Kirchhoff's circuit laws
Encyclopedia
Kirchhoff's circuit laws are two equalities that deal with the conservation of charge
Charge conservation
In physics, charge conservation is the principle that electric charge can neither be created nor destroyed. The net quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved...

and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff
Gustav Kirchhoff
Gustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects...

. Widely used in electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

, they are also called Kirchhoff's rules or simply Kirchhoff's laws (see also Kirchhoff's laws
Kirchhoff's laws
There are several Kirchhoff's laws, all named after Gustav Robert Kirchhoff:* Kirchhoff's circuit laws* Kirchhoff's law of thermal radiation* Kirchhoff's equations* Kirchhoff's three laws of spectroscopy* Kirchhoff's law of thermochemistry-See also:...

for other meanings of that term).

Both circuit rules can be directly derived from Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

, but Kirchhoff preceded Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

and instead generalized work by Georg Ohm
Georg Ohm
Georg Simon Ohm was a German physicist. As a high school teacher, Ohm began his research with the recently-invented electrochemical cell, invented by Italian Count Alessandro Volta. Using equipment of his own creation, Ohm determined that there is a direct proportionality between the potential...

.

## Kirchhoff's current law (KCL)

This law is also called Kirchhoff's first law, Kirchhoff's point rule, Kirchhoff's junction rule (or nodal rule), and Kirchhoff's first rule.

The principle of conservation of electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

implies that:
At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node, or:
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 stated as:

n is the total number of branches with currents flowing towards or away from the node.

This formula is valid for complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

currents:

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).

### Changing charge density

KCL is only valid if the charge density
Charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...

remains constant at the point to which it is applied. Consider the current entering a single plate of a capacitor. If one imagines a closed surface around that single plate, current enters through the surface, but does not exit, thus violating KCL. Certainly, the currents through a closed surface around the entire capacitor will meet KCL since the current entering one plate is balanced by the current exiting the other plate, and that is usually all that is important in circuit analysis, but there is a problem when considering just one plate. Another common example is the current in an antenna
An antenna is an electrical device which converts electric currents into radio waves, and vice versa. It is usually used with a radio transmitter or radio receiver...

where current enters the antenna from the transmitter feeder but no current exits from the other end.(Johnson and Graham, pp.36-37)

Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

introduced the concept of displacement current
Displacement current
In electromagnetism, displacement current is a quantity that is defined in terms of the rate of change of electric displacement field. Displacement current has the units of electric current density, and it has an associated magnetic field just as actual currents do. However it is not an electric...

s to describe these situations. The current flowing into a capacitor plate is equal to the rate of accumulation of charge and hence is also equal to the rate of change of electric flux due to that charge (electric flux is measured in the same units, Coulombs, as electric charge in the SI system of units). This rate of change of flux, , is what Maxwell called displacement current ;

When the displacement currents are included, Kirchhoff's current law once again holds. Displacement currents are not real currents in that they do not consist of moving charges, they should be viewed more as a correction factor to make KCL true. In the case of the capacitor plate, the real current entering the plate is exactly cancelled by a displacement current leaving the plate and heading for the opposite plate.

This can also be expressed in terms of vector field quantities by taking the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

of Ampère's law
Ampère's law
In classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère in 1826, relates the integrated magnetic field around a closed loop to the electric current passing through the loop...

with Maxwell's correction and combining with Gauss's law
Gauss's law
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...

, yielding:

This is simply the charge conservation equation (in integral form, it says that the current flowing out of a closed surface is equal to the rate of loss of charge within the enclosed volume (Divergence theorem
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...

)). Kirchhoff's current law is equivalent to the statement that the divergence of the current is zero, true for time-invariant ρ, or always true if the displacement current is included with J.

### Uses

A matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

version of Kirchhoff's current law is the basis of most circuit simulation software
Electronic circuit simulation
Electronic circuit simulation uses mathematical models to replicate the behavior of an actual electronic device or circuit.Simulation software allows for modeling of circuit operation and is an invaluable analysis tool...

, such as SPICE
SPICE
SPICE is a general-purpose, open source analog electronic circuit simulator.It is a powerful program that is used in integrated circuit and board-level design to check the integrity of circuit designs and to predict circuit behavior.- Introduction :Unlike board-level designs composed of discrete...

.

## Kirchhoff's voltage law (KVL)

This law is also called Kirchhoff's second law, Kirchhoff's loop (or mesh) rule, and Kirchhoff's second rule.

The principle of conservation of energy implies that
The directed sum of the electrical potential differences (voltage) around any closed circuit is zero, or:
More simply, the sum of the emf
Electromotive force
In physics, electromotive force, emf , or electromotance refers to voltage generated by a battery or by the magnetic force according to Faraday's Law, which states that a time varying magnetic field will induce an electric current.It is important to note that the electromotive "force" is not a...

s in any closed loop is equivalent to the sum of the potential drops in that loop, or:
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 emf
Electromotive force
In physics, electromotive force, emf , or electromotance refers to voltage generated by a battery or by the magnetic force according to Faraday's Law, which states that a time varying magnetic field will induce an electric current.It is important to note that the electromotive "force" is not a...

available in that loop.

Similarly to KCL, it can be stated as:

Here, n is the total number of voltages measured. The voltages may also be complex:

This law is based on the conservation of "energy given/taken by potential field" (not including energy taken by dissipation). Given a voltage potential, a charge which has completed a closed loop doesn't gain or lose energy as it has gone back to initial potential level.

This law holds true even when resistance (which causes dissipation of energy) is present in a circuit. The validity of this law in this case can be understood if one realizes that a charge in fact doesn't go back to its starting point, due to dissipation of energy. A charge will just terminate at the negative terminal, instead of positive terminal. This means all the energy given by the potential difference has been fully consumed by resistance which in turn loses the energy as heat dissipation.

To summarize, Kirchhoff's voltage law has nothing to do with gain or loss of energy by electronic components (resistors, capacitors, etc.). It is a law referring to the potential field generated by voltage sources. In this potential field, regardless of what electronic components are present, the gain or loss in "energy given by the potential field" must be zero when a charge completes a closed loop.

### Electric field and electric potential

Kirchhoff's voltage law could be viewed as a consequence of the principle of conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

. Otherwise, it would be possible to build a perpetual motion machine that passed a current in a circle around the circuit.

Considering that electric potential is defined as a line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

over an electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

, Kirchhoff's voltage law can be expressed equivalently as

which states that the line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

of the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

around closed loop C is zero.

In order to return to the more special form, this integral can be "cut in pieces" in order to get the voltage at specific components.

### Limitations

This is a simplification of Faraday's law of induction
Faraday's law of induction dates from the 1830s, and is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators...

for the special case where there is no fluctuating magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

linking the closed loop. Therefore, it practically suffices for explaining circuits containing only resistors and capacitors.

In the presence of a changing magnetic field the electric field is not conservative and it cannot therefore define a pure scalar potential
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...

—the line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

of the electric field around the circuit is not zero. This is because energy is being transferred from the magnetic field to the current (or vice versa). In order to "fix" Kirchhoff's voltage law for circuits containing inductors, an effective potential drop, or electromotive force
Electromotive force
In physics, electromotive force, emf , or electromotance refers to voltage generated by a battery or by the magnetic force according to Faraday's Law, which states that a time varying magnetic field will induce an electric current.It is important to note that the electromotive "force" is not a...

(emf), is associated with each inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

of the circuit, exactly equal to the amount by which the line integral of the electric field is not zero by Faraday's law of induction