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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.

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