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| {{More footnotes|date=October 2013}}
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| '''Kirchhoff's circuit laws''' are two [[Equality (mathematics)|equalities]] that deal with the [[Electric current|current]] and potential difference (commonly known as voltage) in the [[lumped element model]] of [[electrical circuit]]s. They were first described in 1845 by [[Gustav Kirchhoff]].<ref>{{Cite thesis |last=Oldham |first=Kalil T. Swain |title=The doctrine of description: Gustav Kirchhoff, classical physics, and the "purpose of all science" in 19th-century Germany |type=Ph. D. |chapter= |url= |author= |year=2008 |publisher=University of California, Berkeley |accessdate= |docket=3331743 |oclc= |page=52 }}</ref> This generalized the work of [[Georg Ohm]]<!-- 1827 --> and preceded the work of [[James Clerk Maxwell|Maxwell]]<!-- 1873 -->. Widely used in [[electrical engineering]], they are also called Kirchhoff's ''rules'' or simply Kirchhoff's ''laws'' (see also [[Kirchhoff's laws]] for other meanings of that term).
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| Both of Kirchhoff's laws can be understood as corollaries of the [[Maxwell equations]] in the low-frequency limit. They are accurate for DC circuits, and for AC circuits at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuits.
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| == Kirchhoff's current law (KCL) ==
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| [[File:KCL - Kirchhoff's circuit laws.svg|thumb|The current entering any junction is equal to the current leaving that junction. ''i''<sub>2</sub> + ''i''<sub>3</sub> = ''i''<sub>1</sub> + ''i''<sub>4</sub>]]
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| This law is also called '''Kirchhoff's first law''', '''Kirchhoff's point rule''', or '''Kirchhoff's junction rule''' (or nodal rule).
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| The principle of conservation of [[electric charge]] implies that:
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| :At any node (junction) in an [[electrical circuit]], the sum of [[current (electricity)|current]]s flowing into that node is equal to the sum of currents flowing out of that node, or:
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| ::The algebraic sum of currents in a network of conductors meeting at a point is zero.
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| Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be stated as:
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| :<math>\sum_{k=1}^n {I}_k = 0</math>
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| ''n'' is the total number of branches with currents flowing towards or away from the node.
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| This formula is valid for [[Complex number|complex]] currents:
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| :<math>\sum_{k=1}^n \tilde{I}_k = 0</math>
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| The law is based on the conservation of charge whereby the charge (measured in coulombs) is the product of the current (in amperes) and the time (in seconds).
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| === Uses ===
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| A [[Matrix (mathematics)|matrix]] version of Kirchhoff's current law is the basis of most [[Electronic circuit simulation|circuit simulation software]], such as [[SPICE]]. Kirchhoff's current law combined with [[Ohm's Law]] is used in [[nodal analysis]].
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| == Kirchhoff's voltage law (KVL) ==
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| [[File:Kirchhoff voltage law.svg|thumb|200px|The sum of all the voltages around the loop is equal to zero. v<sub>1</sub> + v<sub>2</sub> + v<sub>3</sub> - v<sub>4</sub> = 0]]
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| This law is also called '''Kirchhoff's second law''', '''Kirchhoff's loop (or mesh) rule''', and '''Kirchhoff's second rule'''.
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| The principle of conservation of energy implies that
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| :The directed sum of the electrical [[potential difference]]s (voltage) around any closed network is zero, or:
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| ::More simply, the sum of the [[Electromotive force|emf]]s in any closed loop is equivalent to the sum of the potential drops in that loop, or:
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| :::The algebraic sum of the products of the resistances of the conductors and the currents in them in a closed loop is equal to the total [[Electromotive force|emf]] available in that loop.
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| Similarly to KCL, it can be stated as:
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| :<math>\sum_{k=1}^n V_k = 0</math>
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| Here, ''n'' is the total number of voltages measured. The voltages may also be complex:
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| :<math>\sum_{k=1}^n \tilde{V}_k = 0</math>
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| This law is based on the conservation of energy whereby voltage is defined as the energy per unit charge. The total amount of energy gained per unit charge must equal the amount of energy lost per unit charge. The law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another.
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| ===Generalization=== | |
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| In the low-frequency limit, the voltage drop around any loop is zero. This includes imaginary loops arranged arbitrarily in space – not limited to the loops delineated by the circuit elements and conductors. In the low-frequency limit, this is a corollary of [[Faraday's law of induction]] (which is one of the [[Maxwell equations]]).
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| This has practical application in situations involving "[[static electricity]]".
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| ==Limitations==
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| KCL and KVL both depend on the [[lumped element model]] being applicable to the circuit in question. When the model is not applicable, the laws do not apply.
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| KCL, in its usual form, is dependent on the assumption that current flows only in conductors, and that whenever current flows into one end of a conductor it immediately flows out the other end. This is not a safe assumption for high-frequency AC circuits, where the lumped element model is no longer applicable.<ref name="GSTI">Ralph Morrison, ''Grounding and Shielding Techniques in Instrumentation'' Wiley-Interscience (1986) ISBN 0471838055</ref> It is often possible to improve the applicability of KCL by considering "parasitic capacitances" distributed along the conductors.<ref name="GSTI"/> Significant violations of KCL can occur<ref name="copter"> {{cite web |first=simonjz05, |title = High Voltage Cable Inspection |url = http://www.youtube.com/watch?v=9tzga6qAaBA&t=0m52s |type = video }}</ref><ref name="ncvd">[[Non-contact voltage detector]]</ref> even at 60Hz, which is not a very high frequency.
