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A square waveform can be generated when a op-amp is forced to operate in a saturated region. That is the output of the op-amp is forced to swing repetitively between positive saturation +Vsat  and negative saturation –Vsat, resulting into square-wave output. This square wave generator is also called free running or astable multi vibrator. The output of this circuit can be in positive or negative saturation, depending on whether the differential voltage (vid) is positive or negative, respectively. 




Let us assume that the voltage across the capacitor C is zero volts at the instant the DC supply voltages +Vcc and –Vee are applied. This means that the voltage across the inverting terminal is zero initially. At the same instant however, the voltage V1 at the non inverting terminal is a very small finite value that is a function of the finite offset voltage VooT and the values of the resistors R1 & R2. Thus the differential input voltage vid is equal to the voltage v1 at the non-inverting terminal. Although very small, voltage v1 will start to drive the op-amp into saturation. For example, suppose that the output offset voltage  Voot is positive and that, therefore, voltage v1 is also positive. Since initially the capacitor C acts as a short-circuit, the gain of the op-amp is very large. (A); hence v1 drives the output of the op-amp to its positive saturation +Vsat. With the output voltage of op-amp at +Vsat, the capacitor C starts charging towards +Vsat through resistor R. However, as soon as the voltage v2 across capacitor C is slightly more positive than v1, the output of the op-amp is forced to switch to a negative saturation, -Vsat. With the op-amp’s output voltage in negative saturation,-Vsat, the voltage v1 across R1 is also negative since,

V1= [R1/ (R1+R2)] (-Vsat)

Thus the net differential voltage (vid= v1 – v2) is negative, which holds the output of the op-amp in negative saturation. The output remains in negative saturation until the capacitor C discharges and then recharges to a negative voltage slightly higher than –v1. Now as soon as, the capacitor’s voltage v2 becomes more negative than –v1, the net differential voltage vid becomes positive and then drives the output of the op-amp back to its positive saturation +Vsat. This completes one cycle. With output at +Vsat, voltage v1 at the non inverting input is

v1= [R1/ (R1+R2)] (+Vsat)

The time period T of the output waveform is given by

T= 2RCln [(2R1+R2)/ R2]

Or,  fo = 1/ 2RCln [(2R1+R2)/ R2]

The above equation indicates that the frequency of the output ‘fo’ is not only a function of the RC time constant but also of the relationship between R1 and R2. For example, if R2= 1.16R1, the above equation becomes

fo= 1/ 2RC

The above equation shows that smaller the RC time constant, the higher the output frequency  fo and vice-versa. As with sine wave oscillators, the highest frequency generated by the square wave generator is also set by the slew rate of the op-amp. An attempt to operate the circuit at relatively higher frequencies causes the oscillator’s output to be triangular. In practice, each inverting and non inverting terminal needs a series resistance Rs to prevent excessive differential current flow because the inputs of the op-amp are subjected to large differential voltages. The resistance Rs used should be of 100kΩ or higher.