# Kirchhoff’s current laws for AC/DC (Look a physicist’s guide)

There are two of Kirchhoff’s current laws used for DC and AC. They are presented as two equations that describe the ratio of current and voltage in electric circuits. Electric current is a focused movement of charged particles. Movable charged particles are:

• electrons, in metallic conductors,
• ions in electrolytes and gases

Constant, unchanging (regular) current is usually called direct current, but that name is imprecise because the constant current is direct current, but direct current does not have to be constant. It can change its value. To maintain direct current, there must be a stationary electric field in the conductors. Unlike a stationary electric field, an electrostatic field causes only a short-term current.

### The first Kirchhoff’s current law of DC

Often, three conductors are connected in one place in electric coils. The place where three or more conductors (branches) connect in electrical systems is called a node. In the node, the law of conservation of charge applies. According to the law of conservation of charge, the amount of electric charge that enters a node must be equal to the amount of electric charge that leaves the node.

For example, for the three-branch node shown in the figure, the following applies ${{Q}_{1}}=~{{Q}_{2}}~+~{{Q}_{3}}$ Dividing this equation by t gives: ${{I}_{1}}={{I}_{2}}+{{I}_{3}}$ It means that the sum of currents that have the same direction to the node is equal to the sum of currents that have the direction from the node. That is ${{I}_{1}}+{{I}_{2}}+{{I}_{3}}={{0}_{.}}$
That is, in the general problem with n conductors $\sum\nolimits_{i=1}^{n}{\mathop{I}_{i}}=0$

Where the currents in the formula are represented algebraically. It means that when we look into a specific value of the current, we have to consider a positive or negative sign, depending on the direction of the current relative to the node. This expression represents the mathematical form of Kirchhoff’s first law, which expresses: the algebraic sum of electric currents that meet in a node is equal to zero.

The currents that leave the node have a positive sign, and those that enter the node have a negative sign, although it can be the other way around because it comes down to the same thing. The reference direction is the assumed direction concerning the equation ${{I}_{1}}={{I}_{2}}+{{I}_{3}}$ in which the calculated current can be negative. Namely, if we multiply the expression by -1, the signs of the currents will change, so those that were positive will become negative, and vice versa; the value will remain equal to zero.

### The second Kirchhoff’s current law for DC

In practice, we most often encounter complex connections between sources and resistors, which are called complex electrical circuits. The series connection of the elements connecting the two nodes is called the branch. Therefore, branches are the connections between nodes. The number of branches in a circuit is equal to the number of elements. The same current flows in the branch through all the elements. The loop is a series of branches that form a closed path inside the circuit. A contour (loop) is a closed path along the branches of a circuit. A circuit can be without nodes with only one loop such a circuit is called a simple circuit, otherwise, the circuit is complex.

The figure shows a complex circuit with two nodes A and B, three branches with currents ${{I}_{1}}{{I}_{2}}{{I}_{3}}$ and two independent loops, I and II. A contour is independent if it contains at least one branch that does not belong to other branches. Voltage U_AB can be calculated as the algebraic sum of electric forces from node B to node A over any of the three branches of the circuit. Electric forces are the electromotive forces of individual generators, and the electrical resistance forces, that is, the voltages at the ends of individual resistors are calculated according to Ohm’s law. If this voltage is calculated over branches with current${{I}_{1}}{{I}_{2}}$we have: $\mathop{U}_{AB}=\mathop{E}_{1}-\mathop{R}_{1}\mathop{I}_{1}=\mathop{E}_{2}-\mathop{R}_{2}\mathop{I}_{2} \\$ thus $\mathop{E}_{1}-\mathop{\mathop{E}_{2}-R}_{1}\mathop{I}_{1}+\mathop{R}_{2}\mathop{I}_{2}=0 \\sum{E-\sum{RI=0}}\\$

The last relation represents the mathematical expression of Kirchhoff’s Second Law, which reads: the algebraic sum of all-electric forces in one contour of a complex circuit is equal to zero. The following rules should be observed when applying Kirchhoff’s current laws. The direction of the loop is arbitrarily chosen on the selected contour and marked with an arrow. (in the second law) If the directions of the EM forces coincide with the direction of the marking, they should be considered positive. And if the directions are opposite, then we should consider the contrary.

Those voltages on individual resistors will also have a positive algebraic sign if the referenced current directions are opposite to the bypass direction or a negative sign if their directions coincide.

## Kirchhoff’s current laws for AC

Kirchhoff’s current laws I and II for AC are similar to Kirchhoff’s current laws I and II for DC. When we defined Kirchhoff’s first law for DC, we said the algebraic sum of all currents entering a node must be equal to zero. This one, the definition must be valid at all times, even if currents flow over time change values. Thus, Kirchhoff’s first law also applies to alternating currents of any kind of states, even for simple periodic currents:
$\sum\nolimits_{k=1}^{n}{\mathop{i}_{k}=0} \\$

Kirchhoff’s current laws I and II for AC are similar to Kirchhoff’s current laws I and II for DC. Also, Kirchhoff’s 2nd law must be valid at all times, even if the stresses on the elements along a certain contour are variable, e.g. simple periodic:
$\sum\nolimits_{k=1}^{n}{\mathop{u}_{k}=0} \\ \$
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