matlab: new strip-line port and examples

Signed-off-by: Thorsten Liebig <Thorsten.Liebig@gmx.de>
This commit is contained in:
Thorsten Liebig 2013-10-14 22:47:11 +02:00
parent a28e1b9bb0
commit d48d67c233
5 changed files with 465 additions and 3 deletions

283
matlab/AddStripLinePort.m Normal file
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@ -0,0 +1,283 @@
function [CSX,port] = AddStripLinePort( CSX, prio, portnr, materialname, start, stop, height, dir, evec, varargin )
% [CSX,port] = AddStripLinePort( CSX, prio, portnr, materialname, start, stop, height, dir, evec, varargin )
%
% CSX: CSX-object created by InitCSX()
% prio: priority for excitation and probe boxes
% portnr: (integer) number of the port
% materialname: property for the MSL (created by AddMetal())
% start: 3D start rowvector for port definition
% stop: 3D end rowvector for port definition
% height: height of the stripline (top and bottom)
% dir: direction of wave propagation (choices: 0, 1, 2 or 'x','y','z')
% evec: excitation vector, which defines the direction of the e-field (must be the same as used in AddExcitation())
%
% variable input:
% varargin: optional additional excitations options, see also AddExcitation
% 'ExcitePort' true/false to make the port an active feeding port (default
% is false)
% 'FeedShift' shift to port from start by a given distance in drawing
% units. Default is 0. Only active if 'ExcitePort' is set!
% 'Feed_R' Specifiy a lumped port resistance. Default is no lumped
% port resistance --> port has to end in an ABC.
% 'MeasPlaneShift' Shift the measurement plane from start t a given distance
% in drawing units. Default is the middle of start/stop.
% 'PortNamePrefix' a prefix to the port name
%
% Important: The mesh has to be already set and defined by DefineRectGrid!
%
% example:
% CSX = AddMetal( CSX, 'metal' ); %create a PEC called 'metal'
% start = [0 -width/2 0];
% stop = [length +width/2 0];
% [CSX,port] = AddStripLinePort( CSX, 0, 1, 'metal', start, stop, height, 'x', ...
% [0 0 -1], 'ExcitePort', true, 'Feed_R', 50 )
% Explanation:
% - this defines a stripline in x-direction (dir='x')
% --> the wave travels along the x-direction
% - with an e-field excitation in -z-direction (evec=[0 0 -1])
% - the excitation is active and placed at x=start(1) ('ExcitePort', true)
% - a 50 Ohm lumped port resistance is placed at x=start(1) ('Feed_R', 50)
% - the width-direction is determined by the cross product of the
% direction of propagtion (dir='x') and the excitation vector
% (evec=[0 0 -1]), in this case it is the y-direction
% - the stripline-metal is created in a xy-plane at a height at z=start(3)
% --> The upper and lower reference plane (ground) must be defined by
% the user
%
% Thorsten Liebig <thorsten.liebig@gmx.de> (c) 2013
%
% See also InitCSX DefineRectGrid AddMetal AddMaterial AddExcitation calcPort
%% validate arguments %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%check mesh
if ~isfield(CSX,'RectilinearGrid')
error 'mesh needs to be defined! Use DefineRectGrid() first!';
end
if (~isfield(CSX.RectilinearGrid,'XLines') || ~isfield(CSX.RectilinearGrid,'YLines') || ~isfield(CSX.RectilinearGrid,'ZLines'))
