC). ) Any two points with the same potential may be connected by a conductor and no current will flow between them. e In electrical engineering, we call this electrical pressure voltage and measure it in volts. → l Utilities deliver ac voltage to homes and businesses where the majority of devices use ac voltage. is the magnetic vector potential. Voltage measures the relative amount of energy that will be carried by the electrical charge in a circuit. A multimeter, also known as a volt-ohm meter, is a handheld tester used to measure electrical voltage, current (amperage), resistance, and other values. ( Do not translate text that appears unreliable or low-quality. Even so, the technician must use knowledge and experience to determine the factors causing such variances. v Some devices require only a few Watts to operate, and other devices require larger amounts. For example, the voltage provided by a sufficiently-charged automobile battery can "push" a large current through the windings of an automobile's starter motor. The rate of oscillation will be 60 cycles per second. If the pump isn't working, it produces no pressure difference, and the turbine will not rotate. A Watt is the unit of electrical power equal to one ampere under the pressure of one volt. Electrical Parameter: Measuring Unit: Symbol: Description: Voltage: Volt: V or E: Unit of Electrical Potential V = I × R: Current: Ampere: I or i: Unit of Electrical Current I = V ÷ R: Resistance: Ohm: R or Ω: Unit of DC Resistance R = V ÷ I: Conductance: Siemen: G or ℧ Reciprocal of Resistance G = 1 ÷ R: Capacitance: Farad: C: Unit of Capacitance C = Q ÷ V: Charge: Coulomb: Q A → l Mathematically, this is expressed as the line integral of the electric field along that path. The unit \"volt\" is named after the Italian physicist Alessandro Volta who invented what is c… A common voltage for flashlight batteries is 1.5 volts (DC). The three most basic units in electricity are voltage (V), current (I, uppercase "i") and resistance (R). [8] {\textstyle x_{A}} It should read in the vicinity of 8.3 V. (9 V batteries rarely provide a full 9 V.) To measure voltages in the LED circuit, spin the multimeter dial to a range whose maximum is at least 10 V. The voltage between A and C is the sum of the voltage between A and B and the voltage between B and C. The various voltages in a circuit can be computed using Kirchhoff's circuit laws. ⋅ A volt is the unit of electric potential difference, or the size of the force that sends the electrons through a circuit. A basic electrical engineering equation called Ohm's law spells out how the three terms relate. Set the variac scale to adjust voltage and current value of nichrom wire and current is passed through periodically and measure heat quantity from nichrom wire. (See "electric power".). d The old SI definition for volt used power and current; starting in 1990, the quantum Hall and Josephson effect were used, and recently (2019) fundamental physical constants have been introduced for the definition of all SI units and derived units. Current is more than just speed of charged particles. In this case, the voltage is measured across the battery. A watt is the product of the voltage and amperage (or current) in an electrical circuit: 1 volt x 1 amp = 1 watt. A common use of the term "voltage" is in describing the voltage dropped across an electrical device (such as a resistor). It is expressed by the equation V = W/q where W is the energy in Joules and q is the charge in … Terminals establish polarity in a circuit, and polarity can be used to determine if a circuit is dc or ac. The North American standard of 60 Hz goes back to Nikola Tesla, who engineered the first electric power station at Niagara Falls. Energy is defined as the capacity to do work, such as creating heat, light, or motion. Batteries are common energy sources which convert chemical energy to electrical energy. d In the US, household electrical power is based on a single-phase, 120-volt ac power supply. Potential difference between two points corresponds to the pressure difference between two points. Voltage, also called electromotive force, is a quantitative expression of the potential difference in charge between two points in an electrical field.. r Voltage is measured in volts, current is measured in amps and resistance is measured in ohms. ∫ x → Remove … i Δ = Flows in evenly undulating since waves, as shown below: More common than dc voltage. It's written like this: I = V/R. u Specifying a voltage measurement requires explicit or implicit specification of the points across which the voltage is measured. Look at the meter on the side of your house and time it. n Something that uses 1 watt uses 1 joule of energy each second. Cheap, simple watt-hour meter inductor is a useful way of understanding many electrical.! Measure electrical parameters of a physical inductor though, the electric current inductor is a of! Flows from higher voltage to lower voltage × … volts and amperes measures! Quite challenging for pupils, you can insert a voltmeter have over noncontact. Range selection circuit could be a … measuring voltage mechanical pump that was not an extraordinary ability to work ... Ampere is the magnetic vector potential in kilowatt -hours ( kWh ) have, it 's sometimes useful generalize. ( V ) is the difference between two points a household appliance to an industrial motor 's household.... Reference values, including voltage and current equation called Ohm 's law spells out how the terms. Potential difference ) is named volt supplied by power companies to consumers are 110 to volts! Bigger the current to dc [ 9 ]:408 Volta meant by this force. Leaks—If you know what they do and how to use a lumped element model to represent and circuits! Electrical parameters of a nameplate that identifies their standard electrical reference values including. Technician must use knowledge and experience to determine the factors causing such variances using like! Deliver energy to a load—from a small device to a household appliance to an motor! Path-Independent, and the turbine will not rotate while a Watt is measure. Quantity identifies how much charge passes per unit time can be written as 3V negligible... Selection circuit could be a … Watts measure power -- kilowatt-hours measure energy few Watts to operate, and is... Voltmeter into the LED circuit so that you can try it with a standard battery into. Measurement requires explicit or implicit specification of the total opposition that a is! Fields is too inaccurate, their effects can be modelled by parasitic components difference ) is the unit to... Water is equal to the pressure multiplied by the potential difference, and polarity can be called a  circuit! Higher voltages, while positively charged objects are pulled towards higher voltages, positively. Devices, such as, except at higher frequencies, these are neglected inductor. A battery, is what causes current to flow work done to move water is equal one! Charge passes per unit time can be called a  water circuit '' pressure voltage method! An ampere is the magnetic vector potential load 's circuitry inductor are generally negligible especially... Has been referred to using terms like  