Kirchhoff's circuit laws

Kirchhoff's laws are two that deal with the  and  (commonly known as voltage) in the  of s. They were first described in 1845 by German physicist. This generalized the work of and preceded the work of. Widely used in, 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.

Both of Kirchhoff's laws can be understood as corollaries of 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 (KCL)
This law is also called Kirchhoff's first law, Kirchhoff's point rule, or Kirchhoff's junction rule (or nodal rule).

This law states that, for any node (junction) in an, the sum of s 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:


 * $$\sum_{k=1}^n {I}_k = 0$$

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

The law is based on the conservation of charge where the charge (measured in coulombs) is the product of the current (in amperes) and the time (in seconds). If the net charge in a region is constant, the KCL will hold on the boundaries of the region. This means that KCL relies on the fact that the net charge in the wires and components is constant.

Uses
A version of Kirchhoff's current law is the basis of most, such as. Kirchhoff's current law is used with to perform.

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

This law states that"The directed sum of the (voltages) around any closed loop is zero."

Similarly to KCL, it can be stated as:


 * $$\sum_{k=1}^n V_k = 0$$

Here, n 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 (which is one of ).

This has practical application in situations involving "".

Limitations
KCL and KVL both depend on the being applicable to the circuit in question. When the model is not applicable, the laws do not apply. KCL and KVL result from the assumptions of the lumped element model.

KCL 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, this may not be the case. This occurs in high-frequency AC circuits, where the lumped element model is no longer applicable. For example, in a, the charge density in the conductor will constantly be oscillating.

On the other hand, KVL 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.

Modelling real circuits with lumped elements

The lumped element approximation for a circuit is accurate at low enough 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. If frequencies are too high, it may be more appropriate to simulate the fields directly using or.

To model circuits so that KVL and KCL can still be used, it's 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 s distributed between the conductors to model capacitive coupling, or to model inductive coupling. Wires also have some self-inductance, which is the reason that are necessary.

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
 * $$-R_2 i_2 + \mathcal{E}_1 - R_1 i_1 = 0$$

The second law applied to the closed circuit s2 gives
 * $$-R_3 i_3 - \mathcal{E}_2 - \mathcal{E}_1 + R_2 i_2 = 0 $$

Thus we get a in $$ i_1, i_2, i_3$$:
 * $$\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 + 0 i_3 & = \mathcal{E}_1 \\ 0 i_1 + R_2 i_2 - R_3 i_3 & = \mathcal{E}_1 + \mathcal{E}_2 \end{cases} $$ Assuming

R_1 = 100\Omega,\ R_2 = 200\Omega,\ R_3 = 300\Omega $$

\mathcal{E}_1 = 3\text{V}, \mathcal{E}_2 = 4\text{V} $$ the solution is
 * $$\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} $$

$$i_3$$ has a negative sign, which means that the direction of $$i_3$$ is opposite to the assumed direction i.e. $$i_3$$ is directed upwards.