error 'mesh needs to be defined! Use DefineRectGrid() first!';
end
% check dir
dir = DirChar2Int(dir);
% check evec
if ~(evec(1) == evec(2) == 0) && ~(evec(1) == evec(3) == 0) && ~(evec(2) == evec(3) == 0) || (sum(evec) == 0)
error 'evec must have exactly one component ~= 0'
end
evec0 = evec ./ sum(evec); % evec0 is a unit vector
%set defaults
feed_shift = 0;
feed_R = inf; %(default is open, no resitance)
excite = false;
measplanepos = nan;
PortNamePrefix = '';
excite_args = {};
%% read optional arguments %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for n=1:2:numel(varargin)
if (strcmp(varargin{n},'FeedShift')==1);
feed_shift = varargin{n+1};
if (numel(feed_shift)>1)
error 'FeedShift must be a scalar value'
end
elseif (strcmp(varargin{n},'Feed_R')==1);
feed_R = varargin{n+1};
if (numel(feed_shift)>1)
error 'Feed_R must be a scalar value'
end
elseif (strcmp(varargin{n},'MeasPlaneShift')==1);
measplanepos = varargin{n+1};
if (numel(feed_shift)>1)
error 'MeasPlaneShift must be a scalar value'
end
elseif (strcmp(varargin{n},'ExcitePort')==1);
if ischar(varargin{n+1})
warning('CSXCAD:AddMSLPort','depreceated: a string as excite option is no longer supported and will be removed in the future, please use true or false');
if ~isempty(excite)
excite = true;
else
excite = false;
end
else
excite = varargin{n+1};
end
elseif (strcmpi(varargin{n},'PortNamePrefix'))
PortNamePrefix = varargin{n+1};
else
excite_args{end+1} = varargin{n};
excite_args{end+1} = varargin{n+1};
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% normalize start and stop
nstart = min( [start;stop] );
nstop = max( [start;stop] );
% determine index (1, 2 or 3) of propagation (length of MSL)
idx_prop = dir + 1;
% determine index (1, 2 or 3) of width of MSL
dir = [0 0 0];
dir(idx_prop) = 1;
idx_width = abs(cross(dir,evec0)) * [1;2;3];
% determine index (1, 2 or 3) of height
idx_height = abs(evec0) * [1;2;3];
if (start(idx_height)~=stop(idx_height))
error('openEMS:AddStripLinePort','start/stop in height direction must be equal');
end
% direction of propagation
if stop(idx_prop)-start(idx_prop) > 0
direction = +1;
else
direction = -1;
end
% create the metal/material for the MSL
SL_start = start;
SL_stop = stop;
CSX = AddBox( CSX, materialname, prio, SL_start, SL_stop );
if isnan(measplanepos)
measplanepos = (nstart(idx_prop)+nstop(idx_prop))/2;
else
measplanepos = start(idx_prop)+direction*measplanepos;
end
% calculate position of the voltage probes
mesh{1} = sort(CSX.RectilinearGrid.XLines);
mesh{2} = sort(CSX.RectilinearGrid.YLines);
mesh{3} = sort(CSX.RectilinearGrid.ZLines);
meshlines = interp1( mesh{idx_prop}, 1:numel(mesh{idx_prop}), measplanepos, 'nearest' );
meshlines = mesh{idx_prop}(meshlines-1:meshlines+1); % get three lines (approx. at center)
if direction == -1
meshlines = fliplr(meshlines);
end
SL_w2 = interp1( mesh{idx_width}, 1:numel(mesh{idx_width}), (nstart(idx_width)+nstop(idx_width))/2, 'nearest' );
SL_w2 = mesh{idx_width}(SL_w2); % get e-line at center of MSL (SL_width/2)
v1_start(idx_prop) = meshlines(1);
v1_start(idx_width) = SL_w2;
v1_start(idx_height) = start(idx_height);
v1_stop = v1_start;
v1_stop(idx_height) = v1_start(idx_height);
