## Loi D Ohm Explication Essay

This article is about the law related to electricity. For other uses, see Ohm's acoustic law.

**Ohm's law** states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance,^{[1]} one arrives at the usual mathematical equation that describes this relationship:^{[2]}

where *I* is the current through the conductor in units of amperes, *V* is the voltage measured *across* the conductor in units of volts, and *R* is the resistance of the conductor in units of ohms. More specifically, Ohm's law states that the *R* in this relation is constant, independent of the current.^{[3]}

The law was named after the German physicist Georg Ohm, who, in a treatise published in 1827, described measurements of applied voltage and current through simple electrical circuits containing various lengths of wire. Ohm explained his experimental results by a slightly more complex equation than the modern form above (see History).

In physics, the term *Ohm's law* is also used to refer to various generalizations of the law originally formulated by Ohm. The simplest example of this is:

where **J** is the current density at a given location in a resistive material, **E** is the electric field at that location, and *σ* (sigma) is a material-dependent parameter called the conductivity. This reformulation of Ohm's law is due to Gustav Kirchhoff.^{[4]}

## History

In January 1781, before Georg Ohm's work, Henry Cavendish experimented with Leyden jars and glass tubes of varying diameter and length filled with salt solution. He measured the current by noting how strong a shock he felt as he completed the circuit with his body. Cavendish wrote that the "velocity" (current) varied directly as the "degree of electrification" (voltage). He did not communicate his results to other scientists at the time,^{[5]} and his results were unknown until Maxwell published them in 1879.^{[6]}

Francis Ronalds delineated “intensity” (voltage) and “quantity” (current) for the dry pile – a high voltage source – in 1814 using a gold-leaf electrometer. He found for a dry pile that the relationship between the two parameters was not proportional under certain meteorological conditions.^{[7]}^{[8]}

Ohm did his work on resistance in the years 1825 and 1826, and published his results in 1827 as the book *Die galvanische Kette, mathematisch bearbeitet* ("The galvanic circuit investigated mathematically").^{[9]} He drew considerable inspiration from Fourier's work on heat conduction in the theoretical explanation of his work. For experiments, he initially used voltaic piles, but later used a thermocouple as this provided a more stable voltage source in terms of internal resistance and constant voltage. He used a galvanometer to measure current, and knew that the voltage between the thermocouple terminals was proportional to the junction temperature. He then added test wires of varying length, diameter, and material to complete the circuit. He found that his data could be modeled through the equation

where *x* was the reading from the galvanometer, *l* was the length of the test conductor, *a* depended only on the thermocouple junction temperature, and *b* was a constant of the entire setup. From this, Ohm determined his law of proportionality and published his results.

Ohm's law was probably the most important of the early quantitative descriptions of the physics of electricity. We consider it almost obvious today. When Ohm first published his work, this was not the case; critics reacted to his treatment of the subject with hostility. They called his work a "web of naked fancies"^{[10]} and the German Minister of Education proclaimed that "a professor who preached such heresies was unworthy to teach science."^{[11]} The prevailing scientific philosophy in Germany at the time asserted that experiments need not be performed to develop an understanding of nature because nature is so well ordered, and that scientific truths may be deduced through reasoning alone.^{[12]} Also, Ohm's brother Martin, a mathematician, was battling the German educational system. These factors hindered the acceptance of Ohm's work, and his work did not become widely accepted until the 1840s. Fortunately, Ohm received recognition for his contributions to science well before he died.

In the 1850s, Ohm's law was known as such and was widely considered proved, and alternatives, such as "Barlow's law", were discredited, in terms of real applications to telegraph system design, as discussed by Samuel F. B. Morse in 1855.^{[13]}

While the old term for electrical conductance, the mho (the inverse of the resistance unit ohm), is still used, a new name, the siemens, was adopted in 1971, honoring Ernst Werner von Siemens. The siemens is preferred in formal papers.

