Kirchhoff's circuit laws

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

Kirchhoff's circuit laws are two equalities 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 in principle be derived from Maxwell's equations, but Kirchhoff preceded Maxwell and instead generalized work by Georg Ohm.

Kirchhoff's current law (KCL)

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

This law is also called 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 that does not represent a capacitor plate, the sum of currents flowing towards that point is equal to the sum of currents flowing away from that point.

Adopting the convention that every current flowing towards the point is positive and that every current flowing away is negative (or the other way around), this principle can be stated as:

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

n is the total number of currents flowing towards or away from the point.

This formula is also valid for complex currents:

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

This law is based on the conservation of charge whereby the charge (measured in coulombs) is the product of the current (in amps) and the time (which is measured in seconds).

Changing charge density

Physically speaking, the restriction regarding the "capacitor plate" means that Kirchhoff's current law is only valid if the charge density remains constant in the point that it is applied to. This is normally not a problem because of the strength of electrostatic forces: the charge buildup would cause repulsive forces to disperse the charges.

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 really only required if one wants to apply the current law to a point on a capacitor plate. In circuit analyses, however, the capacitor as a whole is typically treated as a unit, in which case the ordinary current law holds since exactly the current that enters the capacitor on the one side leaves it on the other side.)

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 (Divergence theorem)). 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.

Uses

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

Kirchhoff's voltage law (KVL)

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 voltage drops around any closed circuit is equal to the sum of the voltages generated by any voltage sources associated with the circuit.

By taking the voltage sources as positive, and the voltage drops as negative, the Law can be stated in the form:

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

Here, n is the total number of voltages measured. The voltages may also be complex:

$\sum _{k=1}^{n}{\tilde {V}}_{k}=0$

Checking Kirchoff's voltage law for a simple circuit

For the circuit shown alongside, Kirchhoff's Voltage Law is equivalent to the following statement. If position "d" is attached to "Ground", then a specific voltage value (relative to Ground) can be allocated to each of the points (a), (b) and (c). Also, the sum of the voltage drops across the three resistors is equal to the (closed-circuit) voltage generated across its terminals by the voltage source. The validity of the Law (for this simple circuit) can be checked by using a high-resistance voltmeter to measure the individual voltage drops across the resistors, and checking that the sum of these drops is equal to the voltage across the terminals of the voltage source.

Complications

The detailed mathematics associated with this Law is more complicated if: (a) the voltage source is an AC source and the circuit contains inductors or capacitors; and/or (b) a voltage is induced in the circuit by any external means.

See also

• Voltage
• 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)

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.{{cite book}}: CS1 maint: multiple names: authors list (link)
• 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.