# Two-port

A two-port is a model for an electrical component or an electrical network with four connections, in which two connections are combined into a so-called gate .

A gate exists when the electrical current through both connections of a gate is opposite ( gate condition ). A two-port is therefore a special form of a (general) quadrupole.

The name Vierpol comes from Franz Breisig in 1921 .

The matrices notation common for linear two-ports goes back to Felix Strecker and Richard Feldtkeller from 1929.

A two-port, on the other hand, is a special case of an n- port (which is also referred to as a multi-port).

## General

General two-port

A two-port 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 two-port parameters shown below and the mathematical description with the aid of matrices can only be used for linear two-ports and not for general quadrupoles.

In older specialist literature in particular, the terms two-port and four-port are used synonymously, although the term four-port is implicitly understood as two-port. The gates of a two-port are often referred to as an entrance and an exit.

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

## properties

Two-ports 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 two-port devices are independent of voltage and current. Therefore, the superposition theorem applies to the gate currents and voltages . A two-port, which consists only of the passive linear components resistor , coil , capacitor and transformer (a so-called RLCM two-port ), is always linear itself.

Non - linear two-port networks are networks with at least one non-linear 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 two-port are the subject of the classical quadrupole and modern multi-port theory. The linear two-port equations and thus the matrix representation of the two-port parameters described below apply only to them.

### Current account

If a two-port 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) two-port because it only contains dummy switching elements, it is called a reactance two-port .

### Reversibility

Reversible two-port (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 short-circuited 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 short-circuited gate 2 . If the same voltage is also applied to gate 2 , the same current is generated at the short-circuited gate 1. This means if is. ${\ displaystyle U_ {1}}$${\ displaystyle I_ {2}}$${\ displaystyle U_ {1} = U_ {2}}$${\ displaystyle I_ {2}}$${\ displaystyle I_ {2} = I_ {1}}$${\ displaystyle U_ {1} = U_ {2}}$

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

{\ displaystyle {\ begin {aligned} Z_ {12} & = Z_ {21} \\ Y_ {12} & = Y_ {21} \\ H_ {12} & = - H_ {21} \\ P_ {12} & = - P_ {21} \\\ det \ mathbf {A} & = 1 \\\ det \ mathbf {B} & = 1 \ end {aligned}}}

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

### symmetry

In the case of symmetrical two-ports (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 two-port, it is called asymmetrical .

The following applies to the elements of the two-port equations:

{\ displaystyle {\ begin {aligned} Z_ {11} = Z_ {22} &, Z_ {12} = Z_ {21} \\ Y_ {11} = Y_ {22} &, Y_ {12} = Y_ {21 } \\\ det \ mathbf {H} = 1 &, H_ {12} = - H_ {21} \\\ det \ mathbf {P} = 1 &, P_ {12} = - P_ {21} \\ A_ {11 } = A_ {22} &, \ det \ mathbf {A} = 1 \\ B_ {11} = B_ {22} &, \ det \ mathbf {B} = 1 \ end {aligned}}}

Symmetrical two-port are thus fully characterized by two two-port parameters. Symmetrical two-goals are always reciprocal, but reciprocal two-goals are not always symmetrical.

#### Ground symmetry

A symmetry line can be drawn in the longitudinal direction in the case of two-gate symmetrical or transversely symmetrical. This means that there is no continuous earth line. A typical example is the so-called X-connection of a quadrupole. In contrast, the three-pole terminals used as two-port devices have a continuous earth line and are therefore unbalanced to earth. The property of the ground symmetry has no influence on the two-port parameters. In theory, ideal transformers can be used to convert two-port two-ports balanced to earth into unbalanced ones and vice versa.

### Non-retroactivity

If an output variable that changes (due to load) has no influence on an input variable, the two-port is called non - reactive. Reaction-free two-ports are an “extreme case” of irreversible two-ports.

