Kirchhoff's circuit laws

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

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 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:

$${\displaystyle \sum _{k=1}^{n}{I}_{k}=0}$$

where 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 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.[2][3] This means that the current law relies on the fact that the net charge in the wires and components is constant.

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

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

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

Similarly to Kirchhoff's current law, the voltage law can be stated as:

$${\displaystyle \sum _{k=1}^{n}V_{k}=0}$$

Here, n is the total number of voltages measured.

Derivation of Kirchhoff's voltage law

A similar derivation can be found in The 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, the Maxwell–Faraday equation reveals that

$${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {0} }$$

in the exterior region. If each of the components has a finite volume, then the exterior region is simply connected, and thus the electric field is conservative in that region. Therefore, for any loop in the circuit, we find that

$${\displaystyle \sum _{i}V_{i}=-\sum _{i}\int _{{\mathcal {P}}_{i}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =\oint \mathbf {E} \cdot \mathrm {d} \mathbf {l} =0}$$

where ${\textstyle {\mathcal {P}}_{i}}$ are paths around 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 ${\displaystyle a}$ to ${\displaystyle b}$:

$${\displaystyle V_{a\to b}=-\int _{{\mathcal {P}}_{a\to b}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} }$$

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

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.[4] For example, in a transmission 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 action 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.

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

According to the first law:

$${\displaystyle i_{1}-i_{2}-i_{3}=0}$$

Applying the second law to the closed circuit s₁, and substituting for voltage using Ohm's law gives:

$${\displaystyle -R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}=0}$$

The second law, again combined with Ohm's law, applied to the closed circuit s₂ gives:

$${\displaystyle -R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}=0}$$

This yields a system of linear equations in i₁, i₂, i₃:

$${\displaystyle {\begin{cases}i_{1}-i_{2}-i_{3}&=0\\-R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}&=0\\-R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}&=0\end{cases}}}$$

which is equivalent to

$${\displaystyle {\begin{cases}i_{1}+(-i_{2})+(-i_{3})&=0\\R_{1}i_{1}+R_{2}i_{2}+0i_{3}&={\mathcal {E}}_{1}\\0i_{1}+R_{2}i_{2}-R_{3}i_{3}&={\mathcal {E}}_{1}+{\mathcal {E}}_{2}\end{cases}}}$$

Assuming

$${\displaystyle {\begin{aligned}R_{1}&=100\Omega ,&R_{2}&=200\Omega ,&R_{3}&=300\Omega ,\\{\mathcal {E}}_{1}&=3{\text{V}},&{\mathcal {E}}_{2}&=4{\text{V}}\end{aligned}}}$$

the solution is

$${\displaystyle {\begin{cases}i_{1}={\frac {1}{1100}}{\text{A}}\\[6pt]i_{2}={\frac {4}{275}}{\text{A}}\\[6pt]i_{3}=-{\frac {3}{220}}{\text{A}}\end{cases}}}$$

The current i₃ has a negative sign which means the assumed direction of i₃ was incorrect and i₃ is actually flowing in the direction opposite to the red arrow labeled i₃. The current in R₃ flows from left to right.

• Duality (electrical circuits)
• Faraday's law of induction
• Lumped matter discipline

Wikimedia Commons has media related to Kirchhoff's circuit laws.

• Divider Circuits and Kirchhoff's Laws chapter from Lessons In Electric Circuits Vol 1 DC free ebook and Lessons In Electric Circuits series