  Aims: To constrain the ionization fraction in protoplanetary disks, we
present new high-sensitivity interferometric observations of N$_2$H$^+$ in
three disks surrounding DM Tau, LkCa 15, and MWC 480. Methods: We used the IRAM
PdBI array to observe the N$_2$H$^+$ J=1-0 line and applied a
$\chi^2$-minimization technique to estimate corresponding column densities.
These values are compared, together with HCO$^+$ column densities, to results
of a steady-state disk model with a vertical temperature gradient coupled to
gas-grain chemistry. Results: We report two \ndhp detections for LkCa 15 and DM
Tau at the $5 \sigma$ level and an upper limit for MWC 480. The column density
derived from the data for LkCa 15 is much lower than previously reported. The
[N$_2$H$^+$/HCO$^+$] ratio is on the order of 0.02--0.03. So far, HCO$^+$
remains the most abundant observed molecular ion in disks. Conclusions: All the
observed values generally agree with the modelled column densities of disks at
an evolutionary stage of a few million years (within the uncertainty limits),
but the radial distribution of the molecules is not reproduced well. The low
inferred concentration of N$_2$H$^+$ in three disks around low-mass and
intermediate-mass young stars implies that this ion is not a sensitive tracer
of the overall disk ionization fraction.
