View source for Kirchhoff's circuit laws

{{Short description|Two equalities that deal with the current and potential difference}}
{{For|other laws named after Gustav Kirchhoff|Kirchhoff's laws (disambiguation){{!}}Kirchhoff's laws}}
{{More footnotes|date=November 2017}}
'''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 German physicist [[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. |year=2008 |publisher=University of California, Berkeley |docket=3331743 |page=52 }}</ref> This generalized the work of [[Georg Ohm]]<!-- 1827 --> and preceded the work of [[James Clerk Maxwell]]<!-- 1873 -->. Widely used in [[electrical engineering]], they are also called '''Kirchhoff's rules''' or simply '''Kirchhoff's laws'''. These laws can be applied in time and frequency domains and form the basis for [[Network analysis (electrical circuits)|network analysis]].

Both of Kirchhoff's laws can be understood as corollaries of [[Maxwell's 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.

== Kirchhoff's current law{{anchor|Kirchhoff's current law (KCL)}} ==
[[File:KCL - Kirchhoff's circuit laws.svg|thumb|The current entering any junction is equal to the current leaving that junction. {{math|1=''i''<sub>2</sub> + ''i''<sub>3</sub> = ''i''<sub>1</sub> + ''i''<sub>4</sub>}}]]

This law, also called '''Kirchhoff's first law''', or '''Kirchhoff's junction rule''', states that, for 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 equivalently:

<blockquote>''The algebraic sum of currents in a network of conductors meeting at a point is zero.''</blockquote>

Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be succinctly stated as: <math display="block">\sum_{i=1}^n I_i = 0</math> where {{math|''n''}} is the total number of branches with currents flowing towards or away from the node.

Kirchhoff's circuit laws were originally obtained from experimental results.  However, the current law can be viewed as an extension of the [[Charge conservation|conservation of charge]],  since [[Electric charge|charge]] is the product of current and the time the current has been flowing. If the net charge in a region is constant, the current law will hold on the boundaries of the region.<ref>{{Cite web|url=http://www.ams.jhu.edu/~prashant/KCL_KVL.pdf| title=Kirchoff's current law and Kirchoff's voltage law|last=Athavale|first=Prashant|website=Johns Hopkins University|access-date=6 December 2018}}</ref><ref name=":0">{{Cite web|url=https://feynmanlectures.caltech.edu/II_22.html|title=The Feynman Lectures on Physics Vol. II Ch. 22: AC Circuits|website=feynmanlectures.caltech.edu|access-date=2018-12-06}}</ref> This means that the current law relies on the fact that the net charge in the wires and components is constant.

=== Uses ===
A [[Matrix (mathematics)|matrix]] version of Kirchhoff's current law is the basis of most [[Electronic circuit simulation|circuit simulation software]], such as [[SPICE]]. The current law is used with [[Ohm's law]] to perform [[nodal analysis]].

The current law is applicable to any lumped network irrespective of the nature of the network; whether unilateral or bilateral, active or passive, linear or non-linear.

== Kirchhoff's voltage law ==
[[File:Kirchhoff voltage law.svg|thumb|200px|The sum of all the voltages around a loop is equal to zero. <br />
{{math|1=''v''<sub>1</sub> + ''v''<sub>2</sub> + ''v''<sub>3</sub> + ''v''<sub>4</sub> = 0}}]]

This law, also called '''Kirchhoff's second law''', or '''Kirchhoff's loop rule''', states the following:

<blockquote>''The directed sum of the [[potential difference]]s (voltages) around any closed loop is zero.''</blockquote>

Similarly to Kirchhoff's current law, the voltage law can be stated as: <math display="block">\sum_{i=1}^n V_i = 0</math>

Here, {{mvar|n}} is the total number of voltages measured.

