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.

Figure 1. |
This triggered circuit generates linear, quadratic parabolic, and cubic parabolic pulses, all starting from 0 Vand having equal peak-level magnitudes. |

The circuit incorporates a cascade of three synchronously switched integrators. IC_{3}’s S_{2}D_{2} 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.

The S_{1A}D_{1 }switch in IC_{4} is a reset switch for integrator IC_{2D}. The complementary S_{1B}D_{1} switch serves as a series input switch for integrator IC_{2D}. Similarly, the S_{2A}D_{2} switch is a reset switch for integrator IC_{2C}. The S_{2B}D_{2 }switch is a series input switch for integrator IC_{2C}. The positions of all switches are at logic high at all control inputs: IN_{1} to IN_{4 }of IC_{3 }and IN_{1} and IN_{2} of IC_{4}.

Integrators IC_{2D} and IC_{2C} also have input-grounding switches in IC_{3}, S_{1}D_{1}, and S_{3}D_{3}, 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 0 V 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, T_{1}, 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. T_{1 }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} = 3 V, whereas V_{OUTLPEAK} = V_{OUTQPEAK} = V_{OUTCPEAK} = 5 V.

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.

Figure 2. |
At the midwidth of the pulse, the quadratic parabolic pulse’svoltage level (pink trace) is exactly one-fourth of its peak level. |

A rising edge at the trigger input forces the Integrate signal low, which turns off the reset and grounding switches and turns on the series switches. The integration lasts until V_{OUTQPEAK} = 5 V, forcing the output of IC_{5 }low, which in turn sets Integrate high. Thus, the series switches are off, and the reset and grounding switches are on. The circuit remains in this steady state until the next rising edge at the trigger input. The Analog Devices ADG1213 and ADG1236 switches work well in this design because of their charge injection of 1 pC or less. Figure 2 shows the circuit’s high precision, depicting linear and quadratic-parabolic-pulse shapes.