# The Constant Phase Element (CPE)

### What is a Constant Phase Element?

The Constant Phase Element (CPE) is a non-intuitive circuit element that was discovered (or invented) while looking at the response of real-world systems. In some systems the Nyquist plot (also called the Cole-Cole plot or Complex Impedance Plane plot) was expected to be a semicircle with the center on the x-axis. However, the observed plot was indeed the arc of a circle, but with the center some distance below the x-axis.
These depressed semicircles have been explained by a number of phenomena, depending on the nature of the system being investigated. However, the common thread among these explanations is that some property of the system is not homogeneous or that there is some distribution (dispersion) of the value of some physical property of the system.

#### CPE equations

Mathematically, a CPE’s impedance is given by

1 / Z = Y = Q° ( j omega )n

where Q° has the numerical value of the admittance (1/ |Z|) at omega =1 rad/s. The units of Q° are S•sn (ref 1).
A consequence of this simple equation is that the phase angle of the CPE impedance is independent of the frequency and has a value of -(90*n) degrees. This gives the CPE its name.

When n=1, this is the same equation as that for the impedance of a capacitor, where Q° =C.

1 / Z = Y = j omega Q° = j omega C

When n is close to 1.0, the CPE resembles a capacitor, but the phase angle is not 90°. It is constant and somewhat less than 90° at all frequencies. In some cases, the ‘true’ capacitance can be calculated from Q° and n
The Nyquist (Complex Impedance Plane) Plot of a CPE is a simple one. For a solitary CPE (symbolized here by Q), it is just a straight line which makes an angle of (n*90°) with the x-axis as shown in pink in the Figure. The plot for a resistor (symbolized by R) in parallel with a CPE is shown in green. In this case the center of the semicircle is depressed by an angle of (1-n)*90°

# Reference Electrodes

### Introduction

This Application Note presumes that you have a basic understanding of potentiostat operation. If you are not that knowledgeable concerning electrochemical instrumentation, please read Potentiostat Fundamentals before continuing. Experienced potentiostat users may skip the primer and read on.

It’s only natural that electrochemists concentrate on the working electrode. After all, reactions at the working electrode are what they study. However, the reference electrode shouldn’t be ignored. Its characteristics can greatly influence electrochemical measurements. In some cases, an apparently “good” reference electrode can cause a complete failure of the system.

For reliable reference electrode performance, you should assign a “Lab Master” and treat it very, very carefully so it can serve as a standard for your other reference electrodes. Never use the Lab Master in an actual experiment. The only purpose of the Lab Master is to serve as a check for the other reference electrodes. If a reference electrode is suspected to be bad, you can check the potential versus the Lab Master. You can do that with a voltmeter, or with your Gamry Potentiostat by running and open circuit potential. If the potential difference is less than 2-3 mV, it’s OK. If it’s higher than 5 mV, it needs to be refreshed or discarded.