  Active Plasma Resonance Spectroscopy (APRS) is a well known diagnostic
method, where a radio frequency probe is immersed into a plasma and excites
plasma oscillations. The response of the plasma is recorded as frequency
dependent spectrum, in which resonance peaks occur. By means of a mathematical
model plasma parameters like the electron density or the electron temperature
can be determined from the detected resonances. The majority of all APRS probes
have in common, that they are immersed into the plasma and perturb the plasma
due to the physical presence of the probe. To overcome this problem, the planar
Multipole Resonance Probe (pMRP) was invented, which can be integrated into the
chamber wall of a plasma reactor. Within this paper, the first analytic model
of the pMRP is presented, which is based on a cold plasma description of the
electrons. The general admittance of the probe-plasma system is derived by
means of functional analytic methods and a complete orthonormal set of basis
functions. Explicit spectra for an approximated admittance including a
convergence study are shown. The determined resonance frequencies are in good
agreement with former simulation results.
