An asymptotic theory of rectification and detection

The results of a systematic asymptotic analysis of the basic half wave rectifier and amplitude detector circuits are presented in this paper. A singular perturbation technique is applied directly to the governing differential equation and yields an asymptotic approximation of the output voltage as
\omega RC \rightarrow \infty
. Here,
\omega
is either the source frequency for the rectifier or the carrier frequency for the detector and
RC
is the time constant of the resistor-capacitor filter. The asymptotic result contains transient, ripple, and steady-state information. The latter two components reduce to the standard results when the diode's forward resistance is small, or equivalently, when the parameter
\alpha = V_{0}q/kT
becomes large. Here,
V_{0}
is the peak value of the input voltage,
q
is the charge of an electron, k is the Boltzman constant, and
T
is the temperature. In particular, the ripple component of the rectifier output becomes a sawtooth wave in this limit. This is not an assumption to simplify the analysis but a consequence of the circuit's parameters as born out by the asymptotic analysis.