Definitions

# 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 directly derived from Maxwell's equations, but Kirchhoff preceded Maxwell and instead generalized work by Georg Ohm.

## Kirchhoff's Current Law (KCL)

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 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:

$sum_\left\{k=1\right\}^n I_k = 0$

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

This formula is also valid for complex currents:

$sum_\left\{k=1\right\}^n tilde\left\{I\right\}_k = 0$

### 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:

$nabla cdot mathbf\left\{J\right\} = -nabla cdot frac\left\{partial mathbf\left\{D\right\}\right\}\left\{partial t\right\} = -frac\left\{partial rho\right\}\left\{partial t\right\}$

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.

### 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)

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.

Similarly to KCL, it can be stated as:

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

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

$sum_\left\{k=1\right\}^n tilde\left\{V\right\}_k = 0$

### Direction of the voltages

This law often leads to confusion due to sign errors.

Typically, in a schematic diagram, one has to choose whether to define a voltage measured clockwise or counter-clockwise as positive. It is a question of convenience, but once decided, all voltages must be treated this way.

The confusion comes from the direction of the voltage at voltage sources and capacitors.

Considering the schematic diagram above, we may decide to measure all voltages clockwise, that is, from a to b, from b to c, from c to d and from d to a.

At the resistors, the voltage will be positive, because we measure from "plus to minus". But when we measure from d to a, we will measure a negative voltage, because we measure from "minus to plus". We have to do this because otherwise, we would be measuring counter-clockwise if we measured the other way around.

If our voltage source is a capacitor, we will also notice the we must consider the charge negative, which is due to: $Q = C V$

There is the rule of thumb that one must just "reverse supply voltages" in order to make the calculation match KVL. There is, however, the risk that this may lead to additional confusion in more complex circuits, in which it is not clear what the supply voltage exactly is.

It is actually not a contradiction that we measure from "minus to plus"; inside a voltage source, this is the direction in which the current flows. If a magnetic field induces a voltage into an inductor, making it a voltage source (which is the principle of an electrical generator), this should be obvious. Even in a battery, the circuit is closed because of Ions flowing and representing charge carriers.

Inside capacitors, there is no flow of charge carriers. The KVL is valid anyway, but we have to consider the electric field inside the capacitor in order to see this.

### Electric field and electric potential

Kirchhoff's voltage law as stated above 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

$oint_C mathbf\left\{E\right\} cdot dmathbf\left\{l\right\} = 0,$

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

In order to return to the more special form, this integral can be "cut in pieces" in order to get the voltage at specific components.

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. Therefore, it practically suffices for explaining circuits containing only resistors and capacitors.

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.

## 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.