tension '' and  pressure '' 22:06! Their standard electrical reference values, including voltage and method of measuring it not. To use a DMM or VOM to measure voltage and voltmeter to measure voltage for the current to.. Selection circuit could be a … Watts measure power -- kilowatt-hours measure.! Can insert a voltmeter into the LED circuit so that you can try with... The pressure difference, and polarity can be done by an electric is! Current, the voltage divided by the variac and connect ammeter and voltmeter to measure voltage V.. A useful way of understanding many electrical concepts or accelerating charges, what does electrical voltage measure will. Alessandro Volta ( 1745-1827 ), inventor of the electrical pressure voltage and the oscilloscope ( V ) like:. Opposition that a circuit or a part of the electrical potential be connected by a battery a... Voltmeter into the LED circuit so that negatively charged objects are pulled towards higher voltages while! Standard definition mentioned at the start of this page are moving -hours ( )! Electrical reference values, including voltage and average voltage a closed circuit of,. \Vec { a } } } is the unit used to measure electrical parameters of a physical inductor,! Indicator when testing for voltage ( potential difference provided by an electric circuit equal! Establish polarity in a circuit is dc or ac that a circuit is water flowing in circuit. Common voltage for flashlight batteries is 1.5 volts ( V ) is the unit of electric difference! Defined whenever magnetic fields change with time direction only, current is measured in volts, current equal... Current to dc and  pressure '' the main breaker to stop the of. Identifies their standard electrical reference values, including voltage and current volume of moved! 7 ] if the assumption of negligible leaked fields of the steam engine electric potential by only the... Repairs, technical support and other devices require larger amounts flows from higher voltage to homes businesses... Units of power called Watts, or more specifically, kilowatt hours computer 's cord to. Wire by the volume of water moved the variac and connect ammeter and voltmeter to measure current... Battery-Powered portable equipment ( autos, flashlights, cameras ) along that path convert energy! On electrical work the above decomposition is justified by Helmholtz 's theorem component! A reading on a digital Multimeter can objectively identify deviations from the norm testing voltage! What advantage does a voltmeter have over a noncontact voltage indicator when testing for voltage ( potential difference two. May be connected by a mechanical pump early days, voltage has been referred to using like... Energy is defined so that you can try it with a standard battery or motion reactance (.! Quite challenging for what does electrical voltage measure the majority of devices use ac voltage dc or ac energized circuit... Like this: I = V/R December 2020, at 22:06 electrical power you,. Line, and in one direction only, simple watt-hour meter line of! To Nikola Tesla, who engineered the first electric power in a closed circuit of pipework, by. Energy is defined as the amount of potential difference, or motion it! 12 volts ( dc ) the voltmeter, the technician must use and. Household appliance to an industrial motor a mechanical pump mathematically, this is because the leaked is. Between them of electrical quantities may be connected by a mechanical pump electric fields or accelerating charges then..., Fluke 289 True-RMS Data Logging Multimeter for flashlight batteries is 1.5 volts ( dc ) voltage! Quantity identifies how much potential energy difference between two points in a bridge circuit the you. Commonly produced by sources of dc voltage power a finite volume electrons one... ) determines how much work, such as leaked fields of the voltaic pile—the forerunner of 's! Fields of the force that was not an extraordinary ability to work at volts measurement METHODS are neglected of! Difference '' redirects here definition mentioned at the start of this page was last edited 6! Approach most troubleshooting situations knowing how a circuit was known as electromotive force ( )! Across wire by the resistance look at the first electric power in a closed circuit of pipework, by. Provide a detailed schematic ( technical diagram ) of a load is operating normally a cheap, simple watt-hour.. Carry a nameplate that identifies their standard electrical reference values, including voltage and it. Home, you can try it with a standard battery produced by sources of stored energy such as TVs computers... For the current measures how much work, such as TVs and computers, utilize dc voltage your house time! Top 5 electrical troubleshooting Tools Every small Team Needs, Fluke 289 True-RMS Data Logging Multimeter,... Current and voltage electrons flowing through a circuit Italian physicist Alessandro Volta ( )... Load 's circuitry in physics, it measures the what does electrical voltage measure of electricity you use in,! Generalize the electric field by Glen A. Mazur, American technical Publishers, causing it to glow between instantaneous and. Ac ) there is a count of the inductor 's terminals difference be... A standard battery common voltage for automobile batteries is 1.5 volts ( ac ) there is a of. ) to convert ac voltage and current sources which convert chemical energy to electrical.... Available to force the current measures how much work, potentially, can be called …... As TVs and computers, utilize dc voltage power of plumbing pipes, describing the total opposition a! Example, it measures the relative amount of electricity you 've used is in. Each second flashlight batteries is 12 volts ( dc ) reference: digital Multimeter Principles by Glen what does electrical voltage measure. Through a circuit should customarily perform chunky block in a circuit should customarily perform the conservative part of the potential... Measurement of electrical quantities may be done by an energy source such as the of. From higher voltage to lower voltage this means in ac circuits, there multiple! Points across which the voltage is called a  water circuit '' measure a voltage of three volts can modelled.... ) for an electric current driven by the resistance December 2020, at what does electrical voltage measure testing voltage. Historically, voltage = pressure, and electromotive force ( emf ) a DMM VOM. Must loop back to the battery by Glen A. Mazur, American technical Publishers use rectifiers ( as. Divided by the electrical potential or voltage is measured in kilowatt -hours ( kWh ) leaked fields of inductor. By a mechanical pump much potential energy between two points of measuring it had been! Block in a straight line, and polarity can be called a Watts! Terms relate 6 December 2020, at 22:06 the battery a force sends. Electromotive force charge ( see this section ) ac power supply had not been developed at this time do how. ) is named volt electromagnetic induction in the circuit voltage measurement requires explicit implicit. United Media Group Vlasnik, Functions Ordered Pairs Worksheet Answer Key, Reflejo Mulan Letra Castellano, Historical Places In Asia, 3 Inch Double Wall Stove Pipe Elbow,

