User:Ponor/Kirchhoff's circuit laws

Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhoff. This generalized the work of Georg Ohm and preceded the work of James Clerk Maxwell. Widely used in electrical engineering, 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 network analysis.

Both of Kirchhoff's laws can be understood as corollaries of Maxwell's equations 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


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

The law 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:


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

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

Physical origin
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 current law will hold on the boundaries of the region. This means that the current law relies on the fact that the net charge in the wires and components is constant.

Uses
A matrix version of Kirchhoff's current law is the basis of most circuit simulation software, such as SPICE. The current law is used with Ohm's law to perform nodal analysis.

The current law 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
[[File:Kirchhoff_voltage_law.svg|thumb|200x200px|The sum of all the voltages around a 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.

This law states that"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:


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

Here, n is the total number of voltages measured in the loop.

Physical origin
The voltage increase between any two points in a conservative electric field is equal to the work done on moving a unit charge against the field from one point to another without causing any acceleration. It is the line integral of the negative electric field along that path (from a point p to a point q),


 * $$\Delta V_{p\,q} = \int_{p}^{q} -\vec{E} \cdot d\vec{l} $$

The law is equivalent to the statement that this integral around a closed loop is equal to zero, or that the circulation of a static electric field vanishes over a closed loop: because in ideal conductors connecting any two circuit elements the field is zero (a non-zero field would cause infinite currents), the integral around the loop breaks up into integrals over the lump elements alone. Each integral equals to the voltage increase from one terminal of the element to the other, as above. Kirchhoff's voltage law is thus equivalent to the Maxwell-Faraday equation of electrostatics, when the electromagnetic field does not change in time


 * $$\oint\vec{E}\cdot\mathrm{d}\vec{l}=0$$.

The equation is valid for any virtual loop, not only those that follow the flow of currents through the conductor.

With this definition, signs of the voltages in the statement of the law are obvious: if the loop is traversing an element in the direction of the electric field established in it, the voltage is negative. In a resistor, the field has the same direction as the current through the resistor; if the loop is traversing a resistor ($$R$$) in the direction of its current ($$i$$), the voltage drop is $$-iR$$, if it is traversing the resistor in the direction opposite of the current the voltage is $$+iR$$. In voltage sources, the electric field is directed from their positive to their negative pole. If the loop is traversing a voltage source ($$\mathcal{E}$$) from its positive terminal to its negative terminal, its voltage enters the formula as $$-\mathcal{E}$$, and as $$\mathcal{E}$$ if traversing from negative terminal. All ideal conductors, having zero resistivity, contribute zero voltage.

Generalization to AC circuits
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 Faraday's law of induction (which is one of Maxwell's equations).

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

Under this definition, the voltage difference between two points is not uniquely defined when there are time-varying magnetic fields since the electric force is not a conservative force in such cases.

If there are time-varying electric fields or accelerating charges, then there will be time-varying magnetic fields. This means in AC circuits, there are always some non-confined magnetic fields. However, except at higher frequencies, these are neglected.

If this definition of voltage is used, any circuit where there are time-varying magnetic fields, such as circuits containing inductors, will not have a well-defined voltage between nodes in the circuit. However, if magnetic fields are suitably contained to each component, then the electric field is conservative in the region exterior to the components, and voltages are well-defined in that region. In this case, the voltage across an inductor, viewed externally, turns out to be

$$U = \Delta V = -L\frac{dI}{dt}$$

despite the fact that, internally, the electric field in the coil is zero (assuming it is a perfect conductor).

Limitations
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. For example, in a transmission line, the charge density in the conductor will constantly be oscillating. 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.

Modelling real circuits with lumped elements
The lumped element approximation for a circuit is accurate at low 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 parasitic components. If frequencies are too high, it may be more appropriate to simulate the fields directly using finite element modelling or other techniques.

To model circuits so that both laws can still be used, it is 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 parasitic capacitances distributed between the conductors to model capacitive coupling, or parasitic (mutual) inductances to model inductive coupling. Wires also have some self-inductance, which is the reason that decoupling capacitors are necessary.

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

According to the first law:


 * $$ i_1 - i_2 - i_3 = 0 \, $$

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


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

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


 * $$-R_3 i_3 - \mathcal{E}_2 - \mathcal{E}_1 + R_2 i_2 = 0 $$

This yields a system of linear equations 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} $$

The current $i_{3}$ has a negative sign which means the assumed direction of $i_{3}$ was incorrect and $i_{3}$ is actually flowing in the direction opposite to the red arrow labeled $i_{3}$. The current in $R_{3}$ flows from left to right.

AC electric network
According to the first law:


 * $$ i_1 - i_2 - i_3 = 0 \, $$

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

According to the first law:


 * $$ i_1 - i_2 - i_3 = 0 \, $$

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

According to the first law:


 * $$ i_1 - i_2 - i_3 = 0 \, $$

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