Improved Cole-Cole Parameter Extraction from ... - Semantic Scholar
Improved Cole-Cole Parameter Extraction from Frequency Response using Least Squares Fitting Todd J. Freeborn, Brent Maundy, Ahmed Elwakil* May 21, 2012
*Department of Electrical and Computer Engineering, University of Sharjah
Background • Background
• What is the Cole-Cole impedance model?
• Motivation • Direct • Least Squares • Gain Errors • Three theoretical circuit elements • Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
Background • Background • Motivation
• Constant phase element or fractional capacitor is a generalization of a frequency dependent circuit element using fractional calculus
• Direct • Least Squares • Gain Errors • Simulation
• Fractional calculus is the branch of mathematics concerning integrations and differentiations to non-integer order
• Noise • Experimental Improved Cole-Cole Parameter Extraction
Motivation • Measurement typically requires impedance analyzer
• Background • Motivation • Direct • Gain Errors • Least Squares
• Extract parameters from a Cole plot
• Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
Motivation • Background • Motivation • Direct
• Reduce the information required to accurately extract the impedance parameters • Simplify the hardware required to extract the impedance model from a test sample
• Gain Errors • Least Squares
• Portable and lower cost tools into the hands of researchers to further explore Cole-Cole parameters
• Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
Direct Extraction • Background • Motivation
+
+
+ -
+
+
-
+
• Direct • Gain Errors
-
-
-
• Least Squares • Simulation • Noise
• G1 and G2 are the low and high frequency gains
• Experimental Improved Cole-Cole Parameter Extraction
-
Direct Extraction −4
• Background
• Direct • Gain Errors • Least Squares • Simulation • Noise
Magnitude (dB)
• Motivation
−6 −8 −10 −12 −14 −16 −18
10 3
10
4
Frequency (Hz)
10
5
10
6
• Knowing G1 and G2 both α and τ can be extracted knowing the -3dB frequency and maximum phase frequency points
• Experimental Improved Cole-Cole Parameter Extraction
Gain Errors • Background
• How do errors in the low and high frequency affect the parameter extractions?
• Motivation • Direct • Gain Errors • Least Squares • Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
Gain Errors 12
• Motivation • Direct • Gain Errors • Least Squares • Simulation • Noise
Relative Error (%)
• Background
10 8 6 4 2 0 −2 −4
0
0.5
1
1.5
2
2.5
Relative Error (%)
3
• Propagated errors due to G1 and G2 errors for the case when (G1, G2, α, τ) = (13.89, 1.41, 0.739, 62.34μ)
• Experimental Improved Cole-Cole Parameter Extraction
3.5
Least Squares Extraction • Background • Motivation
• Limitation imposed by accuracy of low and high frequency gain measurements can be overcome using a least squares extraction
• Direct • Gain Errors • Least Squares • Simulation
• Attempts to find the set of parameters (G1, G2, α, τ) that will reduce the squared error to zero
• Noise • Experimental Improved Cole-Cole Parameter Extraction
Simulation Results • Background
• Applying both methods to ideal MATLAB generated datasets yielded relative errors of
• Motivation • Direct • Gain Errors • Least Squares • Simulation
• Least squares show 2 to 4 orders of magnitude lower error
• Noise • Experimental Improved Cole-Cole Parameter Extraction
Simulation Results −4
• Background • Motivation • Direct • Gain Errors • Least Squares
Magnitude (dB)
−6 −8 −10 −12 −14 −16 −18
• Simulation • Noise
10 3
10
4
Frequency (Hz)
10
5
10
6
• Ideal simulation response of an Cole-Cole impedance with (G1, G2, α, τ) = (13.89, 1.41, 0.739, 62.34μ)
• Experimental Improved Cole-Cole Parameter Extraction
Noisy Data • Background
• Noise analysis conducted for (G1, G2, α, τ) = (13.89, 1.41, 0.739, 62.34μ) with 10 dB signal-to-noise ration (SNR) randomly generated for 1000 simulations
• Motivation • Direct • Gain Errors • Least Squares • Simulation
• Non-linear least squares method shows an order of magnitude less error than the direct method, even in the presence of noise
• Noise • Experimental Improved Cole-Cole Parameter Extraction
Noisy Data 0
• Background
• Direct • Gain Errors • Least Squares
Magnitude (dB)
• Motivation
−5
−10
−15
−20
−25 2 10
10 3
• Simulation • Noise
10 4
Frequency (Hz)
10 5
10 6
• Frequency response when (G1, G2, α, τ) = (13.89, 1.41, 0.739, 62.34μ) has 10 dB SNR additive white noise
• Experimental Improved Cole-Cole Parameter Extraction
Experimental −4 −6
Magnitude (dB)
• Background
• Frequency response of an apricot
−8
−10
• Motivation
−12 −14 −16
• Direct • Gain Errors
−18 −20 −22 10
3
10
4
Frequency (Hz)
• Least Squares
10 5
10 6
• Parameters extracted from fruit using both methods
• Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
Experimental • Use the least squares error, calculated as • Background • Motivation • Direct • Gain Errors • Least Squares
to evaluate the fit of the parameters from both extraction methods • Errors for each method from the experimental fruit responses
• Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
• Background • Motivation • Direct • Gain Errors
Questions?