{{math proof|title=Derivation of Kirchhoff's voltage law | proof=
A similar derivation can be found in ''The Feynman Lectures on Physics, Volume II, Chapter 22: AC Circuits''.<ref name=":0" />

Consider some arbitrary circuit. Approximate the circuit with lumped elements, so that time-varying magnetic fields are contained to each component and the field in the region exterior to the circuit is negligible. Based on this assumption, the [[Faraday's law of induction|Maxwell–Faraday equation]] reveals that <math display="block">\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t} = \mathbf{0}</math> in the exterior region. If each of the components has a finite volume, then the exterior region is [[Simply connected space|simply connected]], and thus the electric field is [[Conservative force|conservative]] in that region. Therefore, for any loop in the circuit, we find that <math display="block">\sum_i V_i = - \sum_i \int_{\mathcal{P}_i}\mathbf{E}\cdot\mathrm{d}\mathbf{l} = \oint\mathbf{E}\cdot\mathrm{d}\mathbf{l} = 0</math> where <math display="inline">\mathcal{P}_i</math> are paths around the ''exterior'' of each of the components, from one terminal to another. 

Note that this derivation uses the following definition for the voltage rise from <math>a</math> to <math>b</math>:
<math display="block">V_{a \to b} = -\int_{\mathcal{P}_{a \to b}}\mathbf{E}\cdot \mathrm{d}\mathbf{l}</math>

However, the [[electric potential]] (and thus voltage) can be defined in other ways, such as via the [[Helmholtz decomposition]].
}}

===Generalization===
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 [[Maxwell's equations]]).

This has practical application in situations involving "[[static electricity]]".

== Limitations ==
Kirchhoff's circuit laws are the result of the [[lumped-element model]] and both depend on the model being applicable to the circuit in question. When the model is not applicable, the laws do not apply.

The current law is dependent on the assumption that the net charge in any wire, junction or lumped component is constant. Whenever the electric field between parts of the circuit is non-negligible, such as when two wires are [[Capacitive coupling|capacitively coupled]], this may not be the case. This occurs in 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> For example, in a [[transmission line]], the charge density in the conductor may be constantly changing.
[[File:Transmission line animation3.gif|thumb|300x300px|In a transmission line, the net charge in different parts of the conductor changes with time. In the direct physical sense, this violates KCL.]]
On the other hand, the voltage law relies on the fact that the actions of time-varying magnetic fields are confined to individual components, such as inductors. In reality, the induced electric field produced by an inductor is not confined, but the leaked fields are often negligible.

=== Modelling real circuits with lumped elements ===
The lumped element approximation for a circuit is accurate at low frequencies. At higher frequencies, leaked fluxes and varying charge densities in conductors become significant. To an extent, it is possible to still model such circuits using [[Parasitic element (electrical networks)|parasitic components]]. If frequencies are too high, it may be more appropriate to simulate the fields directly using [[Finite element method|finite element modelling]] or [[Computational electromagnetics|other techniques]].

To model circuits so that both laws can still be used, it is important to understand the distinction between ''physical'' circuit elements and the ''ideal'' lumped elements. For example, a wire is not an ideal conductor. Unlike an ideal conductor, wires can inductively and capacitively couple to each other (and to themselves), and have a finite propagation delay. Real conductors can be modeled in terms of lumped elements by considering [[parasitic capacitance]]s distributed between the conductors to model capacitive coupling, or [[Parasitic inductance|parasitic (mutual) inductances]] to model inductive coupling.<ref name="GSTI" /> Wires also have some self-inductance.

==Example==
[[File:Kirshhoff-example.svg|right|280px]]

Assume an electric network consisting of two voltage sources and three resistors.