In the 1920s, it was discovered that the current through a practical resistor actually has statistical fluctuations, which depend on temperature, even when voltage and resistance are exactly constant; this fluctuation, now known as Johnson–Nyquist noise, is due to the discrete nature of charge. This thermal effect implies that measurements of current and voltage that are taken over sufficiently short periods of time will yield ratios of V/I that fluctuate from the value of R implied by the time average or ensemble average of the measured current; Ohm's law remains correct for the average current, in the case of ordinary resistive materials.

Ohm's work long preceded Maxwell's equations and any understanding of frequency-dependent effects in AC circuits. Modern developments in electromagnetic theory and circuit theory do not contradict Ohm's law when they are evaluated within the appropriate limits.

## Scope

Ohm's law is an empirical law, a generalization from many experiments that have shown that current is approximately proportional to electric field for most materials. It is less fundamental than Maxwell's equations and is not always obeyed. Any given material will break down under a strong-enough electric field, and some materials of interest in electrical engineering are "non-ohmic" under weak fields.^{[14]}^{[15]}

Ohm's law has been observed on a wide range of length scales. In the early 20th century, it was thought that Ohm's law would fail at the atomic scale, but experiments have not borne out this expectation. As of 2012, researchers have demonstrated that Ohm's law works for silicon wires as small as four atoms wide and one atom high.^{[16]}

## Microscopic origins

Main articles: Drude model and Classical and quantum conductivity

The dependence of the current density on the applied electric field is essentially quantum mechanical in nature; (see Classical and quantum conductivity.) A qualitative description leading to Ohm's law can be based upon classical mechanics using the Drude model developed by Paul Drude in 1900.^{[17]}^{[18]}

The Drude model treats electrons (or other charge carriers) like pinballs bouncing among the ions that make up the structure of the material. Electrons will be accelerated in the opposite direction to the electric field by the average electric field at their location. With each collision, though, the electron is deflected in a random direction with a velocity that is much larger than the velocity gained by the electric field. The net result is that electrons take a zigzag path due to the collisions, but generally drift in a direction opposing the electric field.

The drift velocity then determines the electric current density and its relationship to * E* and is independent of the collisions. Drude calculated the average drift velocity from

*= −*

**p***e*where

**E**τ*is the average momentum, −*

**p***e*is the charge of the electron and τ is the average time between the collisions. Since both the momentum and the current density are proportional to the drift velocity, the current density becomes proportional to the applied electric field; this leads to Ohm's law.

## Hydraulic analogy

A hydraulic analogy is sometimes used to describe Ohm's law. Water pressure, measured by pascals (or PSI), is the analog of voltage because establishing a water pressure difference between two points along a (horizontal) pipe causes water to flow. Water flow rate, as in liters per second, is the analog of current, as in coulombs per second. Finally, flow restrictors—such as apertures placed in pipes between points where the water pressure is measured—are the analog of resistors. We say that the rate of water flow through an aperture restrictor is proportional to the difference in water pressure across the restrictor. Similarly, the rate of flow of electrical charge, that is, the electric current, through an electrical resistor is proportional to the difference in voltage measured across the resistor.

Flow and pressure variables can be calculated in fluid flow network with the use of the hydraulic ohm analogy.^{[19]}^{[20]} The method can be applied to both steady and transient flow situations. In the linear laminar flow region, Poiseuille's law describes the hydraulic resistance of a pipe, but in the turbulent flow region the pressure–flow relations become nonlinear.

The hydraulic analogy to Ohm's law has been used, for example, to approximate blood flow through the circulatory system.^{[21]}

## Circuit analysis

In circuit analysis, three equivalent expressions of Ohm's law are used interchangeably:

Each equation is quoted by some sources as the defining relationship of Ohm's law,^{[2]}^{[22]}^{[23]} or all three are quoted,^{[24]} or derived from a proportional form,^{[25]} or even just the two that do not correspond to Ohm's original statement may sometimes be given.^{[26]}^{[27]}

The interchangeability of the equation may be represented by a triangle, where V (voltage) is placed on the top section, the I (current) is placed to the left section, and the R (resistance) is placed to the right. The line that divides the left and right sections indicate multiplication, and the divider between the top and bottom sections indicates division (hence the division bar).

