Editing Ohm's law (section)

==Circuit analysis==
[[File:Ohm law mnemonic principle.svg|thumb|upright|Covering the [[Equation#Parameters and unknowns|unknown]] in the Ohm's law [[mnemonic#Types|image mnemonic]] gives the formula in terms of the remaining parameters]]
[[File:Ohms law wheel WVOA.svg|thumb|right|Ohm's law wheel with international unit symbols]]

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

<math display="block">I = \frac{V}{R} \quad \text{or}\quad V = IR \quad \text{or} \quad R = \frac{V}{I}. </math>

Each equation is quoted by some sources as the defining relationship of Ohm's law,<ref name=Millikan/><ref>{{cite book
 | title = Electric circuits
 |first1=James William|last1=Nilsson |first2=Susan A.|last2=Riedel
  |name-list-style=amp | publisher = Prentice Hall
 | year = 2008
 | isbn = 978-0-13-198925-2
 | page = 29
 | url = https://books.google.com/books?id=sxmM8RFL99wC&q=%22Ohm%27s+law+expresses+the+voltage%22++%22V+%3D+iR%22&pg=PA29
 }}</ref><ref>{{cite book
 | title = Schaum's outline of theory and problems of beginning physics II
 |first1=Alvin M.|last1=Halpern |first2=Erich|last2=Erlbach
  |name-list-style=amp | publisher = McGraw-Hill Professional
 | year = 1998
 | isbn = 978-0-07-025707-8
 | page = 140
 | url = https://books.google.com/books?id=vN2chIay624C&q=%22Ohm%27s+law+that+R%3D+V/I+is+a+constant%22&pg=PA140
 }}</ref>
or all three are quoted,<ref>{{cite book
 | title = Understanding DC circuits
 |first1=Dale R.|last1=Patrick  |first2=Stephen W.|last2=Fardo
  |name-list-style=amp | publisher = Newnes
 | year = 1999
 | isbn = 978-0-7506-7110-1
 | page = 96
 | url = https://books.google.com/books?id=wyC5SFtZskMC&q=%22Ohm%27s+law%22+%22R+%3D%22+%22V+%3D%22+%22I+%3D%22&pg=PA96
 }}</ref> or derived from a proportional form,<ref>{{cite book
 | title = Elementary electrical calculations
 | first = Thomas|last=O'Conor Sloane
 | publisher = D. Van Nostrand Co
 | year = 1909
 | page = [https://archive.org/details/elementaryelect01sloagoog/page/n57 41]
 | url = https://archive.org/details/elementaryelect01sloagoog
 | quote = R= Ohm's law proportional.
 }}</ref>
or even just the two that do not correspond to Ohm's original statement may sometimes be given.<ref>{{cite book
 | title = Electricity treated experimentally for the use of schools and students
 | first = Linnaeus|last=Cumming
 | publisher = Longman's Green and Co
 | year = 1902
 | page = [https://archive.org/details/electricitytrea00cummgoog/page/n242 220]
 | url = https://archive.org/details/electricitytrea00cummgoog
 | quote = V=IR Ohm's law.
 }}</ref><ref>{{cite book
 | title = Building technology
 | edition = 2nd
 | first = Benjamin|last=Stein
 | publisher = John Wiley and Sons
 | year = 1997
 | isbn = 978-0-471-59319-5
 | page = 169
 | url = https://books.google.com/books?id=J_RSbj_KzAQC&q=%22Ohm%27s+law+that+V%3D%22&pg=PA169
 }}</ref>

The interchangeability of the equation may be represented by a triangle, where ''V'' ([[voltage]]) is placed on the top section, the ''I'' ([[electric current|current]]) is placed to the left section, and the ''R'' ([[electrical resistance|resistance]]) is placed to the right. The divider between the top and bottom sections indicates division (hence the division bar).

{{anchor|ohmic}}

===Resistive circuits===
[[Resistor]]s 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 schematic diagrams, a resistor is shown as a long rectangle or 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 ([[direct current|DC]]) or time-varying such as [[alternating current|AC]]. At any instant of time Ohm's law is valid for such circuits.

