Two-port network

A two-port electrical network describes a four-port, are combined with the two terminals on a so-called gate. A gate is present when the electric current to the same through both terminals of a gate is ( Torbedingung ). A two-port network is thus a special form of a two-port network. The quadrupole term dates from the year 1921 by Franz Breisig. The usual two-ports in matrix notation goes to Felix Strecker and Richard Feldtkeller back from the year 1929.

A two-port network is a special case of an n- gate ( also known as a multiport ).

• 3.1 Conversion of the matrices
• 3.2 elementary two-ports 3.2.1 Elemental Längszweitor
• 3.2.2 Elemental Querzweitor
• 3.2.3 Γ - two-port network
• 3.2.4 Mirrored Γ two-port

• 3.3.1 T equivalent circuit
• 3.3.2 π - equivalent circuit

• 5.1 Literature
• 5.2 External links

General

A two-port network is a special form of a two-port network. In a general quadrupole the Torbedingung does not apply, so that the two-port parameters shown in the following and the mathematical description with the help of matrices is only applicable to linear two-ports and not in general four poles.

Especially in older literature, the terms two-port and four-terminal network are used interchangeably, although implicitly the term quadrupole Two goals are understood. Often the gates of a two-port network are also referred to as an input and an output.

The terminal behavior of a linear two-port network is described by its transfer function or its frequency response. From this Zweitorgleichungen can be obtained, from which two-port parameters can be obtained for modeling.

Properties

Two goals can be inferred from the properties of their terminal behavior, that is, as a black box without precise knowledge of their internal structure, as follows, classify:

Linearity

The transfer factors of linear two-ports are independent of voltage and current. Therefore, for the Torströme and voltages of the superposition theorem. A two-port network, which consists only of the linear passive components resistor, coil, capacitor and transformer (called RLCM two-port ), is always self- linearly.

Non-linear two-port networks are non-linear with at least one component and this component itself, such as diodes or transistors. Your transfer behavior depends largely on the size of the Torströme and voltages. An approximately linear description is possible by means of the small-signal theory with continuous curves and small amplitudes.

Only linear two-ports are the subject of the classic four-pole and the modern Mehrtortheorie. Only for them, the linear Zweitorgleichungen and thus the matrix representation of the two-port parameters described below apply.

Current account

Contains a two-port no internal uncontrolled or controlled sources of energy, it is called passive ( eg attenuator ), otherwise active. It follows that the effective output power P2 must be less than the input power P1. Active quadrupoles, as amplifiers, remove power from auxiliary power sources ( power source).

Go into a ( passive ) two port no energy is lost, because it contains only reactive circuit elements, it is called Reaktanzzweitor.

Reversibility

Reversible Two goals (also reciprocal, symmetrical coupling or transmission symmetrical) have in both directions using the same transfer behavior, ie that the ratio of input current and output voltage does not change with short-circuited input when swapping the input and output terminal pair. This property is referred to as reciprocity theorem or Kirchhoff shear reversal rate. Thus, a voltage applied to gate 1 voltage at gate 2 generates a current. If the same voltage is applied to gate 2, the same current is generated at Gate 1. As a result if it is.

Reciprocal two-ports are completely characterized by three two-port parameters, because then the following restrictions apply to the elements of the Zweitorgleichungen:

Reversibility is defined only for linear two-ports. A two-port network, which consists only of the linear passive components resistor, coil, capacitor and transformer ( RLCM two-port ) is always reversible.

Symmetry

For symmetrical two-ports (also called resistance balanced) inputs and outputs are interchangeable with each other. This can often be read out from the circuit. If this does not apply to a two-port, this is referred to as asymmetrical.

The following applies for the elements of Zweitorgleichungen:

Symmetrical Two goals are thus completely characterized by two two-port parameters. Symmetrical Two goals are always reciprocal, reciprocal but two goals are not always symmetric.

Earthing symmetry

When grounding balanced or symmetrical cross two gates can be drawn in the longitudinal direction a line of symmetry. This means that there is no continuous ground line is provided. A typical example is the so-called X - circuit of a two-port circuit. In contrast, the three poles used in practice as a two-port network have a continuous ground wire and are therefore erdungsunsymmetrisch. The property of symmetry of the ground does not affect the two-port parameters. Theoretically, you can grounding symmetrical with the help of ideal transformers Two goals in erdungsunsymmetrische transform and vice versa.

Absence of feedback

Has a changing ( through load) output does not affect an input, it is called the two-port non-reactive. Non-reactive Two goals are a " worst case " nichtumkehrbarer two goals.

