Kirchhoff's circuit laws are two
equalities that deal with the
current and
potential difference
Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge ...
(commonly known as voltage) in the
lumped element model of
electrical circuit
An electrical network is an interconnection of electrical components (e.g., battery (electricity), batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e. ...
s. They were first described in 1845 by German physicist
Gustav Kirchhoff
Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German chemist, mathematician, physicist, and spectroscopist who contributed to the fundamental understanding of electrical circuits, spectroscopy and the emission of black-body ...
. This generalized the work of
Georg Ohm
Georg Simon Ohm (; ; 16 March 1789 – 6 July 1854) was a German mathematician and physicist. As a school teacher, Ohm began his research with the new electrochemical cell, invented by Italian scientist Alessandro Volta. Using equipment of his o ...
and preceded the work of
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
. Widely used in
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, 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.
Both of Kirchhoff's laws can be understood as corollaries of
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
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
This law, also called Kirchhoff's first law, or Kirchhoff's junction rule, states that, for any node (junction) in an
electrical circuit
An electrical network is an interconnection of electrical components (e.g., battery (electricity), batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e. ...
, the sum of
currents 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:
where 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
conservation of charge, since
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.
This means that the current law relies on the fact that the net charge in the wires and components is constant.
Uses
A
matrix
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the m ...
version of Kirchhoff's current law is the basis of most
circuit simulation software, such as
SPICE
In the culinary arts, a spice is any seed, fruit, root, Bark (botany), bark, or other plant substance in a form primarily used for flavoring or coloring food. Spices are distinguished from herbs, which are the leaves, flowers, or stems of pl ...
. The current law is used with
Ohm's law
Ohm's law states that the electric current through a Electrical conductor, conductor between two Node (circuits), points is directly Proportionality (mathematics), proportional to the voltage across the two points. Introducing the constant of ...
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

This law, also called Kirchhoff's second law, or Kirchhoff's loop rule, states the following:
''The directed sum of the potential difference
Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge ...
s (voltages) around any closed loop is zero.''
Similarly to Kirchhoff's current law, the voltage law can be stated as:
Here, is the total number of voltages measured.
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
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
).
This has practical application in situations involving "
static electricity
Static electricity is an imbalance of electric charges within or on the surface of a material. The charge remains until it can move away by an electric current or electrical discharge. The word "static" is used to differentiate it from electric ...
".
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
capacitively coupled, this may not be the case. This occurs in high-frequency AC circuits, where the lumped element model is no longer applicable.
[Ralph Morrison, ''Grounding and Shielding Techniques in Instrumentation'' Wiley-Interscience (1986) ] For example, in a
transmission line
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
, the charge density in the conductor may be constantly changing.

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 components. If frequencies are too high, it may be more appropriate to simulate the fields directly using
finite element modelling or
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
Parasitic capacitance or stray capacitance is the unavoidable and usually unwanted capacitance that exists between the parts of an electronic component or circuit simply because of their proximity to each other. When two electrical conductors a ...
s distributed between the conductors to model capacitive coupling, or
parasitic (mutual) inductances to model inductive coupling.
Wires also have some self-inductance.
Example

Assume an electric network consisting of two voltage sources and three resistors.
According to the first law:
Applying the second law to the closed circuit , and substituting for voltage using Ohm's law gives:
The second law, again combined with Ohm's law, applied to the closed circuit gives:
This yields a
system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables.
For example,
: \begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of th ...
in , , :
which is equivalent to
Assuming
the solution is
The current has a negative sign which means the assumed direction of was incorrect and is actually flowing in the direction opposite to the red arrow labeled . The current in flows from left to right.
See also
*
Duality (electrical circuits)
In electrical engineering, electrical terms are associated into pairs called duals. A dual of a relationship is formed by interchanging voltage and Electric current, current in an expression. The dual expression thus produced is of the same form ...
*
Faraday's law of induction
*
Lumped matter discipline
*
Tellegen's Theorem
References
*
*
*
*
External links
{{Commons category, Kirchhoff's circuit laws
''Divider Circuits and Kirchhoff's Laws''chapter fro
free ebook an
''Lessons In Electric Circuits''series
Circuit theorems
Conservation equations
Eponymous laws of physics
Linear electronic circuits
Voltage
1845 in science
Gustav Kirchhoff