### Resistive circuits

Resistors are circuit elements that impede the passage of electric charge in agreement with Ohm's law, and are designed to have a specific resistance value *R*. In a schematic diagram the resistor is shown as a zig-zag symbol. An element (resistor or conductor) that behaves according to Ohm's law over some operating range is referred to as an *ohmic device* (or an *ohmic resistor*) because Ohm's law and a single value for the resistance suffice to describe the behavior of the device over that range.

Ohm's law holds for circuits containing only resistive elements (no capacitances or inductances) for all forms of driving voltage or current, regardless of whether the driving voltage or current is constant (DC) or time-varying such as AC. At any instant of time Ohm's law is valid for such circuits.

Resistors which are in *series* or in *parallel* may be grouped together into a single "equivalent resistance" in order to apply Ohm's law in analyzing the circuit.

### Reactive circuits with time-varying signals

When reactive elements such as capacitors, inductors, or transmission lines are involved in a circuit to which AC or time-varying voltage or current is applied, the relationship between voltage and current becomes the solution to a differential equation, so Ohm's law (as defined above) does not directly apply since that form contains only resistances having value R, not complex impedances which may contain capacitance ("C") or inductance ("L").

Equations for time-invariantAC circuits take the same form as Ohm's law. However, the variables are generalized to complex numbers and the current and voltage waveforms are complex exponentials.^{[28]}

In this approach, a voltage or current waveform takes the form , where *t* is time, *s* is a complex parameter, and *A* is a complex scalar. In any linear time-invariant system, all of the currents and voltages can be expressed with the same *s* parameter as the input to the system, allowing the time-varying complex exponential term to be canceled out and the system described algebraically in terms of the complex scalars in the current and voltage waveforms.

The complex generalization of resistance is impedance, usually denoted *Z*; it can be shown that for an inductor,

and for a capacitor,

We can now write,

where * V* and

*are the complex scalars in the voltage and current respectively and*

**I***is the complex impedance.*

**Z**This form of Ohm's law, with *Z* taking the place of *R*, generalizes the simpler form. When *Z* is complex, only the real part is responsible for dissipating heat.

In the general AC circuit, *Z* varies strongly with the frequency parameter *s*, and so also will the relationship between voltage and current.

For the common case of a steady sinusoid, the *s* parameter is taken to be , corresponding to a complex sinusoid . The real parts of such complex current and voltage waveforms describe the actual sinusoidal currents and voltages in a circuit, which can be in different phases due to the different complex scalars.

### Linear approximations

See also: Small-signal modeling and Network analysis (electrical circuits) § Small signal equivalent circuit

Ohm's law is one of the basic equations used in the analysis of electrical circuits. It applies to both metal conductors and circuit components (resistors) specifically made for this behaviour. Both are ubiquitous in electrical engineering. Materials and components that obey Ohm's law are described as "ohmic" ^{[29]} which means they produce the same value for resistance (R = V/I) regardless of the value of V or I which is applied and whether the applied voltage or current is DC (direct current) of either positive or negative polarity or AC (alternating current).

In a true ohmic device, the same value of resistance will be calculated from R = V/I regardless of the value of the applied voltage V. That is, the ratio of V/I is constant, and when current is plotted as a function of voltage the curve is *linear* (a straight line). If voltage is forced to some value V, then that voltage V divided by measured current I will equal R. Or if the current is forced to some value I, then the measured voltage V divided by that current I is also R. Since the plot of I versus V is a straight line, then it is also true that for any set of two different voltages V_{1} and V_{2} applied across a given device of resistance R, producing currents I_{1} = V_{1}/R and I_{2} = V_{2}/R, that the ratio (V_{1}−V_{2})/(I_{1}−I_{2}) is also a constant equal to R. The operator "delta" (Δ) is used to represent a difference in a quantity, so we can write ΔV = V_{1}−V_{2} and ΔI = I_{1}−I_{2}. Summarizing, for any truly ohmic device having resistance R, V/I = ΔV/ΔI = R for any applied voltage or current or for the difference between any set of applied voltages or currents.