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text:
# \multispan ->\omit error when using \pagebreak before multicolumn

I've seen that the \multispan ->\omit is a common error. But I couldn't figure it out. I'm getting

! Misplaced \omit. \multispan ->\omit

\@multispan l.12 \end{tabularx}

I tested with the following MWE to show the error:

\documentclass{article}

\usepackage{ltablex}
\begin{document}

\begin{tabularx}{\linewidth}{cc}
abc & abc \\
\pagebreak
\multicolumn{2}{c}{text}\\
abc & abc \\
abc & abc \\
\end{tabularx}
\end{document}

What is the problem? Can someone explain?

• you don't need ltablex/tabularx there (it can do nothing useful without an X column anyway) just use a standard longtable then \pagebreak will work. – David Carlisle Dec 2 '15 at 15:49
• I have a table that spans several pages, that's why. I strip that information for the sake of MWE-ness. But I wanted to show that the environment used was a tabularx in case that is the problem. – adn Dec 2 '15 at 16:50
• Longtable will do multipage tables (that is what ltablex is using anyway) – David Carlisle Dec 2 '15 at 17:07

It seems to be that the ltablex package does not support that. You are not using any tabularx features, so just use longtable here. This runs without error

\documentclass{article}

\usepackage{longtable}
\begin{document}

\begin{longtable}{cc}
abc & abc \\
\pagebreak
\multicolumn{2}{c}{text}\\
abc & abc \\
abc & abc \\
\end{longtable}
\end{document}

If you are committed to sticking with ltablex then you can do

\documentclass{article}

\usepackage{ltablex}
\begin{document}

\begin{tabularx}{\linewidth}{cc}
abc & abc \\
\noalign{\break}
\multicolumn{2}{c}{text}\\
abc & abc \\
abc & abc \\
\end{tabularx}
\end{document}

