Operational amplifier characterization

The operational amplifier (op-amp) is a high-gain, direct-coupled differential amplifier packaged as an IC, such as the classic 741. With external resistors and capacitors it performs amplification, addition, integration, differentiation, comparison and clipping. Because its ideal behaviour is so simple, op-amp circuits are easy to analyse with two rules — and MCQs love to test those rules, gain formulas and the practical parameters like CMRR and slew rate.

Key concepts

Differential amplifier

• The input stage of an op-amp: two matched transistors with a common emitter (or source) resistor or current source.
• Output proportional to the difference of inputs: $V_o = A_d(V_1 - V_2) + A_c\frac{V_1+V_2}{2}$.
• Differential gain $A_d$ should be large; common-mode gain $A_c$ should be small.
• CMRR = $A_d/A_c$, often in dB: $20\log(A_d/A_c)$. A constant-current tail improves CMRR.

Ideal op-amp parameters

Parameter	Ideal	Typical 741
Open-loop gain $A_{OL}$	∞	~2×10⁵
Input impedance	∞	~2 MΩ
Output impedance	0	~75 Ω
Bandwidth	∞	GBW ≈ 1 MHz
CMRR	∞	~90 dB
Slew rate	∞	0.5 V/μs
Offset voltage	0	~1 mV

Practical characteristics: input offset voltage, input bias current, input offset current, slew rate (max rate of change of output, limits large-signal frequency: $f_{max} = SR/(2\pi V_m)$), PSRR, output saturation (output cannot exceed roughly ±(supply − 1 to 2 V)), finite gain-bandwidth product.

Golden rules (with negative feedback)

1. No current flows into either input ($I^+ = I^- = 0$).
2. The voltages of the two inputs are equal ($V^+ = V^-$) — the virtual short.

Inverting amplifier

• Input through $R_1$ to the inverting terminal; feedback $R_f$ from output to inverting terminal; non-inverting terminal grounded.
• Since $V^- = V^+ = 0$, the inverting terminal is a virtual ground (the "negative ground" in the syllabus).
• Current $V_i/R_1 = -V_o/R_f$ → $A_v = -R_f/R_1$. Input impedance $= R_1$.

Non-inverting amplifier

• Input to the non-inverting terminal; $R_1$ from inverting terminal to ground, $R_f$ feedback.
• $V^- = V_o R_1/(R_1+R_f) = V_i$ → $A_v = 1 + R_f/R_1$. Gain is always ≥ 1; input impedance very high.

Voltage follower (buffer): output tied directly to inverting input, $R_f = 0$, $R_1 = \infty$ → $A_v = 1$. Provides impedance isolation (high $Z_{in}$, low $Z_{out}$).

Applications

• Summing amplifier (adder): several inputs through resistors to the virtual ground: $V_o = -R_f\left(\frac{V_1}{R_1} + \frac{V_2}{R_2} + \cdots\right)$. Equal resistors give $-(V_1+V_2+\cdots)$; used in audio mixers and DACs.
• Integrator: capacitor in feedback, resistor at input: $V_o = -\frac{1}{RC}\int V_i\,dt$. Square input → triangular output. A large resistor across C limits DC gain.
• Differentiator: capacitor at input, resistor in feedback: $V_o = -RC\frac{dV_i}{dt}$. Triangular input → square output. Prone to noise at high frequency; a small series resistor is added.
• Comparator: open loop (no feedback); output swings to $+V_{sat}$ if $V^+ > V^-$, else $-V_{sat}$. Zero-crossing detector when reference = 0. Schmitt trigger adds positive feedback for hysteresis (noise immunity).
• Clipping circuits: diodes (or Zeners) in the feedback path or output limit the output to a set level; precision clippers use op-amps to eliminate the diode drop.
• Subtractor (difference amplifier): $V_o = \frac{R_f}{R_1}(V_2 - V_1)$ with matched resistor ratios.

High-yield points

• Ideal op-amp: infinite gain, input impedance, bandwidth and CMRR; zero output impedance.
• Gains: inverting $-R_f/R_1$, non-inverting $1 + R_f/R_1$, voltage follower 1.
• $\text{CMRR} = A_d/A_c$; 741 slew rate ≈ 0.5 V/μs.
• Integrator: $V_o = -\frac{1}{RC}\int V_i\,dt$ (square in, triangle out).
• Full-power bandwidth: $f_{max} = SR/(2\pi V_m)$.

Formulas

$A_{inv} = -\frac{R_f}{R_1},\qquad A_{non\text{-}inv} = 1 + \frac{R_f}{R_1}$

$\text{CMRR (dB)} = 20\log_{10}\left|\frac{A_d}{A_c}\right|,\qquad f_{max} = \frac{SR}{2\pi V_m}$

where SR = slew rate (V/s), $V_m$ = peak output voltage (V).

Worked example

An inverting amplifier has $R_1 = 10$ kΩ, $R_f = 100$ kΩ, supply ±15 V (saturation ±13 V). Input is 0.5 V DC. Find the output. What if input is 2 V?

1. $A_v = -100/10 = -10$.
2. $V_o = -10 \times 0.5 = -5$ V (within saturation).
3. For 2 V input, ideal output = −20 V, but the op-amp saturates at −13 V.

Exam tips

• Inverting gain is $-R_f/R_1$; non-inverting gain is $1 + R_f/R_1$ — never less than 1.
• Virtual ground exists only in the inverting configuration (non-inverting terminal grounded).
• Comparators use no feedback (or positive feedback); linear amplifiers need negative feedback.
• Integrator → R input, C feedback; differentiator → C input, R feedback.
• 741 slew rate ≈ 0.5 V/μs; ideal CMRR = ∞.

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. An op-amp has a slew rate of 0.5 V/μs. Using the full-power bandwidth relation $f_{max} = SR/(2\pi V_p)$, what is the highest frequency at which it can deliver an undistorted sine wave of 10 V peak?

• A. 7.96 kHz
• B. 796 Hz
• C. 50 kHz
• D. 79.6 kHz

Answer: A. $f_{max} = \dfrac{SR}{2\pi V_p} = \dfrac{0.5 \times 10^6}{2\pi \times 10} \approx 7.96$ kHz.

2. Which of the following is NOT a characteristic of an ideal op-amp?

• A. Infinite open-loop voltage gain
• B. Infinite bandwidth
• C. Infinite output impedance
• D. Infinite input impedance

Answer: C. An ideal op-amp has zero output impedance, so it can drive any load without loss. Its gain, input impedance, bandwidth and CMRR are infinite.

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