Twoport
A twoport is a model for an electrical component or an electrical network with four connections, in which two connections are combined into a socalled gate .
A gate exists when the electrical current through both connections of a gate is opposite ( gate condition ). A twoport is therefore a special form of a (general) quadrupole.
The name Vierpol comes from Franz Breisig in 1921 .
The matrices notation common for linear twoports goes back to Felix Strecker and Richard Feldtkeller from 1929.
A twoport, on the other hand, is a special case of an n port (which is also referred to as a multiport).
General
A twoport is a special form of a quadrupole. In the case of a general quadrupole, the gate condition does not have to apply, which means that the twoport parameters shown below and the mathematical description with the aid of matrices can only be used for linear twoports and not for general quadrupoles.
In older specialist literature in particular, the terms twoport and fourport are used synonymously, although the term fourport is implicitly understood as twoport. The gates of a twoport are often referred to as an entrance and an exit.
The clamping behavior of a linear twoport is described by its transfer function or its frequency response . Two port equations can be obtained from this , from which twoport parameters for modeling can be obtained.
properties
Twoports can be identified based on the properties of their clamping behavior, i. H. classify as black box without precise knowledge of its internal structure, as follows:
Linearity
The transmission factors of linear twoport devices are independent of voltage and current. Therefore, the superposition theorem applies to the gate currents and voltages . A twoport, which consists only of the passive linear components resistor , coil , capacitor and transformer (a socalled RLCM twoport ), is always linear itself.
Non  linear twoport networks are networks with at least one nonlinear component and these components themselves, such as diodes or transistors . Their transmission behavior depends essentially on the size of the gate currents and voltages. An approximately linear description is possible by means of the small signal theory with constant characteristic curves and for small amplitudes.
Only linear twoport are the subject of the classical quadrupole and modern multiport theory. The linear twoport equations and thus the matrix representation of the twoport parameters described below apply only to them.
Current account
If a twoport does not contain any internal uncontrolled or controlled energy sources, it is called passive (e.g. attenuator ), otherwise active . This means that the output active power P _{2 must be} less than the input active power P _{1} . Active quadrupoles, such as amplifiers , draw energy from auxiliary energy sources ( power source ).
If no energy is lost in a (passive) twoport because it only contains dummy switching elements, it is called a reactance twoport .
Reversibility
Reversible twoport (also reciprocal , coupling symmetric or transmission symmetric ) have the same transmission behavior in both directions, i.e. H. For example, that the ratio of output current and input voltage does not change when the output is shortcircuited if the input and output terminal pairs are swapped. This property is also known as the reciprocity theorem or Kirchhoff's inverse theorem. Thus, a voltage applied to gate 1 generates a current at the shortcircuited gate 2 . If the same voltage is also applied to gate 2 , the same current is generated at the shortcircuited gate 1. This means if is.
Reciprocal twoports are fully characterized by three twoport parameters, because the following restrictions then apply to the elements of the twoport equations:
Reversibility is only defined for linear twoport. A twoport, which only consists of the passive linear components resistor, coil, capacitor and transformer (RLCM twoport), is always reversible.
symmetry
In the case of symmetrical twoports (also referred to as resistance symmetrical ) inputs and outputs can be interchanged. This can often be read from the circuit. If this does not apply to a twoport, it is called asymmetrical .
The following applies to the elements of the twoport equations:
Symmetrical twoport are thus fully characterized by two twoport parameters. Symmetrical twogoals are always reciprocal, but reciprocal twogoals are not always symmetrical.
Ground symmetry
A symmetry line can be drawn in the longitudinal direction in the case of twogate symmetrical or transversely symmetrical. This means that there is no continuous earth line. A typical example is the socalled Xconnection of a quadrupole. In contrast, the threepole terminals used as twoport devices have a continuous earth line and are therefore unbalanced to earth. The property of the ground symmetry has no influence on the twoport parameters. In theory, ideal transformers can be used to convert twoport twoports balanced to earth into unbalanced ones and vice versa.
Nonretroactivity
If an output variable that changes (due to load) has no influence on an input variable, the twoport is called non  reactive. Reactionfree twoports are an “extreme case” of irreversible twoports.
The following restrictions apply to the parameters of a reactionfree twoport:
This means that the input variables , the output variables , independent.
Twoport equations and parameters
If U _{1 denotes} the voltage and I _{1} the current at the input terminal pair and U _{2} and I _{2} denote the corresponding quantities at the output terminal pair , then two quantities can be calculated from the other two given quantities by a pair of twogate equations . These are generally nonlinear differential equations .
For linear two port , possibly using the symbolic method of alternating current calculation or the Laplace transformation , they transform into a pair of linear equations with four twoport parameters describing the twoport .
Assuming that they exist, these two port equations can be given in the form of matrix equations. Applied currents and voltages are added to these equations as matrices as required. The specified calculation rules are used to determine the matrices for any known twoport, such as a feedback network of an amplifier circuit .
Z characteristic 
: Impedance matrix , exists if the gate currents ( I _{1} and I _{2} ) can be selected independently.
: Open circuit input impedance : Open circuit core impedance backward (reaction resistance) : Open circuit core impedance forward (transfer resistance ) : Open circuit output impedance


