Positive edges trigger parabolic timebase generator

Generate sawtooth, quadratic-parabolic, and cubic-parabolic signals.

Marián Štofka, Slovak University of Technology, Bratislava, Slovakia; Edited by Martin Rowe and Fran Granville -- EDN, July 28, 2011

Nonlinear systems often need to become linear to be useful, and the circuit in Figure 1 provides a nonlinear sawtooth pulse for a PWM (pulse-width modulator) that can compensate for nonlinearities in sensors, controllers, or systems. The circuit outputs a linear sawtooth pulse, a quadratic parabolic pulse, and a cubic parabolic pulse of equal and constant width following an external trigger pulse. All pulses have equal peak amplitudes.

The circuit incorporates a cascade of three synchronously switched integrators. IC₃’s S₂D₂ switch switches the input of integrator IC_2D, the first in the chain, to the source of reference voltage V_REF. You need two integrators employing IC_2D and IC_2C to generate a quadratic parabolic pulse. The third integrator, using IC_2B, lets you simultaneously generate a cubic parabolic pulse. Each integrator has a series input switch and a reset switch that connects in parallel with a respective integrating capacitor.

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The S_1AD₁ switch in IC₄ is a reset switch for integrator IC_2D. The complementary S_1BD₁ switch serves as a series input switch for integrator IC_2C. Similarly, the S_2AD₂ switch is a reset switch for integrator IC_2C. The S_2BD₂ switch is a series input switch for integrator IC_2C. The positions of all switches are at logic high at all control inputs: IN₁ to IN₄ of IC₃ and IN₁ and IN₂ of IC₄.

Integrators IC_2D and IC_2C also have input-grounding switches in IC₃, S₁D₁, and S₃D₃, respectively. The grounding switches ensure that error due to leakage currents of the series switches is approximately 50% less than that of a design not using the grounding switches.

The Integrate logic signal controls all series switches. When the signal is high, it turns on all the reset and grounding switches. Thus, integrators IC_2B, IC_2C, and IC_2D are either integrating their respective analog input signals or resetting to a 0V output. The input of integrator IC_2D switches to the output of precision voltage-reference cell IC_2A. Thus, signal V_OUTL becomes a negative sawtooth pulse. The pulse varies within its duration, T₁, as:

Inverter IC_2A inverts this pulse. IC_2A has a voltage gain of negative one because positive pulses are more common. Integrator IC_2B integrates sawtooth pulse V_OUTL; IC_2B therefore outputs a quadratic parabolic pulse:

The equation describes a pulse that integrator IC_2B simultaneously integrates, producing a cubic parabolic pulse:

V_OUTLPEAK, V_OUTQPEAK, and V_OUTCPEAK are negative or positive voltage peaks at the outputs of their respective integrators. T₁ is the width of the Integrate pulse. Theoretically, to achieve V_REF=V_OUTLPEAK =V_OUTQPEAK=V_OUTCPEAK, you must stagger the integrating time constants of the respective integrators as 1-to-1/2-to-1/3, respectively. In this case, however, V_REF=3V, whereas V_OUTLPEAK=V_OUTQPEAK= V_OUTCPEAK=5V.

You must multiply the 1 in the staggering ratio by 3/5. Considering the time constant of integrator IC_2C, you get a stagger ratio of 6/5-to-1-to-2/3. For the equal values of integrating resistors R_IL=R_IQ=R_IC, this staggering holds true for the values of respective integrating capacitors. The circuit uses a high-quality, SMD (surface-mount-device) ceramic capacitor, C_IQ, with a value of 2.3692 nF. To achieve the necessary precision staggering, C_IL comprises 2.4016-nF, 343-pF, and 79-pF capacitors in parallel. C_IC is a parallel combination of 1067 pF and 499 pF.