## Modelica.Electrical.Analog.Lines

### Information

This package contains lossy and lossless segmented transmission lines, and LC distributed line models.

Main Authors:
  Christoph Clauß <clauss@eas.iis.fhg.de>
  Joachim Haase; <haase@eas.iis.fhg.de>
  André Schneider <schneider@eas.iis.fhg.de>
  Fraunhofer Institute for Integrated Circuits
  Design Automation Department
  Zeunerstraße 38
  D-01069 Dresden

Version:
  $Id: Modelica_Electrical_Analog_Lines.html,v 1.10 2002/12/17 15:10:15 Hans Exp $

Copyright:
  Copyright © 1998-2002, Modelica Association and Fraunhofer-Gesellschaft.
  The Modelica package is free software; it can be redistributed and/or modified under the terms of the Modelica license, see the license conditions and the accompanying disclaimer in the documentation of package Modelica in file "Modelica/package.mo".

## Modelica.Electrical.Analog.Lines.OLine

Lossy Transmission Line

### Information

Lossy Transmission Line. The lossy transmission line OLine consists of segments of lumped resistances and inductances in series and conductances and capacitances that are connected with the reference pin p3. The precision of the model depends on the number N of lumped segments.

References:
  Johnson, B.; Quarles, T.; Newton, A. R.; Pederson, D. O.; Sangiovanni-Vincentelli, A.: SPICE3 Version 3e User's Manual (April 1, 1991). Department of Electrical Engineering and Computer Sciences, University of California, Berkley p. 12, p. 106 - 107

### Parameters

Name | Default | Description
--- | --- | ---
r | 1 | Resistance per meter [Ohm/m]
l | 1 | Inductance per meter [H/m]
g | 1 | Conductance per meter [Siemens/m]
c | 1 | Capacitance per meter [F/m]
length | 1 | Length of line [m]
N | 1 | Number of lumped segments

### Modelica definition

    model OLine "Lossy Transmission Line"
      //extends Interfaces.ThreePol;
      Interfaces.Pin p1;
      Interfaces.Pin p2;
      Interfaces.Pin p3;
      SI.Voltage v13;
      SI.Voltage v23;
      SI.Current i1;
      SI.Current i2;
      parameter Real r(
        final min=Modelica.Constants.small,
        unit="Ohm/m") = 1 "Resistance per meter";
      parameter Real l(
        final min=Modelica.Constants.small,
        unit="H/m") = 1 "Inductance per meter";
      parameter Real g(
        final min=Modelica.Constants.small,
        unit="Siemens/m") = 1 "Conductance per meter";
      parameter Real c(
        final min=Modelica.Constants.small,
        unit="F/m") = 1 "Capacitance per meter";
      parameter SI.Length length(final min=Modelica.Constants.small) = 1 "Length of line";
      parameter Integer N(final min=1) = 1 "Number of lumped segments";
    protected
      Basic.Resistor R[N + 1](R=fill(r*length/(N + 1), N + 1));
      Basic.Inductor L[N + 1](L=fill(l*length/(N + 1), N + 1));
      Basic.Capacitor C[N](C=fill(c*length/(N), N));
      Basic.Conductor G[N](G=fill(g*length/(N), N));
    equation
      v13 = p1.v - p3.v;
      v23 = p2.v - p3.v;
      i1 = p1.i;
      i2 = p2.i;
      connect(p1, R[1].p);
      for i in 1:N loop
        connect(R[i].n, L[i].p);
        connect(L[i].n, C[i].p);
        connect(L[i].n, G[i].p);
        connect(C[i].n, p3);
        connect(G[i].n, p3);
        connect(L[i].n, R[i + 1].p);
      end for;
      connect(R[N + 1].n, L[N + 1].p);
      connect(L[N + 1].n, p2);
    end OLine;

## Modelica.Electrical.Analog.Lines.ULine

Lossy RC Line

### Information

The lossy RC line ULine consists of segments of lumped series resistances and capacitances that are connected with the reference pin p3. The precision of the model depends on the number N of lumped segments.

