From Impedance Measurements Across Five Frequencies to Approximate
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From Impedance across five frequencies to approximate the true resistance and capacitance using Cole-Cole plot extrapolation I. Background and Questions The impedance of an R e //(R i -C m ) circuit model is measured by using alternative current sources with five different frequencies (12KHz, 20KHz, 50KHz, 100KHz and 200KHz) and the induced voltage. Thus, we will have impedance: Z 12K , Z 20K , Z 50K , Z 100K , Z 200K , and the associated phase: φ 12K , φ 20K, φ 50K , φ 100K , φ 200K . The question is from the measurements of Z freq and φ freq , how to find or approximate the true values of R e , R i and C m ? II. Approach Direct Current (Frequency=0Hz) - It’s easy to see that with Direct Current (freq = 0 Hz), the reactance, X cm , is infinity, thus, an open
Transcript
From Impedance across five frequencies to approximate the true resistance and capacitance using Cole-Cole plot extrapolation
I. Background and Questions
The impedance of an Re//(Ri-Cm) circuit model is measured by using alternative current sources with five different frequencies (12KHz, 20KHz, 50KHz, 100KHz and 200KHz) and the induced voltage.
Thus, we will have impedance: Z12K, Z20K, Z50K, Z100K, Z200K, and the associated phase: φ12K, φ20K, φ50K, φ100K, φ200K. The question is from the measurements of Zfreq and φfreq, how to find or approximate the true values of Re, Ri and Cm?
II. Approach
Direct Current (Frequency=0Hz) - It’s easy to see that with Direct Current (freq = 0 Hz), the reactance, Xcm, is infinity, thus, an open circuit. The circuit then reduces to a simple single resistor (Re) circuit, so all current will flow through Re, and the impedance measured will be purely resistance of Re, i.e. Z0Hz = Re.
Alternative Current when frequency approach to infinity –It’s easy to see that with infinity frequency current, the reactance, Xcm, becomes zero. That means a short circuit replaces Cm. The circuit then reduces
to a simple parallel resistor (Re//Ri) circuit. Therefore, the current will flow through both Re and Ri, so Zinf = (Re//Ri) = (1/Re + 1/Ri)-1 = (Re*Ri)/(Re+Ri).Therefore, if Z0Hz and Zinf can be measured, the frequency independent components, i.e. Re and Ri can be calculated:Z0Hz = Re, Zinf = (1/Re + 1/Ri)-1 ,So, Ri can be approximated by (Zinf
-1 – Z0Hz-1)-1 .
Alternative Current when frequency is between 0Hz and infinity – The impedance written in rectangular form is:Zfreq = [(WCm)2ReRi(Re+Ri)+Re]/ [(WCm)2(Re+Ri)+1] + j * -[WCmRe
2]/ [(WCm)2(Re+Ri)+1], when W is angular frequency = 2*pi*freq.When W = 0, Zfreq reduces to Re; when W=inf, Zfreq reduces to (Re*Ri)/(Re+Ri).
The above Zfreq equation can be plotted with the real part on x-axis and the negative of the imaginary part on y-axis across different angular frequency (W), known as Cole-Cole plot.
Cole-Cole plot of 5 frequencies: Re=Ri=100.1 ohm and Cm = 10-8FThe frequency increases from 12KHz, 20KHz, 50KHz, 102KHz to 205KHz, from right to left along the dots, respectively.
According to Cole and Foster, one of the features of Cole-Cole plot is that those dots actually lie on the same circle. Therefore, by simple geometry and algebra calculation, a circle with center (x,y) and radius (r) can be constructed from three of those five dots.
Circle constructed by using 12KHz, 102KHz and 205KHz points, which have resistance of 98.98, 68.87 and 56.6 ohm; and reactance of -7.414, -24.24 and -16.88, respectively.
Ideally, from this circle, the dot on extremely right represents the resistance at DC (0Hz), i.e. Z0Hz, approximate to Re, and the dot on extremely left represents the resistance at infinity frequency, i.e. Zinf, approximate to Re//Ri
= ReRi/(Re+Ri). Thus, Ri can be approximated by (Zinf-1 – Z0Hz
-1)-1.
For impedance measured at other frequency, Zfreq with phase φfreq, it can be decomposed to its rectangular form with Resistance (Rfreq) and Reactance (Xfreq):
Therefore, for each frequency, e.g. W=2*pi*[12KHz, 20KHz, 50KHz, 102KHz,205KHz], one can use either Rfreq or Xfreq with Re and RI to approximate capacitance Cm.
Re estimated from circle: 100.1 OhmRe//Ri estimated from circle: 50.05 OhmRi estimated from circle: 100.1 Ohm
Cm estimated using X: Cm_byX = 1.0e-08 * [1.000 1.000 1.000 0.603 0.151] Farad
Cm estimated using R: Cm_byR : 1.0e-08 * [1.000 1.000 1.000 1.000 1.000] Farad
IMED-4 Measurement of Re//(Ri-Cm) circuit:Re estimated from circle: 100.316 OhmRe//Ri estimated from circle: 50.838 OhmRi estimated from circle: 103.075 Ohm
Cm estimated using X: Cm_byX = 1.0e-08 * [0.9856 0.9859 0.9878 0.5939 0.1421] Farad, for 5 frequencies.
Cm estimated using R: Cm_byR = 1.0e-08 * [0.978 0.980 0.983 0.993 1.032] Farad, for 5 frequencies.
