Voltage

The voltage between two points is a short name for the electrical force that would drive an electric current between those points. Specifically, voltage is equal to energy per unit charge.^[1] In the case of static electric fields, the voltage between two points is equal to the electrical potential difference between those points. In the more general case with electric and magnetic fields that vary with time, the terms are no longer synonymous. ^[2]

Electric potential is the energy required to move a unit electric charge to a particular place in a static electric field. ^[3]

Voltage can be measured by a voltmeter; real voltmeters approximate ideal voltmeters. The unit of measurement is the volt.

Definition

The voltage between two ends of a path is the total energy required to move a small electric charge along that path, divided by the magnitude of the charge. Mathematically this is expressed as the line integral of the electric field and the time rate of change of magnetic field along that path. In the general case, both a static (unchanging) electric field and a dynamic (time-varying) electromagnetic field must be included in determining the voltage between two points.

Historical definitions

Historically this quantity has also been called "tension"^[4] and "pressure". Pressure is now obsolete but tension is still used, for example within the phrase "High Tension" (HT) which is commonly used in thermionic valve (vacuum tube) based electronics.

Hydraulic analogy

A simple analogy for an electric circuit is water flowing in a closed circuit of pipework, driven by a mechanical pump. This can be called a water circuit. Voltage difference between two points corresponds to the water pressure difference between two points. If there is a water pressure difference between two points, then water flow (due to the pump) from the first point to the second will be able to do work, such as driving a turbine. In a similar way, work can be done by the electric current driven by the voltage difference due to an electric battery: for example, the current generated by an automobile battery can drive the starter motor in an automobile. If the pump isn't working, it produces no pressure difference, and the turbine will not rotate. Equally, if the automobile's battery is flat, then it will not turn the starter motor.

This water flow analogy is a useful way of understanding several electrical concepts. In such a system, the work done to move water is equal to the pressure multiplied by the volume of water moved. Similarly, in an electrical circuit, the work done to move electrons or other charge-carriers is equal to "electrical pressure" (an old term for voltage) multiplied by the quantity of electrical charge moved. Voltage is a convenient way of measuring the ability to do work. In relation to "flow", the larger the "pressure difference" between two points (voltage difference or water pressure difference) the greater the flow between them (either electric current or water flow).

There are also analogs for inductance and capacitance using differential equations.

Inductors and pipes or inductors and closed tanks have the same form of differential equations.

Simple applications

Common usage (that "voltage" usually means "voltage difference") is now resumed. Obviously, when using the term "voltage" in the shorthand sense, one must be clear about the two points between which the voltage is specified or measured. When using a voltmeter to measure voltage difference, one electrical lead of the voltmeter must be connected to the first point, one to the second point.

Voltage between two stated points

A common use of the term "voltage" is in specifying how many volts are dropped across an electrical device (such as a resistor). In this case, the "voltage", or more accurately, the "voltage drop across the device", can usefully be understood as the difference between two measurements. The first measurement uses one electrical lead of the voltmeter on the first terminal of the device, with the other voltmeter lead connected to ground. The second measurement is similar, but with the first voltmeter lead on the second terminal of the device. The voltage drop is the difference between the two readings. In practice, the voltage drop across a device can be measured directly and safely using a voltmeter that is isolated from ground, provided that the maximum voltage capability of the voltmeter is not exceeded.

Two points in an electric circuit that are connected by an "ideal conductor," that is, a conductor without resistance and not within a changing magnetic field, have a voltage difference of zero. However, other pairs of points may also have a voltage difference of zero. If two such points are connected with a conductor, no current will flow through the connection.

Addition of voltages

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.

When talking about alternating current (AC) there is a difference between instantaneous voltage and average voltage. Instantaneous voltages can be added for direct current (DC) and AC, but average voltages can be meaningfully added only when they apply to signals that all have the same frequency and phase.

Useful formulas

DC (Direct current) circuits

V=I\times R\;\;\mathrm {(Ohm's} \;\mathrm {Law)}

P=IV=I^{2}R={\frac {V^{2}}{R}}\,\!

V={\sqrt {PR}}

where V = voltage difference (SI unit: volt), I = electric current (SI unit: ampere), R = resistance (SI unit: ohm), P = power (SI unit: watt).