The following restrictions apply to the parameters of a reaction-free two-port:

{\ displaystyle {\ begin {aligned} Z_ {12} & = 0 \\ Y_ {12} & = 0 \\ H_ {12} & = 0 \\ P_ {12} & = 0 \\\ det \ mathbf { A} & = 0 \\\ mathbf {B} & {\ text {is not defined}} \ end {aligned}}}

This means that the input variables , the output variables , independent. ${\ displaystyle U_ {1}}$${\ displaystyle I_ {1}}$${\ displaystyle U_ {2}}$${\ displaystyle I_ {2}}$

## Two-port equations and parameters

Circuit to the impedance matrix

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 two-gate 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 two-port parameters describing the two-port .

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 two-port, such as a feedback network of an amplifier circuit .

 Z characteristic ${\ displaystyle {\ begin {bmatrix} U_ {1} \\ U_ {2} \ end {bmatrix}} = {\ begin {bmatrix} Z_ {11} = \ left. {\ frac {U_ {1}} { I_ {1}}} \ right | _ {I_ {2} = 0} & Z_ {12} = \ left. {\ Frac {U_ {1}} {I_ {2}}} \ right | _ {I_ {1 } = 0} \\ Z_ {21} = \ left. {\ Frac {U_ {2}} {I_ {1}}} \ right | _ {I_ {2} = 0} & Z_ {22} = \ left. {\ frac {U_ {2}} {I_ {2}}} \ right | _ {I_ {1} = 0} \ end {bmatrix}} \ cdot {\ begin {bmatrix} I_ {1} \\ I_ { 2} \ end {bmatrix}}}$ ${\ displaystyle \ mathbf {Z}}$: Impedance matrix , exists if the gate currents ( I 1 and I 2 ) can be selected independently. ${\ displaystyle Z_ {11}}$: Open circuit input impedance : Open circuit core impedance backward (reaction resistance) : Open circuit core impedance forward (transfer resistance ) : Open circuit output impedance ${\ displaystyle Z_ {12}}$ ${\ displaystyle Z_ {21}}$ ${\ displaystyle Z_ {22}}$ Y characteristic ${\ displaystyle {\ begin {bmatrix} I_ {1} \\ I_ {2} \ end {bmatrix}} = {\ begin {bmatrix} Y_ {11} = \ left. {\ frac {I_ {1}} { U_ {1}}} \ right | _ {U_ {2} = 0} & Y_ {12} = \ left. {\ Frac {I_ {1}} {U_ {2}}} \ right | _ {U_ {1 } = 0} \\ Y_ {21} = \ left. {\ Frac {I_ {2}} {U_ {1}}} \ right | _ {U_ {2} = 0} & Y_ {22} = \ left. {\ frac {I_ {2}} {U_ {2}}} \ right | _ {U_ {1} = 0} \ end {bmatrix}} \ cdot {\ begin {bmatrix} U_ {1} \\ U_ { 2} \ end {bmatrix}}}$ ${\ displaystyle \ mathbf {Y}}$: Admittance matrix , exists if the gate voltages ( U 1 and U 2 ) can be selected independently. ${\ displaystyle Y_ {11}}$: Short-circuit input admittance: Negative short-circuit core admittance backwards (retroactive conductance) : Negative short-circuit core admittance forwards (steepness) : Short-circuit output admittance ${\ displaystyle Y_ {12}}$ ${\ displaystyle Y_ {21}}$ ${\ displaystyle Y_ {22}}$ H characteristic ${\ displaystyle {\ begin {bmatrix} U_ {1} \\ I_ {2} \ end {bmatrix}} = {\ begin {bmatrix} H_ {11} = \ left. {\ frac {U_ {1}} { I_ {1}}} \ right | _ {U_ {2} = 0} & H_ {12} = \ left. {\ Frac {U_ {1}} {U_ {2}}} \ right | _ {I_ {1 } = 0} \\ H_ {21} = \ left. {\ Frac {I_ {2}} {I_ {1}}} \ right | _ {U_ {2} = 0} & H_ {22} = \ left. {\ frac {I_ {2}} {U_ {2}}} \ right | _ {I_ {1} = 0} \ end {bmatrix}} \ cdot {\ begin {bmatrix} I_ {1} \\ U_ { 2} \ end {bmatrix}}}$ ${\ displaystyle \ mathbf {H}}$: Hybrid matrix (row-parallel matrix), exists if I 1 and U 2 can be selected independently. ${\ displaystyle H_ {11}}$: Short-circuit input impedance : No-load voltage reaction : Negative short-circuit current translation (or current gain) : No-load output admittance ${\ displaystyle H_ {12}}$ ${\ displaystyle H_ {21}}$ ${\ displaystyle H_ {22}}$ P characteristic ${\ displaystyle {\ begin {bmatrix} I_ {1} \\ U_ {2} \ end {bmatrix}} = {\ begin {bmatrix} P_ {11} = \ left. {\ frac {I_ {1}} { U_ {1}}} \ right | _ {I_ {2} = 0} & P_ {12} = \ left. {\ Frac {I_ {1}} {I_ {2}}} \ right | _ {U_ {1 } = 0} \\ P_ {21} = \ left. {\ Frac {U_ {2}} {U_ {1}}} \ right | _ {I_ {2} = 0} & P_ {22} = \ left. {\ frac {U_ {2}} {I_ {2}}} \ right | _ {U_ {1} = 0} \ end {bmatrix}} \ cdot {\ begin {bmatrix} U_ {1} \\ I_ { 2} \ end {bmatrix}}}$ ${\ displaystyle \ mathbf {P}}$: Inverse hybrid matrix (parallel series matrix), exists if U 1 and I 2 can be selected independently. ${\ displaystyle P_ {11}}$: Open-circuit input admittance : Negative short-circuit current feedback : Open-circuit voltage translation : Short-circuit output impedance ${\ displaystyle P_ {12}}$ ${\ displaystyle P_ {21}}$ ${\ displaystyle P_ {22}}$ A characteristic ${\ displaystyle {\ begin {bmatrix} U_ {1} \\ I_ {1} \ end {bmatrix}} = {\ begin {bmatrix} A_ {11} = \ left. {\ frac {U_ {1}} { U_ {2}}} \ right | _ {I_ {2} = 0} & A_ {12} = \ left. {\ Frac {U_ {1}} {- I_ {2}}} \ right | _ {U_ { 2} = 0} \\ A_ {21} = \ left. {\ Frac {I_ {1}} {U_ {2}}} \ right | _ {I_ {2} = 0} & A_ {22} = \ left . {\ frac {I_ {1}} {- I_ {2}}} \ right | _ {U_ {2} = 0} \ end {bmatrix}} \ cdot {\ begin {bmatrix} U_ {2} \\ -I_ {2} \ end {bmatrix}}}$ ${\ displaystyle \ mathbf {A}}$: Chain matrix, exists if U 2 and I 2 can be selected independently. ${\ displaystyle A_ {11}}$: Reciprocal no-load voltage translation: Short- circuit core impedance forward (reciprocal slope) : No-load core admittance forward (transmission conductance ) : Reciprocal short-circuit current translation ${\ displaystyle A_ {12}}$ ${\ displaystyle A_ {21}}$ ${\ displaystyle A_ {22}}$ B characteristic ${\ displaystyle {\ begin {bmatrix} U_ {2} \\ - I_ {2} \ end {bmatrix}} = {\ begin {bmatrix} B_ {11} = \ left. {\ frac {U_ {2}} {U_ {1}}} \ right | _ {I_ {1} = 0} & B_ {12} = \ left. {\ Frac {U_ {2}} {I_ {1}}} \ right | _ {U_ { 1} = 0} \\ B_ {21} = \ left. {\ Frac {-I_ {2}} {U_ {1}}} \ right | _ {I_ {1} = 0} & B_ {22} = \ left. {\ frac {-I_ {2}} {I_ {1}}} \ right | _ {U_ {1} = 0} \ end {bmatrix}} \ cdot {\ begin {bmatrix} U_ {1} \ \ I_ {1} \ end {bmatrix}}}$ ${\ displaystyle \ mathbf {B}}$: Inverse chain matrix, exists if U 1 and I 1 can be chosen independently. ${\ displaystyle B_ {11}}$: Reciprocal no-load voltage feedback: Negative short-circuit core impedance backwards (feedback resistance) : Negative no-load core admittance backwards (feedback conductance ) : Reciprocal short-circuit current feedback ${\ displaystyle B_ {12}}$ ${\ displaystyle B_ {21}}$ ${\ displaystyle B_ {22}}$

