In classical mechanics, a force acting on a point-like particle can only alter the particle's dynamical state, i.e., its momentum, angular momentum, etc. Quantum field theory, however, allows interactions that can alter other facets of a particle's nature described by non dynamical, discrete quantum numbers. In particular, the action of the weak force is such that it allows the conversion of quantum numbers describing mass and electric charge of both quarks and leptons from one discrete type to another. This is known as a flavour change, or flavour transmutation. Due to their quantum description, flavour states may also undergo quantum superposition.


In atomic physics the principal quantum number of an electron specifies the electron shell in which it resides, which determines the energy level of the whole atom. Analogously, the five flavour quantum numbers (isospin, strangeness, charm, bottomness or topness) can characterize the quantum state of quarks, by the degree to which it exhibits six distinct flavours (u, d, s, c, b, t).


Composite particles can be created from multiple quarks, forming hadrons, such as mesons and baryons, each possessing unique aggregate characteristics, such as different masses, electric charges, and decay modes. A hadron's overall flavour quantum numbers depend on the numbers of constituent quarks of each particular flavour.


All of the various charges discussed above are conserved by the fact that the corresponding charge operators can be understood as "generators of symmetries" that commute with the Hamiltonian. Thus, the eigenvalues of the various charge operators are conserved.


Absolutely conserved flavour quantum numbers in the Standard Model are:


In some theories, such as the grand unified theory, the individual baryon and lepton number conservation can be violated, if the difference between them () is conserved (see chiral anomaly).


Strong interactions conserve all flavours, but all flavour quantum numbers (other than and ) are violated (changed, non-conserved) by electroweak interactions.


If there are two or more particles which have identical interactions, then they may be interchanged without affecting the physics. Any (complex) linear combination of these two particles give the same physics, as long as the combinations are orthogonal, or perpendicular, to each other.


In other words, the theory possesses symmetry transformations such as formula_1, where and are the two fields (representing the various "generations" of leptons and quarks, see below), and is any unitary matrix with a unit determinant. Such matrices form a Lie group called SU(2) (see special unitary group). This is an example of flavour symmetry.


In quantum chromodynamics, flavour is a conserved global symmetry. In the electroweak theory, on the other hand, this symmetry is broken, and flavour changing processes exist, such as quark decay or neutrino oscillations.


All leptons carry a lepton number . In addition, leptons carry weak isospin, , which is − for the three charged leptons (i.e. electron, muon and tau) and + for the three associated neutrinos. Each doublet of a charged lepton and a neutrino consisting of opposite are said to constitute one generation of leptons. In addition, one defines a quantum number called weak hypercharge, , which is −1 for all left-handed leptons. Weak isospin and weak hypercharge are gauged in the Standard Model.


Leptons may be assigned the six flavour quantum numbers: electron number, muon number, tau number, and corresponding numbers for the neutrinos. These are conserved in strong and electromagnetic interactions, but violated by weak interactions. Therefore, such flavour quantum numbers are not of great use. A separate quantum number for each generation is more useful: electronic lepton number (+1 for electrons and electron neutrinos), muonic lepton number (+1 for muons and muon neutrinos), and tauonic lepton number (+1 for tau leptons and tau neutrinos). However, even these numbers are not absolutely conserved, as neutrinos of different generations can mix; that is, a neutrino of one flavour can transform into another flavour. The strength of such mixings is specified by a matrix called the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix).


All quarks carry a baryon number and all anti-quarks have They also all carry weak isospin, The positively charged quarks (up, charm, and top quarks) are called "up-type quarks" and have the negatively charged quarks (down, strange, and bottom quarks) are called "down-type quarks" and have Each doublet of up and down type quarks constitutes one generation of quarks.


For all the quark flavour quantum numbers listed below, the convention is that the flavour charge and the electric charge of a quark have the same sign. Thus any flavour carried by a charged meson has the same sign as its charge. Quarks have the following flavour quantum numbers:


These five quantum numbers, together with baryon number (which is not a flavour quantum number), completely specify numbers of all 6 quark flavours separately (as i.e. an antiquark is counted with the minus sign). They are conserved by both the electromagnetic and strong interactions (but not the weak interaction). From them can be built the derived quantum numbers:


The terms "strange" and "strangeness" predate the discovery of the quark, but continued to be used after its discovery for the sake of continuity (i.e. the strangeness of each type of hadron remained the same); strangeness of anti-particles being referred to as +1, and particles as −1 as per the original definition. Strangeness was introduced to explain the rate of decay of newly discovered particles, such as the kaon, and was used in the Eightfold Way classification of hadrons and in subsequent quark models. These quantum numbers are preserved under strong and electromagnetic interactions, but not under weak interactions.


For first-order weak decays, that is processes involving only one quark decay, these quantum numbers (e.g. charm) can only vary by 1, that is, for a decay involving a charmed quark or antiquark either as the incident particle or as a decay byproduct, likewise, for a decay involving a bottom quark or antiquark Since first-order processes are more common than second-order processes (involving two quark decays), this can be used as an approximate "selection rule" for weak decays.


A special mixture of quark flavours is an eigenstate of the weak interaction part of the Hamiltonian, so will interact in a particularly simple way with the W bosons (charged weak interactions violate flavour). On the other hand, a fermion of a fixed mass (an eigenstate of the kinetic and strong interaction parts of the Hamiltonian) is an eigenstate of flavour. The transformation from the former basis to the flavour-eigenstate/mass-eigenstate basis for quarks underlies the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix). This matrix is analogous to the PMNS matrix for neutrinos, and quantifies flavour changes under charged weak interactions of quarks.


The CKM matrix allows for CP violation if there are at least three generations.


Flavour quantum numbers are additive. Hence antiparticles have flavour equal in magnitude to the particle but opposite in sign. Hadrons inherit their flavour quantum number from their valence quarks: this is the basis of the classification in the quark model. The relations between the hypercharge, electric charge and other flavour quantum numbers hold for hadrons as well as quarks.


Quantum chromodynamics (QCD) contains six flavours of quarks. However, their masses differ and as a result they are not strictly interchangeable with each other. The up and down flavours are close to having equal masses, and the theory of these two quarks possesses an approximate SU(2) symmetry (isospin symmetry).


Under some circumstances (for instance when the quark masses are much smaller than the chiral symmetry breaking scale of 250 MeV), the masses of quarks do not meaningfully contribute to the system's behavior, and can be ignored to zeroth approximation. The simplified behavior of flavour transformations can then be successfully modeled as acting independently on the left- and right-handed parts of each quark field. This approximate description of the flavour symmetry is described by a chiral group .


If all quarks had non-zero but equal masses, then this chiral symmetry is broken to the "vector symmetry" of the "diagonal flavour group" , which applies the same transformation to both helicities of the quarks. This reduction of symmetry is a form of "explicit symmetry breaking". The strength of explicit symmetry breaking is controlled by the current quark masses in QCD.


Even if quarks are massless, chiral flavour symmetry can be spontaneously broken if the vacuum of the theory contains a chiral condensate (as it does in low-energy QCD). This gives rise to an effective mass for the quarks, often identified with the valence quark mass in QCD.


Analysis of experiments indicate that the current quark masses of the lighter flavours of quarks are much smaller than the QCD scale, ΛQCD, hence chiral flavour symmetry is a good approximation to QCD for the up, down and strange quarks. The success of chiral perturbation theory and the even more naive chiral models spring from this fact. The valence quark masses extracted from the quark model are much larger than the current quark mass. This indicates that QCD has spontaneous chiral symmetry breaking with the formation of a chiral condensate. Other phases of QCD may break the chiral flavour symmetries in other ways.


Some of the historical events that led to the development of flavour symmetry are discussed in the article on isospin, the eightfold way (physics) and chiral symmetry. Chief among these would be the November Revolution (physics) in 1974, when the fourth (charm) quark was found.


Reciprocity in electrical networks is a property of a circuit that relates voltages and currents at two points. The reciprocity theorem states that the current at one point in a circuit due to a voltage at a second point is the same as the current at the second point due to the same voltage at the first. The reciprocity theorem is valid for almost all passive networks. The reciprocity theorem is a feature of a more general principle of reciprocity in electromagnetism.


If a current, formula_1, injected into port A produces a voltage, formula_2, at port B and formula_1 injected into port B produces formula_2 at port A, then the network is said to be reciprocal. Equivalently, reciprocity can be defined by the dual situation; applying voltage, formula_5, at port A producing current formula_6 at port B and formula_5 at port B producing current formula_6 at port A. In general, passive networks are reciprocal. Any network that consists entirely of ideal capacitances, inductances (including mutual inductances), and resistances, that is, elements that are linear and bilateral, will be reciprocal. However, passive components that are non-reciprocal do exist. Any component containing ferromagnetic material is likely to be non-reciprocal. Examples of passive components deliberately designed to be non-reciprocal include circulators and isolators.


The transfer function of a reciprocal network has the property that it is symmetrical about the main diagonal if expressed in terms of a z-parameter, y-parameter, or s-parameter matrix. A non-symmetrical matrix implies a non-reciprocal network. A symmetric matrix does not imply a symmetric network.


In some parametisations of networks, the representative matrix is not symmetrical for reciprocal networks. Common examples are h-parameters and ABCD-parameters, but they all have some other condition for reciprocity that can be calculated from the parameters. For h-parameters the condition is formula_9 and for the ABCD parameters it is formula_10. These representations mix voltages and currents in the same column vector and therefore do not even have matching units in transposed elements.


