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Kirchhoff's circuit laws

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Two equalities that deal with the current and potential difference

For other laws named after Gustav Kirchhoff, seeKirchhoff's laws.

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Kirchhoff's circuit laws are twoequalities that deal with thecurrent andpotential difference (commonly known as voltage) in thelumped element model ofelectrical circuits. They were first described in 1845 by German physicistGustav Kirchhoff.^[1] This generalized the work ofGeorg Ohm and preceded the work ofJames Clerk Maxwell. Widely used inelectrical engineering, they are also calledKirchhoff's rules or simplyKirchhoff's laws. These laws can be applied in time and frequency domains and form the basis fornetwork analysis.

Both of Kirchhoff's laws can be understood as corollaries ofMaxwell'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

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The current entering any junction is equal to the current leaving that junction.i₂ +i₃ =i₁ +i₄

This law, also calledKirchhoff's first law, orKirchhoff's junction rule, states that, for any node (junction) in anelectrical circuit, the sum ofcurrents flowing into that node is equal to the sum of currents flowing out of that node; or equivalently:

The algebraic sum of currents in a network of conductors meeting at a point is zero.

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: $\sum _{i=1}^{n}I_{i}=0$ wheren 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 theconservation of charge, sincecharge 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.^[2]^[3] This means that the current law relies on the fact that the net charge in the wires and components is constant.

Uses

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Amatrix version of Kirchhoff's current law is the basis of mostcircuit simulation software, such asSPICE. The current law is used withOhm's law to performnodal 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

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The sum of all the voltages around a loop is equal to zero.
v₁ +v₂ +v₃ +v₄ = 0

This law, also calledKirchhoff's second law, orKirchhoff's loop rule, states the following:

The directed sum of thepotential differences (voltages) around any closed loop is zero.

Similarly to Kirchhoff's current law, the voltage law can be stated as: $\sum _{i=1}^{n}V_{i}=0$

Here,n is the total number of voltages measured.

Derivation of Kirchhoff's voltage law

A similar derivation can be found inThe Feynman Lectures on Physics, Volume II, Chapter 22: AC Circuits.^[3]

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, theMaxwell–Faraday equation reveals that $\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {0}$ in the exterior region. If each of the components has a finite volume, then the exterior region issimply connected, and thus the electric field isconservative in that region. Therefore, for any loop in the circuit, we find that $\sum _{i}V_{i}=-\sum _{i}\int _{{\mathcal {P}}_{i}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =\oint \mathbf {E} \cdot \mathrm {d} \mathbf {l} =0$ where ${\textstyle {\mathcal {P}}_{i}}$ are paths around theexterior of each of the components, from one terminal to another.

Note that this derivation uses the following definition for the voltage rise from $a {\displaystyle a}$ to $b {\displaystyle b}$ : $V_{a\to b}=-\int _{{\mathcal {P}}_{a\to b}}\mathbf {E} \cdot \mathrm {d} \mathbf {l}$

However, theelectric potential (and thus voltage) can be defined in other ways, such as via theHelmholtz decomposition.

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 ofFaraday's law of induction (which is one ofMaxwell's equations).

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

Limitations

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Kirchhoff's circuit laws are the result of thelumped-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 arecapacitively coupled, this may not be the case. This occurs in high-frequency AC circuits, where the lumped element model is no longer applicable.^[4] For example, in atransmission line, the charge density in the conductor may be constantly changing.

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

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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 usingparasitic components. If frequencies are too high, it may be more appropriate to simulate the fields directly usingfinite element modelling orother techniques.

To model circuits so that both laws can still be used, it is important to understand the distinction betweenphysical circuit elements and theideal 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 consideringparasitic capacitances distributed between the conductors to model capacitive coupling, orparasitic (mutual) inductances to model inductive coupling.^[4] Wires also have some self-inductance.

Example

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

According to the first law: $i_{1}-i_{2}-i_{3}=0$ Applying the second law to the closed circuits₁, and substituting for voltage using Ohm's law gives: $-R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}=0$ The second law, again combined with Ohm's law, applied to the closed circuits₂ gives: $-R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}=0$

This yields asystem of linear equations ini₁,i₂,i₃: ${\begin{cases}i_{1}-i_{2}-i_{3}&=0\\-R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}&=0\\-R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}&=0\end{cases}}$ which is equivalent to ${\begin{cases}i_{1}+(-i_{2})+(-i_{3})&=0\\R_{1}i_{1}+R_{2}i_{2}+0i_{3}&={\mathcal {E}}_{1}\\0i_{1}+R_{2}i_{2}-R_{3}i_{3}&={\mathcal {E}}_{1}+{\mathcal {E}}_{2}\end{cases}}$ Assuming ${\begin{aligned}R_{1}&=100\Omega ,&R_{2}&=200\Omega ,&R_{3}&=300\Omega ,\\{\mathcal {E}}_{1}&=3{\text{V}},&{\mathcal {E}}_{2}&=4{\text{V}}\end{aligned}}$ the solution is ${\begin{cases}i_{1}={\frac {1}{1100}}{\text{A}}\\[6pt]i_{2}={\frac {4}{275}}{\text{A}}\\[6pt]i_{3}=-{\frac {3}{220}}{\text{A}}\end{cases}}$

The currenti₃ has a negative sign which means the assumed direction ofi₃ was incorrect andi₃ is actually flowing in the direction opposite to the red arrow labeledi₃. The current inR₃ flows from left to right.

References

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^Oldham, Kalil T. Swain (2008).The doctrine of description: Gustav Kirchhoff, classical physics, and the "purpose of all science" in 19th-century Germany (Ph. D.). University of California, Berkeley. p. 52. Docket 3331743.
^Athavale, Prashant."Kirchoff's current law and Kirchoff's voltage law"(PDF).Johns Hopkins University. Retrieved6 December 2018.
^^a ^b"The Feynman Lectures on Physics Vol. II Ch. 22: AC Circuits".feynmanlectures.caltech.edu. Retrieved2018-12-06.
^^a ^bRalph Morrison,Grounding and Shielding Techniques in Instrumentation Wiley-Interscience (1986)ISBN 0471838055

Paul, Clayton R. (2001).Fundamentals of Electric Circuit Analysis. John Wiley & Sons.ISBN 0-471-37195-5.
Serway, Raymond A.; Jewett, John W. (2004).Physics for Scientists and Engineers (6th ed.). Brooks/Cole.ISBN 0-534-40842-7.
Tipler, Paul (2004).Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman.ISBN 0-7167-0810-8.
Graham, Howard Johnson, Martin (2002).High-speed signal propagation : advanced black magic (10. printing. ed.). Upper Saddle River, NJ: Prentice Hall PTR.ISBN 0-13-084408-X.{{cite book}}: CS1 maint: multiple names: authors list (link)