Ideal and practical op-amp characteristics

Why it matters

The op-amp is the workhorse of signal conditioning: amplifiers, filters, converters and comparators in every instrument are built around it. The ideal model makes design quick, but the measurement errors you see on the bench, such as a few millivolts of output with the input shorted, a sine that turns into a triangle at high frequency, or gain that falls with frequency, come from the practical limits covered here.

Key ideas

The ideal op-amp. A differential amplifier with output v_o = A·(v₊ − v₋) and:

• infinite open-loop gain A,
• infinite input resistance (no input current),
• zero output resistance,
• infinite bandwidth and slew rate,
• zero offset, bias currents and noise; infinite CMRR.

Two golden rules (with negative feedback). (1) No current flows into either input. (2) The output does whatever is needed to make v₊ = v₋ (the virtual short). Rule 2 holds only while there is negative feedback and the output is not saturated. Without feedback the huge A drives the output to one supply rail for almost any input: the op-amp becomes a comparator.

Output limits. The output cannot exceed the supply rails; a classic bipolar op-amp on ±15 V saturates at about ±13 to ±14 V, rail-to-rail types come within tens of mV. It also has a maximum output current (tens of mA).

Practical DC errors.

• Input offset voltage V_os: the differential input needed to bring the output to zero (typically 0.1–5 mV). It appears at the output multiplied by the noise gain (1 + R_f/R₁).
• Input bias current I_B: the average of the two input currents (tens of nA for BJT inputs, pA for FET inputs). Flowing through R_f it gives an output error I_B·R_f.
• Input offset current I_os = |I_B1 − I_B2|: putting a resistor R_comp = R₁ ∥ R_f in the non-inverting input makes the bias-current error depend only on I_os.
• Offsets drift with temperature (µV/°C, pA/°C), so nulling at one temperature is not enough.

Practical AC limits.

• Finite gain and bandwidth: internally compensated op-amps have a dominant pole (a few Hz) so the open-loop gain falls at 20 dB/decade; the unity-gain bandwidth (GBW, f_T) is constant, so closed-loop bandwidth ≈ GBW / noise gain.
• Slew rate SR: the maximum dV_o/dt, set by the current available to charge the compensation capacitor. A sine of amplitude V_p needs SR ≥ 2π·f·V_p; beyond that the output becomes triangular. The full-power bandwidth is f_max = SR/(2π·V_p).
• CMRR (typically 80–100 dB) and PSRR (rejection of supply ripple) are finite.

Typical datasheet values for a general-purpose bipolar op-amp (741 class): A ≈ 2×10⁵, R_in ≈ 2 MΩ, R_o ≈ 75 Ω, GBW ≈ 1 MHz, SR ≈ 0.5 V/µs, V_os ≈ 1 mV, I_B ≈ 80 nA, CMRR ≈ 90 dB. These vary by device: always take them from the datasheet.

Formulas

v_o = A·(v₊ − v₋) — open-loop, valid only inside the output swing limits.

A_CL = A / (1 + A·β) — closed-loop gain with finite A; β = R₁/(R₁ + R_f) for the non-inverting stage.

f_CL = GBW / (1 + R_f/R₁) — closed-loop bandwidth (Hz) for a single-pole op-amp; (1 + R_f/R₁) is the noise gain.

SR = (dv_o/dt)_max — V/µs. f_max = SR / (2π·V_p) — full-power bandwidth for peak output V_p (V).

Δt = ΔV / SR — minimum time for a large output step.

V_o(offset) = V_os·(1 + R_f/R₁) + I_B·R_f — no compensating resistor.

V_o(offset) = V_os·(1 + R_f/R₁) + I_os·R_f — with R_comp = R₁ ∥ R_f at the non-inverting input.

CMRR(dB) = 20·log₁₀|A_d / A_cm|; PSRR = ΔV_os / ΔV_supply (µV/V).

Worked examples

Example 1 (standard: slew rate). An op-amp with SR = 0.5 V/µs must deliver a sine of 10 V peak. What is the highest frequency it can handle without slew distortion? How long does a 5 V output step take?

1. f_max = SR/(2π·V_p) = 0.5×10⁶ V/s / (2π × 10 V) = 7.96 kHz.
2. Δt = ΔV/SR = 5 V / 0.5 V/µs = 10 µs.

Answer: f_max ≈ 7.96 kHz; the 5 V step takes at least 10 µs. Even though the small-signal bandwidth may be far higher, a 10 V output is slew-limited above 8 kHz.

Example 2 (GATE level: DC errors and bandwidth). An inverting amplifier uses R₁ = 10 kΩ and R_f = 100 kΩ. The op-amp has V_os = 2 mV, I_B = 100 nA, I_os = 20 nA, A = 10⁵ and GBW = 1 MHz. Find (a) the output offset with no compensating resistor, (b) the offset with R_comp, (c) the closed-loop bandwidth, (d) the exact magnitude of the gain.

1. Noise gain: 1 + R_f/R₁ = 1 + 10 = 11.
2. (a) V_o = V_os·11 + I_B·R_f = 2 mV × 11 + 100 nA × 100 kΩ = 22 mV + 10 mV = 32 mV.
3. (b) R_comp = 10 kΩ ∥ 100 kΩ = 9.09 kΩ: V_o = 22 mV + I_os·R_f = 22 mV + 20 nA × 100 kΩ = 22 + 2 = 24 mV.
4. (c) f_CL = GBW/11 = 1 MHz/11 = 90.9 kHz.
5. (d) Ideal gain is −10; with β = 1/11, |A_CL| = (R_f/R₁)/(1 + 11/A) = 10/(1 + 11×10⁻⁵) = 9.9989.

Answer: (a) 32 mV, (b) 24 mV, (c) ≈ 91 kHz, (d) |A| ≈ 9.999 (error about 0.011 %). The bandwidth uses the noise gain 11, not the signal gain 10.

Common mistakes

• Applying the virtual short to an op-amp without negative feedback (comparator) or with a saturated output.
• Computing output voltages larger than the supply rails, such as A × v_d = 2000 V; the output saturates.
• Dividing GBW by the inverting signal gain (R_f/R₁) instead of the noise gain (1 + R_f/R₁).
• Confusing slew-rate limiting (large signal, amplitude dependent) with small-signal bandwidth.
• Multiplying the bias-current error by the gain; it is I_B·R_f directly.
• Forgetting that the compensating resistor removes I_B error but not I_os error.

For GATE IN

Expect numericals on slew rate and full-power bandwidth, output offset from V_os and bias currents, closed-loop bandwidth from GBW, gain error due to finite open-loop gain, and CMRR in dB. Conceptual questions test the ideal assumptions and when the virtual short applies. Practise drawing the offset model (a V_os source and two I_B sources at the inputs).

Quick check

1. SR = 1 V/µs. Maximum frequency for a 5 V peak sine?
2. GBW = 2 MHz and a non-inverting gain of 20. Closed-loop bandwidth?
3. V_os = 1 mV in a non-inverting amplifier with gain 101. Output offset from V_os?
4. Why does an open-loop op-amp act as a comparator?

Answers: 1. About 31.8 kHz. 2. 100 kHz. 3. 101 mV. 4. Its huge open-loop gain drives the output to a rail for any differential input above a few µV.