2/28: Op Amps III

in which Julia realizes the limitations of op amps

The two golden rules detailed before are not entirely accurate.

Golden Rule I*: Under negative feedback, the op amp will do everything it can to make V+ - V- = 0, but this will never truly be the case--there will always be a small offset voltage, Vos, between these two inputs.
Golden Rule II*: The op amp draws a tiny amount of current through its inputs (~pA).

Lab 9-1: Op Amp Limitations

A) Slew rate

The circuit diagram for measuring slew rate in a 411 or 741 op amp (H&H).

Part I: Square wave input
We drove the 411 with a 1 kHz square wave and did not observe slew with this excellent op amp, which has a slew rate of 15 V/μs.

The slew of a 411 op amp is quite small, not visible for 1 kHz input frequencies. (Here purple is input, yellow is op amp output; the DC offset was added by the function generator and removed from input by AC coupling.)

A 411 (yellow) accurately follows an input square wave (purple); the offset was caused by a DC offset from the function generator, removed by the AC-coupled oscilloscope.

Part II: Sine input
If we drive with a high enough frequency, we expect the output to greatly decrease (but maintain the same input frequency), as the finite slew rate of the 411 would prevent a rapid climb to the wave's amplitude.  We expect this effect to occur where:

omega*amplitude = 2*pi*f*Vin = 12 x 10^6 V/s

Vin(max) = 5

fmax = 382 kHz

So, for a 5 V input sine wave, we expect the 411 to accurately follow waves of up to 382 kHz frequencies, but to lag behind at greater frequencies.

The circuit can accurately reproduce quite high frequencies (158 kHz here) with no observable phase shift.

Driving the circuit with a sine wave of higher frequency (619 kHz here) causes a larger phase shift, but also a comparable decrease in amplitude of both input (yellow) and output (purple) signals.

The observed decrease in output amplitude corresponded with a decrease in input amplitude at frequencies in excess of 619 kHz, possibly because of physical limitations in the function generator's circuitry that prevented high amplitudes at high-frequency signals.  However, a more significant effect is that of the changing impedance of the oscilloscope probe.  At such a high frequency, the scope probe's impedance is small (it is inversely proportional to frequency), so the voltage may bypass the op amp (which has an impedance of 10^12 Ω) and instead flow through the scope probe to the oscilloscope.

We repeated the experiments with a 741 op amp, which has a "typical" slew rate of 0.5 V/us.  While the 411's slew rate was difficult to perceive, the 741's was quite clear at standard driving frequencies.

The falling slew rate of the 741 (yellow) was not high enough to accurately reproduce a square wave input (purple) at 1.9 kHz.

The falling slew rate is 2.88 V/4.680 us = 0.615 V/us.  This is slightly higher than the expected slew rate.

The rising slew rate was slightly higher.

The rising slew rate is 2.64 V/3.88 us = 0.69 V/us.  This is also slightly higher than the expected slew rate, and higher than the falling slew rate.

Measuring the slew rate from a 741 driven with a 1.9 kHz sine wave.

Calculation for the frequency at which a 741 would no longer accurately follow a sine wave:

max: omega*amplitude = 2*pi*f*Vin = 0.6 x 10^6 V/s

Vin(max) = 5

fmax = (0.6 x 10^6)/(31.4) = 86 kHz

This was in accordance with our experimental findings.

At 86 kHz input sine wave (purple), the 741 produces a triangle wave (yellow).

Part B) Offset voltage (Vos)

We attempted to measure Vos by driving with a 4.00 V, 1 kHz sine wave, but this produced a square wave out of the op amp as it saturated ("railed") because the gain of 10 was too high.  After quite some confusion, we eventually realized we had erroneously switched a few resistors, and we were quickly able to fix this problem.

A railing op amp (yellow) from a large input signal (purple).

A better way to measure: if input is shorted, then the output will be Vos amplified.  Here we measured Vos = 3.2 mV, which is quite reasonable.

Shorting the input (yellow) allowed for measurement of the offset voltage (purple).

To eliminate the bias current here, we ensured that both input terminals of the op amp were running through the same resistance to ground, so we added a 10 kΩ resistor between the V- input and ground.

Part II: Minimize the effect of Vos

Upon viewing a flatline at -10.4 V no matter the input, we assumed our op amp was no longer operating.  A new 741 showed a reasonable signal, and we adjusted the potentiometer so Vout = 0, thus minimizing the offset voltage between V- and V+.

C) Bias Current

Now we removed the extra 10 kΩ resistor to see Ibias.

Suddenly we measured a -0.384 V line with no Vin (measured -384 mV output).

Thus we expect -0.384/(10 kΩ) = -3.8 x 10^-8 = -38 nA, in reasonable agreement with the 741's specs of 80 nA.

Lab 9-2: Op amp integrator

The integrator makes a nice integrating triangle wave when driven with square wave.

Driving with a 1 kHz, 1 Vpp square wave (yellow) causes a triangle wave output (purple), as expected with an integrator.

500 Hz square wave input (yellow) makes less frequent, larger-amplitude oscillations in the output  (yellow).

Driving with a 500 Hz, 2 Vpp square wave makes less frequent oscillations, with greater amplitude.

Removing the 10 MΩ resistor makes the output voltage steadily increase, as the capacitor charges.

Lab 9-4: AC Amplifier: microphone amplifier

To conclude our experiments in electronics, Kathryn and I build an AC amplifier to amplify sound signals--to act as a microphone.  We used a single-supply op amp here (the LM358) so we could easily power it with three AA batteries.  Our circuit amplified signals of less than 20 mV to a maximum of +4.5 V; the input bias voltage goes directly to the output, without amplification, so the DC gain is unity so the signal should be clearly evident over the DC offset.
We tried whistling into the microphone and observed higher pitches produced higher-frequency waves (generally they appeared to be sine waves, as expected when whistling), and louder noises increased the amplitude.