In the case of the existence of the matrices it holds in particular

${\ displaystyle \ mathbf {Y} = \ mathbf {Z} ^ {- 1}}$
${\ displaystyle \ mathbf {P} = \ mathbf {H} ^ {- 1}}$
${\ displaystyle \ mathbf {B} = \ mathbf {A} ^ {- 1}}$

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. ${\ displaystyle \ mathbf {P}}$${\ displaystyle \ mathbf {H '}}$${\ displaystyle \ mathbf {C}}$${\ displaystyle \ mathbf {B}}$${\ displaystyle \ mathbf {A '}}$

### Conversion of the matrices

 ${\ displaystyle \ mathbf {Z}}$ ${\ displaystyle \ mathbf {Y}}$ ${\ displaystyle \ mathbf {H}}$ ${\ displaystyle \ mathbf {P}}$ ${\ displaystyle \ mathbf {A}}$ ${\ displaystyle \ mathbf {B}}$ ${\ displaystyle \ mathbf {Z}}$ ${\ displaystyle {\ begin {bmatrix} Z_ {11} & Z_ {12} \\ Z_ {21} & Z_ {22} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {Y_ {22}} {\ det \ mathbf {Y}}} & {\ dfrac {-Y_ {12}} {\ det \ mathbf {Y}}} \ \ {\ dfrac {-Y_ {21}} {\ det \ mathbf {Y}}} & {\ dfrac {Y_ {11}} {\ det \ mathbf {Y}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {\ det \ mathbf {H}} {H_ {22}}} & {\ dfrac {H_ {12}} {H_ {22}}} \\ {\ dfrac {-H_ {21}} {H_ {22}}} & {\ dfrac {1} {H_ {22}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {1} {P_ {11}}} & {\ dfrac {-P_ {12}} {P_ {11}}} \\ {\ dfrac {P_ {21} } {P_ {11}}} & {\ dfrac {\ det \ mathbf {P}} {P_ {11}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {A_ {11}} {A_ {21}}} & {\ dfrac {\ det \ mathbf {A}} {A_ {21}}} \\ {\ dfrac {1} {A_ {21}}} & {\ dfrac {A_ {22}} {A_ {21}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {-B_ {22}} {B_ {21}}} & {\ dfrac {-1} {B_ {21}}} \\ {\ dfrac {- \ det \ mathbf {B}} {B_ {21}}} & {\ dfrac {-B_ {11}} {B_ {21}}} \ end {bmatrix}}}$ ${\ displaystyle \ mathbf {Y}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {Z_ {22}} {\ det \ mathbf {Z}}} & {\ dfrac {-Z_ {12}} {\ det \ mathbf {Z}}} \ \ {\ dfrac {-Z_ {21}} {\ det \ mathbf {Z}}} & {\ dfrac {Z_ {11}} {\ det \ mathbf {Z}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} Y_ {11} & Y_ {12} \\ Y_ {21} & Y_ {22} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {1} {H_ {11}}} & {\ dfrac {-H_ {12}} {H_ {11}}} \\ {\ dfrac {H_ {21} } {H_ {11}}} & {\ dfrac {\ det \ mathbf {H}} {H_ {11}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {\ det \ mathbf {P}} {P_ {22}}} & {\ dfrac {P_ {12}} {P_ {22}}} \\ {\ dfrac {-P_ {21}} {P_ {22}}} & {\ dfrac {1} {P_ {22}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {A_ {22}} {A_ {12}}} & {\ dfrac {- \ det \ mathbf {A}} {A_ {12}}} \\ {\ dfrac {-1} {A_ {12}}} & {\ dfrac {A_ {11}} {A_ {12}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {-B_ {11}} {B_ {12}}} & {\ dfrac {1} {B_ {12}}} \\ {\ dfrac {\ det \ mathbf {B}} {B_ {12}}} & {\ dfrac {-B_ {22}} {B_ {12}}} \ end {bmatrix}}}$ ${\ displaystyle \ mathbf {H}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {\ det \ mathbf {Z}} {Z_ {22}}} & {\ dfrac {Z_ {12}} {Z_ {22}}} \\ {\ dfrac {-Z_ {21}} {Z_ {22}}} & {\ dfrac {1} {Z_ {22}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {1} {Y_ {11}}} & {\ dfrac {-Y_ {12}} {Y_ {11}}} \\ {\ dfrac {Y_ {21} } {Y_ {11}}} & {\ dfrac {\ det \ mathbf {Y}} {Y_ {11}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} H_ {11} & H_ {12} \\ H_ {21} & H_ {22} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {P_ {22}} {\ det \ mathbf {P}}} & {\ dfrac {-P_ {12}} {\ det \ mathbf {P}}} \ \ {\ dfrac {-P_ {21}} {\ det \ mathbf {P}}} & {\ dfrac {P_ {11}} {\ det \ mathbf {P}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {A_ {12}} {A_ {22}}} & {\ dfrac {\ det \ mathbf {A}} {A_ {22}}} \\ {\ dfrac {-1} {A_ {22}}} & {\ dfrac {A_ {21}} {A_ {22}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {-B_ {12}} {B_ {11}}} & {\ dfrac {1} {B_ {11}}} \\ {\ dfrac {- \ det \ mathbf {B}} {B_ {11}}} & {\ dfrac {-B_ {21}} {B_ {11}}} \ end {bmatrix}}}$ ${\ displaystyle \ mathbf {P}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {1} {Z_ {11}}} & {\ dfrac {-Z_ {12}} {Z_ {11}}} \\ {\ dfrac {Z_ {21} } {Z_ {11}}} & {\ dfrac {\ det \ mathbf {Z}} {Z_ {11}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {\ det \ mathbf {Y}} {Y_ {22}}} & {\ dfrac {Y_ {12}} {Y_ {22}}} \\ {\ dfrac {-Y_ {21}} {Y_ {22}}} & {\ dfrac {1} {Y_ {22}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {H_ {22}} {\ det \ mathbf {H}}} & {\ dfrac {-H_ {12}} {\ det \ mathbf {H}}} \ \ {\ dfrac {-H_ {21}} {\ det \ mathbf {H}}} & {\ dfrac {H_ {11}} {\ det \ mathbf {H}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} P_ {11} & P_ {12} \\ P_ {21} & P_ {22} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {A_ {21}} {A_ {11}}} & {\ dfrac {- \ det \ mathbf {A}} {A_ {11}}} \\ {\ dfrac {1} {A_ {11}}} & {\ dfrac {A_ {12}} {A_ {11}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {-B_ {21}} {B_ {22}}} & {\ dfrac {-1} {B_ {22}}} \\ {\ dfrac {\ det \ mathbf {B}} {B_ {22}}} & {\ dfrac {-B_ {12}} {B_ {22}}} \ end {bmatrix}}}$ ${\ displaystyle \ mathbf {A}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {Z_ {11}} {Z_ {21}}} & {\ dfrac {\ det \ mathbf {Z}} {Z_ {21}}} \\ {\ dfrac {1} {Z_ {21}}} & {\ dfrac {Z_ {22}} {Z_ {21}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {-Y_ {22}} {Y_ {21}}} & {\ dfrac {-1} {Y_ {21}}} \\ {\ dfrac {- \ det \ mathbf {Y}} {Y_ {21}}} & {\ dfrac {-Y_ {11}} {Y_ {21}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {- \ det \ mathbf {H}} {H_ {21}}} & {\ dfrac {-H_ {11}} {H_ {21}}} \\ { \ dfrac {-H_ {22}} {H_ {21}}} & {\ dfrac {-1} {H_ {21}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {1} {P_ {21}}} & {\ dfrac {P_ {22}} {P_ {21}}} \\ {\ dfrac {P_ {11}} {P_ {21}}} & {\ dfrac {\ det \ mathbf {P}} {P_ {21}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} A_ {11} & A_ {12} \\ A_ {21} & A_ {22} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {B_ {22}} {\ det \ mathbf {B}}} & {\ dfrac {-B_ {12}} {\ det \ mathbf {B}}} \ \ {\ dfrac {-B_ {21}} {\ det \ mathbf {B}}} & {\ dfrac {B_ {11}} {\ det \ mathbf {B}}} \ end {bmatrix}}}$ ${\ displaystyle \ mathbf {B}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {Z_ {22}} {Z_ {12}}} & {\ dfrac {- \ det \ mathbf {Z}} {Z_ {12}}} \\ {\ dfrac {-1} {Z_ {12}}} & {\ dfrac {Z_ {11}} {Z_ {12}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {-Y_ {11}} {Y_ {12}}} & {\ dfrac {1} {Y_ {12}}} \\ {\ dfrac {\ det \ mathbf {Y}} {Y_ {12}}} & {\ dfrac {-Y_ {22}} {Y_ {12}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {1} {H_ {12}}} & {\ dfrac {-H_ {11}} {H_ {12}}} \\ {\ dfrac {-H_ {22 }} {H_ {12}}} & {\ dfrac {\ det \ mathbf {H}} {H_ {12}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {- \ det \ mathbf {P}} {P_ {12}}} & {\ dfrac {P_ {22}} {P_ {12}}} \\ {\ dfrac {P_ {11}} {P_ {12}}} & {\ dfrac {-1} {P_ {12}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} {\ dfrac {A_ {22}} {\ det \ mathbf {A}}} & {\ dfrac {-A_ {12}} {\ det \ mathbf {A}}} \ \ {\ dfrac {-A_ {21}} {\ det \ mathbf {A}}} & {\ dfrac {A_ {11}} {\ det \ mathbf {A}}} \ end {bmatrix}}}$ ${\ displaystyle {\ begin {bmatrix} B_ {11} & B_ {12} \\ B_ {21} & B_ {22} \ end {bmatrix}}}$