For the parameters of a non-interacting two-port network, the following restrictions apply:

Thus, the input variables, the output variables.

Zweitorgleichungen and parameters

Denote the voltage U1 and I1 the current at the input terminal pair and U2 and I2 the corresponding quantities at the output terminal pair, then two searched variables from the other two given quantities can be calculated by a pair of Zweitorgleichungen. These are non-linear differential equations in general.

For linear two-ports to go, possibly using the symbolic method of AC circuit analysis, or Laplace transformation, into a pair of linear equations with the two-port four descriptive Zweitorparametern over.

Under the condition of existence, these Zweitorgleichungen specify in the form of matrix equations. Impressed currents and voltages are added depending on these equations as matrices as needed added. The calculation rules provided for the purpose of determining the matrices for any known two-port device, such as a feedback network of an amplifier circuit.

: Input impedance idle: Idle core impedance backward: Idle core impedance forward: Output open circuit impedance

: Input Kurzschlußadmittanz: Short - Kernadmittanz backward ( Rückwirkungsleitwert ): Short - Kernadmittanz forward ( slope): Output Kurzschlußadmittanz

: Short-circuit input impedance: Open-circuit voltage feedback: Short-circuit current gain: Idle output admittance

: Reciprocal voltage translation: Negative, reciprocal slope: Reciprocal core impedance forward: Reciprocal short-circuit current translation

This applies in the case of existence:

The advantage of this notation is that the parameters ( Zxy, etc.) represent known component values ​​and thus are added as numerical values ​​. Now the relationship between the input and output streams, as well as the input and output voltages can be easily read.

Note: Instead of the symbol are also, or instead of the symbol and is also used.

Conversion of the matrices

Elementary two-ports

Elemental Längszweitor

The elementary Längszweitor contains only an impedance in the upper longitudinal axis between the origin of the two-port Poland. There is no connection between the poles in the transverse axis.

Elemental Querzweitor

The elementary Querzweitor contains only an impedance in the transverse axis of the two-port network and contains no components in the longitudinal axis.

Γ two-port

The Γ - two-port network is a synthesis of elementary Querzweitor and elementary Längszweitor. It is formed from the Kettenmatrizen the elementary two-ports as follows:

Mirrored Γ two-port

The mirrored Γ - two-port network is a synthesis of elementary Längszweitor and elementary Querzweitor. It is formed from the Kettenmatrizen the elementary two-ports as follows:

Equivalent circuits

To simplify calculations of complex two gates can be combined using the appropriate two-port parameters of simplified circuits. The equivalent circuits provide no guidance for the physical realization dar.

T equivalent circuit

The T equivalent circuit allows the representation of any two-port network using the equivalent impedances. In reversible two-ports the controlled voltage source is eliminated. It can be synthesized from an elementary Längszweitor Γ and a two-port or equivalent from a two-port mirrored Γ and an elementary Längszweitor. Subsequent composition describes the latter:

π - equivalent circuit

The π - equivalent circuit allows the representation of any two-port network using the Ersatzadmittanzen. In reversible two-ports the controlled current source is eliminated. It can be synthesized from an elementary Querzweitor and a mirrored Γ two-port or according to a elementary Querzweitor Γ and a two-port. Subsequent composition describes the latter:

Interconnect

Two Two goals may, provided that the above Torbedingung is fulfilled at least one goal, to be connected together to form a new two-port network. The parameters of the newly formed two-port network can be calculated from the parameters of the two interconnected two goals. For each connection type, there is a characteristic in which can be calculated very well the interconnection. There are five different ways to interconnect two goals:

Other two-port parameters

Besides the characterization of a two-port by the two-port parameters described above, there are also other forms of representation for particular applications. Thus, a linear two-port device will be described by the so-called S-parameters. This presentation is mainly in the field of high frequency technology common because there the connections of the two-port network is not shorted or have to run empty, but are terminated by its characteristic impedance as a rule.

Between the S- parameters and the above-mentioned Y- parameters of a two-port admittance matrix with the wave impedance ZW following relationship:

With the abbreviation:

Symmetric linear two-ports are described for their application in the theory of filter circuits ( wave parameters theory ) by the so-called wave parameters. The two described the two-port parameters are the wave impedance and the Wellenübertragungsmaß.

Swell

• Lecture - Networks 3 Institute of Fundamentals and Theory of Electrical Engineering, Technical University of Graz ( This subject is referred to as Mehrtortheorie Under this title, therefore, should be able to find other sources. ).
• Lecture - Dynamic networks. Department of Fundamentals of Electrical Engineering and Electronics, Technical University of Dresden