An Analytical Theory of Harmonic Distortion in RC Networks with Voltage-Dependent Capacitance

A Charge-Conserving Small-Nonlinearity Analysis Through the Fifth Harmonic

Abstract

Class II ceramic capacitors, including X7R and X5R MLCCs, are widely used in compact analog and mixed-signal hardware. Their main advantage is high volumetric capacitance, but that advantage comes from nonlinear dielectric materials. As a result, the effective capacitance depends on the applied voltage. In decoupling and bulk filtering applications, this effect is often treated mainly as a loss of capacitance under DC bias [4], [5], [6]. In signal-path circuits, however, voltage-dependent capacitance has a deeper consequence: the capacitor becomes a nonlinear charge-storage element and can generate harmonic distortion [1], [2], [3].

This paper studies the simplest circuit in which the mechanism appears clearly: a first-order RC low-pass network with a voltage-dependent shunt capacitor. Despite its simplicity, this network is a useful model for many practical nodes, including anti-alias filters, ADC input networks, AC-coupling stages, and local analog filters. The central question is practical: why does the distortion peak near the transition region of the filter rather than at the highest signal frequency?

The analysis is based on a charge-conserving capacitor model. The nonlinear capacitor is described by its charge-voltage relation Q(V), not by an instantaneous capacitance value inserted into an otherwise linear current law. The differential capacitance is Cd(V)=dQ/dV, and the capacitor current is i=dQ(V)/dt. This distinction matters because the common shortcut Q=C(V)V can describe a different physical component when C(V) is actually a differential capacitance [7], [8], [9].

The resulting small-nonlinearity RC equation contains nonlinear terms of the form v dv/dt, v^2 dv/dt, v^3 dv/dt, and v^4 dv/dt. This structure explains the frequency dependence of the distortion. At low frequencies, the voltage across the capacitor is large, but its time derivative is small. At high frequencies, the derivative can be large, but the capacitor voltage and the generated harmonics are attenuated by the RC network. The distortion therefore peaks between these two limits.

For the second harmonic, a compact closed-form expression is obtained, with a maximum at Omega=1/sqrt(2), or approximately 0.707 fc. For the direct third-harmonic mechanism, the maximum occurs near 0.458 fc. For the fourth and fifth harmonics, a recursive analytical formulation is given in terms of complex Fourier coefficients and harmonic balance. The theory provides a foundation for analyzing Class II ceramic capacitors in precision analog signal paths, SAR ADC front ends, and other systems where passive-component nonlinearity can limit dynamic range [11], [12], [14].