Non-Foster networks have the potential to provide wideband matching to electrically-small antennas. Impedance data from a model of an electrically-small monopole provides the starting point of this work. A lumped-element model of the monopole is developed, and used to design a matching network using a shunt negative capacitor and L-section sub-network. The resulting network provides a larger bandwidth than that with passive matching. Stability of the network is analyzed under the assumption of ideal negative capacitor, and it is demonstrated that a change in an antenna impedance can induce instability. Transient analysis using Agilent ADS software, calculations based on differential equations using MATLAB, and s-domain analysis are all employed. The network is similarly analyzed when the negative capacitor is replaced by an equivalent J-transform circuit section, allowing for the inclusion of a nonlinear negative resistance. Nonlinear differential equations are developed for general stability analysis of non-Foster networks.