• Least Squares • Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
*Department of Electrical and Computer Engineering, University of Sharjah
Background • Background
• What is the Cole-Cole impedance model?
• Motivation • Direct • Least Squares • Gain Errors • Three theoretical circuit elements • Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
Background • Background • Motivation
• Constant phase element or fractional capacitor is a generalization of a frequency dependent circuit element using fractional calculus
• Direct • Least Squares • Gain Errors • Simulation
• Fractional calculus is the branch of mathematics concerning integrations and differentiations to non-integer order
• Noise • Experimental Improved Cole-Cole Parameter Extraction
Motivation • Measurement typically requires impedance analyzer
• Background • Motivation • Direct • Gain Errors • Least Squares
• Extract parameters from a Cole plot
• Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
Motivation • Background • Motivation • Direct
• Reduce the information required to accurately extract the impedance parameters • Simplify the hardware required to extract the impedance model from a test sample
• Gain Errors • Least Squares
• Portable and lower cost tools into the hands of researchers to further explore Cole-Cole parameters
• Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
Direct Extraction • Background • Motivation
+
+
+ -
+
+
-
+
• Direct • Gain Errors
-
-
-
• Least Squares • Simulation • Noise
• G1 and G2 are the low and high frequency gains
• Experimental Improved Cole-Cole Parameter Extraction
-
Direct Extraction −4
• Background
• Direct • Gain Errors • Least Squares • Simulation • Noise
Magnitude (dB)
• Motivation
−6 −8 −10 −12 −14 −16 −18
10 3
10
4
Frequency (Hz)
10
5
10
6
• Knowing G1 and G2 both α and τ can be extracted knowing the -3dB frequency and maximum phase frequency points
• Experimental Improved Cole-Cole Parameter Extraction
Gain Errors • Background
• How do errors in the low and high frequency affect the parameter extractions?
• Motivation • Direct • Gain Errors • Least Squares • Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
Gain Errors 12
• Motivation • Direct • Gain Errors • Least Squares • Simulation • Noise
Relative Error (%)
• Background
10 8 6 4 2 0 −2 −4
0
0.5
1
1.5
2
2.5
Relative Error (%)
3
• Propagated errors due to G1 and G2 errors for the case when (G1, G2, α, τ) = (13.89, 1.41, 0.739, 62.34μ)
• Experimental Improved Cole-Cole Parameter Extraction
3.5
Least Squares Extraction • Background • Motivation
• Limitation imposed by accuracy of low and high frequency gain measurements can be overcome using a least squares extraction
• Direct • Gain Errors • Least Squares • Simulation
• Attempts to find the set of parameters (G1, G2, α, τ) that will reduce the squared error to zero
• Noise • Experimental Improved Cole-Cole Parameter Extraction
Simulation Results • Background
• Applying both methods to ideal MATLAB generated datasets yielded relative errors of
• Motivation • Direct • Gain Errors • Least Squares • Simulation
• Least squares show 2 to 4 orders of magnitude lower error
• Noise • Experimental Improved Cole-Cole Parameter Extraction
Simulation Results −4
• Background • Motivation • Direct • Gain Errors • Least Squares
Magnitude (dB)
−6 −8 −10 −12 −14 −16 −18
• Simulation • Noise
10 3
10
4
Frequency (Hz)
10
5
10
6
• Ideal simulation response of an Cole-Cole impedance with (G1, G2, α, τ) = (13.89, 1.41, 0.739, 62.34μ)
• Experimental Improved Cole-Cole Parameter Extraction
Noisy Data • Background
• Noise analysis conducted for (G1, G2, α, τ) = (13.89, 1.41, 0.739, 62.34μ) with 10 dB signal-to-noise ration (SNR) randomly generated for 1000 simulations
• Motivation • Direct • Gain Errors • Least Squares • Simulation
• Non-linear least squares method shows an order of magnitude less error than the direct method, even in the presence of noise
• Noise • Experimental Improved Cole-Cole Parameter Extraction
Noisy Data 0
• Background
• Direct • Gain Errors • Least Squares
Magnitude (dB)
• Motivation
−5
−10
−15
−20
−25 2 10
10 3
• Simulation • Noise
10 4
Frequency (Hz)
10 5
10 6
• Frequency response when (G1, G2, α, τ) = (13.89, 1.41, 0.739, 62.34μ) has 10 dB SNR additive white noise
• Experimental Improved Cole-Cole Parameter Extraction
Experimental −4 −6
Magnitude (dB)
• Background
• Frequency response of an apricot
−8
−10
• Motivation
−12 −14 −16
• Direct • Gain Errors
−18 −20 −22 10
3
10
4
Frequency (Hz)
• Least Squares
10 5
10 6
• Parameters extracted from fruit using both methods
• Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
Experimental • Use the least squares error, calculated as • Background • Motivation • Direct • Gain Errors • Least Squares
to evaluate the fit of the parameters from both extraction methods • Errors for each method from the experimental fruit responses
• Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
• Background • Motivation • Direct • Gain Errors
Questions?
• Least Squares • Simulation • Noise • Experimental Improved Cole-Cole Parameter Extraction