v2_start = v1_start;
v2_stop = v1_stop;
v2_start(idx_prop) = meshlines(2);
v2_stop(idx_prop) = meshlines(2);
v3_start = v2_start;
v3_stop = v2_stop;
v3_start(idx_prop) = meshlines(3);
v3_stop(idx_prop) = meshlines(3);
height_vector = [0 0 0];
height_vector(idx_height) = height;
weight = 0.5;
% create the voltage-probes
port.U_filename{1,1} = [PortNamePrefix 'port_ut' num2str(portnr) 'A1'];
CSX = AddProbe( CSX, port.U_filename{1,1}, 0, 'weight', weight );
CSX = AddBox( CSX, port.U_filename{1,1}, prio, v1_start, v1_stop+height_vector);
port.U_filename{1,2} = [PortNamePrefix 'port_ut' num2str(portnr) 'A2'];
CSX = AddProbe( CSX, port.U_filename{1,2}, 0, 'weight', -1*weight );
CSX = AddBox( CSX, port.U_filename{1,2}, prio, v1_start, v1_stop-height_vector);
port.U_filename{2,1} = [PortNamePrefix 'port_ut' num2str(portnr) 'B1'];
CSX = AddProbe( CSX, port.U_filename{2,1}, 0, 'weight', weight );
CSX = AddBox( CSX, port.U_filename{2,1}, prio, v2_start, v2_stop+height_vector );
port.U_filename{2,2} = [PortNamePrefix 'port_ut' num2str(portnr) 'B2'];
CSX = AddProbe( CSX, port.U_filename{2,2}, 0, 'weight', -1*weight );
CSX = AddBox( CSX, port.U_filename{2,2}, prio, v2_start, v2_stop-height_vector );
port.U_filename{3,1} = [PortNamePrefix 'port_ut' num2str(portnr) 'C1'];
CSX = AddProbe( CSX, port.U_filename{3,1}, 0, 'weight', weight );
CSX = AddBox( CSX, port.U_filename{3,1}, prio, v3_start, v3_stop+height_vector );
port.U_filename{3,2} = [PortNamePrefix 'port_ut' num2str(portnr) 'C2'];
CSX = AddProbe( CSX, port.U_filename{3,2}, 0, 'weight', weight );
CSX = AddBox( CSX, port.U_filename{3,2}, prio, v3_start, v3_stop-height_vector );
% calculate position of the current probes
idx = interp1( mesh{idx_width}, 1:numel(mesh{idx_width}), nstart(idx_width), 'nearest' );
i1_start(idx_width) = mesh{idx_width}(idx) - diff(mesh{idx_width}(idx-1:idx))/2;
idx = interp1( mesh{idx_height}, 1:numel(mesh{idx_height}), start(idx_height), 'nearest' );
i1_start(idx_height) = mesh{idx_height}(idx-1) - diff(mesh{idx_height}(idx-2:idx-1))/2;
i1_stop(idx_height) = mesh{idx_height}(idx+1) + diff(mesh{idx_height}(idx+1:idx+2))/2;
i1_start(idx_prop) = sum(meshlines(1:2))/2;
i1_stop(idx_prop) = i1_start(idx_prop);
idx = interp1( mesh{idx_width}, 1:numel(mesh{idx_width}), nstop(idx_width), 'nearest' );
i1_stop(idx_width) = mesh{idx_width}(idx) + diff(mesh{idx_width}(idx:idx+1))/2;
i2_start = i1_start;
i2_stop = i1_stop;
i2_start(idx_prop) = sum(meshlines(2:3))/2;
i2_stop(idx_prop) = i2_start(idx_prop);
% create the curr-probes
weight = direction;
port.I_filename{1} = [PortNamePrefix 'port_it' num2str(portnr) 'A'];
CSX = AddProbe( CSX, port.I_filename{1}, 1, 'weight', weight );
CSX = AddBox( CSX, port.I_filename{1}, prio, i1_start, i1_stop );
port.I_filename{2} = [PortNamePrefix 'port_it' num2str(portnr) 'B'];
CSX = AddProbe( CSX, port.I_filename{2}, 1,'weight', weight );
CSX = AddBox( CSX, port.I_filename{2}, prio, i2_start, i2_stop );
% create port structure
port.LengthScale = 1;
if ((CSX.ATTRIBUTE.CoordSystem==1) && (idx_prop==2))
port.LengthScale = SL_stop(idx_height);
end
port.nr = portnr;
port.type = 'StripLine';
port.drawingunit = CSX.RectilinearGrid.ATTRIBUTE.DeltaUnit;