Resistors which are in ''[[Series and parallel circuits#Series circuits|series]]'' or in ''[[Series and parallel circuits#Parallel circuits|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-invariant]] [[alternating current|AC]] circuits take the same form as Ohm's law. However, the variables are generalized to [[complex number]]s and the current and voltage waveforms are [[complex exponential]]s.<ref>{{cite book | title = Fundamentals of Electrical Engineering | first = Rajendra|last=Prasad | publisher = Prentice-Hall of India | year = 2006 | url = https://books.google.com/books?id=nsmcbzOJU3kC&q=ohm%27s-law+complex+exponentials&pg=PA140 | isbn = 978-81-203-2729-0 }}</ref>

In this approach, a voltage or current waveform takes the form ''Ae''{{sup|''st''}}, where ''t'' is time, ''s'' is a complex parameter, and ''A'' is a complex scalar. In any [[LTI system theory|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 [[electrical impedance|impedance]], usually denoted ''Z''; it can be shown that for an inductor,
<math display="block">Z = sL</math>
and for a capacitor,
<math display="block">Z = \frac{1}{sC}.</math>

We can now write,
<math display="block">V = Z\,I</math>
where ''V'' and ''I'' are the complex scalars in the voltage and current respectively and ''Z'' is the complex impedance.

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 a 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 [[Sine wave|sinusoid]], the ''s'' parameter is taken to be <math>j\omega</math>, corresponding to a complex sinusoid <math>Ae^{\mbox{ } j \omega t}</math>. 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|Network analysis (electrical circuits)#Small signal equivalent circuit}}

Ohm's law is one of the basic equations used in the [[Network analysis (electrical circuits)|analysis of electrical circuits]]. It applies to both metal conductors and circuit components ([[resistor]]s) specifically made for this behaviour. Both are ubiquitous in electrical engineering. Materials and components that obey Ohm's law are described as "ohmic"<ref>Hughes, E, ''Electrical Technology'', pp10, Longmans, 1969.</ref> 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''<sub>1</sub> and ''V''<sub>2</sub> applied across a given device of resistance ''R'', producing currents ''I''<sub>1</sub> = ''V''<sub>1</sub>/''R'' and ''I''<sub>2</sub> = ''V''<sub>2</sub>/''R'', that the ratio (''V''<sub>1</sub> − ''V''<sub>2</sub>)/(''I''<sub>1</sub> − ''I''<sub>2</sub>) is also a constant equal to ''R''. The operator "delta" (Δ) is used to represent a difference in a quantity, so we can write Δ''V'' = ''V''<sub>1</sub> − ''V''<sub>2</sub> and Δ''I'' = ''I''<sub>1</sub> − ''I''<sub>2</sub>. Summarizing, for any truly ohmic device having resistance ''R'', ''V''/''I'' = Δ''V''/Δ''I'' = ''R'' for any applied voltage or current or for the difference between any set of applied voltages or currents.

[[File:FourIVcurves.svg|thumb|400px|The [[Current–voltage characteristic|''I''–''V'' curve]]s of four devices: Two [[resistor]]s, a [[diode]], and a [[Battery (electricity)|battery]]. The two resistors follow Ohm's law: The plot is a straight line through the origin. The other two devices do ''not'' follow Ohm's law.]]

There are, however, components of electrical circuits which do not obey Ohm's law; that is, their relationship between current and voltage (their [[Current–voltage characteristic|''I''–''V'' curve]]) is ''nonlinear'' (or non-ohmic). An example is the [[Diode#Shockley diode equation|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 [[direct current|DC]], resistance,<ref>{{cite book | title = Engineering System Dynamics | first = Forbes T.|last=Brown | publisher = CRC Press | year = 2006 | isbn = 978-0-8493-9648-9 | page = 43 | url = https://books.google.com/books?id=UzqX4j9VZWcC&q=%22chordal+resistance%22&pg=PA43 }}</ref><ref>{{cite book | title = Electromagnetic Compatibility Handbook | first = Kenneth L.|last=Kaiser | publisher = CRC Press | year = 2004 | isbn = 978-0-8493-2087-3 | pages = 13–52 | url = https://books.google.com/books?id=nZzOAsroBIEC&q=%22static+resistance%22+%22dynamic+resistance%22+nonlinear&pg=PT1031 }}</ref> but as seen in the figure the value of total {{math|''V''}} over total {{math|''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 {{math|Δ''V''}} volts or {{math|Δ''I''}} amps centered at that same point along the curve and measuring {{math|Δ''V''/Δ''I''}}. 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.<ref name=horowitz-hill>{{cite book |last1=Horowitz |first1=Paul |author-link=Paul Horowitz |first2=Winfield|last2=Hill | title=The Art of Electronics |edition=2nd |year=1989 |publisher=Cambridge University Press |isbn=978-0-521-37095-0 |page = 13 | url = https://books.google.com/books?id=bkOMDgwFA28C&q=small-signal+%22dynamic+resistance%22&pg=PA13 |author2-link=Winfield Hill }}</ref>