Y characteristic 
: Admittance matrix , exists if the gate voltages ( U _{1} and U _{2} ) can be selected independently.
: Shortcircuit input admittance: Negative shortcircuit core admittance backwards (retroactive conductance) : Negative shortcircuit core admittance forwards (steepness) : Shortcircuit output
admittance 

H characteristic 
: Hybrid matrix (rowparallel matrix), exists if I _{1} and U _{2 can} be selected independently.
: Shortcircuit input impedance : Noload voltage reaction : Negative shortcircuit current translation (or current gain) : Noload output admittance


P characteristic 
: Inverse hybrid matrix (parallel series matrix), exists if U _{1} and I _{2 can} be selected independently.
: Opencircuit input admittance : Negative shortcircuit current feedback : Opencircuit voltage translation : Shortcircuit output impedance


A characteristic 
: Chain matrix, exists if U _{2} and I _{2 can} be selected independently.
: Reciprocal noload voltage translation: Short circuit core impedance forward (reciprocal slope) : Noload core admittance forward (transmission conductance ) : Reciprocal shortcircuit current translation


B characteristic 
: Inverse chain matrix, exists if U _{1} and I _{1 can} be chosen independently.
: Reciprocal noload voltage feedback: Negative shortcircuit core impedance backwards (feedback resistance) : Negative noload core admittance backwards (feedback conductance ) : Reciprocal shortcircuit current feedback

