Kirchhoff's circuit laws

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For other laws named after Gustav Kirchhoff, see Kirchhoff's laws. Not to be confused with Kerckhoffs' principle.

Kirchhoff's circuit laws are a pair of laws that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff. 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 circuit rules can be directly derived from Maxwell's equations, but Kirchhoff preceded Maxwell and instead generalized work by Georg Ohm.

1 Kirchhoff's Current Law (KCL)
2 Kirchhoff's Voltage Law (KVL)
3 See also
4 References
5 External links

[edit] Kirchhoff's Current Law (KCL)

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

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

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

The principle of conservation of electric charge implies that:

At any point in an electrical circuit where charge density is not changing in time, the sum of currents flowing towards that point is equal to the sum of currents flowing away from that point.

An analogy to this principle is:

Two rivers that converge and then later break up into separate rivers. The principle states that the sum of the water flowing in the two upstream rivers is equal to the sum of the water flowing in the two downstream rivers.

A charge density changing in time would mean the accumulation of a net positive or negative charge, which typically cannot happen to any significant degree because of the strength of electrostatic forces: the charge buildup would cause repulsive forces to disperse the charges.

Another way to look at the effect of changing charge density on Kirchhoff's first rule is to go back to the rivers analogy again. Whenever there is a change in the charge density imagine someone going downstream and adding or removing a bucket of water from any of the two downstream rivers. This will change the rate of flow downstream.

However, a charge build-up can occur in a capacitor, where the charge is typically spread over wide parallel plates, with a physical break in the circuit that prevents the positive and negative charge accumulations over the two plates from coming together and cancelling. In this case, the sum of the currents flowing into one plate of the capacitor is not zero, but rather is equal to the rate of charge accumulation. However, if the displacement current dD/dt is included, Kirchhoff's current law once again holds. (This is only required if one wants to apply the current law within the capacitor. In circuit analyses, however, the capacitor as a whole is typically treated as a unit, in which case the ordinary current law holds since the net charge is always zero.)

More technically, Kirchhoff's current law can be found 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). 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.

A matrix version of Kirchhoff's current law is the basis of most circuit simulation software, such as SPICE.

[edit] Kirchhoff's Voltage Law (KVL)

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

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.

The directed sum of the electrical potential differences around any closed circuit must be zero.

KVL may also be stated as " the algebraic sum of various potential drops across an electrical circuit is equal to the electromotive force acting on the circuit"

This statement is equivalent to the statement that a single-valued electric potential can be assigned to each point in the circuit (in the same way that any conservative vector field can be represented as the gradient of a scalar potential).

(This could be viewed as a consequence of the principle of conservation of energy. Otherwise, it would be possible to build a perpetual motion machine that passed a current in a circle around the circuit).

Considering that electric potential is defined as a line integral over an electric field, Kirchhoff's voltage law can be expressed equivalently as

$\oint_C \mathbf{E} \cdot d\mathbf{l} = 0,$

which states that the line integral of the electric field around closed loop C is zero.

$\sum {IR} + \sum \mathcal{E} = 0\,$ (round a loop/mesh)

This is a simplification of Faraday's law of induction for the special case where there is no fluctuating magnetic field linking the closed loop. In the presence of a changing magnetic field the electric field is not conservative and it cannot therefore define a pure scalar potential—the line integral of the electric field around the circuit is not zero. This is because energy is being transferred from the magnetic field to the current (or vice versa). In order to "fix" Kirchhoff's voltage law for circuits containing inductors, an effective potential drop, or electromotive force (emf), is associated with each inductance of the circuit, exactly equal to the amount by which the line integral of the electric field is not zero by Faraday's law of induction.

[edit] See also

Electronics Portal

Kirchhoff's law of thermal radiation
Faraday's law of induction
Gauss's law
Maxwell's equations
Conservation of energy
Mesh analysis
SPICE (Simulation Program with Integrated Circuits Emphasis)

[edit] References

Paul, Clayton R. (2001). Fundamentals of Electric Circuit Analysis. John Wiley & Sons. ISBN 0-471-37195-5.
Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 0-534-40842-7.
Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.