Ideal & Non-Ideal Op-Amps — Gain-Bandwidth Product & Phase Margin

Ideal & Non-Ideal Op-Amps — Gain-Bandwidth Product & Phase Margin

1. Ideal Op-Amp Rules

Two golden rules for circuits with negative feedback:

1. Virtual short: $V_+ = V_-$ (the differential input voltage is driven to zero by feedback).
2. Virtual open: $I_+ = I_- = 0$ (no current flows into the input terminals).

Inverting amplifier: $A_v = -R_F/R_{in}$. Non-inverting: $A_v = 1 + R_F/R_1$. Summing: $V_{out} = -R_F(V_1/R_1 + V_2/R_2 + \cdots)$.

2. Gain-Bandwidth Product

A real op-amp has a single-pole open-loop gain model:

$$A_{OL}(s) = \frac{A_0}{1 + s/\omega_p} \approx \frac{\omega_T}{s} \quad (\text{for } \omega \gg \omega_p)$$

where $\omega_T = A_0 \omega_p$ is the unity-gain angular frequency and GBW $= \omega_T / 2\pi$. For a closed-loop gain of $|A_{CL}|$, the closed-loop $-3$ dB bandwidth is:

$$f_{-3\text{dB}} = \frac{\text{GBW}}{|A_{CL}|}$$

For example, an LM741 (GBW = 1 MHz) at gain 100 V/V has bandwidth $f_{-3\text{dB}} = 10$ kHz. An OPA627 (GBW = 16 MHz) at the same gain reaches 160 kHz.

3. Phase Margin & Compensation

With a capacitive load $C_L$, the op-amp output impedance $R_o$ creates a second pole at:

$$f_{p2} = \frac{1}{2\pi R_o C_L}$$

If $f_{p2}$ falls below the unity-gain frequency $f_T$, the phase at $f_T$ drops below $-180°$ and the circuit oscillates. Phase margin:

$$PM = 180° + \angle A_{OL}(jf_T) = 90° - \arctan\left(\frac{f_T}{f_{p2}}\right)$$

Stability requires $PM > 0°$; practical systems need $PM \geq 45°$ for acceptable step response (overshoot $< 16\%$).

4. Slew Rate Limitation

The slew rate SR (V/μs) limits large-signal bandwidth independently of GBW. Full-power bandwidth:

$$f_{FP} = \frac{SR}{2\pi V_{out,pk}}$$

An output swing of ±10 V at 100 kHz requires $SR \geq 2\pi \times 10 \times 10^5 \approx 6.3$ V/μs. The LM741's SR = 0.5 V/μs is catastrophically insufficient; use the OPA355 (SR = 200 V/μs).