1. The Differential Amplifier: Heart of Every Op-Amp

Before discussing op-amps we need to understand the differential amplifier (the "diff pair" or "diff amp") because every modern op-amp's input stage is one and its key properties propagate up.

1.1 What it does, in one sentence

Two inputs $V_1, V_2$. Output is proportional to the difference $V_1 - V_2$. Common-mode signals (the same value on both inputs) are rejected. That is the whole game.

Twin-microphone analogy. Imagine two microphones held an inch apart, each connected to one input of a diff amp. They both pick up the ambient noise of the room: traffic outside, the hum of the air-conditioning, mains coupling on the cables, the dog barking next door. Subtract one mic's signal from the other and that common-mode noise vanishes, because both mics heard it the same way. Now hold a tuning fork right next to mic 1, far from mic 2. The fork hits one diaphragm hard and the other only weakly. Subtract: the fork survives, the room noise dies. Differential is signal; common-mode is noise. The diff amp's whole purpose is to throw away the part both wires have in common and keep the part that distinguishes them.

This idea is so useful that it powers far more than op-amps. Every USB cable, every Ethernet pair, every HDMI lane, every PCIe lane carries data differentially exactly because mains hum, ESD spikes, and stray RF couple equally onto both wires of a pair, and the receiver's diff amp throws all of it away.

1.2 Topology: BJT version

Two BJTs (matched as closely as possible, ideally fabricated side by side on the same die so they share temperature and process variations) with their emitters tied together and dropped through a "tail" current source $I_{EE}$ to the negative supply. Inputs go to the bases, collectors are loaded by resistors $R_C$ from the positive supply.

                 V_CC
            │            │
           [R_C]        [R_C]
            │            │
       V_o1 ●            ● V_o2     ← differential output: V_o1 - V_o2
            │            │
            C            C
   V_1 ─── B[Q1]     [Q2]B ─── V_2  ← differential input
            E            E
            ●─────●──────●
                  │
                ─┴─
                (I_EE)              ← tail current source
                  │
                 V_EE

The single trick: the tail current is fixed. It cannot change. The two transistors have to share whatever the tail source delivers. Mathematically, $I_{C1} + I_{C2} \approx I_{EE}$ at all times.

Suppose both inputs rise by the same amount, $\Delta V$ (a common-mode disturbance). Each transistor's $V_{BE}$ would naively want to rise; the emitter current of each would naively want to grow. But the sum of the two currents has to stay at $I_{EE}$. So the emitters simply float up by the same $\Delta V$ as the bases, $V_{BE}$ on each transistor stays exactly where it was, both currents stay at $I_{EE}/2$, both collector voltages stay put, and the differential output $V_{o1} - V_{o2}$ does not move at all. The common-mode disturbance has been completely rejected.

Now suppose one input rises by $+\delta/2$ and the other falls by $-\delta/2$ (a pure differential disturbance of amplitude $\delta$). The transistor with the higher base wants more current; the one with the lower base wants less. The tail current is steered between them. One collector pulls more current through its $R_C$, dropping the collector voltage; the other collector lets up. The differential output $V_{o1} - V_{o2}$ swings hard.

That is the whole intuition. Differential signals steer the tail current and produce a big output. Common-mode signals only float the emitters and produce nothing.

1.3 DC analysis: balanced and beautifully temperature-stable

At equilibrium with both bases at the same DC voltage, $I_{C1} = I_{C2} = I_{EE}/2$. The drop across each $R_C$ is the same, so both collector voltages sit at $V_{CC} - (I_{EE}/2) R_C$. The differential output is exactly zero.

This balance is intrinsically temperature-stable. Both transistors are biased identically and sit at the same die temperature, so any drift in their $V_{BE}$ from temperature, in their $\beta$, or in $I_S$, drifts the same way on both sides and cancels in the difference. The differential output stays at zero across temperature even though each individual collector voltage might drift by tens of millivolts. The diff amp converts temperature drift from a multiplicative gain error into a small offset that mostly cancels. This is the single biggest reason every modern analog IC uses a diff pair as the input stage.

1.4 Differential and common-mode gain, derivation

Let the inputs be $V_1 = V_{cm} + v_d/2$ and $V_2 = V_{cm} - v_d/2$, splitting any input into a common-mode part and a differential part. Use the small-signal hybrid-pi model from Chapter 5 ($g_m = I_C/V_T$, where $V_T \approx 26$ mV at room temperature, and $r_\pi = \beta/g_m$).