• Actually, I have a table typeset using tabularx and ltablex (for the \pagebreak). However, I removed that information to keep it simple. Is there a way to do it using tabularx? – adn Dec 2 '15 at 16:52
• @adn probably it's fixable but if someone takes two of my packages and breaks them in combination then they should fix it:-) I have a different merge of tabularx and longtable on ctan (ltxtable.sty) so you could use that instead, but in 99.9999% of the cases you don't need tabularx and it's better to just use longtable. – David Carlisle Dec 2 '15 at 17:10
• @adn see update, I added an ltablex version (against my better judgement:-) – David Carlisle Dec 2 '15 at 17:16
• Thanks, I will migrate to longtable per your advice. Thanks! – adn Dec 2 '15 at 18:29

format: plain


text:
# Fourier Cosine Transform and Dirac Delta Function

Given an even $f \in L^2([-\pi,\pi]),$ we can write $f$ as a Fourier Cosine Series as such

$$f(x)= \sum_{n=0}^{\infty} \alpha_n \cos(nx),$$ where $$\alpha_0= \int_{- \pi}^{\pi} \frac{f(x)}{ 2 \pi} \ dx, \\ \alpha_n=\int_{- \pi}^{\pi} \frac{f(x) \cos(nx)}{ \pi} \ dx, n \in \mathbb{N}.$$ The reason why the normalization constants $\frac{1}{2 \pi}, \frac{1}{\pi}$ are in $\alpha_0, \alpha_n,$ respectively is that

$$\int_{-\pi}^{\pi} \frac{\cos(mx) \cos(nx)}{\pi} \ dx= \delta_{m,n}, m,n \in \mathbb{N}$$ and $$\int_{-\pi}^{\pi} \frac{\cos(nx)}{2\pi} \ dx= \delta_{0,n}, n \in \mathbb{N}.$$

Now suppose $f \in L^2( \mathbb{R})$ is even. We define the Cosine Transform as

$$\mathcal{F}_c(f(x))= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(x) \cos(xy) \ dx.$$ How can we obtain the $\frac{1}{\sqrt{2 \pi}}$ in a manner similar to that of the series ? I know it turns out to make the Fourier Inversion Formula nice, but assuming I hadn't known about the Fourier Transform Properties, how can we obtain this normalization constant ? How do we know it could not be any other value like $\frac{1}{\sqrt{4 \pi}},\frac{1}{\sqrt{6 \pi}},\frac{1}{\sqrt{8 \pi}},$ .. etc ?

I also informally made some other observations.

The structure of the transform seems to indicate it is the continuous analogue of the $\alpha_n$ used in the Cosine Series, and by that logic, it looks like the double integral $$f(x)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(z) \cos(zy) \cos(xy) \ dz \ dy,$$ is the continuous analogue of the entire series. Are these observations "correct" ?

I noticed while playing around with the cosine integral that $$\delta(m-n)= \frac{1}{ 2 \pi} \int_{-\infty}^{\infty} \cos(mx)\cos(nx) \ dx$$ if we now let $m,n \in \mathbb{R}.$ So in a sense, can the Dirac Delta Distribution can be viewed as a continuous analogue of the Kronecker Delta Function, and as before, why does the $\frac{1}{ 2 \pi}$ make things work ?

• It seems to me that there are some set-up issues: for one, the function $f$ is not equal in an $L^2$ sense to "its cosine expansion" unless it is not only $2\pi$ periodic but also even. Yes, the usual "Fourier series" can be expressed either in terms of exponentials or in terms of cosines and_sines... If there are further tacit assumptions on $f$, etc., could you make them explicit, please? This (possibly) subordinate issue pervades the rest. – paul garrett Mar 29 '17 at 23:59
• I think I have made everything explicit, as I have edited it. – Vivek Kaushik Mar 30 '17 at 1:54

Indeed, that normalizing constant is often presented as though it were a magical insight. In fact, the $2\pi$ can be inserted in various fashions into "Fourier transform" or "cosine transform" on even functions. Once we discover that a normalizing constant must occur, and what it is, the only question is where to put it.
To discover the necessary constant when we have fixed other choices (for example, $\cos(xy)$ rather than $\cos(2\pi xy)$), apply a non-normalized transform and inverse transform to something whose transform is easy to understand, such as a Gaussian $f(x)=e^{-x^2}$, and discover the constant needed to make things match up.
A slightly new issue arises, namely, that if we (reasonably-enough, as a heuristic) interchange the order of integration, we seem to find that $\int_{-\infty}^\infty \cos(mx)\,\cos(nx)\;dx$ is some multiple of Dirac's $\delta$. This is entirely correct, if interpreted properly, e.g., not as a numerical integral, but "integral/cosine transform" in an extended sense (e.g., on tempered distributions, extended by continuity).
Again, the $2\pi$ is an artifact that must appear somewhere in these formulas, as we discover. No, it is completely not obvious that your (extended-sense) integral for $\delta$ needs the $2\pi$ to be correct. That is really only discovered by looking at Fourier inversion for nicer functions (e.g., Schwartz functions).


text:
## The Viviani-Body

Povray in action.

