Kirchhoff's circuit laws

Last updated

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. [1] 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.

Contents

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

The current entering any junction is equal to the current leaving that junction. i2 + i3 = i1 + i4 KCL - Kirchhoff's circuit laws.svg
The current entering any junction is equal to the current leaving that junction. i2 + i3 = i1 + i4

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

This 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:

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

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. [2] [3] 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

The sum of all the voltages around a loop is equal to zero.
v1 + v2 + v3 +v4 = 0 Kirchhoff voltage law.svg
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:

Here, n is the total number of voltages measured.

Derivation of Kirchhoff's voltage law
A similar derivation can be found in The Feynman Lectures on Physics, Volume II, Chapter 22: AC Circuits. [3]

Consider some arbitrary circuit. Approximate the circuit with lumped elements, so that (time-varying) magnetic fields are contained to each component and the field in the region exterior to the circuit is negligible. Based on this assumption, the Maxwell-Faraday equation reveals that

in the exterior region. If each of the components have a finite volume, then the exterior region is simply connected, and thus the electric field is conservative in that region. Therefore, for any loop in the circuit, we find that

where are paths around the exterior of each of the components, from one terminal to another.

Generalization

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".

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

In a transmission line, the net charge in different parts of the conductor changes with time. In the direct physical sense, this violates KCL. Transmission line animation3.gif
In a transmission line, the net charge in different parts of the conductor changes with time. In the direct physical sense, this violates KCL.

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. [4] Wires also have some self-inductance, which is the reason that decoupling capacitors are necessary.

Example

Kirshhoff-example.svg

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

According to the first law:

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

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

This yields a system of linear equations in :

which is equivalent to

Assuming

the solution is

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

See also

Related Research Articles

Electrical network Assemblage of connected electrical elements

An electrical network is an interconnection of electrical components or a model of such an interconnection, consisting of electrical elements. An electrical circuit is a network consisting of a closed loop, giving a return path for the current. Linear electrical networks, a special type consisting only of sources, linear lumped elements, and linear distributed elements, have the property that signals are linearly superimposable. They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, and transient response.

Inductor Passive two-terminal electrical component that stores energy in its magnetic field

An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil around a core.

Voltage difference in the electric potential between two points in space

Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named volt. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for volt uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by V, but more often simply as V, for instance in the context of Ohm's or Kirchhoff's circuit laws.

Transmission line specialized cable or other structure designed to carry alternating current of radio frequency

In radio-frequency engineering, a transmission line is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

Electrical impedance intensive physical property

Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term complex impedance may be used interchangeably.

Ohms law relationship between voltage and current across an ideal resistor

Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:

Electrical resistance and conductance opposition to the passage of an electric current

The electrical resistance of an object is a measure of its opposition to the flow of electric current. The inverse quantity is electrical conductance, and is the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical conductance is measured in siemens (S).

Electromotive force scalar physical quantity

Electromotive force, is the electrical action produced by a non-electrical source. A device that converts other forms of energy into electrical energy, such as a battery or generator, provides an emf as its output. Sometimes an analogy to water "pressure" is used to describe electromotive force.

In electric and electronic systems, reactance is the opposition of a circuit element to a change in current or voltage, due to that element's inductance or capacitance. The notion of reactance is similar to electric resistance, but it differs in several respects.

Inductance electrical property

In electromagnetism and electronics, inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current through a conductor creates a magnetic field around the conductor, whose strength depends on the magnitude of the current. A change in current causes a change in the magnetic field. From Faraday's law of induction, any change in magnetic field through a circuit induces an electromotive force (EMF) (voltage) in the conductors; this is known as electromagnetic induction. So the changing current induces a voltage in the conductor. This induced voltage is in a direction which tends to oppose the change in current, so it is called a back EMF. Due to this back EMF, a conductor's inductance opposes any increase or decrease in electric current through it.

Series and parallel circuits the two basic ways of connecting the components of an electrical circuit

Components of an electrical circuit or electronic circuit can be connected in series, parallel, or series-parallel. The two simplest of these are called series and parallel and occur frequently. Components connected in series are connected along a single conductive path, so the same current flows through all of the components but voltage is dropped (lost) across each of the resistances. In a series circuit, the sum of the voltages consumed by each individual resistance is equal to the source voltage. Components connected in parallel are connected along multiple paths so that the current can split up; the same voltage is applied to each component.

Joule heating, also known as Ohmic heating and resistive heating, is the process by which the passage of an electric current through a conductor produces heat.

A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials like iron, although there may be air gaps or other materials in the path. Magnetic circuits are employed to efficiently channel magnetic fields in many devices such as electric motors, generators, transformers, relays, lifting electromagnets, SQUIDs, galvanometers, and magnetic recording heads.

Hydraulic analogy

The electronic–hydraulic analogy is the most widely used analogy for "electron fluid" in a metal conductor. Since electric current is invisible and the processes at play in electronics are often difficult to demonstrate, the various electronic components are represented by hydraulic equivalents. Electricity was originally understood to be a kind of fluid, and the names of certain electric quantities are derived from hydraulic equivalents. As with all analogies, it demands an intuitive and competent understanding of the baseline paradigms.

Nodal analysis method of determining the voltage (potential difference) between "nodes" (points where elements or branches connect) in an electrical circuit in terms of the branch currents

In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage between "nodes" in an electrical circuit in terms of the branch currents.

Mathematical methods are integral to the study of electronics.

Voltage drop is the decrease of electrical potential along the path of a current flowing in an electrical circuit. Voltage drops in the internal resistance of the source, across conductors, across contacts, and across connectors are undesirable because some of the energy supplied is dissipated. The voltage drop across the electrical load is proportional to the power available to be converted in that load to some other useful form of energy.

Current divider

In electronics, a current divider is a simple linear circuit that produces an output current (IX) that is a fraction of its input current (IT). Current division refers to the splitting of current between the branches of the divider. The currents in the various branches of such a circuit will always divide in such a way as to minimize the total energy expended.

Electromagnetism is the study of forces between charged particles, electromagnetic fields, electric (scalar) potentials, magnetic vector potentials, the behavior of conductors and insulators in fields, circuits, magnetism, and electromagnetic waves. An understanding of electromagnetism is important for practical applications like electrical engineering and chemistry. In addition, concepts taught in courses on electromagnetism provide a basis for more advanced material in physics, such as quantum field theory and general relativity. This article focuses on a conceptual understanding of the topics rather than the details of the mathematics involved.

Performance and modelling of AC transmission

Performance modelling is the abstraction of a real system into a simplified representation to enable the prediction of performance. The creation of a model can provide insight into how a proposed or actual system will or does work. This can, however, point towards different things to people belonging to different fields of work.

References

  1. Oldham, Kalil T. Swain (2008). The doctrine of description: Gustav Kirchhoff, classical physics, and the "purpose of all science" in 19th-century Germany (Ph. D.). University of California, Berkeley. p. 52. Docket 3331743.
  2. Athavale, Prashant. "Kirchoff's current law and Kirchoff's voltage law" (PDF). Johns Hopkins University. Retrieved 6 December 2018.
  3. 1 2 "The Feynman Lectures on Physics Vol. II Ch. 22: AC Circuits". www.feynmanlectures.caltech.edu. Retrieved 2018-12-06.
  4. 1 2 Ralph Morrison, Grounding and Shielding Techniques in Instrumentation Wiley-Interscience (1986) ISBN   0471838055