An example of reciprocity can be demonstrated using an asymmetrical resistive attenuator. An asymmetrical network is chosen as the example because a symmetrical network is fairly self-evidently reciprocal.


Injecting six amps into port 1 of this network produces 24 volts at port 2.


Injecting six amps into port 2 produces 24 volts at port 1.


Hence, the network is reciprocal. In this example, the port that is not injecting current is left open circuit. This is because a current generator applying zero current is an open circuit. If, on the other hand, one wished to apply voltages and measure the resulting current, then the port to which the voltage is not applied would be made short circuit. This is because a voltage generator applying zero volts is a short circuit.


Reciprocity of electrical networks is a special case of Lorentz reciprocity, but it can also be proven more directly from network theorems. This proof shows reciprocity for a two-node network in terms of its admittance matrix, and then shows reciprocity for a network with an arbitrary number of nodes by an induction argument. A linear network can be represented as a set of linear equations through nodal analysis. These equations can be expressed in the form of an admittance matrix,


If we further require that network is made up of passive, bilateral elements, then


since the admittance connected between nodes "j" and "k" is the same element as the admittance connected between nodes "k" and "j". The matrix is therefore symmetrical. For the case where formula_17 the matrix reduces to,


From which it can be seen that,


which is synonymous with the condition for reciprocity. In words, the ratio of the current at one port to the voltage at another is the same ratio if the ports being driven and measured are interchanged. Thus reciprocity is proven for the case of formula_17.


For the case of a matrix of arbitrary size, the order of the matrix can be reduced through node elimination. After eliminating the "s"th node, the new admittance matrix will have the form,


It can be seen that this new matrix is also symmetrical. Nodes can continue to be eliminated in this way until only a 2×2 symmetrical matrix remains involving the two nodes of interest. Since this matrix is symmetrical it is proved that reciprocity applies to a matrix of arbitrary size when one node is driven by a voltage and current measured at another. A similar process using the impedance matrix from mesh analysis demonstrates reciprocity where one node is driven by a current and voltage is measured at another.


The Extra Element Theorem (EET) is an analytic technique developed by R. D. Middlebrook for simplifying the process of deriving driving point and transfer functions for linear electronic circuits. Much like Thévenin's theorem, the extra element theorem breaks down one complicated problem into several simpler ones.


Driving point and transfer functions can generally be found using Kirchhoff's circuit laws. However several complicated equations may result that offer little insight into the circuit's behavior. Using the extra element theorem, a circuit element (such as a resistor) can be removed from a circuit and the desired driving point or transfer function found. By removing the element that most complicates the circuit (such as an element that creates feedback), the desired function can be easier to obtain. Next two correctional factors must be found and combined with the previously derived function to find the exact expression.


The general form of the extra element theorem is called the N-extra element theorem and allows multiple circuit elements to be removed at once.


The (single) extra element theorem expresses any transfer function as a product of the transfer function with that element removed and a correction factor. The correction factor term consists of the impedance of the extra element and two driving point impedances seen by the extra element: The double null injection driving point impedance and the single injection driving point impedance. Because an extra element can be removed in general by either short-circuiting or open-circuiting the element, there are two equivalent forms of the EET:


Where the Laplace-domain transfer functions and impedances in the above expressions are defined as follows: is the transfer function with the extra element present. is the transfer function with the extra element open-circuited. is the transfer function with the extra element short-circuited. is the impedance of the extra element. is the single-injection driving point impedance "seen" by the extra element. is the double-null-injection driving point impedance "seen" by the extra element.


The extra element theorem incidentally proves that any electric circuit transfer function can be expressed as no more than a bilinear function of any particular circuit element.


is found by making the input to the system's transfer function zero (short circuit a voltage source or open circuit a current source) and determining the impedance across the terminals to which the extra element will be connected with the extra element absent. This impedance is same as the Thévenin's equivalent impedance.


is found by replacing the extra element with a second test signal source (either current source or voltage source as appropriate). Then, is defined as the ratio of voltage across the terminals of this second test source to the current leaving its positive terminal when the output of the system's transfer function is nulled for any value of the primary input to the system's transfer function.


In practice, can be found from working backwards from the facts that the output of the transfer function is made zero and that the primary input to the transfer function is unknown. Then using conventional circuit analysis techniques to express both the voltage across the extra element test source's terminals, , and the current leaving the extra element test source's positive terminals, , and calculating formula_3. Although computation of is an unfamiliar process for many engineers, its expressions are often much simpler than those for because the nulling of the transfer function's output often leads to other voltages/currents in the circuit being zero, which may allow exclusion of certain components from analysis.


Special case with transfer function as a self-impedance.


As a special case, the EET can be used to find the input impedance of a network with the addition of an element designated as "extra". In this case, is same as the impedance of the input test current source signal made zero or equivalently with the input open circuited. Likewise, since the transfer function output signal can be considered to be the voltage at the input terminals, is found when the input voltage is zero i.e. the input terminals are short-circuited. Thus, for this particular application the EET can be written as:


Computing these three terms may seem like extra effort, but they are often easier to compute than the overall input impedance.


Consider the problem of finding formula_9 for the circuit in Figure 1 using the EET (note all component values are unity for simplicity). If the capacitor (gray shading) is denoted the extra element then


Calculating the impedance seen by the capacitor with the input shorted,


Calculating the impedance seen by the capacitor with the input open,


This problem was solved by calculating three simple driving point impedances by inspection.


The EET is also useful for analyzing single and multi-loop feedback amplifiers. In this case the EET can take the form of the asymptotic gain model.


The star-mesh transform, or star-polygon transform, is a mathematical circuit analysis technique to transform a resistive network into an equivalent network with one less node. The equivalence follows from the Schur complement identity applied to the Kirchhoff matrix of the network.


The equivalent impedance betweens nodes A and B is given by:


where formula_2 is the impedance between node A and the central node being removed.


The transform replaces "N" resistors with formula_3 resistors. For formula_4, the result is an increase in the number of resistors, so the transform has no general inverse without additional constraints.


It is possible, though not necessarily efficient, to transform an arbitrarily complex two-terminal resistive network into a single equivalent resistor by repeatedly applying the star-mesh transform to eliminate each non-terminal node.


Source transformation is the process of simplifying a circuit solution, especially with mixed sources, by transforming voltage sources into current sources, and vice versa, using Thévenin's theorem and Norton's theorem respectively.


Performing a source transformation consists of using Ohm's law to take an existing voltage source in series with a resistance, and replacing it with a current source in parallel with the same resistance, or vice versa. The transformed sources are considered identical and can be substituted for one another in a circuit.


Source transformations are not limited to resistive circuits. They can be performed on a circuit involving capacitors and inductors as well, by expressing circuit elements as impedances and sources in the frequency domain. In general, the concept of source transformation is an application of Thévenin's theorem to a current source, or Norton's theorem to a voltage source. However, this means that source transformation is bound by the same conditions as Thevenin's theorem and Norton's theorem; namely that the load behaves linearly, and does not contain dependent voltage or current sources.


Source transformations are easy to compute using Ohm's law. If there is a voltage source in series with an impedance, it is possible to find the value of the equivalent current source in parallel with the impedance by dividing the value of the voltage source by the value of the impedance. The converse also holds: if a current source in parallel with an impedance is present, multiplying the value of the current source with the value of the impedance provides the equivalent voltage source in series with the impedance. A visual example of a source transformation can be seen in Figure 1.


The transformation can be derived from the uniqueness theorem. In the present context, it implies that a black box with two terminals must have a unique well-defined relation between its voltage and current. It is readily to verify that the above transformation indeed gives the same V-I curve, and therefore the transformation is valid.


In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis. Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point.


A Hopf bifurcation is also known as a Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Aleksandr Andronov and Eberhard Hopf.


The limit cycle is orbitally stable if a specific quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical. Otherwise it is unstable and the bifurcation is subcritical.


The normal form of a Hopf bifurcation is:


Write: formula_2 The number "α" is called the first Lyapunov coefficient.


Hopf bifurcations occur in the Lotka–Volterra model of predator–prey interaction (known as paradox of enrichment), the Hodgkin–Huxley model for nerve membrane, the Selkov model of glycolysis, the Belousov–Zhabotinsky reaction, the Lorenz attractor, the Brusselator and Classical electromagnetism.


The phase portrait illustrating the Hopf bifurcation in the Selkov model is shown on the right.


In railway vehicle systems, Hopf bifurcation analysis is notably important. Conventionally a railway vehicle's stable motion at low speeds crosses over to unstable at high speeds. One aim of the nonlinear analysis of these systems is to perform an analytical investigation of bifurcation, nonlinear lateral stability and hunting behavior of rail vehicles on a tangent track, which uses the Bogoliubov method.


The appearance or the disappearance of a periodic orbit through a local change in the stability properties of a fixed point is known as the Hopf bifurcation. The following theorem works for fixed points with one pair of conjugate nonzero purely imaginary eigenvalues. It tells the conditions under which this bifurcation phenomenon occurs.


Theorem (see section 11.2 of ). Let formula_6 be the Jacobian of a continuous parametric dynamical system evaluated at a steady point formula_7. Suppose that all eigenvalues of formula_6 have negative real part except one conjugate nonzero purely imaginary pair formula_9. A "Hopf bifurcation" arises when these two eigenvalues cross the imaginary axis because of a variation of the system parameters.