There are, however, components of electrical circuits which do not obey Ohm's law; that is, their relationship between current and voltage (their I–V curve) is *nonlinear* (or non-ohmic). An example is the p-n junction diode (curve at right). As seen in the figure, the current does not increase linearly with applied voltage for a diode. One can determine a value of current (I) for a given value of applied voltage (V) from the curve, but not from Ohm's law, since the value of "resistance" is not constant as a function of applied voltage. Further, the current only increases significantly if the applied voltage is positive, not negative. The ratio *V*/*I* for some point along the nonlinear curve is sometimes called the *static*, or *chordal*, or DC, resistance,^{[30]}^{[31]} but as seen in the figure the value of total *V* over total *I* varies depending on the particular point along the nonlinear curve which is chosen. This means the "DC resistance" V/I at some point on the curve is not the same as what would be determined by applying an AC signal having peak amplitude ΔV volts or ΔI amps centered at that same point along the curve and measuring ΔV/ΔI. However, in some diode applications, the AC signal applied to the device is small and it is possible to analyze the circuit in terms of the *dynamic*, *small-signal*, or *incremental* resistance, defined as the one over the slope of the V–I curve at the average value (DC operating point) of the voltage (that is, one over the derivative of current with respect to voltage). For sufficiently small signals, the dynamic resistance allows the Ohm's law small signal resistance to be calculated as approximately one over the slope of a line drawn tangentially to the V-I curve at the DC operating point.^{[32]}

## Temperature effects

Ohm's law has sometimes been stated as, "for a conductor in a given state, the electromotive force is proportional to the current produced." That is, that the resistance, the ratio of the applied electromotive force (or voltage) to the current, "does not vary with the current strength ." The qualifier "in a given state" is usually interpreted as meaning "at a constant temperature," since the resistivity of materials is usually temperature dependent. Because the conduction of current is related to Joule heating of the conducting body, according to Joule's first law, the temperature of a conducting body may change when it carries a current. The dependence of resistance on temperature therefore makes resistance depend upon the current in a typical experimental setup, making the law in this form difficult to directly verify. Maxwell and others worked out several methods to test the law experimentally in 1876, controlling for heating effects.^{[33]}

## Relation to heat conductions

See also: Conduction (heat)

Ohm's principle predicts the flow of electrical charge (i.e. current) in electrical conductors when subjected to the influence of voltage differences; Jean-Baptiste-Joseph Fourier's principle predicts the flow of heat in heat conductors when subjected to the influence of temperature differences.

The same equation describes both phenomena, the equation's variables taking on different meanings in the two cases. Specifically, solving a heat conduction (Fourier) problem with *temperature* (the driving "force") and *flux of heat* (the rate of flow of the driven "quantity", i.e. heat energy) variables also solves an analogous electrical conduction (Ohm) problem having *electric potential* (the driving "force") and *electric current* (the rate of flow of the driven "quantity", i.e. charge) variables.

The basis of Fourier's work was his clear conception and definition of thermal conductivity. He assumed that, all else being the same, the flux of heat is strictly proportional to the gradient of temperature. Although undoubtedly true for small temperature gradients, strictly proportional behavior will be lost when real materials (e.g. ones having a thermal conductivity that is a function of temperature) are subjected to large temperature gradients.

A similar assumption is made in the statement of Ohm's law: other things being alike, the strength of the current at each point is proportional to the gradient of electric potential. The accuracy of the assumption that flow is proportional to the gradient is more readily tested, using modern measurement methods, for the electrical case than for the heat case.

