Radiotherapy is a favorable method of destroying cancerous cells, yet methods for quantifying radiation damage in DNA could use improvement. Current methods for determining DNA damage due to radiation involve separating different conformations of plasmid DNA. Irradiated samples of plasmid DNA will produce different conformations with different mobilities, varied by the amount of damage, and can be separated with a technique called agarose gel electrophoresis. Gel electrophoresis is a time-consuming process with a sizable degree of systematic error. It would be a substantial advancement to medical physics if an alternative method to electrophoresis were developed. One promising method is the electrochemical impedance spectroscopy (EIS) of plasmid DNA in solution. In this work, we develop an understanding of the EIS properties of electrolytes combined with well-established theories of linear time-invariant (LTI) circuit theory to investigate aqueous solutions of pBR322 plasmid DNA. Experimental data on sodium chloride, potassium phosphate, Tris-EDTA, and Tris with pBR322 solutions justify utilizing a Randles equivalent-circuit model. The augmentation of a standard Randles model with a constant-phase element (CPE) provides an excellent fit between model and data. A bulk resistance element, while theoretically a zero-phase (real) element, was adjusted to exhibit similar properties to a CPE. This new circuit model provides a normalized root-mean-squared error (NRMSE) fit as high as 98.9% in some cases. These working models offer much-needed insights into the properties of plasmid DNA in solution.