If R1 is removed, Z1 = R2 & Z2 = 1/(sC1). Vout/Vin = -1/sC1R2). For the ideal integrator, when s = 0 Vout/Vin = -R2 / R1. With R1 removed, when s= 0 Vout/Vin = infinity. Any small input signal will produce an infinite output. An infinite output voltage is impossible. Therefore, the output saturates close to the voltage of the op-amp positive or negative power supply (L+ or L-).
The input in the time domain (f(t)) is a unit step. Taking the Laplace transform, F(s) = 1/s. The transfer function for the circuit (G(s)) is -1/(sR2C1). The output function in the frequency domain is found by multiplying F(s) and G(s), giving -1/(s2R2C1). Taking the inverse laplace transform, the output function in the time domain is found to be G(t) = -t/(R2C1). The step occurs over 500 us. R2 = 100k ohms and C1 = .01uF. Plugging in these values, the peak to peak amplitude of the triangle wave output is .5 V. This is confirmed by our scope printout and by our MultiSim simulation.
The manufacturer's slew rate specification for the LM411 is 15 Volts / usec. With a feedback resistor of 30 k, the voltage transition from +10V to -10V took 960ns, giving a slew rate of 20V/0.96usec = 20.83. Using a 10k feedback resistor, the slew rate was measured to be 10/1.44 = 6.94 V/usec. The input resistor in both cases was 10k. This data is interesting because it seems reasonable that since the slew rate is the ratio of the voltage drop to the time required for the transition, the slew rate should be independent of the gain. For the -3 gain circuit, the slew rate was far higher than for the circuit of -1 gain. This is probably due to the point at which the time measurements were made for the smaller gain circuit. Looking at the oscilloscope printout, the circuit overshot the target voltage of -5 V on the transition and oscillated slightly before stabilizing at -5V. Our measurement was taken from the start of the transition to where the output voltage appeared stable. Had we measured to the minimum peak voltage value, the time would have been about 0.7 usec, resulting in a slew rate of 14.28 V/usec, which is in much greater accordance with the manufacturer's specification.
(So anyways, we probably just measured our time wrong because it was not clear exactly how the slew rate measurement was supposed to be made.)
An amplifier with a gain of 1 might be useful as buffers to separate parts of circuits. Since the input impedance of an Op Amp is very high, the Op Amp will not draw large currents from the input circuit and load it down. That way, the intended operating characteristics of the input circuit can be maintained and separated from the output circuit. This could be useful perhaps when designing an audio amplifier designed to drive large speakers.
The comparator circuit implemented with an Op Amp operates by setting a threshold voltage using a voltage divider between Vcc and Gnd with resistors R1 and R2. Since Vcc in this case is 5V and R1=R2=10k, the V+ terminal of the Op Amp is at 2.5V. Since the Op Amp attempts to drive the two inputs to the same voltage, a voltage less than 2.5V at the V- terminal will cause the Op Amp output voltage to saturate at positive supply rail since there is no feedback resistance. If V- is greater than 2.5, the output voltage saturates at the opposite supply rail.
Inside the comparator, we can assume that the output is connected to an internal switch connected to ground. If V+ > V-, the switch opens, and no current flows from Vcc through R3 to ground, and the output is essentially pulled up to Vcc. When the input changes and V+ < V-, the switch closes and makes short circuit, pulling the output node to ground. As a result, current flows from Vcc through R3 to ground, and power is essentially wasted by R3. As a result, it is desirable to make R3 large in order to minimize power waste, but a larger R3 would require the load circuit to have an even higher impedance to actually realize the full 5V Vcc on its inputs.
The Schmitt trigger circuit solves the problem that a simple comparator exhibits when presented with jittery input with multiple zero crossings. It does this by establishing low and high voltage thresholds for the transition points. Inside the IC, the output is connected to an internal switch connected to ground. If Vi > V+, the switch closes, and the output Vo is pulled to ground (0V). As a result, the terminal voltage at V+ changes, since instead of just a voltage divider between two 10k resistors, it is between a R2 (10k) and R1 in parallel with R3 (5k). The voltage at V+ is now 5/3 V, requiring that the voltage at Vi drop below 5/3 V to toggle the output state. Assume now that Vi indeed drops past 5/3 volts. The switch opens, and the output node is no longer pulled to ground. The voltage at the output node is now equal to V+ since no current flows through R3. The V+ terminal voltage and Vout return to 5/2V by way of the 10k resistors R2 and R1. That means that the voltage must increase above 5/2 volts to cause a transition to the low output state.
For some reason the experimentally measured transition threshold voltages measured do not correspond with the theoretical values. Also the output voltage of the Schmitt trigger does not seem to be 2.5 volts, but rather something far larger. The low output state however does seem to go to 0V, as expected. We don't know how to explain these irregularities. The resistors were each verified to be 10k.
The DC offset on the signal generator is required to push the triangle wave up above 0V so that the transitions at the edges of the threshold range happen as the triangle wave changes direction. The DC offset simply adds a DC voltage to the original triangle wave signal. Also the DC coupling is required on the oscilloscope so that it measures absolute voltages and does not take the average of the maximum and minimum voltages as the zero point (AC coupling).
The relaxation oscillator is essentially a Schmitt trigger circuit with the Vin terminal connected to a resistor R3 and a capacitor C. The RC combination ensures that the voltage at the Vin terminal cannot change instantaneously, meaning that it takes time for Vin to transition across the threshold voltage set at the V+ terminal. The analysis graph above shows clearly the Schmitt trigger-like transitions of the V+ threshold voltage (green). Refer to the explanation of the Schmitt trigger circuit for an explanation. When the V+ terminal changes from the low voltage state (~1.2V) to the high voltage state (~2V), the voltage across the R3 resistor changes, causing more current to flow through it. Since the voltage across the capacitor cannot change instantaneously, the capacitor 'fills up' with the time constant defined by RC in theory until no current flows through it (stable state). However, before it reaches a stable state, the voltage at the V- terminal of the capacitor will pass the threshold voltage for the Schmitt trigger, and the V+ voltage and thus the threshold voltage will drop again. Since the state is now V+ < V-, the voltage at the output is 0, causing the capacitor to discharge with the same RC time constant. As soon as the voltage across the capacitor drops below the lower threshold voltage, the Schmitt trigger flips, and the process repeats.
The measured time per transition from the above oscilloscope image shows a period of about ~6 uSec. The calculated period was 8.06x10^-6, or ~8 uSec. The theoretical and measured periods are essentially the same, and the variation in the experimentally obtained value could easily be a result of resistor tolerances.