Op Amp Example Circuits: Solving Gain and Slew Rate Constraints

When studying op amp example circuits, textbook problems often assume ideal components with infinite bandwidth and instantaneous output swings. On the bench, however, physics gets in the way. If you design an amplifier based solely on DC gain equations, your high-frequency AC signals will distort. This walkthrough tackles a classic non-inverting amplifier design problem, exposing the exact algebraic steps, the most common exam trap, and a concrete component selection matrix to get you from theory to a working breadboard.

The Problem Statement: Sensor Amplification

Method Selection and The Slew Rate Trap

To solve this, we apply the Ideal Op-Amp Golden Rules (infinite input impedance, zero output impedance, and the virtual short circuit between inputs) combined with Large-Signal Bandwidth limits. According to All About Circuits, the virtual short allows us to set the closed-loop gain purely by the external resistor ratio.

Step-by-Step Algebraic Solution

We will break the math down into five discrete steps, ensuring no algebraic leaps.

1. Convert Decibels to Linear Gain:
   The formula for voltage gain in dB is $A_{dB} = 20 \log_{10}(A_{V/V})$.
   $40 = 20 \log_{10}(A_{V/V})$
   $2 = \log_{10}(A_{V/V})$
   $A_{V/V} = 10^2 = 100 \text{ V/V}$.
2. Calculate Peak Output Voltage:
   $V_{out(peak)} = A_{V/V} \times V_{in(peak)}$
   $V_{out(peak)} = 100 \times 20 \text{ mV} = 2000 \text{ mV} = 2.0 \text{ V peak}$.
3. Calculate Minimum Gain-Bandwidth Product (GBWP):
   For a non-inverting amplifier, the required GBWP is the closed-loop gain multiplied by the operating frequency.
   $GBWP_{min} = A_{V/V} \times f$
   $GBWP_{min} = 100 \times 50 \text{ kHz} = 5,000,000 \text{ Hz} = 5.0 \text{ MHz}$.
4. Calculate Minimum Slew Rate (SR):
   The maximum rate of change of a sine wave $V(t) = V_{peak} \sin(2\pi ft)$ occurs at the zero-crossing. The derivative is $2\pi f V_{peak}$.
   $SR_{min} = 2 \pi \times 50,000 \text{ Hz} \times 2.0 \text{ V}$
   $SR_{min} = 628,318 \text{ V/s} = 0.628 \text{ V/}\mu\text{s}$.
5. Select E24 Standard Resistors:
   The non-inverting gain equation is $A = 1 + (R_f / R_g)$.
   $100 = 1 + (R_f / R_g) \implies R_f / R_g = 99$.
   Choosing $R_g = 1.0 \text{ k}\Omega$ (a standard E24 value) requires $R_f = 99 \text{ k}\Omega$. Since 99k is not an E24 value, we select the closest standard value: $R_f = 100 \text{ k}\Omega$.
   Recalculated Gain: $1 + (100/1) = 101 \text{ V/V}$ (40.08 dB). This is well within standard engineering tolerances.

Sanity Check and Bench Verification

Before ordering parts, we must verify the math and define how to test it on the bench.

• Order of Magnitude: A 100x multiplier turning 20 mV into 2.0 V is logically sound and keeps the output well within standard ±5V or ±15V supply rails.
• Unit Check: Slew rate is correctly expressed in V/µs (industry standard), and GBWP in MHz. The derivative of $V \sin(\omega t)$ correctly yields $V\omega \cos(\omega t)$, confirming the $2\pi f V_{peak}$ formula.
• Independent Verification: Build the circuit on a breadboard. Inject a 50 kHz, 40 mV peak-to-peak sine wave from a function generator. Probe the output with an oscilloscope.
  • If the waveform looks like a triangle wave, your op-amp is slew-rate limited.
  • If it is a clean sine wave but the amplitude is only 1.5 V peak instead of 2.0 V, your op-amp is GBWP limited (the open-loop gain has rolled off at 50 kHz).

Component Decision Tree: Picking the Exact IC

We need an op-amp with a GBWP > 5.0 MHz and an SR > 0.628 V/µs. Applying a standard 20% engineering safety margin, our target specs are GBWP ≥ 6.0 MHz and SR ≥ 0.75 V/µs. Let us evaluate common bench and production ICs against these hard limits.