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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 anal... |
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... |
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 ci... |
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... |
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 ... |
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. Re... |
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... |
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 ter... |
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-lik... |
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 ohmi... |
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 ... |
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 derive... |
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 quantitie... |
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 rat... |
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_... |
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 integ... |
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... |
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 abo... |
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-anisotr... |
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... |
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 b... |
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 ina... |
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 f... |
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 tw... |
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 t... |
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 ... |
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 dependen... |
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... |
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 artifici... |
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... |
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 amplifi... |
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... |
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 inc... |
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 (em... |
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 ci... |
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, ... |
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 decrease... |
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 purp... |
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... |
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 diffe... |
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 condition... |
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 voltag... |
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 Mil... |
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 ... |
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 imperes... |
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