Measuring plasma parameters, e.g. electron density and electron temperature, is an important procedure to verify the stability and behavior of a plasma process. For this purpose, the multipole resonance probe (MRP) represents a promising design. However, the influence of the probe on the plasma through its physical presence makes it unattractive for most processes in industrial applications. A solution, suitable for industrial applications, is the planar version of the MRP (pMRP). It combines the design and benefits of the spherical MRP and can be mounted into the chamber wall. To analyze the resonance behavior of the pMRP the cold plasma model is coupled to the Poisson equation and an analytical expression for the admittance of the probe-plasma system can be derived by means of functional analytic methods. It is adjusted to the design of the pMRP in cylindrical geometry and the corresponding spectra are compared to full 3D electromagnetic field simulations. Based on the Finite Integration Technique, the commercial software CST Microwave Studio is utilized for these simulations. The resulting complex reflection coefficient is evaluated, obtained by the frequency domain solver in conjunction with a tetrahedral mesh and the Drude model.