2.1 Fundamental AC impedance

Alternating current impedance (AC impedance), also called electrochemical impedance spectroscopy (EIS), started from the prototypical experiment of faradaic impedance measurement, which is the resistance and capacitance at the surface of the electrode in an electrochemical cell. It is the measurement of the response of calculated impedance at each frequency with applied a small sinusoidal potential (or current) at fixed frequency. In Figure 1, a small sinusoidal potential signal is supplied by the waveform generator and then applied to the electrochemical cell through a potentiostat. The produced output signal of current (or potential) is converted through the i/E converter and lock-in amplifier to get the information of angular frequency ω, phase angle ψ, and a rotating vector (phasor).(1)

• 2.1.1 Fundamental AC circuits

  
  As illustrated in electrochemical methods and applications, The electrochemical cell is considered as an equivalent circuit with applying a sinusoidal signal to measure the response. For a purely sinusoidal voltage, it can be presented as
  
  

  e = E sin ωt

  
  
  where ω is the angular frequency (2π times the conventional frequency in Hz), E is amplitude, and t is time. And resulting produced a current 𝐼ሶ which is a phasor as well as ሶ the potential 𝐸, it is illustrated as
  
  

  i = I sin (ωt + ψ)

  
  
  The phasors 𝐼 and 𝐸, are separated by a phase angle, ψ. (see Figure 2) (1)
  



• 2.1.1 Fundamental AC circuits

  
  As illustrated in electrochemical methods and applications, a simple circuit through a resistor with the phase angle of zero, its sinusoidal voltage is represented as e = E sin ωt, and the responded current is (E/R) sin ωt which obeys the Ohm’s law (see Figure 3(a)). While the voltage and current across the capacitor, the relation is described as i = (E/Xc) sin (ωt + π/2), where Xc is the capacitive reactance, 1/ωC, and resulting the current leads the voltage as shown in Figure 3(b). In general, the impedance is divided into two parts, which are the real parts ZRe and imaginary parts Zim (multiplied by j= −1) and thus is represented as Z(ω) = ZRe – jZIm. (1)
  



• 2.1.2 Circuit Elements

  
  The common existed in an equivalent electrical circuit are resistor, inductor, capacitor. In Table 1, it is illustrated that the circuit elements and its symbol, the impedance, and the relation between current and voltage. The impedance of a resistor is independent of frequency with real part. The impedance of a inductor increases as frequency increases. It is with imaginary part, the through current is phase shifted 900 to the voltage. The impedance of capacitor decreases as the frequency increases with an imaginary impedance component. The flowed through current is phase shifted –900 to the voltage.
  

• 2.1.3 Nyquist plot and Bode plot for EIS

  
  The EIS response can be represented as either a Nyquist plot, Zim vs. ZRe for different values of ω (see Figure 4 (A)), or a Bode plot, log 𝑧 and phase angle ψ are both plotted against log ω (see Figure 4(B). In Nyquist plot, it is plotted with higher frequency at left side of the plot, and it is for lower frequency on the right region. In Figure 4(A), the capacitance line in the Nyquist plot in the lower frequency region inclines constantly by an angle between 0° and 45°. ZCPE = 1/T(jω)γ, where T represents a pure capacitor only when γ = 1, and γ is related to α by α = (1 – γ)90°. So, α = 0 and γ = 1 represents a perfect capacitor, and lower γ values directly reflect the roughness of the electrode. (1)
  
  



• 2.1.4 Equivalent circuit of a cell

  
  For the real system of the NCGCE/solution interface (see Figure 5(a)), It has been illustrated its performance by an equivalent circuit of resistors and capacitors which flowed current with the same amplitude and phase angle. The most common used model is called the Randles equivalent circuit as shown in Figure 5(b). The GCE/Nafion interface is represented by a constant phase element (CPE) in parallel connection with a charge- transfer resistance (Rct), and pure capacitor (Cdl). The parallel combination of bulk resistance (Rb) and bulk capacitance (Cb) represents the bulk dielectric properties of Nafion. To maintain the electro neutrality within the Nafion film, the cations/anions extrude into the Nafion film. Thus, the circuit is accommodated by the inclusion of additional Rfs and Cfs in parallel combination. Finally, the circuit is terminated with the Rs in serial connection, representing the solution resistance and uncompensated potential drop
  
  



• 2.1.5 Impedance plot for an electrochemical system

  
  As published, in the Nyquist plot (see Figure 6), the first semicircle appearing at a very high frequency domain represents the dielectric properties of Nafion while the second semicircle represents Rct and CPE. The straight line after these two semicircles arises due to the mass-transfer limitation of the redox that couples with a typical characteristic of 45° with respect to Z’ axis. At low-frequency domain, a vertical line is observed due to the capacitance of the film. Apart from ZD, all other components in the circuit are assumed independent of frequency.
  
  

Reference

Please click the link below

• 1.Jyh-Myng Zen,* Hsieh-Hsun Chung, Govindasamy Ilangovan and Annamalai Senthil Kumar, Analyst, 2000, 125, 1139–1146
• 2.Jyh-Myng Zen,* Govindasamy Ilangovan, and Jia-Jen Jou, Anal. Chem. 1999, 71, 2797-2805