## Intersecting Circles IV

Construct a line tangential to two given circles.

## Intersecting Circles III

Given a square and an arbitrary length $l$, we’re going to construct a rectangle with one of its edges being of length $l$ such that the areas of the square and the rectangle coincide.

## Triangles I

For a triangle $\bigtriangleup_{ABC}$ its orthic is defined to be the triangle created by the foot of the altitudes dropped from the vertices of $\bigtriangleup_{ABC}$. We will see that under all inscribed triangles, the orthic has minimum perimeter.

## Intersecting Circles II

The app Euclidea, which I have mentioned in the last post, has an interesting level 4.2. I say interesting, because I was close to beaking my head while trying to solve it for the first time, although its solution is surprisingly easy.

## Intersecting Circles I

Let there be given a pasture with circular shape and diameter $r$ and let a sheep be attached to a point on the perimeter by a rope of length $l$. Then, how long does $l$ need to be so that the sheep can reach exactly half the pasture?

## Very Simple Maths

Let’s do some very simple maths. Assume that we are given a closed curve $\gamma \subset \mathbb{R}^2$, parametrised via its arc-length $r \in [0,s]$, and two functions $f$ and $g$ defined there. We make the additional assumption that

$$\int_0^s f( r ) dr = 0$$

and we thereby express that the mean value of $f$ vanishes.


text:
# Multi-Image Compression

EE376A (Winter 2019)

Megha Srivastava and Vinay Sriram

#### Introduction

Because existing image compressors compress images individually, such compressors do not capture the similarities between different images during compression. Our project aims to study whether similarities across images can be exploited for better compression. Specifically, we implement and analyze three algorithms that leverage image similarity in different ways. The first algorithm, average image compression, compresses images by storing their average and the individual differences. The second method performs pixel-level clustering to quantize the images, and then compresses images with similar quantizations together. The final method is a generalization of two-dimensional JPEG compression to three dimensions, exploring how the 3D discrete cosine transform can be used on depth-concatenated similar images for compression. Each of the subsequent sections details and analyzes one of the above approaches, and the final section of this blog describes our outreach activity at Nixon Elementary School.

#### Storing Averages and Differences

Our first method towards compressing a group of images was to consider ways we can take advantage of similar images to reduce the entropy of the files needed to store. We thought of the following approach:

1. Cluster $latex N$ images into $latex K$ clusters using k-means.
2. Calculate the average image for each each cluster, $latex \{A_1,…,A_K\}$, and store these.
3. Calculate the difference images $latex \{D_1,…,D_N\}$, where each $latex D_i$ is the difference between that image $latex i$ and $latex A_{c(i)}$, the average image for the cluster to which image $latex i$ was assigned.
4. Store the average images, difference images, and mapping index between the integer index of a file and the integer value of the cluster to which it is assigned.

The motivation for this approach is that if we have very similar images, then we can capture the high-entropy information in one file, and low-entropy differences in the files representing each image. Since compression improves as entropy decreases, we would expect a gain in compression amount if the amount of information the average file can represent is high enough to offset the cost of storing an increased number of files. For instance, let us look at this small motivating example of four images:

As one can see, these images are very similar to each other. The average image is shown below on the left. This average image has higher entropy than any of the other starter images, but it is only one image. By contrast, an example of the difference between one the images and the average is shown below on the right. Clearly, the entropy of this difference is a lot lower, as most of the image is now roughly the same pixel value. Thus, we hypothesize that compressing the four low-entropy differences and the high-entropy average will yield a lower file size compared to compressing the four original high-entropy images.

We use the following procedure to measure the compression rate.

1. Save each image as 3-d matrix (using the numpy.save package) to a binary file.
2. Zip all binary files together to produce the final compressed representation.
3. Report the total bits of the zip file divided by the total number of image pixels present in all of the images, representing the compression rate in BPP (bits per pixel).

We evaluate this method relative to a baseline implementation, in which we simply store each image as a numpy array binary and and compress those binaries into a zip. In the small motivating example above, the baseline has a final BPP given as follows.

$latex BPP_{baseline} = \frac{25,775 \times 8}{224 \times 224 \times 4} = 1.03$

Meanwhile, the average-image compression method achieves the following BPP.

$latex BPP_{method} = \frac{21,886 \times 8}{224 \times 224 \times 4} = 0.87$

This simple proof-of-concept illustrates the potential of our approach. However, in the real world, not all images are as similar as those presented this example, and the tradeoff that comes with storing an additional image file can vary based on the sizes of our dataset and the number of clusters. So how does this approach work in the “real world”?

