RF Toolbox 2.6

Modeling a High-Speed Backplane (Part 3: Rational Function Model to a Simulink® Model)

This demo shows how to use Simulink® to simulate a differential high-speed backplane channel. The demo first reads a Touchstone® data file that contains single-ended 4-port S-parameters for a differential high-speed backplane and converts them to 2-port differential S-parameters. It computes the transfer function of the differential circuit and uses the RATIONALFIT function to fit a closed-form rational function model to the circuit's transfer function. Then, the demo converts the poles and residues of the rational function model into the numerators and denominators of the Laplace Transform S-Domain transfer functions that it uses to build the Simulink model of the rational function.

To run this demo, you must have Simulink installed.

Contents

Read the Single-Ended 4-Port S-Parameters and Convert Them to Differential 2-Port S-Parameters

Read a Touchstone data file into an RFDATA.DATA object using the object's READ method. The parameters in this data file are the 50-ohm S-parameters of a single-ended 4-port passive circuit, measured at 1496 frequencies ranging from 50 MHz to 15 GHz. Then, get the single-ended 4-port S-parameters from the data object, and use the matrix conversion function S2SDD to convert them to differential 2-port S-parameters.

FileName = 'default.s4p';
SingleEndedData = read(rfdata.data, FileName);
SingleEndedSparams = SingleEndedData.S_Parameters;
Freq = SingleEndedData.Freq;
DifferentialSparams = s2sdd(SingleEndedSparams);

Compute the Transfer Function and Its Rational Function Representation

First, use the S2TF function to compute the differential transfer function. Then, use the RATIONALFIT function to compute the closed form of the transfer function and store it in an RFMODEL.RATIONAL object. The RATIONALFIT function fits a rational function model to the specified data over the specified frequencies.

DifferentialTransFunc = s2tf(DifferentialSparams);
FittingTolerance = -30;          % Rational fitting tolerance in dB
DelayFactor = 0.9;               % Delay factor
RationalFuncModel = ...
    rationalfit(Freq, DifferentialTransFunc, FittingTolerance, [], DelayFact
or)
nPoles = length(RationalFuncModel.A);
disp(sprintf('The derived rational function contains %d poles.', nPoles));

RationalFuncModel =

     Name: 'Rational Function'
        A: [22x1 double]
        C: [22x1 double]
        D: 0
    Delay: 6.0172e-009

The derived rational function contains 22 poles.

Get the Numerator and Denominator of the Laplace Transform S-Domain Transfer Functions

This demo uses Laplace Transform S-Domain transfer functions to represent the backplane in the Simulink model. Convert the poles and corresponding residues of the rational function model into numerator and denominator form for use in the Laplace Transform transfer function blocks. Each transfer function block represents either one real pole and the corresponding real residue, or a pair of complex conjugate poles and residues, so the transfer function block always has real coefficients. For this demo, the rational function model contains 2 real poles/residues and 10 pairs of complex poles/residues, so the Simulink model contains 12 transfer function blocks.

A = RationalFuncModel.A; C = RationalFuncModel.C;
den = cell(size(A)); num = cell(size(A));
k = 1;                          % Index of poles and residues
n = 0;                          % Index of numerators and denominators
while k <= nPoles
    if isreal(A(k))             % Real poles
        n = n + 1;
        num{n} = C(k);
        den{n} = [1, -A(k)];
        k = k + 1;
    else                        % Complex poles
        n = n + 1;
        real_a = real(A(k)); imag_a = imag(A(k));
        real_c = real(C(k)); imag_c = imag(C(k));
        num{n} = [2*real_c, -2*(real_a*real_c+imag_a*imag_c)];
        den{n} = [1, -2*real_a, real_a^2+imag_a^2];
        k = k + 2;
    end
end
den = den(1:n); num = num(1:n);

Build the Simulink Model of the Backplane

Build a Simulink model of the backplane using the Laplace Transform transfer functions. Then, connect a random source to the input of the backplane and a scope to its input and output.

ModelName = fliplr(strtok(fliplr(tempname), filesep));
simulink_rfmodel_build_rational_system_helper(ModelName , numel(num))
simulink_rfmodel_add_source_sink_helper(ModelName)

Figure 1. Simulink model for a rational function

Simulate the Simulink Model of the Rational Function

When you simulate the model, the Scope shows the impact of the differential backplane on the random input signal.

set_param([ModelName,'/Rational Model Output'], 'Open', 'on')
set(findall(0, 'Type', 'Figure', 'Name', 'Rational Model Output'), ...
    'Position', [200, 216, 901, 442])
sim(ModelName);

Close the Model

close_system(ModelName, 0)