References:
  Johnson, B.; Quarles, T.; Newton, A. R.; Pederson, D. O.; Sangiovanni-Vincentelli, A.: SPICE3 Version 3e User's Manual (April 1, 1991). Department of Electrical Engineering and Computer Sciences, University of California, Berkley p. 22, p. 124

### Parameters

Name | Default | Description
--- | --- | ---
r | 1 | Resistance per meter [Ohm/m]
c | 1 | Capacitance per meter [F/m]
length | 1 | Length of line [m]
N | 1 | Number of lumped segments

### Modelica definition

    model ULine "Lossy RC Line"
      //extends Interfaces.ThreePol;
      Interfaces.Pin p1;
      Interfaces.Pin p2;
      Interfaces.Pin p3;
      SI.Voltage v13;
      SI.Voltage v23;
      SI.Current i1;
      SI.Current i2;
      parameter Real r(
        final min=Modelica.Constants.small,
        unit="Ohm/m") = 1 "Resistance per meter";
      parameter Real c(
        final min=Modelica.Constants.small,
        unit="F/m") = 1 "Capacitance per meter";
      parameter SI.Length length(final min=Modelica.Constants.small) = 1 "Length of line";
      parameter Integer N(final min=1) = 1 "Number of lumped segments";
    protected
      Basic.Resistor R[N + 1](R=fill(r*length/(N + 1), N + 1));
      Basic.Capacitor C[N](C=fill(c*length/(N), N));
    equation
      v13 = p1.v - p3.v;
      v23 = p2.v - p3.v;
      i1 = p1.i;
      i2 = p2.i;
      connect(p1, R[1].p);
      for i in 1:N loop
        connect(R[i].n, R[i + 1].p);
      end for;
      for i in 1:N loop
        connect(R[i].n, C[i].p);
      end for;
      for i in 1:N loop
        connect(C[i].n, p3);
      end for;
      connect(R[N + 1].n, p2);
    end ULine;

## Modelica.Electrical.Analog.Lines.TLine1

Lossless transmission line with characteristic impedance Z0 and transmission delay TD

### Information

    Lossless transmission line with characteristic impedance Z0 and transmission delay TD

      The lossless transmission line TLine1 is a two Port. Both port branches
      consist of a resistor with characteristic impedance Z0 and a controled voltage
      source that takes into consideration the transmission delay TD.
      For further details see Branin's article.
      The model parameters can be derived from inductance and
      capacitance per length (L' resp. C'), i. e.
      Z0 = sqrt(L'/C') and TD = sqrt(L'*C')*length_of_line. Resistance R'
      and conductance C' per meter are assumed to be zero.

    References:
      Branin Jr., F. H.: Transient Analysis of Lossless Transmission Lines.
      Proceedings of the IEEE 55(1967), 2012 - 2013

      Hoefer, E. E. E.; Nielinger, H.: SPICE : Analyseprogramm fuer elektronische
      Schaltungen. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.

### Parameters

Name | Default | Description
--- | --- | ---
Z0 | 1 | Characteristic impedance [Ohm]
TD | 1 | Transmission delay [s]

### Modelica definition

    model TLine1
      "Lossless transmission line with characteristic impedance Z0 and transmission delay TD"

      extends Modelica.Electrical.Analog.Interfaces.TwoPort;

      parameter Modelica.SIunits.Resistance Z0=1 "Characteristic impedance";
      parameter Modelica.SIunits.Time TD=1 "Transmission delay";

    protected
      Modelica.SIunits.Voltage er;
      Modelica.SIunits.Voltage es;
    equation

      assert(Z0 > 0, "Z0 has to be positive");
      assert(TD > 0, "TD has to be positive");
      i1 = (v1 - es)/Z0;
      i2 = (v2 - er)/Z0;
      es = 2*delay(v2, TD) - delay(er, TD);
      er = 2*delay(v1, TD) - delay(es, TD);

    end TLine1;

## Modelica.Electrical.Analog.Lines.TLine2

Lossless transmission line with characteristic impedance Z0, frequency F and normalized length NL

### Information

    Lossless transmission line with characteristic impedance Z0, frequency F and normalized length NL

      The lossless transmission line TLine2 is a two Port. Both port branches
      consist of a resistor with the value of the characteristic impedance Z0
      and a controled voltage source that takes into consideration
      the transmission delay.
      For further details see Branin's article.
      Resistance R' and conductance C' per meter are assumed to be zero.
      The characteristic impedance Z0 can be derived from inductance and
      capacitance per length (L' resp. C'), i. e. Z0 = sqrt(L'/C').
      The normalized length NL is equal to the length of the line divided
      by the wavelength corresponding to the frequency F, i. e. the
      transmission delay TD is the quotient of NL and F.

