Swap a resistor for a capacitor
Across this rung you have built a whole family from two sentences. Tie the + input down and the virtual ground gave you the inverting amplifier (gain -Rf/Rin); move the signal to the + pin for the non-inverting amplifier (gain 1 + Rf/Rin); shrink its feedback resistors to a wire for the unity buffer; and pour several inputs into one virtual ground for the summing and difference amplifiers. Every one obeyed the same two golden rules. Now we make one small, audacious change: replace a feedback resistor with a capacitor — and the circuit stops doing algebra and starts doing calculus.
Why a capacitor changes everything is hidden in one fact you met in the AC rung: a capacitor is a tiny rechargeable bucket whose current depends on how fast its voltage changes, I = C times dV/dt. Push a steady current into it and its voltage climbs a steady ramp. Seen as a frequency-dependent resistor, its reactance Xc = 1/(2 times pi times f times C) falls as frequency rises — a fat impedance to slow signals, near a short to fast ones. Pair that lopsided behaviour with a resistor's flat impedance and the gain itself becomes frequency-dependent, governed by the time constant RC.
The integrator: a bucket filling at a steady rate
Build the integrator: input through Rin to the minus pin, a capacitor C as the feedback element, plus pin to ground. The minus pin is a virtual ground at 0 V, so the input current is simply Iin = Vin/Rin. That current pours into C, and since a capacitor's voltage is its accumulated current divided by C, the output (sitting across C below the virtual ground) becomes Vout = -(1/RC) times the running integral of Vin. The minus sign is the familiar inversion; the 1/RC sets how fast the bucket fills.
Make it concrete. Take Rin = 100 k and C = 100 nF, so RC = 100,000 times 100 times 10^-9 = 0.01 s = 10 ms. Hold the input at a constant +1 V. The output does not jump to a value — it ramps, sliding downward at a rate of -Vin/RC = -1/0.01 = -100 V/s, which is -0.1 V every millisecond. After 10 ms it has reached -1 V, after 20 ms -2 V, and it would keep going until it hit a rail. A constant input gives a straight ramp; that is the signature of an integrator.
Now feed it shapes. A square wave (constant up, then constant down) integrates into a triangle wave — up-ramp, down-ramp — the classic waveform-shaper hidden inside countless function generators. In the frequency world the same circuit is an active low-pass filter: because the capacitor's reactance falls with frequency, the gain (Xc/Rin) falls steadily with frequency, smoothly attenuating fast wiggles while passing slow ones. An integrator and a gentle low-pass are two faces of one circuit — averaging and smoothing are just integration in disguise.
The differentiator: the mirror image
Swap the two parts around and you get the differentiator: now the capacitor is the input element and a resistor Rf is the feedback. The current into the virtual ground is the capacitor's current, C times dVin/dt, and that current flows through Rf, so Vout = -RC times dVin/dt. The output no longer reports how much the input is — it reports how fast it is changing. A flat input gives zero out; a steep change gives a tall spike.
Reuse RC = 10 ms (Rf = 100 k, C = 100 nF). Feed in a ramp climbing at +100 V/s; the output sits at a steady -RC times 100 = -0.01 times 100 = -1 V for as long as the ramp climbs. A triangle wave — up-ramp then down-ramp — therefore differentiates into a square wave, flipping sign each time the slope flips. In the frequency world this is an active high-pass filter: gain (Rf/Xc) rises with frequency, the exact mirror of the integrator. That rising gain is also the differentiator's curse, as the honest section will show.
INTEGRATOR DIFFERENTIATOR
what swaps feedback R -> cap C input R -> cap C
output -(1/RC) x integral(Vin dt) -RC x dVin/dt
steady input a ramp (constant slope) zero (nothing changing)
square wave in triangle wave out narrow spikes out
triangle wave in parabola-ish out square wave out
frequency view gain FALLS with f gain RISES with f
= active low-pass = active high-pass
real-world snag drifts to a rail amplifies noise, can ring
the usual fix big R across C, + reset small R with C, small C on Rf
Both examples use RC = 100k x 100nF = 10 ms.What feedback buys
Step back from the circuits and look at the deal you have been making all rung. In every one, the closed-loop gain came out as a ratio of two components, and the op-amp's own gigantic open-loop gain vanished from the answer. The reason is negative feedback: the closed-loop gain equals the ideal ratio times 1/(1 + 1/T), where T is the loop gain — the spare gain circulating around the loop. When T is huge, that correction term is essentially 1, and the gain is the ratio, full stop. Feedback spends surplus gain to buy accuracy.
And accuracy is only the first thing in the basket. The same surplus loop gain also buys stability — the gain barely budges as temperature, ageing, and part-to-part spread shove the raw op-amp around. It buys low distortion, because the loop corrects the amplifier's own nonlinear kinks on the fly. It buys tailored impedance — a near-infinite input resistance at the + pin and a near-zero output resistance, which is exactly what made the buffer worth building. And it flattens and widens the bandwidth for a given gain. One mechanism, a whole shopping basket of virtues — that is why negative feedback is the spine of analog design.
Nothing is free, and the bill comes due at high frequency. Loop gain T is not constant: it falls as frequency climbs, set by the gain-bandwidth product — a 1 MHz part configured for a gain of 100 runs out of loop gain by 10 kHz, and above that every virtue above fades together. Worse, the very feedback that corrects can turn vicious: if the signal arrives back at the minus pin too late — too much phase shift around the loop — the correction reinforces instead of opposing, and the amplifier breaks into oscillation. Keeping that delay in check is the whole craft of amplifier stability, the headline subject of the next rung.
Honest limits, and the op-amp with no feedback
The ideal integrator hides a flaw that bites hard in real life. A real op-amp has a tiny input offset voltage and a small input bias current, and a perfect integrator integrates those too — so even with the input grounded, the output slowly creeps until it parks against a rail. The honest fix is to bridge a large resistor across the capacitor: it bleeds off the offset and caps the DC gain at -R(big)/Rin, at the price of no longer integrating the very lowest frequencies. Precision integrators add a reset switch to dump the capacitor on command. The textbook formula is the starting point, not the finished circuit.
The differentiator has the opposite vice. Because its gain rises with frequency, it amplifies high-frequency noise mercilessly — the faint hiss an integrator would have smoothed away, a differentiator blows up. And that rising gain eventually collides with the op-amp's falling open-loop gain, which is a recipe for ringing or outright oscillation. The standard cure is to stop the gain rising forever: a small resistor in series with the input capacitor, plus a small capacitor across the feedback resistor, flatten the response above a chosen frequency, trading ideal high-frequency differentiation for a circuit that actually behaves.
Step further back and the usual honest fences apply to all of these. The golden rules hold only with negative feedback and inside the chip's real limits — the output cannot pass the supply rails, and it cannot move faster than the slew rate. These RC circuits are also unusually fussy about layout: a breadboard's stray capacitance and pickup can swamp a small integrator current or feed a differentiator exactly the noise it loves, and a simulation that omits those parasitics will look perfect while the bench misbehaves. On fast or precise work the layout is itself a circuit element.