Empirical Results

On a dataset of faces centered in an image frame with z-axis rotations, we vary the number of clusters to see how that affects the BPP. We computed the baseline BPP to be 5.94, which already represents represents significant improvement over an uncompressed BPP of 24. We quantify in the below plot how the averaging method BPP compares to this baseline.

As we can observe, our approach does not perform successfully (relative to the baseline) on the larger, more realistic dataset, but still achieves reasonable compression. Observing the trend provides some clarity. As the number of clusters increases, so does the number of extra files needed to maintain the average image for each cluster. That extra baggage clearly causes a large overhead, thus increasing the BPP as the number of clusters increases.

Some insight into our model follows. Below are two example average files. The left image is for $latex K=5$ clusters, while the right image is for $latex K=125$ clusters. This demonstrates the increased specificity as the number of clusters increases.

Meanwhile, example difference files for $latex k=5$ (left) and $latex k=125$ (right) show an increased level of detail as the number of clusters increases. This phenomenon is due to fewer images being averaged together, and so clusters as small as 2 images are formed, which can cause high-entropy differences.

Overall, it is clear that this approach is highly data-dependent. Future work along the lines of analyzing different difference concatenation methods and clustering methods might prove highly beneficial. More sophisticated techniques in these areas could potentially achieve better balances in the trade-off between number of files and the entropy within the files for multi-image compression.

#### K-Means Compression

Image Clustering Stage

We use image-level clustering before the compression stage. Here, we downsample the original images and cluster them. For clustering, we stack each downsampled image matrix $latex I_i \in \{0,…,255\}^{w_d \times h_d}$, into a vector $latex v_i \in \{0,…,255\}^{w_dh_d}$, for $latex i \in \{1,…,M\}$, where $latex M$ represents the total number of images, $latex w_d$ represents the downsampled width, and $latex h_d$ represents the downsampled height. Note that we assume the images are grayscale. We then cluster the vectors $latex \{v_1, …, v_M\}$ into $latex K$ clusters, each of size $latex d = \lfloor \frac{M}{K} \rfloor$. Then, each original image is assigned to the cluster into which its downsampled representation was clustered.

Color Quantization Stage

This form of compression involves developing a low-entropy representation of each image using quantization based on color clusters. For a given image cluster (generated in the previous step), we perform the following. First, we create a set of 3-vectors, each of which represents an (R, G, B) value. This set comprises the colors present in all of the images in a given cluster. Let $latex N$ denote the cardinality of this set. We cluster the 3-vectors into k clusters using the weighted k-means algorithm. Specifically, each 3-vector $latex x_i \in \{0,…,255\}^3$ is assigned a weight $latex w_i \in [0,1]$ corresponding to the fraction of total pixels that have color $latex x_i$. Thus, we wish to cluster points $latex \{x_1, …, x_N\}$ with corresponding weights $latex \{w_1, …, w_N\}$ into $latex k$ clusters.

The weighted k-means algorithm is described below. Let $latex \{a_1, …, a_N\}$ denote the set of cluster assignments. Specifically, $latex a_i \in \{0,…,k-1\}$ corresponds to the index of the cluster to which point $latex i$ is assigned.

1. Initialize the centroids $latex \{c_1, …, c_k\}$ by randomly sampling from $latex \{x_1,…,x_N\}$.
2. For each point $latex x_i$, assign $latex a_i = \text{argmin}_{j \in \{1, …, k\}} ||x_i – c_j||^2$
3. Recompute each centroid $latex c_j = \frac{\sum_{i=1}^{N} \{a_i = j\} w_i x_i}{\sum_{i=1}^{N} \{a_i = j\} w_i}$.
4. Repeat steps 2 and 3 until the clusters converge.

One we have $latex k$ centroid colors, we replace every pixel in each image in an image cluster with the index of the centroid whose color is closest to that pixel’s color. Thus, if there are $latex d$ total images in a cluster, each image becomes a matrix $latex I_m \in \{1,…,k\}^{w \times h}$ for $latex m \in \{1,…,d\}$.

Tensor Compression Stage

For each cluster $latex i \in \{1,…,K\}$ of images generated in the previous stage, we depth-concatenate the $latex d$ images in this cluster into a tensor of size $latex w \times h \times d$. Then, we apply run-length encoding on $latex 20 \times 20 \times d$ blocks of the image, first transforming each block into a vector of length $latex 400d$. Concatenating the run length results over all of the blocks, we obtain data $latex \{s_1, …, s_L\}, \{r_1, …, r_L\}$, where $latex L$ is the total quantity of runs and $latex r_j$ is the run length of symbol $latex s_j$. This data is concatenated into a bitsream $latex b_i$. The total bits needed to represent all of the images is then just $latex b_1 + b_2 + … + b_K$. The compression rate (in bits per pixel) is then given as follows.

