Kirchhoff's circuit laws are two approximate equalities that deal with the current and voltage in electrical circuits. They were first described in 1845 by Gustav Kirchhoff. This generalized the work of Georg Ohm and preceded the work of Maxwell. 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).

Both of Kirchhoff's laws can be understood as corollaries of the Maxwell equations in the low-frequency limit -- conventionally called "DC" circuits. They serve as first approximations for AC circuits.

## Kirchhoff's current law (KCL)

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

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

The principle of conservation of electric charge 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: $\sum_{k=1}^n {I}_k = 0$

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

This formula is valid for complex currents: $\sum_{k=1}^n \tilde{I}_k = 0$

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).

### Limitations

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 AC circuits. It may be possible to salvage the form of KCL by considering "parasitic capacitances" distributed along the conductors. However, this greatly detracts from the simplicity of KCL and invalidates the notion of topological circuit diagram as discussed below. Significant violations of KCL can occur even at 60Hz, which is not a very high frequency.

As another way of saying almost the same thing, KCL is valid only if the total electric charge, $Q$
, 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.

Formally, from the volume integral of the current continuity equation, $\int_V \nabla \cdot \mathbf{J} \, dV= -\frac {d}{d t}Q,$
where $\mathbf{J}$
is the current density vector and $V$
is the volume of the region

Converting the volume integral to a surface integral using the divergence theorem $\int_V \nabla \cdot \mathbf{J} \, dV=$  $\; S$ $(\mathbf{J}\cdot\mathbf{n}) \ dS$

Hence,  $\; S$ $(\mathbf{J}\cdot\mathbf{n}) \, dS = -\frac {d}{d t}Q$

The right-hand side vanishes if $Q$
is independent of time. If practically all of $\mathbf{J}$
is contained within small regions, conducting wires for instance, then the left-hand side can be interpreted as a sum of discrete currents and KCL is recovered, providing that $dQ / dt = 0$
.

A particular case where KCL does not hold is 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 where current enters the antenna from the transmitter feeder but no current exits from the other end (Johnson and Graham, pp. 36–37).

Maxwell introduced the concept of displacement currents 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, $\psi \$
, is what Maxwell called displacement current $I_\mathrm D$
; $I_\mathrm D = \frac {d \psi}{d t}$

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 of Ampère's law with Maxwell's correction and combining with Gauss's law, yielding: $\nabla \cdot \mathbf{J} = -\nabla \cdot \frac{\partial \mathbf{D}}{\partial t} = -\frac{\partial \rho}{\partial t}$

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). 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 version of Kirchhoff's current law is the basis of most circuit simulation software, such as SPICE. Kirchhoff's current law combined with Ohm's Law is used in nodal analysis.

## Kirchhoff's voltage law (KVL)

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

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

Similarly to KCL, it can be stated as: $\sum_{k=1}^n V_k = 0$

Here, n is the total number of voltages measured. The voltages may also be complex: $\sum_{k=1}^n \tilde{V}_k = 0$

This law is based one of the Maxwell equations, namely the Maxwell-Faraday law of induction, which tells us that the voltage drop around any closed loop is equal to the rate-of-change of the flux threading the loop. The amount of flux depends on the area of the loop and on the magnetic field strength. KVL says the loop voltage is zero. The Maxwell equations tell us that the loop voltage will be small if the area of the loop is small, the magnetic field is weak, and/or the magnetic field is slowly changing.

Routine engineering techniques -- such as the use of coaxial cable and twisted pairs -- can be used to minimize stray magnetic fields and minimize the area of vulnerable loops. In this way things can be arranged so that KVL becomes a good approximation, even in situations where it otherwise would not have been.

### Limitations

KVL is based on the assumption that there is no fluctuating magnetic field linking the closed loop. This is not a safe assumption for AC circuits. In the presence of a changing magnetic field the electric field is not a conservative vector field. Therefore it cannot be the gradient of any potential. That is to say, the line integral of the electric field around the loop is not zero, which directly contradicts KVL.

It may be possible to salvage the form of KVL by considering "parasitic inductances" (including mutual inductances) distributed along the conductors. These are treated as imaginary circuit elements that produce a voltage drop equal to the rate-of-change of the flux. However, this greatly detracts from the simplicity of KVL and invalidates the notion of topological circuit diagram.

### Generalization

In the DC 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).

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

## Topological circuit diagrams

The approximations that lead to Kirchhoff's circuit laws are part of a package that also leads to topological circuit diagrams, i.e. the idea that the physical and geometrical layout of the circuit does not matter; the only thing that matters is the topology as determined by the conductors and circuit elements connected to the nodes. These can be treated as the arcs and nodes of formal graph theory. In other words, Kirchhoff's laws say it suffices to use a circuit diagram that is purely schematic. This is a very useful, powerful simplification.

This works fine in the DC limit, but it is only a first approximation for AC circuits. For high-power, high-precision, and/or high-frequency work, the deviations from Kirchhoff's laws cannot be neglected. The physical and geometrical layout of the circuit matters, because it determines the magnitude of the parasitic capacitances and inductances.

## Example

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

According to the first law we have $i_1 - i_2 - i_3 = 0 \,$

The second law applied to the closed circuit s1 gives $-R_2 i_2 + \epsilon_1 - R_1 i_1 = 0$

The second law applied to the closed circuit s2 gives $-R_3 i_3 - \epsilon_2 - \epsilon_1 + R_2 i_2 = 0$

Thus we get a linear system of equations in $i_1, i_2, i_3$
: $\begin{cases} i_1 - i_2 - i_3 & = 0 \\ -R_2 i_2 + \epsilon_1 - R_1 i_1 & = 0 \\ -R_3 i_3 - \epsilon_2 - \epsilon_1 + R_2 i_2 & = 0 \\ \end{cases}$

Assuming $R_1 = 100,\ R_2 = 200,\ R_3 = 300,\ \epsilon_1 = 3,\ \epsilon_2 = 4$

the solution is $\begin{cases} i_1 = \frac{1}{1100} \text{ or } 0.\bar{90}\text{mA}\\ i_2 = \frac{4}{275} \text{ or } 14.\bar{54}\text{mA}\\ i_3 = - \frac{3}{220} \text{ or } -13.\bar{63}\text{mA}\\ \end{cases}$ $i_3$
has a negative sign, which means that the direction of $i_3$
is opposite to the assumed direction (the direction defined in the picture).