The question asks for the minimum acceptable slew rate for an operational amplifier (op-amp) used in an audio pre-amplifier circuit. We are given the maximum signal frequency and the maximum output voltage swing required from the op-amp.

Understanding Slew Rate in Op-Amps

Slew rate (SR) is a critical parameter for op-amps, especially when dealing with high-frequency signals or large output voltage swings. It represents the maximum rate of change of the output voltage per unit of time. If the required rate of change of the output signal is faster than the op-amp's slew rate, the op-amp will not be able to reproduce the signal accurately, leading to distortion.

For a sinusoidal signal, the maximum rate of change occurs when the signal crosses the zero-voltage point. The output voltage of a sinusoidal signal can be described as:

\(v_{out}(t) = V_{peak} \sin(2 \pi f t)\)

The rate of change of the output voltage is the derivative with respect to time:

\(\frac{dv_{out}(t)}{dt} = \frac{d}{dt} (V_{peak} \sin(2 \pi f t))\)

\(\frac{dv_{out}(t)}{dt} = V_{peak} (2 \pi f) \cos(2 \pi f t)\)

The maximum rate of change occurs when \(|\cos(2 \pi f t)| = 1\), which is \(2 \pi f V_{peak}\).

Therefore, the minimum slew rate required for an op-amp to faithfully reproduce a sinusoidal signal of frequency \(f\) and peak voltage \(V_{peak}\) without slew-induced distortion is given by the formula:

\(SR \ge 2 \pi f V_{peak}\)

Calculating Minimum Slew Rate for Audio Pre-amplifier

We are given the following information for the audio pre-amplifier:

• Maximum frequency (\(f\)) = 20 kHz
• Maximum output swing (V_{peak}) = 10 V peak

First, let's convert the frequency to Hz:

\(f = 20 \text{ kHz} = 20 \times 10^3 \text{ Hz} = 20,000 \text{ Hz}\)

Now, we can use the formula to calculate the minimum required slew rate (SR):

\(SR \ge 2 \pi f V_{peak}\)

\(SR \ge 2 \pi (20,000 \text{ Hz}) (10 \text{ V})\)

\(SR \ge 400,000 \pi \text{ V/s}\)

To compare this value with the given options, we need to convert V/s to V/μs. We know that 1 μs = 10\(^{-6}\) s. Therefore, 1 V/s = 10\(^{-6}\) V/μs.

\(SR \ge 400,000 \pi \times 10^{-6} \text{ V/μs}\)

\(SR \ge 0.4 \pi \text{ V/μs}\)

Let's calculate the numerical value:

\(SR \ge 0.4 \times 3.14159 \text{ V/μs}\)

\(SR \ge 1.256636 \text{ V/μs}\)

Rounding this value to three decimal places, we get approximately 1.257 V/μs.

The minimum acceptable slew rate for the op-amp used in this audio pre-amplifier application must be at least 1.257 V/μs to accurately reproduce the 20 kHz signal with a 10 V peak swing.

Comparing Calculated Slew Rate with Options

Let's compare our calculated minimum slew rate with the provided options:

• Option 1: 2.514 V/μs
• Option 2: 1.257 V/μs
• Option 3: 12.568 V/μs
• Option 4: 25.136 V/μs

Our calculated minimum slew rate is approximately 1.257 V/μs. This matches option 2.

Conclusion on Minimum Slew Rate for Audio

For an audio pre-amplifier designed to handle signals up to 20 kHz with a maximum output peak voltage of 10 V, the operational amplifier must have a slew rate of at least 1.257 V/μs to avoid slew-rate-induced distortion. Selecting an op-amp with a higher slew rate would provide better performance margins, but 1.257 V/μs is the theoretical minimum.

Revision Table: Op-amp Slew Rate Calculation

Additional Information on Op-amp Slew Rate and Bandwidth

The slew rate is a parameter that limits the op-amp's performance at high frequencies and large amplitudes. It is related to the op-amp's ability to charge and discharge internal capacitances.

• Full Power Bandwidth: The full power bandwidth (\(f_{FPB}\)) of an op-amp is the maximum frequency at which the op-amp can produce its maximum output voltage swing without slew rate limiting. It is related to the slew rate and maximum peak output voltage by the formula: \(f_{FPB} = \frac{SR}{2 \pi V_{peak}}\). In this problem, we were essentially asked to calculate the frequency where \(f = \frac{SR}{2 \pi V_{peak}}\) for a given frequency limit and voltage swing, to find the required SR.
• Small Signal Bandwidth: This is typically the unity-gain bandwidth (\(f_T\)) of the op-amp, which is the frequency at which the open-loop gain drops to 1 (0 dB). The small-signal bandwidth is determined by the op-amp's internal frequency compensation and is generally much higher than the full power bandwidth, especially for larger output swings. Slew rate limiting occurs when the output voltage rate of change demanded by the input signal exceeds the SR, regardless of whether the op-amp is operating within its small-signal bandwidth.
• Distortion: If the op-amp's slew rate is insufficient, the output signal will be distorted, often appearing as a triangular-like waveform instead of a clean sine wave at the peaks, particularly for high frequencies and large amplitudes.