For the differential part, the symmetry of the circuit lets us treat one half as a "common-emitter with grounded emitter" stage (because the AC potential at the joined emitter point is zero by symmetry, even though there is no physical ground there). Each transistor sees an effective emitter ground and produces:

$A_d = \frac{v_{o1}}{v_d} = -g_m R_C$

(matching the gain of an ordinary CE stage from Chapter 5). Taken differentially across both collectors, the gain doubles: $V_{o1} - V_{o2} = -g_m R_C \cdot v_d$.

For the common-mode part, both emitters move together. The joined emitter sees the tail source's output impedance $R_{EE}$ (finite, not infinite for a real source: a single transistor current mirror typically gives $R_{EE}$ around 100 kΩ to 1 MΩ). A small common-mode rise $v_{cm}$ pulls a small additional current $v_{cm}/(2 R_{EE})$ through each transistor (factor of 2 because the two transistors share the load), which drops $R_C$ by $-R_C \cdot v_{cm}/(2 R_{EE})$. So:

$A_{cm} = -\frac{R_C}{2 R_{EE}}$

Small but nonzero.

The figure of merit is the ratio of these:

$\text{CMRR} = \left|\frac{A_d}{A_{cm}}\right| = g_m \cdot 2 R_{EE}$

Bigger tail-source impedance means better CMRR. That is why op-amp designers spend transistors lavishly on the tail source: cascoded current mirrors, Wilson mirrors, even cascoded Wilson mirrors, all to push $R_{EE}$ above 10 MΩ. CMRR for general-purpose op-amps runs 80 dB to 100 dB; for instrumentation-grade parts (LT1167, AD8421), 120 dB or more. A 1 V common-mode signal in a 120 dB CMRR amp couples into the output as if it were 1 µV of differential signal: completely rejected.

1.5 Why CMRR matters in real life

• Long sensor cables pick up 50/60 Hz mains hum on both wires equally. Run a strain-gauge bridge cable across a noisy factory floor and both wires of the differential pair pick up the same hum. The diff amp at the receiver sees only the strain-gauge difference and rejects the mains.
• ECG and EEG. Heart signal is a few millivolts; muscle twitch and mains pickup are 100x larger but common-mode. The instrumentation amp at the front end (LT1789, AD620, INA126) suppresses common-mode by 110 dB and the heart signal survives.
• Audio mics on long cables. Balanced lines (XLR connectors) carry signal differentially; cable-pickup noise is common-mode and rejected at the input. This is why every professional studio mic uses a balanced cable.
• Differential pairs in USB, Ethernet, HDMI, PCIe, DDR. Same physics. Noise from the other side of the board is common-mode; signal is differential.
• Hardware security: tamper meshes. A protective mesh wrapped around a secure chip is a long differential pair. Anything that touches it (a probe, a drilled hole) creates a differential disturbance distinguishable from common-mode environmental noise.

1.6 The FET version and the JFET-input op-amp

Replace the BJTs with JFETs or MOSFETs and the same topology works. Instead of $I_C = I_S e^{V_{BE}/V_T}$ you have $I_D = (k/2)(V_{GS} - V_t)^2$, but the steered-tail-current intuition is identical. The headline benefit: gate current is essentially zero (femtoamps), so the input bias current of a FET-input op-amp is many orders of magnitude smaller than a BJT-input op-amp's.

This is why the TL072 (JFET input) dominates audio: a 50 fA bias current cannot disturb a high-impedance pickup or a guitar volume pot. The OPA627 is a higher-spec JFET-input op-amp beloved in audiophile gear. The LMC6041 is a CMOS-input op-amp with bias currents in the femtoamps, used in pH meters and ion-selective electrode amplifiers, where loading the source by a single nanoampere would destroy the measurement.

1.7 Cascading diff stages with level translators

A single diff stage gives you maybe 100 to 1000 of voltage gain. Op-amps need 100,000 or more. So we cascade: diff stage to diff stage to common-emitter to output buffer.

Between stages the DC operating point shifts. The collectors of stage 1 sit at, say, $+10$ V; stage 2 wants its bases near 0 V. A level shifter drops the DC by a fixed amount without disturbing the AC. Common implementations are emitter followers with a Zener clamp, a current source pulling against a resistor, or a string of forward-biased diodes. Inside an integrated op-amp the level-shifter is usually implemented with a PNP follower that lifts the signal to the right DC range for the next stage.

The 741 op-amp's input is a diff pair (PNPs for level-shift convenience), whose output drives a second diff stage acting as a cascode for high gain, which drives a class-AB push-pull output stage. We will lift the lid in the next section.