AC (Alternating current) circuits

V={\frac {P}{I\;\cos \phi }}

V={\frac {\sqrt {P\;Z}}{\sqrt {\cos \phi }}}\!\

V={\frac {I\;R}{\cos \phi }}

Where V=voltage, I=current, R=resistance, P=true power, Z=impedance, φ=phase difference between I and V.

AC conversions

V_{avg}=0.637\,V_{pk}={\frac {2}{\pi }}V_{pk}={\frac {\omega }{\pi }}\int _{0}^{\pi /\omega }V_{pk}\sin(\omega t-kx){\rm {d}}x\!\

V_{rms}=0.707\,V_{pk}={\frac {1}{\sqrt {2}}}V_{pk}=V_{pk}{\sqrt {\langle \sin ^{2}(\omega t-kx)\rangle }}\!\

V_{pk}=0.5\,V_{ppk}\!\

V_{avg}=0.319\,V_{ppk}\!\

V_{rms}=0.354\,V_{ppk}={\frac {1}{2{\sqrt {2}}}}V_{ppk}\!\

V_{avg}=0.900\,V_{rms}={\frac {2{\sqrt {2}}}{\pi }}V_{rms}\!\

Where V_pk=peak voltage, V_ppk=peak-to-peak voltage, V_avg=average voltage over a half-cycle, V_rms=effective (root mean square) voltage, and we assumed a sinusoidal wave of the form $V_{pk}\sin(\omega t-kx)$ , with a period $T=2\pi /\omega$ , and where the angle brackets (in the root-mean-square equation) denote a time average over an entire period.

Total voltage

Voltage sources and drops in series:

V_{T}=V_{1}+V_{2}+V_{3}+...+V_{n}\!\

Voltage sources and drops in parallel:

V_{T}=V_{1}=V_{2}=V_{3}=...=V_{n}\!\

Where $n\!\$ is the nth voltage source or drop

Voltage drops

Across a resistor (Resistor R):

V_{R}=IR_{R}\!\

Across a capacitor (Capacitor C):

V_{C}=IX_{C}\!\

Across an inductor (Inductor L):

V_{L}=IX_{L}\!\

Where V=voltage, I=current, R=resistance, X=reactance.

Measuring instruments

Instruments for measuring voltage differences include the voltmeter, the potentiometer, and the oscilloscope. The voltmeter works by measuring the current through a fixed resistor, which, according to Ohm's Law, is proportional to the voltage difference across the resistor. The potentiometer works by balancing the unknown voltage against a known voltage in a bridge circuit. The cathode-ray oscilloscope works by amplifying the voltage difference and using it to deflect an electron beam from a straight path, so that the deflection of the beam is proportional to the voltage difference.

Safety

Voltages as low as 50 volts can lead to a lethal electric shock under certain circumstances. Electrical safety is discussed in the articles on high voltage and electric shock.

References

^ "To find the electric potential difference between two points A and B in an electric field, we move a test charge q₀ from A to B, always keeping it in equilibrium, and we measure the work W_AB that must be done by the agent moving the charge. The electric potential difference is defined from V_B − V_A = W_AB/q₀" Halliday, D. and Resnick, R. (1974). Fundamentals of Physics. New York: John Wiley & Sons. p. 465.
^ Demetrius T. Paris and F. Kenneth Hurd, Basic Electromagnetic Theory, Mc Graw Hill, New York 1969, ISBN 0-48470-8 page 546
^ Griffiths, D. (1999). Introduction to Electrodynamics. Upper Saddle River, NJ: Prentice-Hall.
^ CollinsLanguage.com

External links

[1] "To find the electric potential difference between two points A and B in an electric field, we move a test charge q₀ from A to B, always keeping it in equilibrium, and we measure the work W_AB that must be done by the agent moving the charge. The electric potential difference is defined from V_B − V_A = W_AB/q₀" Halliday, D. and Resnick, R. (1974). Fundamentals of Physics. New York: John Wiley & Sons. p. 465.

[2] Demetrius T. Paris and F. Kenneth Hurd, Basic Electromagnetic Theory, Mc Graw Hill, New York 1969, ISBN 0-48470-8 page 546

[3] Griffiths, D. (1999). Introduction to Electrodynamics. Upper Saddle River, NJ: Prentice-Hall.

[4] CollinsLanguage.com

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