### Elementary two-goal

#### Elementary longitudinal second

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

${\ displaystyle {\ begin {bmatrix} A_ {l} \ end {bmatrix}} = {\ begin {bmatrix} 1 & Z_ {l} \\ 0 & 1 \ end {bmatrix}}}$

#### Elementary cross-two

The elementary transverse two-port only contains an impedance in the transverse axis of the two-port and does not contain any components in the longitudinal axis.

${\ displaystyle {\ begin {bmatrix} A_ {q} \ end {bmatrix}} = {\ begin {bmatrix} 1 & 0 \\ Y_ {q} & 1 \ end {bmatrix}}}$

#### Γ-two-port

The Γ-two-port is a synthesis of elementary transverse two-port and elementary longitudinal two-port. It is formed from the chain matrices of the elementary two-port as follows:

{\ displaystyle {\ begin {aligned} {\ begin {bmatrix} A _ {\ Gamma} \ end {bmatrix}} = {\ begin {bmatrix} A_ {q} \ end {bmatrix}} {\ begin {bmatrix} A_ {l} \ end {bmatrix}} & = {\ begin {bmatrix} 1 & 0 \\ Y_ {q} & 1 \ end {bmatrix}} {\ begin {bmatrix} 1 & Z_ {l} \\ 0 & 1 \ end {bmatrix}} \\ & = {\ begin {bmatrix} 1 & Z_ {l} \\ Y_ {q} & (Y_ {q} Z_ {l} +1) \ end {bmatrix}} \ end {aligned}}}

#### Mirrored Γ two-port

The mirrored Γ-two-port is a synthesis of elementary longitudinal two-port and elementary transverse two-port. It is formed from the chain matrices of the elementary two-port as follows:

{\ displaystyle {\ begin {aligned} {\ begin {bmatrix} A _ {} \ end {bmatrix}} = {\ begin {bmatrix} A_ {l} \ end {bmatrix}} {\ begin {bmatrix} A_ {q } \ end {bmatrix}} & = {\ begin {bmatrix} 1 & Z_ {l} \\ 0 & 1 \ end {bmatrix}} {\ begin {bmatrix} 1 & 0 \\ Y_ {q} & 1 \ end {bmatrix}} \\ & = {\ begin {bmatrix} (Y_ {q} Z_ {l} +1) & Z_ {l} \\ Y_ {q} & 1 \ end {bmatrix}} \ end {aligned}}}

### Equivalent circuits

To simplify calculations, complex two-port circuits can be combined to form simplified circuits using appropriate two-port parameters. The equivalent circuits do not represent a guide for physical implementation.