Routh–Hurwitz criterion (section I.13 of ) gives necessary conditions so that a Hopf bifurcation occurs. Let us see how one can use concretely this idea.


Let formula_10 be Sturm series associated to a characteristic polynomial formula_11. They can be written in the form:


The coefficients formula_13 for formula_14 in formula_15 correspond to what is called Hurwitz determinants. Their definition is related to the associated Hurwitz matrix.


Proposition 1. If all the Hurwitz determinants formula_13 are positive, apart perhaps formula_17 then the associated Jacobian has no pure imaginary eigenvalues.


Proposition 2. If all Hurwitz determinants formula_13 (for all formula_14 in formula_20 are positive, formula_21 and formula_22 then all the eigenvalues of the associated Jacobian have negative real parts except a purely imaginary conjugate pair.


The conditions that we are looking for so that a Hopf bifurcation occurs (see theorem above) for a parametric continuous dynamical system are given by this last proposition.


Consider the classical Van der Pol oscillator written with ordinary differential equations:


The Jacobian matrix associated to this system follows:


The characteristic polynomial (in formula_25) of the linearization at (0,0) is equal to:


The Sturm polynomials can be written as (here formula_29):


The above proposition 2 tells that one must have:


Because 1 > 0 and −1 < 0 are obvious, one can conclude that a Hopf bifurcation may occur for Van der Pol oscillator if formula_32.


In electrical engineering and science, an equivalent circuit refers to a theoretical circuit that retains all of the electrical characteristics of a given circuit. Often, an equivalent circuit is sought that simplifies calculation, and more broadly, that is a simplest form of a more complex circuit in order to aid analysis. In its most common form, an equivalent circuit is made up of linear, passive elements. However, more complex equivalent circuits are used that approximate the nonlinear behavior of the original circuit as well. These more complex circuits often are called "macromodels" of the original circuit. An example of a macromodel is the Boyle circuit for the 741 operational amplifier.


One of linear circuit theory's most surprising properties relates to the ability to treat any two-terminal circuit no matter how complex as behaving as only a source and an impedance, which have either of two simple equivalent circuit forms:


However, the single impedance can be of arbitrary complexity (as a function of frequency) and may be irreducible to a simpler form.


In linear circuits, due to the superposition principle, the output of a circuit is equal to the sum of the output due to its DC sources alone, and the output from its AC sources alone. Therefore, the DC and AC response of a circuit is often analyzed independently, using separate DC and AC equivalent circuits which have the same response as the original circuit to DC and AC currents respectively. The composite response is calculated by adding the DC and AC responses:


This technique is often extended to small-signal nonlinear circuits like tube and transistor circuits, by linearizing the circuit about the DC bias point Q-point, using an AC equivalent circuit made by calculating the equivalent "small signal" AC resistance of the nonlinear components at the bias point.


Linear four-terminal circuits in which a signal is applied to one pair of terminals and an output is taken from another, are often modeled as two-port networks. These can be represented by simple equivalent circuits of impedances and dependent sources. To be analyzed as a two port network the currents applied to the circuit must satisfy the "port condition": the current entering one terminal of a port must be equal to the current leaving the other terminal of the port. By linearizing a nonlinear circuit about its operating point, such a two-port representation can be made for transistors: see hybrid pi and h-parameter circuits.


In three phase power circuits, three phase sources and loads can be connected in two different ways, called a "delta" connection and a "wye" connection. In analyzing circuits, sometimes it simplifies the analysis to convert between equivalent wye and delta circuits. This can be done with the wye-delta transform.


Equivalent circuits can be used to electrically describe and model either a) continuous materials or biological systems in which current does not actually flow in defined circuits, or, b) distributed reactances, such as found in electrical lines or windings, that do not represent actual discrete components. For example, a cell membrane can be modelled as a capacitance (i.e. the lipid bilayer) in parallel with resistance-DC voltage source combinations (i.e. ion channels powered by an ion gradient across the membrane).


In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time-harmonic electric current densities (sources) and the resulting electromagnetic fields in Maxwell's equations for time-invariant linear media under certain constraints. Reciprocity is closely related to the concept of Hermitian operators from linear algebra, applied to electromagnetism.


Reciprocity is useful in optics, which (apart from quantum effects) can be expressed in terms of classical electromagnetism, but also in terms of radiometry.


There is also an analogous theorem in electrostatics, known as Green's reciprocity, relating the interchange of electric potential and electric charge density.


Forms of the reciprocity theorems are used in many electromagnetic applications, such as analyzing electrical networks and antenna systems. For example, reciprocity implies that antennas work equally well as transmitters or receivers, and specifically that an antenna's radiation and receiving patterns are identical. Reciprocity is also a basic lemma that is used to prove other theorems about electromagnetic systems, such as the symmetry of the impedance matrix and scattering matrix, symmetries of Green's functions for use in boundary-element and transfer-matrix computational methods, as well as orthogonality properties of harmonic modes in waveguide systems (as an alternative to proving those properties directly from the symmetries of the eigen-operators).


Specifically, suppose that one has a current density formula_1 that produces an electric field formula_2 and a magnetic field formula_3, where all three are periodic functions of time with angular frequency ω, and in particular they have time-dependence formula_4. Suppose that we similarly have a second current formula_5 at the same frequency ω which (by itself) produces fields formula_6 and formula_7. The Lorentz reciprocity theorem then states, under certain simple conditions on the materials of the medium described below, that for an arbitrary surface "S" enclosing a volume "V":


Equivalently, in differential form (by the divergence theorem):


This general form is commonly simplified for a number of special cases. In particular, one usually assumes that formula_1 and formula_5 are localized (i.e. have compact support), and that there are no incoming waves from infinitely far away. In this case, if one integrates throughout space then the surface-integral terms cancel (see below) and one obtains:


This result (along with the following simplifications) is sometimes called the Rayleigh-Carson reciprocity theorem, after Lord Rayleigh's work on sound waves and an extension by John R. Carson (1924; 1930) to applications for radio frequency antennas. Often, one further simplifies this relation by considering point-like dipole sources, in which case the integrals disappear and one simply has the product of the electric field with the corresponding dipole moments of the currents. Or, for wires of negligible thickness, one obtains the applied current in one wire multiplied by the resulting voltage across another and vice versa; see also below.


Another special case of the Lorentz reciprocity theorem applies when the volume "V" entirely contains "both" of the localized sources (or alternatively if "V" intersects "neither" of the sources). In this case:


Above, Lorentz reciprocity was phrased in terms of an externally applied current source and the resulting field. Often, especially for electrical networks, one instead prefers to think of an externally applied voltage and the resulting currents. The Lorentz reciprocity theorem describes this case as well, assuming ohmic materials (i.e. currents that respond linearly to the applied field) with a 3×3 conductivity matrix σ that is required to be symmetric, which is implied by the other conditions below. In order to properly describe this situation, one must carefully distinguish between the externally "applied" fields (from the driving voltages) and the "total" fields that result (King, 1963).


More specifically, the formula_14 above only consisted of external "source" terms introduced into Maxwell's equations. We now denote this by formula_15 to distinguish it from the "total" current produced by both the external source and by the resulting electric fields in the materials. If this external current is in a material with a conductivity σ, then it corresponds to an externally applied electric field formula_16 where, by definition of σ:


Moreover, the electric field formula_18 above only consisted of the "response" to this current, and did not include the "external" field formula_16. Therefore, we now denote the field from before as formula_20, where the "total" field is given by formula_21.


Now, the equation on the left-hand side of the Lorentz reciprocity theorem can be rewritten by moving the σ from the external current term formula_15 to the response field terms formula_20, and also adding and subtracting a formula_24 term, to obtain the external field multiplied by the "total" current formula_25:


For the limit of thin wires, this gives the product of the externally applied voltage (1) multiplied by the resulting total current (2) and vice versa. In particular, the Rayleigh-Carson reciprocity theorem becomes a simple summation:


where "V" and "I" denote the complex amplitudes of the AC applied voltages and the resulting currents, respectively, in a set of circuit elements (indexed by "n") for two possible sets of voltages formula_28 and formula_29.


Most commonly, this is simplified further to the case where each system has a "single" voltage source "V", at formula_30 and formula_31. Then the theorem becomes simply


The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator formula_33 relating formula_14 and formula_18 at a fixed frequency formula_36 (in linear media):


For any Hermitian operator formula_33 under an inner product formula_43, we have formula_44 by definition, and the Rayleigh-Carson reciprocity theorem is merely the vectorial version of this statement for this particular operator formula_45: that is, formula_46. The Hermitian property of the operator here can be derived by integration by parts. For a finite integration volume, the surface terms from this integration by parts yield the more-general surface-integral theorem above. In particular, the key fact is that, for vector fields formula_39 and formula_40, integration by parts (or the divergence theorem) over a volume "V" enclosed by a surface "S" gives the identity:


This identity is then applied twice to formula_50 to yield formula_51 plus the surface term, giving the Lorentz reciprocity relation.


