The connection between the constant phase element (CPE) and the depressed arcs often observed in the Nyquist plane is critically analyzed using the analytic continuation principle enunciated earlier by the author. A major inconsistency is pointed out in a model presently used to explain these depressed arcs. A new model is advanced which removes this inconsistency and also has a clear physical basis. A set of exact analytic results are reported which may be used as diagnostic criteria for testing experimental impedance data against different models. Several Nyquist and Bode phase plots are presented as illustrations. Experimental systems where the proposed model will be a good candidate are also discussed. (C) 2008 Elsevier B.V. All rights reserved.