port.v_delta = diff(meshlines)*port.LengthScale;
port.i_delta = diff( meshlines(1:end-1) + diff(meshlines)/2 )*port.LengthScale;
port.direction = direction;
port.excite = 0;
port.measplanepos = abs(v2_start(idx_prop) - start(idx_prop))*port.LengthScale;
% port
% create excitation (if enabled) and port resistance
meshline = interp1( mesh{idx_prop}, 1:numel(mesh{idx_prop}), start(idx_prop) + feed_shift*direction, 'nearest' );
ex_start(idx_prop) = mesh{idx_prop}(meshline) ;
ex_start(idx_width) = nstart(idx_width);
ex_start(idx_height) = nstart(idx_height);
ex_stop(idx_prop) = ex_start(idx_prop);
ex_stop(idx_width) = nstop(idx_width);
ex_stop(idx_height) = nstop(idx_height);
port.excite = 0;
if excite
port.excite = 1;
CSX = AddExcitation( CSX, [PortNamePrefix 'port_excite_1_' num2str(portnr)], 0, evec, excite_args{:} );
CSX = AddBox( CSX, [PortNamePrefix 'port_excite_1_' num2str(portnr)], prio, ex_start, ex_stop+height_vector );
CSX = AddExcitation( CSX, [PortNamePrefix 'port_excite_2_' num2str(portnr)], 0, -evec, excite_args{:} );
CSX = AddBox( CSX, [PortNamePrefix 'port_excite_2_' num2str(portnr)], prio, ex_start, ex_stop-height_vector );
end
%% MSL resitance at start of MSL line
ex_start(idx_prop) = start(idx_prop);
ex_stop(idx_prop) = ex_start(idx_prop);
if (feed_R > 0) && ~isinf(feed_R)
CSX = AddLumpedElement( CSX, [PortNamePrefix 'port_resist_' int2str(portnr)], idx_height-1, 'R', 2*feed_R );
CSX = AddBox( CSX, [PortNamePrefix 'port_resist_' int2str(portnr)], prio, ex_start, ex_stop+height_vector );
CSX = AddBox( CSX, [PortNamePrefix 'port_resist_' int2str(portnr)], prio, ex_start, ex_stop-height_vector );
elseif isinf(feed_R)
% do nothing --> open port
elseif feed_R == 0
%port "resistance" as metal
CSX = AddBox( CSX, materialname, prio, ex_start, ex_stop+height_vector );
CSX = AddBox( CSX, materialname, prio, ex_start, ex_stop-height_vector );
else
error('openEMS:AddMSLPort','MSL port with resitance <= 0 it not possible');
end
end

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@ -50,7 +50,7 @@ if isempty(port)
return;
end
if (strcmpi(port.type,'MSL') || strcmpi(port.type,'Coaxial'))
if (strcmpi(port.type,'MSL') || strcmpi(port.type,'Coaxial') || strcmpi(port.type,'StripLine'))
port = calcTLPort( port, SimDir, f, varargin{:});
return
elseif strcmpi(port.type,'WaveGuide')

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@ -48,7 +48,7 @@ if (iscell(port))
return;
end
if ((strcmpi(port.type,'MSL')~=1) && (strcmpi(port.type,'Coaxial')~=1))
if ((strcmpi(port.type,'MSL')~=1) && (strcmpi(port.type,'Coaxial')~=1) && (strcmpi(port.type,'StripLine')~=1))
error('openEMS:calcTLPort','error, type is not a transmission line port');
end
@ -82,8 +82,18 @@ for n=1:2:numel(varargin)
end
end
if (strcmpi(port.type,'StripLine')==1)
U1 = ReadUI( port.U_filename(:,1), SimDir, f, UI_args{:} );
U2 = ReadUI( port.U_filename(:,1), SimDir, f, UI_args{:} );
U = U1;
for n=1:3
U.TD{n}.val = U1.TD{n}.val+U2.TD{n}.val;
U.FD{n}.val = U1.FD{n}.val+U2.FD{n}.val;
end
else
U = ReadUI( port.U_filename, SimDir, f, UI_args{:} );
end
% read time domain data (multiples files)
U = ReadUI( port.U_filename, SimDir, f, UI_args{:} );