    References:
      Branin Jr., F. H.: Transient Analysis of Lossless Transmission Lines.
      Proceedings of the IEEE 55(1967), 2012 - 2013

      Hoefer, E. E. E.; Nielinger, H.: SPICE : Analyseprogramm fuer elektronische
      Schaltungen. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.

### Parameters

Name | Default | Description
--- | --- | ---
Z0 | 1 | Characteristic impedance [Ohm]
F | 1 | Frequency [Hz]
NL | 1 | Normalized length [m]

### Modelica definition

    model TLine2
      "Lossless transmission line with characteristic impedance Z0, frequency F and normalized length NL"

      extends Modelica.Electrical.Analog.Interfaces.TwoPort;

      parameter Modelica.SIunits.Resistance Z0=1 "Characteristic impedance";
      parameter Modelica.SIunits.Frequency F=1 "Frequency";
      parameter Modelica.SIunits.Length NL=1 "Normalized length";

    protected
      Modelica.SIunits.Voltage er;
      Modelica.SIunits.Voltage es;
      Modelica.SIunits.Time TD;
    equation

      assert(Z0 > 0, "Z0 has to be positive");
      assert(NL > 0, "NL has to be positive");
      assert(F > 0, "F  has to be positive");
      TD = NL/F;
      i1 = (v1 - es)/Z0;
      i2 = (v2 - er)/Z0;
      es = 2*delay(v2, TD) - delay(er, TD);
      er = 2*delay(v1, TD) - delay(es, TD);

    end TLine2;

## Modelica.Electrical.Analog.Lines.TLine3

Lossless transmission line with characteristic impedance Z0 and frequency F

### Information

    Lossless transmission line with characteristic impedance Z0 and frequency F

      The lossless transmission line TLine3 is a two Port. Both port branches
      consist of a resistor with value of the characteristic impedance Z0
      and a controled voltage source that takes into consideration
      the transmission delay.
      For further details see Branin's article.
      Resistance R' and conductance C' per meter are assumed to be zero.
      The characteristic impedance Z0 can be derived from inductance and
      capacitance per length (L' resp. C'), i. e. Z0 = sqrt(L'/C').
      The length of the line is equal to a quarter of the wavelength
      corresponding to the frequency F, i. e. the
      transmission delay is the quotient of 4 and F.
      In this case, the caracteristic impedance is called natural impedance.

      References:
        Branin Jr., F. H.: Transient Analysis of Lossless Transmission Lines.
        Proceedings of the IEEE 55(1967), 2012 - 2013

        Hoefer, E. E. E.; Nielinger, H.: SPICE : Analyseprogramm fuer elektronische
        Schaltungen. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.

### Parameters

Name | Default | Description
--- | --- | ---
Z0 | 1 | Natural impedance [Ohm]
F | 1 | Frequency [Hz]

### Modelica definition

    model TLine3
      "Lossless transmission line with characteristic impedance Z0 and frequency F"

      extends Modelica.Electrical.Analog.Interfaces.TwoPort;

      parameter Modelica.SIunits.Resistance Z0=1 "Natural impedance";
      parameter Modelica.SIunits.Frequency F=1 "Frequency";

    protected
      Modelica.SIunits.Voltage er;
      Modelica.SIunits.Voltage es;
      Modelica.SIunits.Time TD;
    equation

      assert(Z0 > 0, "Z0 has to be positive");
      assert(F > 0, "F  has to be positive");
      TD = 1/F/4;
      i1 = (v1 - es)/Z0;
      i2 = (v2 - er)/Z0;
      es = 2*delay(v2, TD) - delay(er, TD);
      er = 2*delay(v1, TD) - delay(es, TD);

    end TLine3;