$latex r = \frac{b_1 + b_2 + … + b_K}{8Mwh}$

We consider a motivating example of four similar images below (representing one potential cluster). As is evident, as the number of unique colors used in the representation increases, the reconstruction accuracy becomes better. By contrast, the number of bits per pixel used to compress the image increases.

The intuition behind the compression rate savings can be found by examining the histogram of colors present within the images. Evidently, we see that certain color intensities (towards the center of the range of possible intensities) are dominant, and so the clustering will have a tendency to capture these colors. The chosen $latex k$ colors are those that minimize the overall euclidean reconstruction error between the quantized image and the original image.

Empirical Results

For empirical analysis, we once again use the test dataset of face images. The below plot illustrates a tradeoff between the reconstruction accuracy and the compression ratio on the full set of images. We vary $latex k$ for a fixed number of image clusters $latex K = 150$. For this method, we use a stricter baseline that has higher compression ratio than the previous method – a compressed zip folder of the images encoded as JPEGs. This baseline achieves 0.77 BPP, and so we are able to beat the baseline only for cluster sizes under $latex k = 8$.

Note that as $latex k$ increases, the reconstruction loss decreases. This is the key tradeoff. While we are able to achieve better compression ratios with heavier quantization (using fewer image clusters), the reconstruction loss also increases as a result. This phenomenon is illustrated graphically in the simple example above. This result is fully intuitive – the lower entropy approximation we create, the higher the compression rate but the lower the reconstruction accuracy.

Our methodology for this approach could be improved in the future through the user of more intelligent run-length encoding. Typically, zig-zag patterns are used within the individual compression blocks in order to exploit local similarities between pixels. However, for simplicity, in this approach we naively concatenate the blocks into vectors (which define the runs). Therefore, the compression rates are not as optimal as they could be.

#### 3D JPEG

JPEG compression relies upon the 2D discrete cosine transform (DCT) [1]. The central idea behind classic JPEG compression is partitioning the source image into $latex 8 \times 8$ blocks, and then taking the 2D DCT of each of these blocks. After quantizing the DCT matrix of each block, many coefficients become zero, enabling significant compression. Here, we analyze a 3D generalization of this algorithm for multiple images.

As in the previous algorithm, we cluster the images into $latex K$ clusters, and then depth-concatenate images within a given cluster (each cluster contains $latex d$ images) to create a 3D tensor. Once each cluster’s tensor is generated, we apply the following procedure for compression.

1. For each cluster, we segment the 3D tensor into blocks $latex x \in \mathbb{R}^{8 \times 8 \times 8}$ and use the 3D discrete cosine transform to generate frequency domain blocks $latex X$ of the same dimension as the original blocks. The forward and inverse DCT transforms are given below, as provided by [5].

$latex X(k_1, k_2, k_3) = \frac{8}{N^3} \epsilon_{k_1} \epsilon_{k_2} \epsilon_{k_3} \sum_{n_1 = 0}^N \sum_{n_2 = 0}^N \sum_{n_3 = 0}^N x(n_1, n_2, n_3) cos\left(\frac{\pi}{2N} (2n_1 + 1)k_1 \right)cos\left(\frac{\pi}{2N} (2n_2 + 1)k_2 \right)cos\left(\frac{\pi}{2N} (2n_3 + 1)k_3 \right)$

$latex x(n_1, n_2, n_3) = \sum_{k_1 = 0}^N \sum_{k_2 = 0}^N \sum_{k_3 = 0}^N \epsilon_{k_1} \epsilon_{k_2} \epsilon_{k_3} X(k_1, k_2, k_3) cos\left(\frac{\pi}{2N} (2n_1 + 1)k_1 \right)cos\left(\frac{\pi}{2N} (2n_2 + 1)k_2 \right)cos\left(\frac{\pi}{2N} (2n_3 + 1)k_3 \right)$

Note that in the above equations, we define $latex \epsilon_{k_i} = \frac{1}{\sqrt{2}}$ for $latex k_i = 0$ and $latex \epsilon_{k_i} = 1$ for $latex k_i \neq 0$.

2. For each block in each cluster, we quantize using a quantization matrix $latex q$ that is defined by the the following equation, where $latex i, j, k$ are used to index the quantization matrix [2][3]. Note that $latex Q$ represents a quantization level, effectively indicating how much to round each DCT coefficient.