According to the first law:
<math display="block"> i_1 - i_2 - i_3 = 0</math>
Applying the second law to the closed circuit {{math|''s''<sub>1</sub>}}, and substituting for voltage using Ohm's law gives:
<math display="block">-R_2 i_2 + \mathcal{E}_1 - R_1 i_1 = 0</math>
The second law, again combined with Ohm's law, applied to the closed circuit {{math|''s''<sub>2</sub>}} gives:
<math display="block">-R_3 i_3 - \mathcal{E}_2 - \mathcal{E}_1 + R_2 i_2 = 0</math>

This yields a [[system of linear equations]] in {{math|''i''<sub>1</sub>}}, {{math|''i''<sub>2</sub>}}, {{math|''i''<sub>3</sub>}}:
<math display="block">\begin{cases}
                                   i_1 - i_2 - i_3 & = 0 \\
                -R_2 i_2 + \mathcal{E}_1 - R_1 i_1 & = 0 \\
-R_3 i_3 - \mathcal{E}_2 - \mathcal{E}_1 + R_2 i_2 & = 0
\end{cases}</math>
which is equivalent to
<math display="block">\begin{cases}
  i_1 + (- i_2) + (- i_3) & = 0 \\
R_1 i_1 + R_2 i_2 + 0 i_3 & = \mathcal{E}_1 \\
0 i_1 + R_2 i_2 - R_3 i_3 & = \mathcal{E}_1 + \mathcal{E}_2
\end{cases}</math>
Assuming
<math display="block">\begin{align}
  R_1 &= 100\Omega, & R_2 &= 200\Omega, & R_3 &= 300\Omega, \\
 \mathcal{E}_1 &= 3\text{V}, & \mathcal{E}_2 &= 4\text{V}
\end{align}</math>
the solution is
<math display="block">\begin{cases}
i_1 =  \frac{1}{1100}\text{A} \\[6pt]
i_2 =  \frac{4}{275}\text{A} \\[6pt]
i_3 = -\frac{3}{220}\text{A}
\end{cases}</math>

The current {{math|''i''<sub>3</sub>}} has a negative sign which means the assumed direction of {{math|''i''<sub>3</sub>}} was incorrect and {{math|''i''<sub>3</sub>}} is actually flowing in the direction opposite to the red arrow labeled {{math|''i''<sub>3</sub>}}. The current in {{math|''R''<sub>3</sub>}} flows from left to right.

== See also ==
{{Portal|Electronics}}
* [[Duality (electrical circuits)]]
* [[Faraday's law of induction]]
* [[Lumped matter discipline]]
* [[Tellegen's Theorem]]

== References ==
{{reflist}}
{{Refbegin}}
*{{cite book|title=Fundamentals of Electric Circuit Analysis|author=Paul, Clayton R.|publisher=John Wiley & Sons|year=2001| isbn=0-471-37195-5}}
*{{cite book|title=Physics for Scientists and Engineers (6th ed.)|author1=Serway, Raymond A.|author2=Jewett, John W.| publisher=Brooks/Cole|year=2004|isbn=0-534-40842-7|url-access=registration|url=https://archive.org/details/physicssciengv2p00serw }}
*{{cite book|title=Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.)| author=Tipler, Paul|publisher=W. H. Freeman|year=2004|isbn=0-7167-0810-8}}
*{{cite book|title=High-speed signal propagation : advanced black magic|last=Graham|first=Howard Johnson, Martin| publisher=Prentice Hall PTR|year=2002| isbn=0-13-084408-X|edition=10. printing.|location=Upper Saddle River, NJ}}
{{Refend}}

==External links==
{{Commons category|Kirchhoff's circuit laws}}
* [http://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_6.html ''Divider Circuits and Kirchhoff's Laws''] chapter from [http://www.ibiblio.org/kuphaldt/electricCircuits/DC/index.html ''Lessons In Electric Circuits Vol 1 DC''] free ebook and [http://www.ibiblio.org/kuphaldt/electricCircuits/ ''Lessons In Electric Circuits''] series

[[Category:Circuit theorems]]
[[Category:Conservation equations]]
[[Category:Eponymous laws of physics]]
[[Category:Linear electronic circuits]]
[[Category:Voltage]]
[[Category:1845 in science]]
[[Category:Gustav Kirchhoff]]