Conditions and proof of Lorenz reciprocity using Maxwell's equations and vector operations


We shall prove a general form of the electromagnetic reciprocity theorem due to Lorenz which states that fields formula_52 and formula_53 generated by two different sinusoidal current densities respectively formula_54 and formula_55 of the same frequency, satisfy the condition


Let us take a region in which dielectric constant and permeability may be functions of position but not of time. Maxwell's equations, written in terms of the total fields, currents and charges of the region describe the electromagnetic behavior of the region. The two curl equations are:


Under steady constant frequency conditions we get from the two curl equations the Maxwell's equations for the Time-Periodic case:


It must be recognized that the symbols in the equations of this article represent the complex multipliers of formula_59, giving the in-phase and out-of-phase parts with respect to the chosen reference. The complex vector multipliers of formula_59 may be called "vector phasors" by analogy to the complex scalar quantities which are commonly referred to as "phasors".


An equivalence of vector operations shows that


formula_61 for every vectors formula_62 and formula_63.


If we apply this equivalence to formula_64 and formula_65 we get:


If products in the Time-Periodic equations are taken as indicated by this last equivalence, and added,


This now may be integrated over the volume of concern,


From the divergence theorem the volume integral of formula_69 equals the surface integral of formula_70 over the boundary.


This form is valid for general media, but in the common case of linear, isotropic, time-invariant materials, formula_72 is a scalar independent of time. Then generally as physical magnitudes formula_73 and formula_74.


In an exactly analogous way we get for vectors formula_6 and formula_3 the following expression:


Subtracting the two last equations by members we get


The cancellation of the surface terms on the right-hand side of the Lorentz reciprocity theorem, for an integration over all space, is not entirely obvious but can be derived in a number of ways.


Another simple argument would be that the fields goes to zero at infinity for a localized source, but this argument fails in the case of lossless media: in the absence of absorption, radiated fields decay inversely with distance, but the surface area of the integral increases with the square of distance, so the two rates balance one another in the integral.


Instead, it is common (e.g. King, 1963) to assume that the medium is homogeneous and isotropic sufficiently far away. In this case, the radiated field asymptotically takes the form of planewaves propagating radially outward (in the formula_81 direction) with formula_82 and formula_83 where "Z" is the impedance formula_84 of the surrounding medium. Then it follows that formula_85, which by a simple vector identity equals formula_86. Similarly, formula_87 and the two terms cancel one another.


The above argument shows explicitly why the surface terms can cancel, but lacks generality. Alternatively, one can treat the case of lossless surrounding media by taking the limit as the losses (the imaginary part of ε) go to zero. For any nonzero loss, the fields decay exponentially with distance and the surface integral vanishes, regardless of whether the medium is homogeneous. Since the left-hand side of the Lorentz reciprocity theorem vanishes for integration over all space with any non-zero losses, it must also vanish in the limit as the losses go to zero. (Note that we implicitly assumed the standard boundary condition of zero incoming waves from infinity, because otherwise even an infinitesimal loss would eliminate the incoming waves and the limit would not give the lossless solution.)


One case in which ε is "not" a symmetric matrix is for magneto-optic materials, in which case the usual statement of Lorentz reciprocity does not hold (see below for a generalization, however). If we allow magneto-optic materials, but restrict ourselves to the situation where material "absorption is negligible", then ε and μ are in general 3×3 complex Hermitian matrices. In this case, the operator formula_99 is Hermitian under the "conjugated" inner product formula_100, and a variant of the reciprocity theorem still holds:


where the sign changes come from the formula_102 in the equation above, which makes the operator formula_33 anti-Hermitian (neglecting surface terms). For the special case of formula_104, this gives a re-statement of conservation of energy or Poynting's theorem (since here we have assumed lossless materials, unlike above): the time-average rate of work done by the current (given by the real part of formula_105) is equal to the time-average outward flux of power (the integral of the Poynting vector). By the same token, however, the surface terms do not in general vanish if one integrates over all space for this reciprocity variant, so a Rayleigh-Carson form does not hold without additional assumptions.


The fact that magneto-optic materials break Rayleigh-Carson reciprocity is the key to devices such as Faraday isolators and circulators. A current on one side of a Faraday isolator produces a field on the other side but "not" vice versa.


For a combination of lossy and magneto-optic materials, and in general when the ε and μ tensors are neither symmetric nor Hermitian matrices, one can still obtain a generalized version of Lorentz reciprocity by considering formula_106 and formula_107 to exist in "different systems."


In particular, if formula_106 satisfy Maxwell's equations at ω for a system with materials formula_109, and formula_107 satisfy Maxwell's equations at ω for a system with materials formula_111, where "T" denotes the transpose, then the equation of Lorentz reciprocity holds. This can be further generalized to bi-anisotropic materials by transposing the full 6×6 susceptibility tensor.


For nonlinear media, no reciprocity theorem generally holds. Reciprocity also does not generally apply for time-varying ("active") media; for example, when ε is modulated in time by some external process. (In both of these cases, the frequency ω is not generally a conserved quantity.)


A closely related reciprocity theorem was articulated independently by Y. A. Feld and C. T. Tai in 1992 and is known as Feld-Tai reciprocity or the Feld-Tai lemma. It relates two time-harmonic localized current sources and the resulting magnetic fields:


However, the Feld-Tai lemma is only valid under much more restrictive conditions than Lorentz reciprocity. It generally requires time-invariant linear media with an isotropic homogeneous impedance, i.e. a constant scalar μ/ε ratio, with the possible exception of regions of perfectly conducting material.


More precisely, Feld-Tai reciprocity requires the Hermitian (or rather, complex-symmetric) symmetry of the electromagnetic operators as above, but also relies on the assumption that the operator relating formula_18 and formula_114 is a constant scalar multiple of the operator relating formula_115 and formula_116, which is true when ε is a constant scalar multiple of μ (the two operators generally differ by an interchange of ε and μ). As above, one can also construct a more general formulation for integrals over a finite volume.


Apart from quantal effects, classical theory covers near-, middle-, and far-field electric and magnetic phenomena with arbitrary time courses. Optics refers to far-field nearly-sinusoidal oscillatory electromagnetic effects. Instead of paired electric and magnetic variables, optics, including optical reciprocity, can be expressed in polarization-paired radiometric variables, such as spectral radiance, traditionally called specific intensity.


This is sometimes called the Helmholtz reciprocity (or reversion) principle. When the wave propagates through a material acted upon by an applied magnetic field, reciprocity can be broken so this principle will not apply. Similarly, when there are moving objects in the path of the ray, the principle may be entirely inapplicable. Historically, in 1849, Sir George Stokes stated his optical reversion principle without attending to polarization.


Like the principles of thermodynamics, this principle is reliable enough to use as a check on the correct performance of experiments, in contrast with the usual situation in which the experiments are tests of a proposed law.


The simplest statement of the principle is 'if I can see you, then you can see me.' The principle was used by Gustav Kirchhoff in his derivation of his law of thermal radiation and by Max Planck in his analysis of his law of thermal radiation.


For ray-tracing global illumination algorithms, incoming and outgoing light can be considered as reversals of each other, without affecting the bidirectional reflectance distribution function (BRDF) outcome.


Whereas the above reciprocity theorems were for oscillating fields, Green's reciprocity is an analogous theorem for electrostatics with a fixed distribution of electric charge (Panofsky and Phillips, 1962).


In particular, let formula_117 denote the electric potential resulting from a total charge density formula_118. The electric potential satisfies Poisson's equation, formula_119, where formula_120 is the vacuum permittivity. Similarly, let formula_121 denote the electric potential resulting from a total charge density formula_122, satisfying formula_123. In both cases, we assume that the charge distributions are localized, so that the potentials can be chosen to go to zero at infinity. Then, Green's reciprocity theorem states that, for integrals over all space:


This theorem is easily proven from Green's second identity. Equivalently, it is the statement that formula_125, i.e. that formula_126 is a Hermitian operator (as follows by integrating by parts twice).


The Miller theorem refers to the process of creating equivalent circuits. It asserts that a floating impedance element, supplied by two voltage sources connected in series, may be split into two grounded elements with corresponding impedances. There is also a dual Miller theorem with regards to impedance supplied by two current sources connected in parallel. The two versions are based on the two Kirchhoff's circuit laws.


Miller theorems are not only pure mathematical expressions. These arrangements explain important circuit phenomena about modifying impedance (Miller effect, virtual ground, bootstrapping, negative impedance, etc.) and help in designing and understanding various commonplace circuits (feedback amplifiers, resistive and time-dependent converters, negative impedance converters, etc.). The theorems are useful in 'circuit analysis' especially for analyzing circuits with feedback and certain transistor amplifiers at high frequencies.


There is a close relationship between Miller theorem and Miller effect: the theorem may be considered as a generalization of the effect and the effect may be thought as of a special case of the theorem.


The Miller theorem establishes that in a linear circuit, if there exists a branch with impedance "Z", connecting two nodes with nodal voltages "V1" and "V2", we can replace this branch by two branches connecting the corresponding nodes to ground by impedances respectively Z/(1 − K) and KZ/(K − 1), where K = V2/V1. The Miller theorem may be proved by using the equivalent two-port network technique to replace the two-port to its equivalent and by applying the source absorption theorem. This version of the Miller theorem is based on Kirchhoff's voltage law; for that reason, it is named also "Miller theorem for voltages".


The Miller theorem implies that an impedance element is supplied by two arbitrary (not necessarily dependent) voltage sources that are connected in series through the common ground. In practice, one of them acts as a main (independent) voltage source with voltage "V1" and the other – as an additional (linearly dependent) voltage source with voltage formula_1. The idea of the Miller theorem (modifying circuit impedances seen from the sides of the input and output sources) is revealed below by comparing the two situations – without and with connecting an additional voltage source V2.