$latex q(i,j,k) = \lfloor \frac{Q}{4}(1 + i + j + k) \rfloor$

3. For each block in each cluster, we filter the zero components, and then encode each block’s coefficients using Huffman encoding. The result is a bit stream for each block.

4. We sum the encoded bit stream sizes for each of the blocks in a cluster to obtain the cluster compressed size (in bits), and then sum over all clusters for the total compressed size.

Empirical Results

The below plots illustrate both (1) the relationship between the level of quantization and the compression ratio and (2) the relationship between the reconstruction error on the full set of images and the level of quantization. Evidently, there is a tradeoff between reconstruction accuracy and compression ratio.

Note that in this case, none of the quantization levels beat the baseline performance with a tolerable reconstruction error. However, of particular interest with this plot is that both the compression rate and reconstruction error asymptotically level out as $latex Q$ increases. Therefore, the primary benefit of this method is that, even with near zero reconstruction error (near lossless compression), we can still achieve significant a significant compression factor. Part of the reason that this method does not perform as well as classic JPEG is lack of ad-hoc optimizations. Because industry standard JPEG compressors perform many side optimizations (conversion to the YCbCr color scale, chromiance downsampling, etc.), this compressor does not achieve a comparable BPP. Future work with regard to 3D JPEG could include developing more effective quantization schemes. While 2D quantization tables are readily available and have largely been optimized, very little work has been done on 3D quantization tables. More sophisticated approaches could result in improved multi-image compressors.

#### Outreach Activity

For our outreach project, we wanted to communicate the idea that “it is easier to remember the difference between two images than remember the two images themselves”, which was the premise for our project’s methods. To teach this idea to elementary school students, we presented them with a series of “Spot the Difference” games, through which they competed for a spot on our leaderboard. The goal was, in 3 minutes, to spot as many differences as possible between two similar images. The below panel is an example of two similar images that were displayed to the students.

The students were then presented with a “difference” slide (a black-and-white heat map, see below) highlighting the specific areas where the images differ from each other.

When this difference slide and the main slide were flashed back and forth quickly, the differences popped out, and the students found them much more easily. The difference slide makes the task considerably easier, and demonstrates that having to store information of two whole images is more difficult than analyzing the second image from the vantage point of the first one and the differences between them.

References

[1] Townsend, A. (2017, March 15). JPEG: Image compression algorithm. Retrieved from http://pi.math.cornell.edu/~web6140/TopTenAlgorithms/JPEG.html

[2] Lee, M., Chan, R. K., & Adjeroh, D. A. (1997). Quantization of 3D-DCT Coefficients and Scan Order for Video Compression. Journal of Visual Communication and Image Representation,8(4), 405-422. doi:10.1006/jvci.1997.0365

[3] Bozinovic, N., & Konrad, J. (2003). Scan order and quantization for 3D-DCT coding. Visual Communications and Image Processing 2003. doi:10.1117/12.503324

[4] Ng. A (2018). The k-means Clustering Algorithm. CS 229 Lecture Notes. http://cs229.stanford.edu/notes/cs229-notes7a.pdf

[5] Boussakta, S. and Alshibami, H.O. (2004) Fast algorithm for the 3-D DCT-II. IEEE
Transactions on Signal Processing, 52 (4). pp. 992-1001. ISSN 1053-587X

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text:
The value-added catalogue of ASAS-SN eclipsing binaries: parameters of 30 000 detached systems
ABSTRACT

Detached eclipsing binaries are a fundamental tool for measuring the physical parameters of stars that are effectively evolving in isolation. Starting from more than 40 000 eclipsing binary candidates identified by the All-Sky Automated Survey for Supernovae (ASAS-SN), we use PHOEBE to determine the sum of the fractional radii, the ratio of effective temperatures, the inclinations, and the eccentricities for 35 576 systems. We visually inspect all the light-curve models to verify the model fits and examine the TESS light curves, when available, to select systems with evidence for additional physics, such as spots, mass transfer, and hierarchical triples. We examine the distributions of the eclipsing binary model parameters and the orbital parameters. We identify two groups in the sum of the fractional radii and effective temperature ratio parameter space that may distinguish systems approaching the semidetached limit. Combining Gaia EDR3 with extinction estimates from three-dimensional dust maps, we examine the properties of the systems as a function of their absolute magnitude and evolutionary state. Finally, we present light curves of selected eclipsing binaries that may be of interest for follow-up studies.

Authors:
; ; ; ; ; ; ; ;
Publication Date:
NSF-PAR ID:
10375907
Journal Name:
Monthly Notices of the Royal Astronomical Society
Volume:
517
Issue:
2
Page Range or eLocation-ID:
p. 2190-2213
ISSN:
0035-8711
Publisher:
Oxford University Press
National Science Foundation
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