#### T equivalent circuit

T equivalent circuit

The T equivalent circuit enables any two-port circuit to be represented using the equivalent impedances. The controlled voltage source is not required for reversible two-port devices. It can be synthesized from an elementary longitudinal two-port and a Γ-two-port or correspondingly from a mirrored Γ-two-port and an elementary longitudinal two-port. The following composition describes the latter:

{\ displaystyle {\ begin {aligned} {\ begin {bmatrix} A_ {T} \ end {bmatrix}} = {\ begin {bmatrix} A _ {\ Gamma} \ end {bmatrix}} {\ begin {bmatrix} A_ {l} \ end {bmatrix}} & = {\ begin {bmatrix} (Y_ {q} Z_ {l1} +1) & Z_ {l1} \\ Y_ {q} & 1 \ end {bmatrix}} {\ begin { bmatrix} 1 & Z_ {l2} \\ 0 & 1 \ end {bmatrix}} \\ & = {\ begin {bmatrix} (Y_ {q} Z_ {l1} +1) & {Z_ {l2} Y_ {q} Z_ {l1 } + Z_ {l2} + Z_ {l1}} \\ Y_ {q} & {Y_ {q} Z_ {l2} +1} \ end {bmatrix}} \ end {aligned}}}

#### π equivalent circuit

π equivalent circuit

The π equivalent circuit enables any two-port to be represented using the equivalent admittances. The controlled power source is not required for reversible two-port devices. It can be synthesized from an elementary transverse two-port and a mirrored Γ-two-port or accordingly from an elementary transverse two-port and a Γ-two-port. The following composition describes the latter:

{\ displaystyle {\ begin {aligned} {\ begin {bmatrix} A _ {\ Pi} \ end {bmatrix}} = {\ begin {bmatrix} A _ {\ Gamma} \ end {bmatrix}} {\ begin {bmatrix} A_ {q} \ end {bmatrix}} & = {\ begin {bmatrix} 1 & Z_ {l} \\ Y_ {q1} & (Y_ {q1} Z_ {l} +1) \ end {bmatrix}} {\ begin {bmatrix} 1 & 0 \\ Y_ {q2} & 1 \ end {bmatrix}} \\ & = {\ begin {bmatrix} {1 + Z_ {l} Y_ {q2}} & Z_ {l} \\ {Y_ {q1} + Y_ {q1} Z_ {l} Y_ {q2} + Y_ {q2}} & {Y_ {q1} Z_ {l} +1} \ end {bmatrix}} \ end {aligned}}}

### Interconnect

Two two-port gates can be interconnected to form a new two-port 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 two-port can be calculated from the parameters of the two interconnected two-ports. 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 two-goals:

Description type presentation Mathematical description
Series connection ${\ displaystyle \ mathbf {Z_ {ges}} = \ mathbf {Z_ {1}} + \ mathbf {Z_ {2}}}$
Parallel connection ${\ displaystyle \ mathbf {Y_ {ges}} = \ mathbf {Y_ {1}} + \ mathbf {Y_ {2}}}$
Hybrid connection or
series-parallel connection
${\ displaystyle \ mathbf {H_ {ges}} = \ mathbf {H_ {1}} + \ mathbf {H_ {2}}}$
inverse hybrid connection or
parallel series connection
${\ displaystyle \ mathbf {P_ {ges}} = \ mathbf {P_ {1}} + \ mathbf {P_ {2}}}$
or or
${\ displaystyle \ mathbf {H '_ {ges}} = \ mathbf {H' _ {1}} + \ mathbf {H '_ {2}}}$

${\ displaystyle \ mathbf {C_ {ges}} = \ mathbf {C_ {1}} + \ mathbf {C_ {2}}}$
Derailleur ${\ displaystyle \ mathbf {A_ {ges}} = \ mathbf {A_ {1}} \ cdot \ mathbf {A_ {2}}}$
or or
${\ displaystyle \ mathbf {B_ {ges}} = \ mathbf {B_ {2}} \ cdot \ mathbf {B_ {1}}}$

