PASSIVE COMPONENTS Fig. 4: The predicted instantaneous change in capacitance of a 10 nF, 50V, X7R capacitor resulting from an applied 40Vp sine wave. Fig. 5: Ideal and actual current through a capacitor considering voltage coefficient effects Because the cubic equation for C(V) extracted in figure 3 is only defined for voltages greater than zero, the absolute value function must be incorporated to accurately model the capacitance value for both positive and negative voltages. This approach was found to be much more accurate than attempting to fit a polynomial over the entire range of possible voltages (both positive and negative). A Sallen-Key 1 kHz lowpass filter was simulated using polynomial nonlinear capacitor models for 10 nF and 22 nF, 50V X7R capacitors. Figure 7 shows the simulation schematic incorporating X7R capacitors CX1 and CX2. TINA was used to perform a Fourier Spectrum analysis of the output of the filter for a 500 Hz, 1 Vrms input signal. A 65536-sample FFT was performed on an 80 ms sample of the output signal to produce the spectrum. In order to avoid the use of windowing, the time steps of the simulation were constrained and the input signal was chosen to be a coherent frequency. Fig. 6: SPICE netlist for a nonlinear capacitor. The values for the coefficients were extracted from manufacturer data given in Figure 3. Any harmonics in the output spectrum are the result of the capacitors’ voltage coefficient because the distortion characteristic of the operational amplifier (op amp) is not modelled below the full power bandwidth limitation. Figure 8 shows the simulated output spectrum of the circuit when using the modeled X7R capacitors (blue) compared to idealized capacitors (red). The simulation shows a large number of predominantly odd order harmonics are produced when using X7R capacitors in the filter circuit, which agrees with the results from the first article. As expected, the simulation using ideal capacitors shows only a spur at the fundamental frequency. The amplitude of the harmonics in the simulation, on average, is 10 dB lower than that measured in the actual circuit (using capacitors with more pronounced voltage coefficients than those chosen for the simulation model). Conclusion In ceramic capacitors, the relative permittivity of the dielectric is changed by the intensity of the applied electric field, giving rise to a substantial voltage coefficient. This effect is worst in high-K dielectric types and smaller package sizes. As a result, the value of the capacitor is changing instantaneously due to the applied signal, causing distortion in the current waveform. We demonstrated this by producing a SPICE model for a capacitor that replicates the voltage coefficient of a typical 50V X7R capacitor. The X7R capacitor models produced a large number of harmonics when used to simulate a Sallen-Key lowpass filter. In wide dynamic range applications where a substantial voltage may appear across a capacitor, it is best to select C0G, polypropylene film, or silvered mica capacitors to avoid excess distortion. Fig. 7: Simulation schematic of a Sallen-Key 1 kHz lowpass filter with X7R capacitors shown as CX1 and CX2. Fig. 8: TINA Fourier spectrum analysis of a 500 Hz sine wave applied to the filter circuit using nonlinear capacitor models (top) and ideal capacitor models (bottom). 36 Electronic Engineering Times Europe January 2014 www.electronics-eetimes.com

EETE JAN 2014

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