If "V2" were zero (there was not a second voltage source or the right end of the element with impedance "Z" was just grounded), the input current flowing through the element would be determined, according to Ohm's law, only by "V1"


and the input impedance of the circuit would be


As a second voltage source is included, the input current depends on both the voltages. According to its polarity, "V2" is subtracted from or added to "V1"; so, the input current decreases/increases


and the input impedance of the circuit seen from the side of the input source accordingly increases/decreases


So, the Miller theorem expresses the fact that connecting a second voltage source with proportional voltage formula_1 in series with the input voltage source changes the effective voltage, the current and respectively, the circuit impedance seen from the side of the input source. Depending on the polarity, "V2" acts as a supplemental voltage source helping or opposing the main voltage source to pass the current through the impedance.


Besides by presenting the combination of the two voltage sources as a new composed voltage source, the theorem may be explained by "combining the actual element and the second voltage source into a new virtual element with dynamically modified impedance". From this viewpoint, "V2" is an additional voltage that artificially increases/decreases the voltage drop "Vz" across the impedance "Z" thus decreasing/increasing the current. The proportion between the voltages determines the value of the obtained impedance (see the tables below) and gives in total six groups of typical applications.


The circuit impedance, seen from the side of the output source, may be defined similarly, if the voltages "V1" and "V2" are swapped and the coefficient "K" is replaced by 1/"K"


Most frequently, the Miller theorem may be observed in, and implemented by, an arrangement consisting of an element with impedance "Z" connected between the two terminals of a grounded general linear network. Usually, a voltage amplifier with gain of formula_8 serves as such a linear network, but also other devices can play this role: a man and a potentiometer in a potentiometric null-balance meter, an electromechanical integrator (servomechanisms using potentiometric feedback sensors), etc.


In the amplifier implementation, the input voltage "Vi" serves as "V1" and the output voltage "Vo" – as "V2". In many cases, the input voltage source has some internal impedance formula_9 or an additional input impedance is connected that, in combination with "Z", introduces a feedback. Depending on the kind of amplifier (non-inverting, inverting or differential), the feedback can be positive, negative or mixed.


The Miller amplifier arrangement has two aspects:


The introduction of an impedance that connects amplifier input and output ports adds a great


deal of complexity in the analysis process. Miller theorem helps reduce the


complexity in some circuits particularly with feedback by converting them to simpler equivalent circuits. But Miller theorem is not only an effective tool for creating equivalent circuits; it is also a powerful tool for designing and understanding circuits based on "modifying impedance by additional voltage". Depending on the polarity of the output voltage versus the input voltage and the proportion between their magnitudes, there are six groups of typical situations. In some of them, the Miller phenomenon appears as desired (bootstrapping) or undesired (Miller effect) unintentional effects; in other cases it is intentionally introduced.


Applications based on subtracting "V2" from "V1".


In these applications, the output voltage "Vo" is inserted with an opposite polarity in respect to the input voltage "Vi" travelling along the loop (but in respect to ground, the polarities are the same). As a result, the effective voltage across, and the current through, the impedance decrease; the input impedance increases.


Increased impedance is implemented by a non-inverting amplifier with gain of 0 < Av < 1. The (magnitude of) output voltage is less than the input voltage "Vi" and partially neutralizes it. Examples are imperfect voltage followers (emitter, source, cathode follower, etc.) and amplifiers with series negative feedback (emitter degeneration), whose input impedance is moderately increased.


Infinite impedance uses a non-inverting amplifier with Av = 1. The output voltage is equal to the input voltage "Vi" and completely neutralizes it. Examples are potentiometric null-balance meters and op-amp followers and amplifiers with series negative feedback (op-amp follower and non-inverting amplifier) where the circuit input impedance is enormously increased. This technique is referred to as bootstrapping and is intentionally used in biasing circuits, input guarding circuits, etc.


Negative impedance obtained by current inversion is implemented by a non-inverting amplifier with Av > 1. The current changes its direction, as the output voltage is higher than the input voltage. If the input voltage source has some internal impedance formula_9 or if it is connected through another impedance element, a positive feedback appears. A typical application is the negative impedance converter with current inversion (INIC) that uses both negative and positive feedback (the negative feedback is used to realize a non-inverting amplifier and the positive feedback – to modify the impedance).


Applications based on adding "V2" to "V1".


In these applications, the output voltage "Vo" is inserted with the same polarity in respect to the input voltage "Vi" travelling along the loop (but in respect to ground, the polarities are opposite). As a result, the effective voltage across and the current through the impedance increase; the input impedance decreases.


Decreased impedance is implemented by an inverting amplifier having some moderate gain, usually 10 < Av < 1000. It may be observed as an undesired Miller effect in common-emitter, common-source and "common-cathode" amplifying stages where effective input capacitance is increased. Frequency compensation for general purpose operational amplifiers and transistor Miller integrator are examples of useful usage of the Miller effect.


Zeroed impedance uses an inverting (usually op-amp) amplifier with enormously high gain Av → ∞. The output voltage is almost equal to the voltage drop "VZ" across the impedance and completely neutralizes it. The circuit behaves as a short connection and a virtual ground appears at the input; so, it should not be driven by a constant voltage source. For this purpose, some circuits are driven by a constant current source or by a real voltage source with internal impedance: current-to-voltage converter (transimpedance amplifier), capacitive integrator (named also current integrator or charge amplifier), resistance-to-voltage converter (a resistive sensor connected in the place of the impedance "Z").


The rest of them have additional impedance connected in series to the input: voltage-to-current converter (transconductance amplifier), inverting amplifier, summing amplifier, inductive integrator, capacitive differentiator, resistive-capacitive integrator, capacitive-resistive differentiator, inductive-resistive differentiator, etc. The inverting integrators from this list are examples of useful and desired applications of the Miller effect in its extreme manifestation.


In all these "op-amp inverting circuits with parallel negative feedback", the input current is increased to its maximum. It is determined only by the input voltage and the input impedance according to Ohm's law; it does not depend on the impedance "Z".


Negative impedance with voltage inversion is implemented by applying both negative and positive feedback to an op-amp amplifier with a differential input. The input voltage source has to have internal impedance formula_9 > 0 or it has to be connected through another impedance element to the input. Under these conditions, the input voltage "Vi" of the circuit changes its polarity as the output voltage exceeds the voltage drop "VZ" across the impedance ("Vi" = "Vz" – "Vo" < 0).


A typical application is a negative impedance converter with voltage inversion (VNIC). It is interesting that the circuit input voltage has the same polarity as the output voltage, although it is applied to the inverting op-amp input; the input source has an opposite polarity to both the circuit input and output voltages.


The original Miller effect is implemented by capacitive impedance connected between the two nodes. Miller theorem generalizes Miller effect as it implies arbitrary impedance Z connected between the nodes. It is supposed also a constant coefficient K; then the expressions above are valid. But modifying properties of Miller theorem exist even when these requirements are violated and this arrangement can be generalized further by dynamizing the impedance and the coefficient.


Non-linear element. Besides impedance, Miller arrangement can modify the IV characteristic of an arbitrary element. The circuit of a diode log converter is an example of a non-linear virtually zeroed resistance where the logarithmic forward IV curve of a diode is transformed to a vertical straight line overlapping the Y axis.


Not constant coefficient. If the coefficient K varies, some exotic virtual elements can be obtained. A is an example of such a virtual element where the resistance RL is modified so that to mimic inductance, capacitance or inversed resistance.


There is also a dual version of Miller theorem that is based on Kirchhoff's current law ("Miller theorem for currents"): if there is a branch in a circuit with impedance Z connecting a node, where two currents I1 and I2 converge to ground, we can replace this branch by two conducting the referred currents, with imperespectively equal to (1 + α)Z and (1 + α)Z/α, where α = I2/I1. The dual theorem may be proved by replacing the two-port network by its equivalent and by applying the source absorption theorem.


Dual Miller theorem actually expresses the fact that connecting a second current source producing proportional current formula_12 in parallel with the main input source and the impedance element changes the current flowing through it, the voltage and accordingly, the circuit impedance seen from the side of the input source. Depending on the direction, "I2" acts as a supplemental current source helping or opposing the main current source "I1" to create voltage across the impedance. The combination of the actual element and the second current source may be thought as of a new virtual element with dynamically modified impedance.


Dual Miller theorem is usually implemented by an arrangement consisting of two voltage sources supplying the grounded impedance "Z" through floating impedances (see Fig. 3). The combinations of the voltage sources and belonging impedances form the two current sources – the main and the auxiliary one. As in the case of the main Miller theorem, the second voltage is usually produced by a voltage amplifier. Depending on the kind of the amplifier (inverting, non-inverting or differential) and the gain, the circuit input impedance may be virtually increased, infinite, decreased, zero or negative.


List of specific applications based on Miller theorems.


Below is a list of circuit solutions, phenomena and techniques based on the two Miller theorems.


In electrical engineering, the maximum power transfer theorem states that, to obtain "maximum" external power from a source with a finite internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Moritz von Jacobi published the maximum power (transfer) theorem around 1840; it is also referred to as "Jacobi's law".


The theorem results in maximum "power" transfer across the circuit, and not maximum "efficiency". If the resistance of the load is made larger than the resistance of the source then efficiency is higher, since a higher percentage of the source power is transferred to the load, but the "magnitude" of the load power is lower since the total circuit resistance increases.