In the case of the existence of the matrices it holds in particular
The advantage of this notation is that the parameters ( Z _{xy} etc.) represent known component values and are therefore given as numerical values. The relationship between the input and output currents as well as the input and output voltages can now be easily read.
Note: Instead of the symbol , or and instead of the symbol are also used.
Conversion of the matrices
Elementary twogoal
Elementary longitudinal second
The elementary longitudinal twoport only contains an impedance in the upper longitudinal axis between the poles of the origin of the twoport. There is no connection between the poles in the transverse axis.
Elementary crosstwo
The elementary transverse twoport only contains an impedance in the transverse axis of the twoport and does not contain any components in the longitudinal axis.
Γtwoport
The Γtwoport is a synthesis of elementary transverse twoport and elementary longitudinal twoport. It is formed from the chain matrices of the elementary twoport as follows:
Mirrored Γ twoport
The mirrored Γtwoport is a synthesis of elementary longitudinal twoport and elementary transverse twoport. It is formed from the chain matrices of the elementary twoport as follows:
Equivalent circuits
To simplify calculations, complex twoport circuits can be combined to form simplified circuits using appropriate twoport parameters. The equivalent circuits do not represent a guide for physical implementation.
T equivalent circuit
The T equivalent circuit enables any twoport circuit to be represented using the equivalent impedances. The controlled voltage source is not required for reversible twoport devices. It can be synthesized from an elementary longitudinal twoport and a Γtwoport or correspondingly from a mirrored Γtwoport and an elementary longitudinal twoport. The following composition describes the latter:
π equivalent circuit
The π equivalent circuit enables any twoport to be represented using the equivalent admittances. The controlled power source is not required for reversible twoport devices. It can be synthesized from an elementary transverse twoport and a mirrored Γtwoport or accordingly from an elementary transverse twoport and a Γtwoport. The following composition describes the latter:
Interconnect
Two twoport gates can be interconnected to form a new twoport gate, provided that the above gate condition is met on at least one gate. The Brune test is used to check the permissibility of the interconnection . The parameters of the newly created twoport can be calculated from the parameters of the two interconnected twoports. For each type of interconnection there is a characteristic with which the interconnection can be calculated particularly well . There are a total of five different ways of connecting twogoals:
Swap two ports
It is often helpful to transform one twoport through another. For this, the two ports A and T are given in chain parameters, T is also invertible and should not be changed. When shifted through T, A now becomes A ', so that the behavior of the total second does not change. The following must therefore apply:
The transformer coupling where only the main diagonal of T is occupied and the gyratory coupling where the main diagonal of T is zero play a special role here . Simple impedances can be expressed in chain parameters using the matrices opposite. In the case of transformer or gyratory coupling, A and A 'are one of these two matrices in terms of shape, i.e. a single impedance Z on one side of T can be expressed by a single impedance Z' on the other side. With the gyratory coupling, parallel connections become series connections and vice versa. In this way it is possible to project entire networks through a twoport.
Further twoport parameters
In addition to the characterization of a twoport using the twoport parameters described above, there are also other forms of representation for special purposes. A linear twoport can also be described by socalled scattering parameters . This form of representation is particularly common in the field of highfrequency technology , since the connections of the twoport do not have to be shortcircuited or empty, but are usually closed by their wave impedance.
The following relationship exists between the S parameters and the abovementioned Y parameters of the admittance matrix of a twoport with the wave impedance Z _{W} :
with the abbreviation:
For their application in the theory of filter circuits (wave parameter theory), symmetrical linear twoports are described by the socalled wave parameters . The two parameters describing the twoport are the wave impedance and the wave transmission factor .
Numerical CAD systems in electronics
The practical evaluation and processing of the above matrices from complex elements requires the use of computers. The computeraided numerical evaluation and further processing of the above equations have been reported since the 1970s , and from the 1980s onwards , complex numerical CAD systems (e.g. SuperCompact) gradually gained acceptance in industry. There is no fundamental difference between highfrequency and microwave electronics and electronics at lower frequencies. The linear dependencies between current and voltage are described for all frequencies by the above six quadrupole forms and fully equivalent by s and t parameters (an s parameter chain form), as long as the terms of current and voltage are used as "substitute quantities" instead of electromagnetic fields and other physical terms can be taken. In many cases this is possible up to the microwave range without having to solve the field equations for the electromagnetic fields using a significantly higher amount of computing effort. The transition is always fluid.
Symbolic CAD systems for computeraided formula derivation
Also of interest is the symbolic processing of the above matrix equations and, based on this, the computeraided formula derivation for signal analysis in linear electronics and highfrequency electronics. A special addition for a mathematics program converts the above eight fourpole parameter representations plus the wave parameter shape and supplements them with numerous other models (active elements such as individual transistors and standard circuits as well as passive elements such as lines, transformers, couplers, directional lines: all with their respective model parameters) symbolically and  so as far as this is logically possible  from any kind to any other kind. The above networking of quadrupoles is largely automated and supplemented by other commands such as B. the deembedding of an embedded quadrupole, the above pushing a quadrupole through another quadrupole, various terminal interchanges et cetera. As a result, the model parameters and / or the quadrupole parameters are dependent on formulas for the usual report sizes, which only appear as numbers in the numerical CAD systems: Formula sets for voltage and current amplification, input and output impedances, reflection factors, gain variables or stability factor up to towards the generalized scattering parameters according to Kurokawa, i.e. for quantities that are standard in numerical systems. Such a system is fascinating and is also very suitable for working out new relationships, developing ideas or proving or refuting assumptions.
Possibilities and limits of the CAD systems
Despite the enormous possibilities, these symbolic as well as numeric CAD systems always remain only arithmetic assistants that always have to be managed with a great deal of expertise, because they consist of mathematical theories, models and many lines of code. The quality of all calculation results always depends on the measurement.
literature
 LorenzPeter Schmidt, Gerd Schaller, Siegfried Martius: Fundamentals of electrical engineering. Volume 3: Networks. Pearson Studium, Munich 2006, ISBN 3827371074 .
 Richard Feldtkeller : Introduction to the fourpole theory of electrical communications engineering. Hirzel, Stuttgart 1976, ISBN 3777603198 .
 Handheld electrical telecommunications dictionary . Essay on the quadrupole theory by Zuhrt / Matthes, 2nd edition. 3rd volume, pp. 18371868.
 Wolfgang Kretz: Collection of formulas for the quadrupole theory (with a short introduction). Oldenbourg, Munich 1967.
 Eugen Philippow: Fundamentals of electrical engineering. Academic publishing company Geest & Portig, Leipzig 1966.
 Heinrich Schröder: Electrical communications engineering. Volume I, Verlag für RadioFotoKinotechnik, BerlinBorsigwalde 1966.
 ClausChristian Timmermann : High Frequency Electronics with CAD, Vol. 1, Introduction to Lines , FourPole, Transistor Models and Simulation with Numerical and Symbolic CAD Systems, Vol. 1 , PROFUNDVerlag, 1997 and 2005, ISBN 978393265121 2
 George D. Vendelin, Anthony M. Pavio, Ulrich L. Rohde. Microwave Circuit Design Using Linear and Nonlinear Techniques , 2nd Edition, Wiley, 2005, ISBN 9780471414797
 Lecture  Networks 3. Institute for Fundamentals and Theory of Electrical Engineering, Graz University of Technology (This subject is called multigate theory. Therefore, further sources should be found under this title).
 Lecture  Dynamic Networks. Institute for Fundamentals of Electrical Engineering and Electronics, Technical University of Dresden
Web links
 Ronny Harbich: Passive two goals . 2005, accessed January 18, 2010.
Individual evidence
 ↑ Gerhard Wunsch : History of Systems Theory (= Scientific Pocket Books: Texts and Studies . Volume 296 ). AkademieVerlag, Leipzig 1985, DNB 850752914 , p. 49 .
 ↑ ClausChristian Timmermann, High Frequency Electronics with CAD, Vol. 1, Introduction to Lines, FourPole, Transistor Models and Simulation with Numerical and Symbolic CAD Systems, Vol. 1, Appendix STWOP: Symbolic 2Gate Analysis, Appendix TWOP: Numerical 2 Goal analysis. , Pp. 117126