Operational Amplifiers (Op-Amps) & Linear Analog Computing

1. Ideal Op-Amp Characteristics vs. Practical 741 IC

An Operational Amplifier (Op-Amp) is a direct-coupled, high-gain differential voltage amplifier. The ideal op-amp model provides an indispensable mathematical abstraction for circuit analysis:

2. The Golden Rules & Virtual Ground Principle

In any linear op-amp circuit operating under negative feedback where output remains unclipped between the supply rails ($-V_{EE} < V_o < +V_{CC}$), two fundamental axioms (the Golden Rules) govern behavior:

1. Axiom 1 (Virtual Short): The voltage difference between the inverting input ($-$) and non-inverting input ($+$) is infinitesimally small:
   $$V_d = V^+ - V^- = \frac{V_o}{A_{OL}} \xrightarrow{A_{OL} \to \infty} 0 \implies V^- = V^+$$
   When the non-inverting terminal is connected to physical ground ($V^+ = 0$), the inverting terminal is held at ground potential ($V^- = 0$) by the feedback loop. This node is termed a Virtual Ground: it has the potential of ground ($0\,\text{V}$), yet no current can flow directly to physical earth through the terminal.
2. Axiom 2 (Zero Input Bias Current): Because input impedance is infinite ($R_{in} \to \infty$), no current enters either input terminal:
   $$I^+ = I^- = 0$$
   All incoming signal currents are forced entirely into the feedback loop.

1. Inverting Amplifier Configuration

In the inverting amplifier, input voltage $v_{in}$ is applied through resistor $R_1$ to the inverting terminal ($-$), with feedback resistor $R_f$ connected from output $v_o$ back to the inverting terminal. The non-inverting terminal ($+$) is tied to ground ($V^+ = 0$).

By Axiom 1, $V^- = V^+ = 0\,\text{V}$ (virtual ground). Applying Kirchhoff's Current Law at the inverting summing node:

Since $I^- = 0$, $I_1 = I_f$:

Solving for the closed-loop voltage gain $A_v$:

The input impedance seen by the source is strictly $Z_{in} = R_1$. Output impedance is $Z_{out} \approx 0$.

2. Non-Inverting Amplifier Configuration

In the non-inverting amplifier, input voltage $v_{in}$ is applied directly to the high-impedance non-inverting terminal ($+$), so $V^+ = v_{in}$. Resistors $R_1$ and $R_f$ form a voltage divider from output $v_o$ to ground, feeding the inverting node.

By Axiom 1, $V^- = V^+ = v_{in}$. Applying KCL at the inverting node:

Thus, the non-inverting gain is always positive and strictly $\ge 1$:

Input impedance is extremely high ($Z_{in} \approx R_{in} (1 + A_{OL}\beta) \sim 10^{9}\,\Omega$ for BJT), preventing any loading on preceding stages.

3. Voltage Follower (Unity-Gain Buffer)

Setting $R_f = 0$ (direct short from output to inverting terminal) and $R_1 = \infty$ (open circuit):

The voltage follower provides unity voltage gain, zero phase shift, astronomical input impedance ($Z_{in} \to \infty$), and near-zero output impedance ($Z_{out} \to 0$). It acts as the universal impedance-matching buffer, isolating sensitive high-impedance sensors (such as pH glass electrodes or piezoelectric transducers) from low-impedance measuring loads.

1. Inverting Summing Amplifier (Analog Adder)

Multiple input voltages $v_1, v_2, \dots, v_n$ are connected through respective resistors $R_1, R_2, \dots, R_n$ to the common inverting virtual ground node. Applying KCL:

Solving for output voltage $v_o$:

If all input resistors are chosen equal ($R_1 = R_2 = \dots = R_n = R$):

By selecting binary-weighted resistor values ($R, 2R, 4R, 8R$), the summing amplifier operates as a digital-to-analog converter (DAC).

2. Difference Subtractor Amplifier

The difference amplifier rejects common-mode voltages and amplifies the differential voltage between two signals $v_1$ and $v_2$. Applying superposition or voltage division:

At non-inverting terminal ($+$): $V^+ = v_2 \left( \frac{R_4}{R_3 + R_4} \right)$.

At inverting terminal ($-$), using virtual short $V^- = V^+$ and nodal analysis:

Setting the bridge balance condition $\frac{R_2}{R_1} = \frac{R_4}{R_3}$:

The circuit precisely subtracts $v_1$ from $v_2$ with differential gain $A_d = R_2 / R_1$. Any common-mode noise present equally on both leads cancels out.

3. Three-Op-Amp Instrumentation Amplifier

While the single op-amp difference amplifier is simple, its input impedances are low and asymmetrical ($Z_{in1} = R_1$, $Z_{in2} = R_3 + R_4$).

The Three-Op-Amp Instrumentation Amplifier overcomes this by placing two non-inverting buffers ($A_1, A_2$) before the differential stage ($A_3$). Differential gain is set by a single variable gain resistor $R_G$:

Common-mode signals pass through the input buffers with unity gain ($A_{cm,1} = 1$) while differential signals are heavily amplified, yielding common-mode rejection ratios exceeding $120\,\text{dB}$ with gigohm input impedances.

1. Ideal Op-Amp Integrator

Replacing the feedback resistor $R_f$ with a capacitor $C$ in an inverting configuration yields an integrator. By virtual ground ($V^- = 0$):

Equating currents $i_{in} = i_f$ and integrating from $t = 0$ to $t$:

In Laplace transform domain:

The DC Drift/Saturation Problem: At DC ($s \to 0$ or $\omega = 0$), capacitor impedance is infinite ($X_C \to \infty$). The op-amp operates open-loop with astronomical DC gain $A_{OL} \sim 10^5$. Any minuscule input offset voltage $V_{os}$ or bias current $I_B$ charges the capacitor continually, driving the op-amp into saturation at $\pm V_{sat}$ within seconds!

2. Practical (Lossy) Integrator

To prevent DC saturation, a large stabilizing resistor $R_f$ (typically $10 R$ to $100 R$) is placed in parallel with $C$. The transfer function becomes:

At DC ($\omega = 0$), the gain is securely clamped to a finite value $A_{dc} = - R_f / R$.

The lower corner frequency separating amplifier behavior from true integration is:

For input frequencies $f \gg f_L$ (at least $10 f_L$), the capacitor impedance dominates $R_f$, and the circuit performs pure integration with a $-20\,\text{dB/decade}$ roll-off.

3. Ideal vs. Practical Differentiator

Inverting the positions—placing $C$ at input and $R$ in feedback—produces an ideal differentiator:

The Noise and Instability Problem: Gain $|H(j\omega)|$ increases linearly with frequency at $+20\,\text{dB/decade}$ indefinitely! High-frequency thermal noise, radio pickup, and power transients are amplified enormously, overwhelming the desired signal and causing parasitic oscillations.

Practical Differentiator: A small resistor $R_1$ is added in series with $C$, and a small capacitor $C_f$ is placed in parallel with $R_f$. The high-frequency cutoff is set to:

For frequencies $f \ll f_H$, the circuit differentiates cleanly; for frequencies $f > f_H$, the response rolls off at $-20\,\text{dB/decade}$, guaranteeing total high-frequency noise immunity and closed-loop stability.