If the load resistance is smaller than the source resistance, then most of the power ends up being dissipated in the source, and although the total power dissipated is higher, due to a lower total resistance, it turns out that the amount dissipated in the load is reduced.


The theorem states how to choose (so as to maximize power transfer) the load resistance, once the source resistance is given. It is a common misconception to apply the theorem in the opposite scenario. It does "not" say how to choose the source resistance for a given load resistance. In fact, the source resistance that maximizes power transfer from a voltage source is always zero, regardless of the value of the load resistance.


The theorem can be extended to alternating current circuits that include reactance, and states that maximum power transfer occurs when the load impedance is equal to the complex conjugate of the source impedance.


Recent expository articles illustrate how the fundamental mathematics of the maximum power theorem also applies to other physical situations, such as:


The theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient since, when the impedances were matched, the power lost as heat in the battery would always be equal to the power delivered to the motor.


In 1880 this assumption was shown to be false by either Edison or his colleague Francis Robbins Upton, who realized that maximum efficiency was not the same as maximum power transfer.


To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be (or should be) made as close to zero as possible. Using this new understanding, they obtained an efficiency of about 90%, and proved that the electric motor was a practical alternative to the heat engine.


The condition of maximum power transfer does not result in maximum efficiency.


If we define the efficiency as the ratio of power dissipated by the load, "R", to power developed by the source, "V", then it is straightforward to calculate from the above circuit diagram that


The efficiency is only 50% when maximum power transfer is achieved, but approaches 100% as the load resistance approaches infinity, though the total power level tends towards zero.


Efficiency also approaches 100% if the source resistance approaches zero, and 0% if the load resistance approaches zero. In the latter case, all the power is consumed inside the source (unless the source also has no resistance), so the power dissipated in a short circuit is zero.


A related concept is reflectionless impedance matching.


In radio frequency transmission lines, and other electronics, there is often a requirement to match the source impedance (at the transmitter) to the load impedance (such as an antenna) to avoid reflections in the transmission line that could overload or damage the transmitter.


In the diagram opposite, power is being transferred from the source, with voltage and fixed source resistance , to a load with resistance , resulting in a current . By Ohm's law, is simply the source voltage divided by the total circuit resistance:


The power dissipated in the load is the square of the current multiplied by the resistance:


The value of for which this expression is a maximum could be calculated by differentiating it, but it is easier to calculate the value of for which the denominator


is a minimum. The result will be the same in either case. Differentiating the denominator with respect to :


For a maximum or minimum, the first derivative is zero, so


In practical resistive circuits, and are both positive, so the positive sign in the above is the correct solution.


To find out whether this solution is a minimum or a maximum, the denominator expression is differentiated again:


This is always positive for positive values of formula_16 and formula_17, showing that the denominator is a minimum, and the power is therefore a maximum, when


The above proof assumes fixed source resistance formula_16. When the source resistance can be varied, power transferred to the load can be increased by reducing formula_20. For example, a 100 Volt source with an formula_20 of formula_22 will deliver 250 watts of power to a formula_22 load; reducing formula_20 to formula_25 increases the power delivered to 1000 watts.


Note that this shows that maximum power transfer can also be interpreted as the load voltage being equal to one-half of the Thevenin voltage equivalent of the source.


The power transfer theorem also applies when the source and/or load are not purely resistive.


A refinement of the maximum power theorem says that any reactive components of source and load should be of equal magnitude but opposite sign. ("See below for a derivation.")


Physically realizable sources and loads are not usually purely resistive, having some inductive or capacitive components, and so practical applications of this theorem, under the name of complex conjugate impedance matching, do, in fact, exist.


If the source is totally inductive (capacitive), then a totally capacitive (inductive) load, in the absence of resistive losses, would receive 100% of the energy from the source but send it back after a quarter cycle.


The resultant circuit is nothing other than a resonant LC circuit in which the energy continues to oscillate to and fro. This oscillation is called reactive power.


Power factor correction (where an inductive reactance is used to "balance out" a capacitive one), is essentially the same idea as complex conjugate impedance matching although it is done for entirely different reasons.


For a fixed reactive "source", the maximum power theorem maximizes the real power (P) delivered to the load by complex conjugate matching the load to the source.


For a fixed reactive "load", power factor correction minimizes the apparent power (S) (and unnecessary current) conducted by the transmission lines, while maintaining the same amount of real power transfer.


This is done by adding a reactance to the load to balance out the load's own reactance, changing the reactive load impedance into a resistive load impedance.


In this diagram, AC power is being transferred from the source, with phasor magnitude of voltage formula_26 (positive peak voltage) and fixed source impedance formula_27 (S for source), to a load with impedance formula_28 (L for load), resulting in a (positive) magnitude formula_29 of the current phasor formula_30. This magnitude formula_29 results from dividing the magnitude of the source voltage by the magnitude of the total circuit impedance:


The average power formula_33 dissipated in the load is the square of the current multiplied by the resistive portion (the real part) formula_34 of the load impedance formula_28:


where formula_37 and formula_34 denote the resistances, that is the real parts, and formula_39 and formula_40 denote the reactances, that is the imaginary parts, of respectively the source and load impedances formula_27 and formula_28.


To determine, for a given source voltage formula_43 and impedance formula_44 the value of the load impedance formula_45 for which this expression for the power yields a maximum, one first finds, for each fixed positive value of formula_34, the value of the reactive term formula_40 for which the denominator


is a minimum. Since reactances can be negative, this is achieved by adapting the load reactance to


and it remains to find the value of formula_34 which maximizes this expression. This problem has the same form as in the purely resistive case, and the maximizing condition therefore is formula_52


describe the complex conjugate of the source impedance, denoted by formula_55 and thus can be concisely combined to:


Commensurate line circuits are electrical circuits composed of transmission lines that are all the same length; commonly one-eighth of a wavelength. Lumped element circuits can be directly converted to distributed-element circuits of this form by the use of Richards' transformation. This transformation has a particularly simple result; inductors are replaced with transmission lines terminated in short-circuits and capacitors are replaced with lines terminated in open-circuits. Commensurate line theory is particularly useful for designing distributed-element filters for use at microwave frequencies.


It is usually necessary to carry out a further transformation of the circuit using Kuroda's identities. There are several reasons for applying one of the Kuroda transformations; the principal reason is usually to eliminate series connected components. In some technologies, including the widely used microstrip, series connections are difficult or impossible to implement.


The frequency response of commensurate line circuits, like all distributed-element circuits, will periodically repeat, limiting the frequency range over which they are effective. Circuits designed by the methods of Richards and Kuroda are not the most compact. Refinements to the methods of coupling elements together can produce more compact designs. Nevertheless, the commensurate line theory remains the basis for many of these more advanced filter designs.


Commensurate lines are transmission lines that are all the same electrical length, but not necessarily the same characteristic impedance ("Z"0). A commensurate line circuit is an electrical circuit composed only of commensurate lines terminated with resistors or short- and open-circuits. In 1948, Paul I. Richards published a theory of commensurate line circuits by which a passive lumped element circuit could be transformed into a distributed element circuit with precisely the same characteristics over a certain frequency range.


Electrical length can also be expressed as the phase change between the start and the end of the line. Phase is measured in angle units. formula_1, the mathematical symbol for an angle variable, is used as the symbol for electrical length when expressed as an angle. In this convention λ represents 360°, or 2π radians.


The advantage of using commensurate lines is that the commensurate line theory allows circuits to be synthesised from a prescribed frequency function. While any circuit using arbitrary transmission line lengths can be analysed to determine its frequency function, that circuit cannot necessarily be easily synthesised starting from the frequency function. The fundamental problem is that using more than one length generally requires more than one frequency variable. Using commensurate lines requires only one frequency variable. A well developed theory exists for synthesising lumped-element circuits from a given frequency function. Any circuit so synthesised can be converted to a commensurate line circuit using Richards' transformation and a new frequency variable.


Richards' transformation transforms the angular frequency variable, ω, according to,


or, more usefully for further analysis, in terms of the complex frequency variable, "s",


Comparing this transform with expressions for the driving point impedance of stubs terminated, respectively, with a short circuit and an open circuit,


it can be seen that (for θ < π/2) a short circuit stub has the impedance of a lumped inductance and an open circuit stub has the impedance of a lumped capacitance. Richards' transformation substitutes inductors with short circuited UEs and capacitors with open circuited UEs.


When the length is λ/8 (or θ=π/4), this simplifies to,


"L" and "C" are conventionally the symbols for inductance and capacitance, but here they represent respectively the characteristic impedance of an inductive stub and the characteristic admittance of a capacitive stub. This convention is used by numerous authors, and later in this article.


Richards' transformation can be viewed as transforming from a s-domain representation to a new domain called the Ω-domain where,


If Ω is normalised so that Ω=1 when ω=ωc, then it is required that,


and the length in distance units becomes,


Any circuit composed of discrete, linear, lumped components will have a transfer function "H"("s") that is a rational function in "s". A circuit composed of transmission line UEs derived from the lumped circuit by Richards' transformation will have a transfer function "H"("j"Ω) that is a rational function of precisely the same form as "H"("s"). That is, the shape of the frequency response of the lumped circuit against the "s" frequency variable will be precisely the same as the shape of the frequency response of the transmission line circuit against the "j"Ω frequency variable and the circuit will be functionally the same.