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| In other words, KCL is valid only if the total [[electric charge]], <math>\scriptstyle Q </math>, remains constant in the region being considered. In practical cases this is always so when KCL is applied at a geometric point. When investigating a finite region, however, it is possible that the [[charge density]] within the region may change. Since charge is conserved, this can only come about by a flow of charge across the region boundary. This flow represents a net current, and KCL is violated.
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| KVL is based on the assumption that there is no fluctuating [[magnetic field]] linking the closed loop.
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| This is not a safe assumption for high-frequency (short-wavelength) AC circuits.<ref name="GSTI"/>
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| In the presence of a changing magnetic field the electric field is not a [[conservative vector field]].
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| Therefore the electric field can not be the gradient of any [[potential]].
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| That is to say, the [[line integral]] of the electric field around the loop is not zero, directly contradicting KVL.
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| It is often possible to improve the applicability of KVL by considering "parasitic inductances" (including mutual
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| inductances) distributed along the conductors.<ref name="GSTI"/> These are treated as imaginary
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| circuit elements that produce a voltage drop equal to the rate-of-change of the flux.
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| ==Example==
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| [[File:Kirshhoff-example.svg|right|260px]]
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| Assume an electric network consisting of two voltage sources and three resistors:
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| According to the first law we have
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| :<math> i_1 - i_2 - i_3 = 0 \, </math>
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| The second law applied to the closed circuit ''s''<sub>1</sub> gives
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| :<math>-R_2 i_2 + \epsilon_1 - R_1 i_1 = 0</math>
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| The second law applied to the closed circuit ''s''<sub>2</sub> gives
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| :<math>-R_3 i_3 - \epsilon_2 - \epsilon_1 + R_2 i_2 = 0 </math>
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| Thus we get a linear system of equations in <math> i_1, i_2, i_3</math>:
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| :<math>\begin{cases}
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| i_1 - i_2 - i_3 & = 0 \\
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| -R_2 i_2 + \epsilon_1 - R_1 i_1 & = 0 \\ | |
| -R_3 i_3 - \epsilon_2 - \epsilon_1 + R_2 i_2 & = 0 \\
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| \end{cases}
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| </math>
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| Assuming
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| :<math>
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| R_1 = 100,\ R_2 = 200,\ R_3 = 300\text{ (ohms)};\ \epsilon_1 = 3,\ \epsilon_2 = 4\text{ (volts)}
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| </math>
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| the solution is
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| :<math>\begin{cases}
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| i_1 = \frac{1}{1100} \text{ or } 0.\bar{90}\text{ mA}\\
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| i_2 = \frac{4}{275} \text{ or } 14.\bar{54}\text{ mA}\\
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| i_3 = - \frac{3}{220} \text{ or } -13.\bar{63}\text{ mA}\\
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| \end{cases}
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| </math>
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| <math>i_3</math> has a negative sign, which means that the direction of <math>i_3</math> is opposite to the assumed direction (the direction defined in the picture).
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| == See also ==
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| {{Portal|Electronics}}
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| * [[Faraday's law of induction]]
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| * [[Kirchhoff's laws]]
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| * [[Lumped matter discipline]]
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| == References ==
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| {{reflist}}
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| {{Refbegin}}
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| *{{cite book | author=Paul, Clayton R. | title=Fundamentals of Electric Circuit Analysis | publisher=John Wiley & Sons | year=2001 | isbn=0-471-37195-5}}
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| *{{cite book | author=Serway, Raymond A.; Jewett, John W. | title=Physics for Scientists and Engineers (6th ed.) | publisher=Brooks/Cole | year=2004 | isbn=0-534-40842-7}}
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| *{{cite book | author=Tipler, Paul | title=Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.) | publisher=W. H. Freeman | year=2004 | isbn=0-7167-0810-8}}
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| *{{cite book|last=Graham|first=Howard Johnson, Martin|title=High-speed signal propagation : advanced black magic|year=2002|publisher=Prentice Hall PTR|location=Upper Saddle River, NJ|isbn=0-13-084408-X|edition=10. printing.}}
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| {{Refend}}
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| == External links ==
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| {{Commons category|Kirchhoff's circuit laws}}
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| * [http://academicearth.org/lectures/basic-circuit-analysis-method-kvl-and-kcl-mmethod MIT video lecture] on the KVL and KCL methods
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| * [http://ocw.mit.edu/courses/physics/8-02-electricity-and-magnetism-spring-2002/lecture-notes/lecsup41.pdf Faraday's Law - Most Physics College Books have it WRONG!] by Walter H. G. Lewin, Ph.D., MIT
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| {{DEFAULTSORT:Kirchhoff's Circuit Laws}}
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| [[Category:Circuit theorems]]
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| [[Category:Conservation equations]]
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