Electrochemical Impedence Spectroscopy EIS - Revision history

Cmditradmin: /* Nyquist Plots */

2020-06-04T18:07:37Z

Nyquist Plots

← Older revision		Revision as of 11:07, 4 June 2020
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	==== Nyquist Plots ====		==== Nyquist Plots ====
	At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.		At this point it is useful to discuss the most common ways to present EIS spectra, and how to glean useful information from them. First is the Complex-Impedance Plane representation, or Nyquist Plot, in which the data from each frequency point is plotted by the imaginary part on the ordinate and the real part on the abscissa. It is a common convention in the electrochemistry community to plot -ZIm (also found as -Z’’ or -Zj’’) on the y-axis so the data fit into the first quadrant of a graph. Although this type of plot is valuable for identifying how many characteristic features are exhibited by a system, all frequency information is inherently lost. To compensate, one should always annotate the frequencies of crucial data points like high and low real axis intercepts, and the characteristic frequency of an arc, ωc. This characteristic frequency is that which exhibits a maximum in -ZIm for a feature. An example of a Nyquist plot for the circuit in Fig. 4(a) with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF is shown in Figure 5.
	[[File:nyquist2.png\|thumb]]		[[File:nyquist2.png\|thumb\|Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]
	Fig 5. Nyquist plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF
	The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:		The values listed above were known a priori since the EIS spectrum was collected from an actual circuit containing these elements; however, these values can be obtained easily with a model fitting program or a known EEC model. The high-frequency intercept yields Rs, the value of frequency-independent contributions, most commonly the ohmic resistance of the electrolyte. The low-frequency intercept gives Rs+Rr, and, therefore Rr, after subtracting Rs. This is the characteristic impedance of the feature. The characteristic capacitance of this feature is found using Rr and ωc with the following formula:
	(9) C=1/(R_r ω_c )		(9) C=1/(R_r ω_c )

Cmditradmin: /* Equivalent Electrical Circuit Elements */

2020-06-04T18:07:00Z

Equivalent Electrical Circuit Elements

← Older revision		Revision as of 11:07, 4 June 2020
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	In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. <br />		In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. <br />

	[[File:eis2.png\|thumb]Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.		[[File:eis2.png\|thumb\|Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.]
	] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n		] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
	(b)		(b)
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	(b) [[File:eis3.png\|thumb]]		(b) [[File:eis3.png\|thumb\|Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.]]
	Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.

	==== EEC Model Example ====		==== EEC Model Example ====

Cmditradmin: /* EEC Model Example */

2020-06-04T18:06:16Z

EEC Model Example

← Older revision		Revision as of 11:06, 4 June 2020
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	(b)		(b)
	[[File:eis3.png\|thumb]]		[[File:eis3.png\|thumb\|Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.]]
	Fig. 4 (a) Graphical and (b) mathematical representations of an RRC circuit, or Simplified Randles cell.
	This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:		This circuit is known as the Simplified Randles Cell, which can be used to model processes with a single electrochemical reaction, such as iron corrosion in an anaerobic aqueous environment. Further details on relating this EEC model to a kinetic model can be found in Example 10.1 of Ref. 3. If we further manipulate the equation in Fig. 4(b), we can separate the expression into its real and imaginary parts:
	(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )		(8) Z_Re=R_s+R_r/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 ) Z_Im=(-jωC_dl 〖R_r〗^2)/(1+ω^2 〖R_r〗^2 〖C_dl〗^2 )

Cmditradmin: /* Bode Plots */

2020-06-04T18:05:33Z

Bode Plots

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	<br />		<br />

	[[File:bode2.png\|thumb~~\|left~~\|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]		[[File:bode2.png\|thumb\|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

	<br />		<br />

Cmditradmin: /* Bode Plots */

2020-06-04T18:03:19Z

Bode Plots

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	[[File:bode2.png\|thumb\|left\|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]		[[File:bode2.png\|thumb\|left\|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

			<br />
	As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area.<br />		As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area.<br />

Cmditradmin: /* Bode Plots */

2020-06-04T18:02:41Z

Bode Plots

← Older revision		Revision as of 11:02, 4 June 2020
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	<br />		<br />

	[[File:bode2.png\|thumb\|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]		[[File:bode2.png\|thumb\|left\|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

Cmditradmin: /* Bode Plots */

2020-06-04T18:01:03Z

Bode Plots

← Older revision		Revision as of 11:01, 4 June 2020
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	[[File:bode2.png\|thumb\|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]		[[File:bode2.png\|thumb\|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

	~~Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF~~
	As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area.<br />		As a final note on EIS, it is commonplace to find area-specific impedance data, which is achieved by multiplying the real and imaginary parts of the impedance by the electrode cross-sectional area.<br />

Cmditradmin: /* Bode Plots */

2020-06-04T18:00:30Z

Bode Plots

← Older revision		Revision as of 11:00, 4 June 2020
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	<br />		<br />

	[[File:bode2.png\|thumb]]		[[File:bode2.png\|thumb\|Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF]]

	Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF		Fig. 6 Bode Plot of a Simple Randles Cell with values Rs=47 Ω, Rr=467 Ω, and Cdl=860nF

Cmditradmin: /* Equivalent Electrical Circuit Elements */

2020-06-04T17:58:51Z

Equivalent Electrical Circuit Elements

← Older revision		Revision as of 10:58, 4 June 2020
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	In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. <br />		In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. <br />

	[[File:eis2.png\|thumb]] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n		[[File:eis2.png\|thumb]Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.
			] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
	(b)		(b)
	Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.		Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.

Cmditradmin: /* Equivalent Electrical Circuit Elements */

2020-06-04T17:57:47Z

Equivalent Electrical Circuit Elements

← Older revision		Revision as of 10:57, 4 June 2020
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	In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. <br />		In developing EEC models for an EIS spectrum, impedance relations are treated with the same rules as resistors in circuit combinations. Graphical and mathematical representations for a circuit comprised of elements in series and parallel are given in Figs. 2 and 3, respectively. <br />

	〖 Z〗_eq=Z_1+Z_2+⋯+Z_n		[[File:eis2.png\|thumb]] 〖 Z〗_eq=Z_1+Z_2+⋯+Z_n
	(b)		(b)
	Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.		Fig. 2 (a) Graphical and (b) mathematical representation of circuit elements in series.
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	(b)		(b) [[File:eis3.png\|thumb]]
	Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.		Fig. 3 (a) Graphical and (b) mathematical representation of circuit elements in parallel.