However, infinity in the Ω domain is transformed to ω=π/4"k" in the "s" domain. The entire frequency response is squeezed down to this finite interval. Above this frequency, the same response is repeated in the same intervals, alternately in reverse. This is a consequence of the periodic nature of the tangent function. This multiple passband result is a general feature of all distributed-element circuits, not just those arrived at through Richards' transformation.


A UE connected in cascade is a two-port network that has no exactly corresponding circuit in lumped elements. It is functionally a fixed delay. There are lumped-element circuits that can approximate a fixed delay such as the Bessel filter, but they only work within a prescribed passband, even with ideal components. Alternatively, lumped-element all-pass filters can be constructed that pass all frequencies (with ideal components), but they have constant delay only within a narrow band of frequencies. Examples are the lattice phase equaliser and bridged T delay equaliser.


There is consequently no lumped circuit that Richard's transformation can transform into a cascade-connected line, and there is no reverse transformation for this element. Commensurate line theory thus introduces a new element of "delay", or "length".


Two or more UEs connected in cascade with the same "Z"0 are equivalent to a single, longer, transmission line. Thus, lines of length "n"θ for integer "n" are allowable in commensurate circuits. Some circuits can be implemented "entirely" as a cascade of UEs: impedance matching networks, for instance, can be done this way, as can most filters.


Kuroda's identities are a set of four equivalent circuits that overcome certain difficulties with applying Richards' transformations directly. The four basic transformations are shown in the figure. Here the symbols for capacitors and inductors are used to represent open-circuit and short-circuit stubs. Likewise, the symbols "C" and "L" here represent respectively the susceptance of an open circuit stub and the reactance of a short circuit stub, which, for θ=λ/8, are respectively equal to the characteristic admittance and characteristic impedance of the stub line. The boxes with thick lines represent cascade connected commensurate lengths of line with the marked characteristic impedance.


The first difficulty solved is that all the UEs are required to be connected together at the same point. This arises because the lumped-element model assumes that all the elements take up zero space (or no significant space) and that there is no delay in signals between the elements. Applying Richards' transformation to convert the lumped circuit into a distributed circuit allows the element to now occupy a finite space (its length) but does not remove the requirement for zero distance between the interconnections. By repeatedly applying the first two Kuroda identities, UE lengths of the lines feeding into the ports of the circuit can be moved between the circuit components to physically separate them.


A second difficulty that Kuroda's identities can overcome is that series connected lines are not always practical. While series connection of lines can easily be done in, for instance, coaxial technology, it is not possible in the widely used microstrip technology and other planar technologies. Filter circuits frequently use a ladder topology with alternating series and shunt elements. Such circuits can be converted to all shunt components in the same step used to space the components with the first two identities.


The third and fourth identities allow characteristic impedances to be scaled down or up respectively. These can be useful for transforming impedances that are impractical to implement. However, they have the disadvantage of requiring the addition of an ideal transformer with a turns ratio equal to the scaling factor.


In the decade after Richards' publication, advances in the theory of distributed circuits took place mostly in Japan. K. Kuroda published these identities in 1955 in his Ph.D thesis. However, they did not appear in English until 1958 in a paper by Ozaki and Ishii on stripline filters.


Coupling elements together with impedance transformer lines is not the most compact design. Other methods of coupling have been developed, especially for band-pass filters that are far more compact. These include parallel lines filters, interdigital filters, hairpin filters, and the semi-lumped design combline filters.


A generator in electrical circuit theory is one of two ideal elements: an ideal voltage source, or an ideal current source. These are two of the fundamental elements in circuit theory. Real electrical generators are most commonly modelled as a non-ideal source consisting of a combination of an ideal source and a resistor. Voltage generators are modelled as an ideal voltage source in series with a resistor. Current generators are modelled as an ideal current source in parallel with a resistor. The resistor is referred to as the internal resistance of the source. Real world equipment may not perfectly follow these models, especially at extremes of loading (both high and low) but for most purposes they suffice.


The two models of non-ideal generators are interchangeable, either can be used for any given generator. Thévenin's theorem allows a non-ideal current source model to be converted to a non-ideal voltage source model and Norton's theorem allows a non-ideal voltage source model to be converted to a non-ideal current source model. Both models are equally valid, but the voltage source model is more applicable to when the internal resistance is low (that is, much lower than the load impedance) and the current source model is more applicable when the internal resistance is high (compared to the load).


Symbols commonly used for ideal sources are shown in the figure. Symbols do vary from region to region and time period to time period. Another common symbol for a current source is two interlocking circles.


The model used to represent "h"-parameters is shown in the figure. "h"-parameters are frequently used in transistor data sheets to specify the device. The "h"-parameters are defined as the matrix


where the voltage and current variables are as shown in the figure. The circuit model using dependent generators is just an alternative way of representing this matrix.


The superposition theorem is a derived result of the superposition principle suited to the network analysis of electrical circuits. The superposition theorem states that for a linear system (notably including the subcategory of time-invariant linear systems) the response (voltage or current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by their internal impedances.


To ascertain the contribution of each individual source, all of the other sources first must be "turned off" (set to zero) by:


This procedure is followed for each source in turn, then the resultant responses are added to determine the true operation of the circuit. The resultant circuit operation is the superposition of the various voltage and current sources.


The superposition theorem is very important in circuit analysis. It is used in converting any circuit into its Norton equivalent or Thevenin equivalent.


The theorem is applicable to linear networks (time varying or time invariant) consisting of independent sources, linear dependent sources, linear passive elements (resistors, inductors, capacitors) and linear transformers.


Superposition works for voltage and current but not power. In other words, the sum of the powers of each source with the other sources turned off is not the real consumed power. To calculate power we first use superposition to find both current and voltage of each linear element and then calculate the sum of the multiplied voltages and currents.


However, if the linear network is operating in steady-state and each external independent source has a different frequency, then superposition can be applied to compute the average power or active power. If at least two independent sources have the same frequency (for example in power systems, where many generators operate at 50 Hz or 60 Hz), then superposition can't be used to determine average power.


The electric circuit superposition theorem is analogous to Dalton's law of partial pressure which can be stated as the total pressure exerted by an ideal gas mixture in a given volume is the algebraic sum of all the pressures exerted by each gas if it were alone in that volume.


In electrical circuit theory, a port is a pair of terminals connecting an electrical network or circuit to an external circuit, as a point of entry or exit for electrical energy. A port consists of two nodes (terminals) connected to an outside circuit which meets the "port condition" - the currents flowing into the two nodes must be equal and opposite.


The use of ports helps to reduce the complexity of circuit analysis. Many common electronic devices and circuit blocks, such as transistors, transformers, electronic filters, and amplifiers, are analyzed in terms of ports. In multiport network analysis, the circuit is regarded as a "black box" connected to the outside world through its ports. The ports are points where input signals are applied or output signals taken. Its behavior is completely specified by a matrix of parameters relating the voltage and current at its ports, so the internal makeup or design of the circuit need not be considered, or even known, in determining the circuit's response to applied signals.


The concept of ports can be extended to waveguides, but the definition in terms of current is not appropriate and the possible existence of multiple waveguide modes must be accounted for.


Any node of a circuit that is available for connection to an external circuit is called a pole (or terminal if it is a physical object). The port condition is that a pair of poles of a circuit is considered a port if and only if the current flowing into one pole from outside the circuit is equal to the current flowing out of the other pole into the external circuit. Equivalently, the algebraic sum of the currents flowing into the two poles from the external circuit must be zero.


It cannot be determined if a pair of nodes meets the port condition by analysing the internal properties of the circuit itself. The port condition is dependent entirely on the external connections of the circuit. What are ports under one set of external circumstances may well not be ports under another. Consider the circuit of four resistors in the figure for example. If generators are connected to the pole pairs (1, 2) and (3, 4) then those two pairs are ports and the circuit is a box attenuator. On the other hand, if generators are connected to pole pairs (1, 4) and (2, 3) then those pairs are ports, the pairs (1, 2) and (3, 4) are no longer ports, and the circuit is a bridge circuit.


It is even possible to arrange the inputs so that "no" pair of poles meets the port condition. However, it is possible to deal with such a circuit by splitting one or more poles into a number of separate poles joined to the same node. If only one external generator terminal is connected to each pole (whether a split pole or otherwise) then the circuit can again be analysed in terms of ports. The most common arrangement of this type is to designate one pole of an "n"-pole circuit as the common and split it into "n"−1 poles. This latter form is especially useful for unbalanced circuit topologies and the resulting circuit has "n"−1 ports.


In the most general case, it is possible to have a generator connected to every pair of poles, that is, "n"C2 generators, then every pole must be split into "n"−1 poles. For instance, in the figure example (c), if the poles 2 and 4 are each split into two poles each then the circuit can be described as a 3-port. However, it is also possible to connect generators to pole pairs , , and making generators in all and the circuit has to be treated as a 6-port.


Any two-pole circuit is guaranteed to meet the port condition by virtue of Kirchhoff's current law and they are therefore one-ports unconditionally. All of the basic electrical elements (inductance, resistance, capacitance, voltage source, current source) are one-ports, as is a general impedance.


Study of one-ports is an important part of the foundation of network synthesis, most especially in filter design. Two-element one-ports (that is RC, RL and LC circuits) are easier to synthesise than the general case. For a two-element one-port Foster's canonical form or Cauer's canonical form can be used. In particular, LC circuits are studied since these are lossless and are commonly used in filter design.