I = ReadUI( port.I_filename, SimDir, f, UI_args{:} );
% store the original frequency domain waveforms

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@ -0,0 +1,91 @@
% example demonstrating the use of a stripline terminated by a resistance
% (c) 2013 Thorsten Liebig
close all
clear
clc
%% setup the simulation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
physical_constants;
unit = 1e-6; % specify everything in um
SL_length = 50000;
SL_width = 520;
SL_height = 500;
substrate_thickness = SL_height;
substrate_epr = 3.66;
f_max = 7e9;
Air_Spacer = 20000;
%% setup FDTD parameters & excitation function %%%%%%%%%%%%%%%%%%%%%%%%%%%%
FDTD = InitFDTD();
FDTD = SetGaussExcite( FDTD, f_max/2, f_max/2 );
BC = {'MUR' 'MUR' 'MUR' 'MUR' 'MUR' 'MUR'};
FDTD = SetBoundaryCond( FDTD, BC );
%% setup CSXCAD geometry & mesh %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
CSX = InitCSX();
resolution = c0/(f_max*sqrt(substrate_epr))/unit /50; % resolution of lambda/50
mesh.x = SmoothMeshLines( [-SL_length/2 0 SL_length/2], resolution, 1.5 ,0 );
mesh.y = SmoothMeshLines( [0 SL_width/2+[-resolution/3 +resolution/3*2]/4], resolution/4 , 1.5 ,0);
mesh.y = SmoothMeshLines( [-10*SL_width -mesh.y mesh.y 10*SL_width], resolution, 1.3 ,0);
mesh.z = linspace(0,substrate_thickness,5);
mesh.z = sort(unique([mesh.z -mesh.z]));
%% substrate
CSX = AddMaterial( CSX, 'RO4350B' );
CSX = SetMaterialProperty( CSX, 'RO4350B', 'Epsilon', substrate_epr );
start = [mesh.x(1), mesh.y(1), mesh.z(1)];
stop = [mesh.x(end), mesh.y(end), mesh.z(end)];
CSX = AddBox( CSX, 'RO4350B', 0, start, stop );
%% add air spacer
mesh.x = [mesh.x mesh.x(1)-Air_Spacer mesh.x(end)+Air_Spacer];
mesh.y = [mesh.y mesh.y(1)-Air_Spacer mesh.y(end)+Air_Spacer];
mesh.z = [mesh.z mesh.z(1)-Air_Spacer mesh.z(end)+Air_Spacer];
mesh = SmoothMesh(mesh, c0/f_max/unit/20);
CSX = DefineRectGrid( CSX, unit, mesh );
%% SL port
CSX = AddMetal( CSX, 'PEC' );
portstart = [ -SL_length/2, -SL_width/2, 0];
portstop = [ 0, SL_width/2, 0];
[CSX,port{1}] = AddStripLinePort( CSX, 999, 1, 'PEC', portstart, portstop, SL_height, 'x', [0 0 -1], 'ExcitePort', true, 'Feed_R', 50, 'MeasPlaneShift', SL_length/3);
portstart = [+SL_length/2, -SL_width/2, 0];
portstop = [0 , SL_width/2, 0];
[CSX,port{2}] = AddStripLinePort( CSX, 999, 2, 'PEC', portstart, portstop, SL_height, 'x', [0 0 -1], 'MeasPlaneShift', SL_length/3, 'Feed_R', 50);
% bottom PEC plane
CSX = AddBox(CSX, 'PEC', 999, [-SL_length/2 -10*SL_width -SL_height],[+SL_length/2 +10*SL_width -SL_height]);
% top PEC plane
CSX = AddBox(CSX, 'PEC', 999, [-SL_length/2 -10*SL_width SL_height],[+SL_length/2 +10*SL_width SL_height]);
%% write/show/run the openEMS compatible xml-file
Sim_Path = ['tmp_' mfilename];
Sim_CSX = 'stripline.xml';
[status, message, messageid] = rmdir( Sim_Path, 's' ); % clear previous directory
[status, message, messageid] = mkdir( Sim_Path ); % create empty simulation folder
WriteOpenEMS( [Sim_Path '/' Sim_CSX], FDTD, CSX );
CSXGeomPlot( [Sim_Path '/' Sim_CSX] );
RunOpenEMS( Sim_Path, Sim_CSX );
%% post-processing
close all
f = linspace( 1e6, f_max, 1601 );
port = calcPort( port, Sim_Path, f, 'RefImpedance', 50);
s11 = port{1}.uf.ref./ port{1}.uf.inc;
