How to find the Transfer function of a Simulink output plot?

Hello My friends

I have a lot of figures and I want to find the transfer functions as I=Imax cos(ωt+ϴ) and V=Vmax cos(ωt+ϴ)

here is the values of V and I

Can you Advise?

I attached one of my plots.

I really appreciate your assistance.

Answer 1

[Sam Chak]

Hi @Atefeh

Think you want to apply the curve-fitting method to fit the sinusoidal model to the time-series data.

load('V_I.mat')
Vtime  = Vabc3.Time;
Vdat1  = Vabc3.Data(:,1);
ta     = Vtime(1:1001);
Va     = Vdat1(1:1001);
tb     = Vtime(1002:2100);
Vb     = Vdat1(1002:2100);
tc     = Vtime(2101:3001);
Vc     = Vdat1(2101:3001);

figure(1)
plot(Vtime, Vdat1), ylim([-1.5 1.5]), grid on
xlabel('Time (seconds)'), ylabel('V_{abc}')

[Vafit, gof1] = fit(ta, Va, 'sin1')

Vafit = 
     General model Sin1:
     Vafit(x) =  a1*sin(b1*x+c1)
     Coefficients (with 95% confidence bounds):
       a1 =       0.998  (0.998, 0.998)
       b1 =       314.2  (314.2, 314.2)
       c1 =       1.031  (1.031, 1.031)

gof1 = struct with fields:
           sse: 4.5659e-13
       rsquare: 1.0000
           dfe: 998
    adjrsquare: 1.0000
          rmse: 2.1389e-08

figure(2)
plot(Vafit, ta, Va), ylim([-1.5 1.5]), grid on
xlabel('Time (seconds)'), ylabel('V_{a}')

[Vbfit, gof1] = fit(tb, Vb, 'sin1')

Vbfit = 
     General model Sin1:
     Vbfit(x) =  a1*sin(b1*x+c1)
     Coefficients (with 95% confidence bounds):
       a1 =      0.1984  (0.1983, 0.1985)
       b1 =       314.2  (314.1, 314.2)
       c1 =       7.268  (7.266, 7.27)

gof1 = struct with fields:
           sse: 8.6937e-04
       rsquare: 1.0000
           dfe: 1096
    adjrsquare: 1.0000
          rmse: 8.9063e-04

figure(3)
plot(Vbfit, tb, Vb), ylim([-0.3 0.3]), grid on
xlabel('Time (seconds)'), ylabel('V_{b}')

[Vcfit, gof1] = fit(tc, Vc, 'sin1')

Vcfit = 
     General model Sin1:
     Vcfit(x) =  a1*sin(b1*x+c1)
     Coefficients (with 95% confidence bounds):
       a1 =      0.9971  (0.9958, 0.9984)
       b1 =       314.2  (314.1, 314.2)
       c1 =       7.304  (7.291, 7.317)

gof1 = struct with fields:
           sse: 0.1770
       rsquare: 0.9996
           dfe: 898
    adjrsquare: 0.9996
          rmse: 0.0140

figure(4)
plot(Vcfit, tc, Vc), ylim([-1.5 1.5]), grid on
xlabel('Time (seconds)'), ylabel('V_{c}')

[Atefeh]

Dear Sam

Is it possible to find the transfer function of a sinosidal wave without separating it to 3 parts?

[Sam Chak]

Hi @Atefeh

Do you mean a single mathematical function that describes all 3 sine waves?

[Atefeh]

Yes, Is this possible/feasobale?

Answer 2

[Sam Chak]

Hi @Atefeh

I'm not an expert, but I have friends who are, and one of them said this is possible with this math function:

where

, and .

If you find the proposed math function useful, don't forget voting 👍 for the answer.

load('V_I.mat')
Vtime  = Vabc3.Time;
Vdat1  = Vabc3.Data(:,1);
ta     = Vtime(1:1001);
Va     = Vdat1(1:1001);
tb     = Vtime(1002:2100);
Vb     = Vdat1(1002:2100);
tc     = Vtime(2101:3001);
Vc     = Vdat1(2101:3001);

subplot(211)
plot(Vtime, Vdat1, 'color', '#528AFA'), ylim([-1.5 1.5]), grid on
title('Plot from Data')
xlabel('Time (seconds)'), ylabel('V_{abc}')

[Vafit, gof1] = fit(ta, Va, 'sin1');
vaAmp = Vafit.a1; vaFreq = Vafit.b1; vaPhi = Vafit.c1;
[Vbfit, gof1] = fit(tb, Vb, 'sin1');
vbAmp = Vbfit.a1; vbFreq = Vbfit.b1; vbPhi = Vbfit.c1;
[Vcfit, gof1] = fit(tc, Vc, 'sin1');
vcAmp = Vcfit.a1; vcFreq = Vcfit.b1; vcPhi = Vcfit.c1;

t  = linspace(0, 0.3, 3001);
t1 = 0.1;
t2 = 0.21;