Linear two port networks have been widely studied and a large number of ways of representing them have been developed. One of these representations is the z-parameters which can be described in matrix form by;


where "Vn" and "In" are the voltages and currents respectively at port "n". Most of the other descriptions of two-ports can likewise be described with a similar matrix but with a different arrangement of the voltage and current column vectors.


Common circuit blocks which are two-ports include amplifiers, attenuators and filters.


In general, a circuit can consist of any number of ports—a multiport. Some, but not all, of the two-port parameter representations can be extended to arbitrary multiports. Of the voltage and current based matrices, the ones that can be extended are z-parameters and y-parameters. Neither of these are suitable for use at microwave frequencies because voltages and currents are not convenient to measure in formats using conductors and are not relevant at all in waveguide formats. Instead, s-parameters are used at these frequencies and these too can be extended to an arbitrary number of ports.


Circuit blocks which have more than two ports include directional couplers, power splitters, circulators, diplexers, duplexers, multiplexers, hybrids and directional filters.


RF and microwave circuit topologies are commonly unbalanced circuit topologies such as coaxial or microstrip. In these formats, one pole of each port in a circuit is connected to a common node such as a ground plane. It is assumed in the circuit analysis that all these commoned poles are at the same potential and that current is sourced to or sunk into the ground plane that is equal and opposite to that going into the other pole of any port. In this topology a port is treated as being just a single pole. The corresponding balancing pole is imagined to be incorporated into the ground plane.


The one-pole representation of a port will start to fail if there are significant ground plane loop currents. The assumption in the model is that the ground plane is perfectly conducting and that there is no potential difference between two locations on the ground plane. In reality, the ground plane is not perfectly conducting and loop currents in it will cause potential differences. If there is a potential difference between the commoned poles of two ports then the port condition is broken and the model is invalid.


The idea of ports can be (and is) extended to waveguide devices, but a port can no longer be defined in terms of circuit poles because in waveguides the electromagnetic waves are not guided by electrical conductors. They are, instead guided by the walls of the waveguide. Thus, the concept of a circuit conductor pole does not exist in this format. Ports in waveguides consist of an aperture or break in the waveguide through which the electromagnetic waves can pass. The bounded plane through which the wave passes is the definition of the port.


Waveguides have an additional complication in port analysis in that it is possible (and sometimes desirable) for more than one waveguide mode to exist at the same time. In such cases, for each physical port, a separate port must be added to the analysis model for each of the modes present at that physical port.


The concept of ports can be extended into other energy domains. The generalised definition of a port is a place where energy can flow from one element or subsystem to another element or subsystem. This generalised view of the port concept helps to explain why the port condition is so defined in electrical analysis. If the algebraic sum of the currents is not zero, such as in example diagram (c), then the energy delivered from an external generator is not equal to the energy entering the pair of circuit poles. The energy transfer at that place is thus more complex than a simple flow from one subsystem to another and does not meet the generalised definition of a port.


An equivalent impedance is an equivalent circuit of an electrical network of impedance elements which presents the same impedance between all pairs of terminals as did the given network. This article describes mathematical transformations between some passive, linear impedance networks commonly found in electronic circuits.


There are a number of very well known and often used equivalent circuits in linear network analysis. These include resistors in series, resistors in parallel and the extension to series and parallel circuits for capacitors, inductors and general impedances. Also well known are the Norton and Thévenin equivalent current generator and voltage generator circuits respectively, as is the Y-Δ transform. None of these are discussed in detail here; the individual linked articles should be consulted.


One-element networks are trivial and two-element, two-terminal networks are either two elements in series or two elements in parallel, also trivial. The smallest number of elements that is non-trivial is three, and there are two 2-element-kind non-trivial transformations possible, one being both the reverse transformation and the topological dual, of the other.


Example 3 shows the result is a Π-network rather than an L-network. The reason for this is that the shunt element has more capacitance than is required by the transform so some is still left over after applying the transform. If the excess were instead, in the element nearest the transformer, this could be dealt with by first shifting the excess to the other side of the transformer before carrying out the transform.


"spooky action at a distance" on the assumption of QM's completeness).


The no-communication theorem states that, within the context of quantum mechanics, it is not possible to transmit classical bits of information by means of carefully prepared mixed or pure states, whether entangled or not. The theorem disallows all communication, not just faster-than-light communication, by means of shared quantum states. The theorem disallows not only the communication of whole bits, but even fractions of a bit. This is important to take note of, as there are many classical radio communications encoding techniques that can send arbitrarily small fractions of a bit across arbitrarily narrow, noisy communications channels. In particular, one may imagine that there is some ensemble that can be prepared, with small portions of the ensemble communicating a fraction of a bit; this, too, is not possible.


The theorem is built on the basic presumption that the laws of quantum mechanics hold. Similar theorems may or may not hold for other related theories, such as hidden variable theories. The no-communication theorem is not meant to constrain other, non-quantum-mechanical theories.


The basic assumption entering into the theorem is that a quantum-mechanical system is prepared in an initial state, and that this initial state is describable as a mixed or pure state in a Hilbert space "H". The system then evolves over time in such a way that there are two spatially distinct parts, "A" and "B", sent to two distinct observers, Alice and Bob, who are free to perform quantum mechanical measurements on their portion of the total system (viz, A and B). The question is: is there any action that Alice can perform on A that would be detectable by Bob making an observation of B? The theorem replies 'no'.


An important assumption going into the theorem is that neither Alice nor Bob is allowed, in any way, to affect the preparation of the initial state. If Alice were allowed to take part in the preparation of the initial state, it would be trivially easy for her to encode a message into it; thus neither Alice nor Bob participates in the preparation of the initial state. The theorem does not require that the initial state be somehow 'random' or 'balanced' or 'uniform': indeed, a third party preparing the initial state could easily encode messages in it, received by Alice and Bob. Simply, the theorem states that, given some initial state, prepared in some way, there is no action that Alice can take that would be detectable by Bob.


The proof proceeds by defining how the total Hilbert space "H" can be split into two parts, "H""A" and "H""B", describing the subspaces accessible to Alice and Bob. The total state of the system is assumed to be described by a density matrix σ. This appears to be a reasonable assumption, as a density matrix is sufficient to describe both pure and mixed states in quantum mechanics. Another important part of the theorem is that measurement is performed by applying a generalized projection operator "P" to the state σ. This again is reasonable, as projection operators give the appropriate mathematical description of quantum measurements. After a measurement by Alice, the state of the total system is said to have "collapsed" to a state "P"(σ).


The proof of the theorem is commonly illustrated for the setup of Bell tests in which two observers Alice and Bob perform local observations on a common bipartite system, and uses the statistical machinery of quantum mechanics, namely density states and quantum operations.


Alice and Bob perform measurements on system S whose underlying Hilbert space is


It is also assumed that everything is finite-dimensional to avoid convergence issues. The state of the composite system is given by a density operator on "H". Any density operator σ on "H" is a sum of the form:


where "Ti" and "Si" are operators on "H""A" and "H""B" respectively. For the following, it is not required to assume that "Ti" and "Si" are state projection operators: "i.e." they need not necessarily be non-negative, nor have a trace of one. That is, σ can have a definition somewhat broader than that of a density matrix; the theorem still holds. Note that the theorem holds trivially for separable states. If the shared state σ is separable, it is clear that any local operation by Alice will leave Bob's system intact. Thus the point of the theorem is no communication can be achieved via a shared entangled state.


Alice performs a local measurement on her subsystem. In general, this is described by a quantum operation, on the system state, of the following kind


where "V""k" are called Kraus matrices which satisfy


means that Alice's measurement apparatus does not interact with Bob's subsystem.


Supposing the combined system is prepared in state σ and assuming, for purposes of argument, a non-relativistic situation, immediately (with no time delay) after Alice performs her measurement, the relative state of Bob's system is given by the partial trace of the overall state with respect to Alice's system. In symbols, the relative state of Bob's system after Alice's operation is


where formula_8 is the partial trace mapping with respect to Alice's system.


From this it is argued that, statistically, Bob cannot tell the difference between what Alice did and a random measurement (or whether she did anything at all).


The scallop theorem is a consequence of the subsequent forces applied to the organism as it swims from the surrounding fluid. For an incompressible Newtonian fluid with density formula_1 and viscosity formula_2, the flow satisfies the Navier–Stokes equations


where formula_4 denotes the velocity of the swimmer. However, at the low Reynolds number limit, the inertial terms of the Navier-Stokes equation on the left-hand side tend to zero. This is made more apparent by nondimensionalizing the Navier–Stokes equation. By defining a characteristic velocity and length, formula_5 and formula_6, we can cast our variables to dimensionless form:


By plugging back into the Navier-Stokes equation and performing some algebra, we arrive at a dimensionless form:


where formula_9 is the Reynolds number, formula_10. In the low Reynolds number limit (as formula_11), the LHS tends to zero and we arrive at a dimensionless form of Stokes equations. Redimensionalizing yields


The proof of the scallop theorem can be represented in a mathematically elegant way. To do this, we must first understand the mathematical consequences of the linearity of Stokes equations. To summarize, the linearity of Stokes equations allows us to use the reciprocal theorem to relate the swimming velocity of the swimmer to the velocity field of the fluid around its surface (known as the swimming gait), which changes according to the periodic motion it exhibits. This relation allows us to conclude that locomotion is independent of swimming rate. Subsequently, this leads to the discovery that reversal of periodic motion is identical to the forward motion due to symmetry, allowing us to conclude that there can be no net displacement.