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ficient curves provided by the capacitor manufacturers in order to illustrate how the capacitance changes for a given input signal. In figure 3, a cubic equation for C(V) of a 10 nF, 50V, X7R capacitor was fitted to numerical data (R2 value .99982) from a manufacturer’s datasheet. The curve from the manufacturer is intended to show the change in capacitor value for an applied DC voltage. However, we can presume that complex AC signals are composed of extremely short instances of DC voltages. Also, the assumption is made that the voltage coefficient only depends on the magnitude of the applied voltage and not its polarity. Hence, the change in capacitance should be the same for both positive and negative applied voltages. This allows us to use a DC voltage coefficient curve to predict the resulting instantaneous Table 1: Barium titanate content of selected ceramic capacitor dielectric types 4 changes in capacitance. Figure 4 shows the predicted effect of an applied 50 Vpk, 1 kHz sine wave on the capacitance of a 10 nF, 50V X7R capacitor. The capacitance value dips over the period of the sine wave, reaching a minimum of 8.56 nF at the maximum applied voltage of 50V. The effect of the voltage coefficient on the current waveform in the capacitor can be produced by inserting into equation 4 the cubic equation for C(V) extracted in figure 3. Figure 5 compares the ideal current waveform of a 10 nF capacitor for a 50 Vpk, 1 kHz sine wave to the actual waveform when including the effects of the voltage coefficient. The voltage coefficient distorts the current waveform into a more triangular shape, indicating the introduction of odd harmonics into the signal path. In the previous example, a 50 Vpk sine wave was chosen such that the distortion of the current waveform would be visibly noticeable. However, these effects begin at much lower voltages. Simulating with non-ideal capacitors The effect on the current waveform may seem miniscule, but the degree to which it degrades the total harmonic distortion of a circuit can be surprising. In order to prove this, a SPICE model for a polynomial non-linear capacitor (polycap) was modified to incorporate the cubic equation for voltage coefficient, C(V). This model approximates a nonlinear capacitor using a controlled current source whose output current is defined by the polynomial equation for C(V), as well as the derivative of the applied voltage with respect to time, dV/dt. The time derivative is determined by applying a copy of the applied voltage across a known capacitance CREF, and measuring the resulting current. The model accepts four parameters: C0, C1, C2, and C3. These can be positive or negative and define the capacitor’s voltage coefficient equation. Fig. 3: An equation for voltage coefficient of capacitance is extracted from manufacturer data Your Global Link to the Electronics World Test & Measurement www.tm-eetimes.com TEST & MEASUREMENT LEDLighting www.ledlighting-eetimes.com www.electronics-eetimes.com Electronic Engineering Times Europe January 2014 35


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