${\ displaystyle \ mathbf {A '_ {ges}} = \ mathbf {A' _ {2}} \ cdot \ mathbf {A '_ {1}}}$

### Swap two ports

Moving a two-gate through another.
Chain matrix (A matrix) of simple impedances in chain parameters (not Z parameters; Z is only the name of the impedance here)

It is often helpful to transform one two-port 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:

${\ displaystyle \ mathbf {A} '= \ mathbf {T} ^ {- 1} \ cdot \ mathbf {A} \ cdot \ mathbf {T}}$

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 two-port.

## Further two-port parameters

In addition to the characterization of a two-port using the two-port parameters described above, there are also other forms of representation for special purposes. A linear two-port can also be described by so-called scattering parameters . This form of representation is particularly common in the field of high-frequency technology , since the connections of the two-port do not have to be short-circuited or empty, but are usually closed by their wave impedance.

The following relationship exists between the S parameters and the above-mentioned Y parameters of the admittance matrix of a two-port with the wave impedance Z W :

${\ displaystyle Y_ {11} = {\ frac {1} {Z_ {w}}} {\ frac {1-S_ {11} + S_ {22} - \ Delta _ {S}} {1 + S_ {11 } + S_ {22} + \ Delta _ {S}}}}$
${\ displaystyle Y_ {12} = {\ frac {1} {Z_ {w}}} {\ frac {-2S_ {12}} {1 + S_ {11} + S_ {22} + \ Delta _ {S} }}}$
${\ displaystyle Y_ {21} = {\ frac {1} {Z_ {w}}} {\ frac {-2S_ {21}} {1 + S_ {11} + S_ {22} + \ Delta _ {S} }}}$
${\ displaystyle Y_ {22} = {\ frac {1} {Z_ {w}}} {\ frac {1 + S_ {11} -S_ {22} - \ Delta _ {S}} {1 + S_ {11 } + S_ {22} + \ Delta _ {S}}}}$

with the abbreviation:

${\ displaystyle \ Delta _ {S} = {\ det \ mathbf {S}} = S_ {11} S_ {22} -S_ {12} S_ {21}}$

For their application in the theory of filter circuits (wave parameter theory), symmetrical linear two-ports are described by the so-called wave parameters . The two parameters describing the two-port 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 computer-aided 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. Super-Compact) gradually gained acceptance in industry. There is no fundamental difference between high-frequency 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 computer-aided formula derivation

Also of interest is the symbolic processing of the above matrix equations and, based on this, the computer-aided formula derivation for signal analysis in linear electronics and high-frequency electronics. A special addition for a mathematics program converts the above eight four-pole 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

• Lorenz-Peter Schmidt, Gerd Schaller, Siegfried Martius: Fundamentals of electrical engineering. Volume 3: Networks. Pearson Studium, Munich 2006, ISBN 3-8273-7107-4 .
• Richard Feldtkeller : Introduction to the four-pole theory of electrical communications engineering. Hirzel, Stuttgart 1976, ISBN 3-7776-0319-8 .
• Handheld electrical telecommunications dictionary . Essay on the quadrupole theory by Zuhrt / Matthes, 2nd edition. 3rd volume, pp. 1837-1868.
• 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 Radio-Foto-Kinotechnik, Berlin-Borsigwalde 1966.
• Claus-Christian Timmermann : High Frequency Electronics with CAD, Vol. 1, Introduction to Lines , Four-Pole, Transistor Models and Simulation with Numerical and Symbolic CAD Systems, Vol. 1 , PROFUND-Verlag, 1997 and 2005, ISBN 978-3-932651-21 -2
• George D. Vendelin, Anthony M. Pavio, Ulrich L. Rohde. Microwave Circuit Design Using Linear and Nonlinear Techniques , 2nd Edition, Wiley, 2005, ISBN 978-0-471-41479-7
• Lecture - Networks 3. Institute for Fundamentals and Theory of Electrical Engineering, Graz University of Technology (This subject is called multi-gate 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