## Other versions

Ohm's law, in the form above, is an extremely useful equation in the field of electrical/electronic engineering because it describes how voltage, current and resistance are interrelated on a "macroscopic" level, that is, commonly, as circuit elements in an electrical circuit. Physicists who study the electrical properties of matter at the microscopic level use a closely related and more general vector equation, sometimes also referred to as Ohm's law, having variables that are closely related to the V, I, and R scalar variables of Ohm's law, but which are each functions of position within the conductor. Physicists often use this continuum form of Ohm's Law:^{[34]}

where "**E**" is the electric field vector with units of volts per meter (analogous to "V" of Ohm's law which has units of volts), "**J**" is the current density vector with units of amperes per unit area (analogous to "I" of Ohm's law which has units of amperes), and "ρ" (Greek "rho") is the resistivity with units of ohm·meters (analogous to "R" of Ohm's law which has units of ohms). The above equation is sometimes written^{[35]} as **J** = **E** where "σ" (Greek "sigma") is the conductivity which is the reciprocal of ρ.

The voltage between two points is defined as:^{[36]}

with the element of path along the integration of electric field vector **E**. If the applied **E** field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage V in the usual convention of being opposite in direction to the field (see figure), and with the understanding that the voltage V is measured differentially across the length of the conductor allowing us to drop the Δ symbol, the above vector equation reduces to the scalar equation:

Since the **E** field is uniform in the direction of wire length, for a conductor having uniformly consistent resistivity ρ, the current density **J** will also be uniform in any cross-sectional area and oriented in the direction of wire length, so we may write:^{[37]}

Substituting the above 2 results (for *E* and *J* respectively) into the continuum form shown at the beginning of this section:

The electrical resistance of a uniform conductor is given in terms of resistivity by:^{[37]}

where *l* is the length of the conductor in SI units of meters, *a* is the cross-sectional area (for a round wire *a* = *πr*^{2} if *r* is radius) in units of meters squared, and ρ is the resistivity in units of ohm·meters.

After substitution of *R* from the above equation into the equation preceding it, the continuum form of Ohm's law for a uniform field (and uniform current density) oriented along the length of the conductor reduces to the more familiar form:

A perfect crystal lattice, with low enough thermal motion and no deviations from periodic structure, would have no resistivity,^{[38]} but a real metal has crystallographic defects, impurities, multiple isotopes, and thermal motion of the atoms. Electrons scatter from all of these, resulting in resistance to their flow.

The more complex generalized forms of Ohm's law are important to condensed matter physics, which studies the properties of matter and, in particular, its electronic structure. In broad terms, they fall under the topic of constitutive equations and the theory of transport coefficients.

### Magnetic effects

If an external **B**-field is present and the conductor is not at rest but moving at velocity **v**, then an extra term must be added to account for the current induced by the Lorentz force on the charge carriers.

In the rest frame of the moving conductor this term drops out because **v**= 0. There is no contradiction because the electric field in the rest frame differs from the **E**-field in the lab frame: **E′** = **E** + **v**×**B**. Electric and magnetic fields are relative, see Lorentz transformation.

If the current **J** is alternating because the applied voltage or **E**-field varies in time, then reactance must be added to resistance to account for self-inductance, see electrical impedance. The reactance may be strong if the frequency is high or the conductor is coiled.

### Conductive fluids

In a conductive fluid, such as a plasma, there is a similar effect. Consider a fluid moving with the velocity in a magnetic field . The relative motion induces an electric field which exerts electric force on the charged particles giving rise to an electric current. The equation of motion for the electron gas, with a number density, is written as

where , and are the charge, mass and velocity of the electrons, respectively. Also, is the frequency of collisions of the electrons with ions which have a velocity field . Since, the electron has a very small mass compared with that of ions, we can ignore the left hand side of the above equation to write

where we have used the definition of the current density, and also put which is the electrical conductivity. This equation can also be equivalently written as

where is the electrical resistivity. It is also common to write instead of

*not*follow Ohm's law.

