How to unwrap the phase of a complex electrical impedance signal

Hi everyone,

I have a problem with unwrapping a phase. The real and imaginary parts of the phase are named as 'real_part' and 'imaginary_part, respectively, and attached to this question. When I plot the phase using atan2 function, I see a big jump in phase data. I tried to unwrap it using Matlab's unwrap function but it also did not work. The results are as shown:

Here is the code that I used:

phase = atan2(imaginary_part, real_part) * 180/pi;

phase_unwrapped = unwrap(atan2(imaginary_part, real_part)) * 180/pi;

fig_Phase = figure('Visible', true, 'Units', 'inches', 'Position', [0 0 20 20]);

subplot(1,2,1);semilogx(Frequency,phase,'b','LineWidth',2, 'DisplayName', 'Wrapped Phase');hold on

axis square; legend();

xlabel('Frequency (Hz)');

ylabel('Phase (degree)');

set(gca,'fontsize',28);box on;

xticks([1e0 1e1 1e2 1e3 1e4 1e5 1e6]);

subplot(1,2,2);semilogx(Frequency,phase_unwrapped,'r','LineWidth',2, 'DisplayName', 'Unwrapped Phase');

axis square; legend();

xlabel('Frequency (Hz)');

ylabel('Phase (degree)');

set(gca,'fontsize',28);box on;

xticks([1e0 1e1 1e2 1e3 1e4 1e5 1e6]);

hold off;

Could anyone help me fix this problem? The phase cannot be less than -90 degrees since this is a measuremental data. The phase jump occurs due to some simple operations.

[David Goodmanson]

Hello Easa, is there anything that you do not like about the unwrapped result, other than your thought that the phase should not be less than -90? At the low end, the phase seems to be making an effort to stay above -270, which is the same as staying above +90.

[Easa AliAbbasi]

Dear David, thank you for the reply. Please see the following plot where I used atan for plotting the phase:

As you can see, the phase reaches -80 and then jumps to +90. Since this phase is for a physical measurement (a combination of resistor and capacitor), it cannot be less than -90 or even a positive value. It must be between 0 and -90. I am trying to remove the jump and attach the phase value at 50 Hz to its value at 100 kHz to have a smooth phase graph like the following:

[Mathieu NOE]

I wonder how the data has been measured

frequency vector is not supplied so I figured out from the figures

the code seems fine to me but I have a doubt about the data . neither the modulus or the phase ressembles a simple RC low pass (first order) transfer function (if the data IS a transfer function)

at higher frequencies, the modulus roll of rate is about 20 dB/decade which indicates a first order filter but the phase does not match the modulus (phase is related to the modulus slope)

the phase reaches -270 ° where it should normally only reach - 90° so I suspect the reason is not in the code and certainlynot because of atan2 or atan functions.

have you made sure that your test equipement is calibrated (no extra phase roll due to the equipment itself ?)

Frequency = logspace(0,6,100);
phase_rad = atan2(imaginary_part, real_part);
phase_deg = phase_rad * 180/pi;
phase_deg_unwrapped = unwrap(phase_rad) * 180/pi;

mod_dB = 10*log10(real_part.^2 + imaginary_part.^2); % modulus in dB


figure(1)
subplot(2,1,1);semilogx(Frequency,mod_dB,'b','LineWidth',2, 'DisplayName', 'modulus in dB');
xlabel('Frequency (Hz)');
ylabel('Modulus (dB)');
set(gca,'fontsize',15);box on;
subplot(2,1,2);semilogx(Frequency,phase_deg,'b','LineWidth',2, 'DisplayName', 'Wrapped Phase');
xlabel('Frequency (Hz)');
ylabel('Phase (degree)');
set(gca,'fontsize',15);box on;

figure(2)
subplot(2,1,1);semilogx(Frequency,mod_dB,'b','LineWidth',2, 'DisplayName', 'modulus in dB');
xlabel('Frequency (Hz)');
ylabel('Modulus (dB)');
set(gca,'fontsize',15);box on;
subplot(2,1,2);semilogx(Frequency,phase_deg_unwrapped,'r','LineWidth',2, 'DisplayName', 'Unwrapped Phase');
xlabel('Frequency (Hz)');
ylabel('Phase (degree)');
set(gca,'fontsize',15);box on;

[Easa AliAbbasi]

Dear @Mathieu NOE,

Thank you for the response. This data is not a direct measurement. Three impedances are measured and after some mathematical operations, this impedance is calculated. All measurements are performed carefully and the measurement device was calibrated before every meaasurement. Moreover, the measurements are repeated 10 times and their mean values are used for the mathematical operations.

[Mathieu NOE]

hello again

maybe the error is somewhere in the intermediate computations - have you somehow checked each individual measurement vs theory ?

Answer 1

[Sayan]

Hi Easa AliAbbasi,

I understand from your issue that the "Phase" plot of your readings are not as expected.

I do not think there is any issue with the code.However,you mentioned in the comments that the "real_part" of the reading is of the resistance and the "imaginary_part" is of the capacitance reactance value, which implies that the "real_part" should always be positive and the "imaginary_part" should always be negative. I have these two observations about the data.

• From the 29th row of "real_part.mat" the value becomes negetive, but the resistance value cannot be negetive.
• From the 45th row of the "imaginary_part.mat" the value becomes positive, but the capacitive reactance value cannot be positive.

Those are the reasons that I think the issue is with the measured data. Whenever there is a change of sign in the data, the "atan2" and "unwrap" functions provide unexpected results.

You can find more information on "atan2" and "unwrap" in the following documentations.

• "atan2" : https://www.mathworks.com/help/releases/R2021b/matlab/ref/atan2.html#buct8h0-4
• "unwrap" : https://www.mathworks.com/help/releases/R2021b/matlab/ref/unwrap.html#mw_1884ee59-182e-4ef6-a323-ee5a45b76693

Hope this helps in resolving the issue.