s21 = port{2}.uf.ref./ port{1}.uf.inc;
plot(f/1e9,20*log10(abs(s11)),'k-','LineWidth',2);
hold on;
grid on;
plot(f/1e9,20*log10(abs(s21)),'r--','LineWidth',2);
legend('S_{11}','S_{21}');
ylabel('S-Parameter (dB)','FontSize',12);
xlabel('frequency (GHz) \rightarrow','FontSize',12);
ylim([-50 2]);

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@ -0,0 +1,78 @@
% example demonstrating the use of a stripline terminated by the pml
% (c) 2013 Thorsten Liebig
close all
clear
clc
%% setup the simulation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
physical_constants;
unit = 1e-6; % specify everything in um
SL_length = 50000;
SL_width = 520;
SL_height = 500;
substrate_thickness = SL_height;
substrate_epr = 3.66;
f_max = 7e9;
%% setup FDTD parameters & excitation function %%%%%%%%%%%%%%%%%%%%%%%%%%%%
FDTD = InitFDTD();
FDTD = SetGaussExcite( FDTD, f_max/2, f_max/2 );
BC = {'PML_8' 'PML_8' 'PMC' 'PMC' 'PEC' 'PEC'};
FDTD = SetBoundaryCond( FDTD, BC );
%% setup CSXCAD geometry & mesh %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
CSX = InitCSX();
resolution = c0/(f_max*sqrt(substrate_epr))/unit /50; % resolution of lambda/50
mesh.x = SmoothMeshLines( [-SL_length/2 0 SL_length/2], resolution, 1.5 ,0 );
mesh.y = SmoothMeshLines( [0 SL_width/2+[-resolution/3 +resolution/3*2]/4], resolution/4 , 1.5 ,0);
mesh.y = SmoothMeshLines( [-10*SL_width -mesh.y mesh.y 10*SL_width], resolution, 1.3 ,0);
mesh.z = linspace(0,substrate_thickness,5);
mesh.z = sort(unique([mesh.z -mesh.z]));
CSX = DefineRectGrid( CSX, unit, mesh );
%% substrate
CSX = AddMaterial( CSX, 'RO4350B' );
CSX = SetMaterialProperty( CSX, 'RO4350B', 'Epsilon', substrate_epr );
start = [mesh.x(1), mesh.y(1), mesh.z(1)];
stop = [mesh.x(end), mesh.y(end), mesh.z(end)];
CSX = AddBox( CSX, 'RO4350B', 0, start, stop );
%% SL port
CSX = AddMetal( CSX, 'PEC' );
portstart = [ mesh.x(1), -SL_width/2, 0];
portstop = [ 0, SL_width/2, 0];
[CSX,port{1}] = AddStripLinePort( CSX, 999, 1, 'PEC', portstart, portstop, SL_height, 'x', [0 0 -1], 'ExcitePort', true, 'FeedShift', 10*resolution, 'MeasPlaneShift', SL_length/3);
portstart = [mesh.x(end), -SL_width/2, 0];
portstop = [0 , SL_width/2, 0];
[CSX,port{2}] = AddStripLinePort( CSX, 999, 2, 'PEC', portstart, portstop, SL_height, 'x', [0 0 -1], 'MeasPlaneShift', SL_length/3 );
%% write/show/run the openEMS compatible xml-file
Sim_Path = ['tmp_' mfilename];
Sim_CSX = 'stripline.xml';
[status, message, messageid] = rmdir( Sim_Path, 's' ); % clear previous directory
[status, message, messageid] = mkdir( Sim_Path ); % create empty simulation folder
WriteOpenEMS( [Sim_Path '/' Sim_CSX], FDTD, CSX );
CSXGeomPlot( [Sim_Path '/' Sim_CSX] );
RunOpenEMS( Sim_Path, Sim_CSX );
%% post-processing
close all
f = linspace( 1e6, f_max, 1601 );
port = calcPort( port, Sim_Path, f, 'RefImpedance', 50);
s11 = port{1}.uf.ref./ port{1}.uf.inc;
s21 = port{2}.uf.ref./ port{1}.uf.inc;
plot(f/1e9,20*log10(abs(s11)),'k-','LineWidth',2);
hold on;
grid on;
plot(f/1e9,20*log10(abs(s21)),'r--','LineWidth',2);
legend('S_{11}','S_{21}');
ylabel('S-Parameter (dB)','FontSize',12);
xlabel('frequency (GHz) \rightarrow','FontSize',12);
ylim([-50 2]);