As originally stated in terms of DC resistive circuits only, **Thévenin's theorem** holds that:

- Any linearelectrical network with voltage and current sources and resistances only can be replaced at terminals A-B by an equivalent voltage source V
_{th}in series connection with an equivalent resistance R_{th}. - The equivalent voltage V
_{th}is the voltage obtained at terminals A-B of the network with terminals A-B open circuited. - The equivalent resistance R
_{th}is the resistance that the circuit between terminals A and B would have if all ideal voltage sources in the circuit were replaced by a short circuit and all ideal current sources were replaced by an open circuit. - If terminals A and B are connected to one another, the current flowing from A to B will be V
_{th}/R_{th}. This means that R_{th}could alternatively be calculated as V_{th}divided by the short-circuit current between A and B when they are connected together.

- Any linearelectrical network with voltage and current sources and resistances only can be replaced at terminals A-B by an equivalent voltage source V

In circuit theory terms, the theorem allows any one-port network to be reduced to a single voltage source and a single impedance.

The theorem also applies to frequency domain AC circuits consisting of reactive and resistiveimpedances. It means the theorem applies for AC in an exactly same way to DC except that resistances are generalized to impedances.

The theorem was independently derived in 1853 by the German scientist Hermann von Helmholtz and in 1883 by Léon Charles Thévenin (1857–1926), an electrical engineer with France's national Postes et Télégraphes telecommunications organization.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}

Thévenin's theorem and its dual, Norton's theorem, are widely used to make circuit analysis simpler and to study a circuit's initial-condition and steady-state response.^{[7]}^{[8]} Thévenin's theorem can be used to convert any circuit's sources and impedances to a **Thévenin equivalent**; use of the theorem may in some cases be more convenient than use of Kirchhoff's circuit laws.^{[6]}^{[9]}

## Calculating the Thévenin equivalent[edit]

The equivalent circuit is a voltage source with voltage *V*_{Th} in series with a resistance *R*_{Th}.

The Thévenin-equivalent voltage *V*_{Th} is the open-circuit voltage at the output terminals of the original circuit. When calculating a Thévenin-equivalent voltage, the voltage divider principle is often useful, by declaring one terminal to be *V*_{out} and the other terminal to be at the ground point.

The Thévenin-equivalent resistance *R*_{Th} is the resistance measured across points A and B "looking back" into the circuit. The resistance is measured after replacing all voltage- and current-sources with their internal resistances. That means an ideal voltage source is replaced with a short circuit, and an ideal current source is replaced with an open circuit. Resistance can then be calculated across the terminals using the formulae for series and parallel circuits. This method is valid only for circuits with independent sources. If there are dependent sources in the circuit, another method must be used such as connecting a test source across A and B and calculating the voltage across or current through the test source.

The replacements of voltage and current sources do what the sources would do if their values were set to zero. A zero valued voltage source would create a potential difference of zero volts between its terminals, regardless of the current that passes through it; its replacement, a short circuit, does the same thing. A zero valued current source passes zero current, regardless of the voltage across it; its replacement, an open circuit, does the same thing.

### Example[edit]

In the example, calculating the equivalent voltage:

(notice that *R*_{1} is not taken into consideration, as above calculations are done in an open-circuit condition between A and B, therefore no current flows through this part, which means there is no current through R_{1} and therefore no voltage drop along this part)

Calculating equivalent resistance ( is the total resistance of two parallel resistors):

## Conversion to a Norton equivalent[edit]

Main article: Norton's theorem

A Norton equivalent circuit is related to the Thévenin equivalent by

## Practical limitations[edit]

- Many circuits are only linear over a certain range of values, thus the Thévenin equivalent is valid only within this linear range.
- The Thévenin equivalent has an equivalent I–V characteristic only from the point of view of the load.
- The power dissipation of the Thévenin equivalent is not necessarily identical to the power dissipation of the real system. However, the power dissipated by an external resistor between the two output terminals is the same regardless of how the internal circuit is implemented.

## A proof of the theorem[edit]

The proof involves two steps. The first step is to use superposition theorem to construct a solution. Then, uniqueness theorem is employed to show that the obtained solution is unique. It is noted that the second step is usually implied in literature.