Vaf = vaAmp*sin(vaFreq*t + vaPhi);
Vbf = vbAmp*sin(vbFreq*t + vbPhi);
Vcf = vcAmp*sin(vcFreq*t + vcPhi);

fun1 = sign(t - t1).*(Vaf/4 - Vbf/4); % sigma1
fun2 = Vaf/4 + Vbf/4 + Vcf/2 - fun1;
fun3 = Vaf/4 + Vbf/4 - Vcf/2 - fun1;
f      = fun2 - sign(t - t2).*fun3;     % <-- this math function

subplot(212)
plot(t, f, 'color', '#FA477A'), ylim([-1.5 1.5]), grid on
title('Plot using Math function')
xlabel('Time (seconds)'), ylabel('V')

[Atefeh]

Dear Sam

I really appreciate your time.It was so useful to me.

Wish you all the best

Atefeh

[Sam Chak]

You are welcome, @Atefeh. Have a nice weekend!

[Atefeh]

Dear Sam

Would you mind telling me the reference of the attached formulas?

[Sam Chak]

Hi @Atefeh

The provided signum-based formula does describe the behavior of the piecewise function. However, I find it unnecessary to cite the formula, as it involves a clever but unintuitive manipulation of the sign functions to separate the sub-functions at different intervals in the domain.

Moreover, I have provided a more intuitively understandable formula in my Answer below. If you still wish to cite, consider referencing the special function used in the formulation, acknowledging the mathematical properties of that function.

Answer 3

[Sam Chak]

Hi @Atefeh

Given the piecewise function defined for

I propose a more intuitively understandable Heaviside-based formula, which I affectionately refer to as the Piecewise Function Put Together (PFPT) formula:

with .

In this formulation, the 1st term represents throughout the entire range of x. The 2nd term is activated by the Heaviside step function when x equals , plotting and canceling out from ​ onwards. Upon reaching , the Heaviside function triggers the 3rd term, plotting and canceling out from onwards.

If you appreciate the explanation of the PFPT formula, don't forget to vote 👍 for the answer as a token of appreciation.

load('V_I.mat')
Vtime  = Vabc3.Time;
Vdat1  = Vabc3.Data(:,1);
ta     = Vtime(1:1001);
Va     = Vdat1(1:1001);
tb     = Vtime(1002:2100);
Vb     = Vdat1(1002:2100);
tc     = Vtime(2101:3001);
Vc     = Vdat1(2101:3001);

subplot(211)
plot(Vtime, Vdat1, 'color', '#528AFA'), ylim([-1.5 1.5]), grid on
title('Plot from Data')
xlabel('Time (seconds)'), ylabel('V_{abc}')

[Vafit, gof1] = fit(ta, Va, 'sin1');
vaAmp = Vafit.a1; vaFreq = Vafit.b1; vaPhi = Vafit.c1;
[Vbfit, gof1] = fit(tb, Vb, 'sin1');
vbAmp = Vbfit.a1; vbFreq = Vbfit.b1; vbPhi = Vbfit.c1;
[Vcfit, gof1] = fit(tc, Vc, 'sin1');
vcAmp = Vcfit.a1; vcFreq = Vcfit.b1; vcPhi = Vcfit.c1;

t  = linspace(0, 0.3, 3001);
t1 = 0.1;
t2 = 0.21;

Vaf = vaAmp*sin(vaFreq*t + vaPhi);  % f1, function at the 1st interval, t  < t1
Vbf = vbAmp*sin(vbFreq*t + vbPhi);  % f2, function at the 2nd interval, t1 < t < t2
Vcf = vcAmp*sin(vcFreq*t + vcPhi);  % f3, function at the 3rd interval, t2 < t

%% Old Piecewise Function formula
% fun1 = sign(t - t1).*(Vaf/4 - Vbf/4); % sigma1
% fun2 = Vaf/4 + Vbf/4 + Vcf/2 - fun1;
% fun3 = Vaf/4 + Vbf/4 - Vcf/2 - fun1;
% fold = fun2 - sign(t - t2).*fun3;     % <-- this math function

%% New Piecewise Function Put Together (PFPT) formula
fnew = Vaf + (Vbf - Vaf).*heaviside(t - t1) + (Vcf - Vbf).*heaviside(t - t2);

subplot(212)
plot(t, fnew, 'color', '#FA477A'), ylim([-1.5 1.5]), grid on
title('Plot using PFPT formula')
xlabel('Time (seconds)'), ylabel('V')

[Atefeh]

Dear Sam

I really appreciate your time, it was more understandable.

Thanks a bunch