By using superposition of specific configurations, it can be shown that for any linear "black box" circuit which contains voltage sources and resistors, its voltage is a linear function of the corresponding current as follows

Here, the first term reflects the linear summation of contributions from each voltage source, while the second term measures the contributions from all the resistors. The above expression is obtained by using the fact that the voltage of the black box for a given current is identical to the linear superposition of the solutions of the following problems: (1) to leave the black box open circuited but activate individual voltage source one at a time and, (2) to short circuit all the voltage sources but feed the circuit with a certain ideal voltage source so that the resulting current exactly reads (Alternatively, one can use an ideal current source of current ). Moreover, it is straightforward to show that and are the single voltage source and the single series resistor in question.

As a matter of fact, the above relation between and is established by superposition of some particular configurations. Now, the uniqueness theorem guarantees that the result is general. To be specific, there is one and only one value of once the value of is given. In other words, the above relation holds true independent of what the "black box" is plugged to.

## See also[edit]

## References[edit]

## Bibliography[edit]

- Brenner, Egon; Javid, Mansour (1959). "Chapter 12 - Network Functions".
*Analysis of Electric Circuits*. McGraw-Hill. pp. 268–269. - Brittain, J.E. (March 1990). "Thevenin's theorem".
*IEEE Spectrum*.**27**(3): 42. doi:10.1109/6.48845. Retrieved 1 February 2013. - Dorf, Richard C.; Svoboda, James A. (2010). "Chapter 5 - Circuit Theorems".
*Introduction to Electric Circuits*(8th ed.). Hoboken, NJ: John Wiley & Sons. pp. 162–207. ISBN 978-0-470-52157-1. - Dwight, Herbert B. (1949). "Sec. 2 - Electric and Magnetic Circuits". In Knowlton, A.E.
*Standard Handbook for Electrical Engineers*(8th ed.). McGraw-Hill. p. 26. - Elgerd, Olle I. (2007). "Chapter 10, Energy System Transients - Surge Phenomena and Symmetrical Fault Analysis".
*Electric Energy Systems Theory: An Introduction*. Tata McGraw-Hill. pp. 402–429. ISBN 978-0070192300. - Helmholtz, H. (1853). "Über einige Gesetze der Vertheilung elektrischer Ströme in körperlichen Leitern mit Anwendung auf die thierisch-elektrischen Versuche (Some laws concerning the distribution of electrical currents in conductors with applications to experiments on animal electricity)".
*Annalen der Physik und Chemie*.**89**(6): 211–233. - Johnson, D.H. (2003a). "Origins of the equivalent circuit concept: the voltage-source equivalent"(PDF).
*Proceedings of the IEEE*.**91**(4): 636–640. doi:10.1109/JPROC.2003.811716. - Johnson, D.H. (2003b). "Origins of the equivalent circuit concept: the current-source equivalent"(PDF).
*Proceedings of the IEEE*.**91**(5): 817–821. doi:10.1109/JPROC.2003.811795. - Thévenin, L. (1883a). "Extension de la loi d'Ohm aux circuits électromoteurs complexes (Extension of Ohm's law to complex electromotive circuits)".
*Annales Télégraphiques*. 3^{e}series.**10**: 222–224. - Thévenin, L. (1883b). "Sur un nouveau théorème d'électricité dynamique (On a new theorem of dynamic electricity)".
*Comptes rendus hebdomadaires des séances de l'Académie des Sciences*.**97**: 159–161. - Wenner, F. (1926). "Sci. Paper S531, A principle governing the distribution of current in systems of linear conductors". Washington, D.C.: Bureau of Standards.

## External links[edit]

- Original circuit
- The equivalent voltage
- The equivalent resistance
- The equivalent circuit

**^**Helmholtz**^**Thévenin (1883a)**^**Thévenin (1883b)**^**Johnson (2003a)**^**Brittain- ^
^{a}^{b}Dorf **^**Brenner**^**Elgerd**^**Dwight

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