AC, Filters & Op-Amps: Make Circuits Behave by Frequency

1. AC and reactance

A DC supply pushes charge one way at a constant rate. An AC supply instead swings back and forth, and the cleanest mathematical form is a sine wave: v(t) = V_peak·sin(2πft). Three numbers describe it. The peak (V_peak) is the maximum it reaches; the frequency f (in hertz) is how many complete cycles happen each second, with the period T = 1/f the time for one cycle; and the phase tells you where in the cycle you are at a given instant. Because a sine spends as much time positive as negative, its average is zero — which is why we quote an AC voltage as an RMS (root‑mean‑square) value, the DC equivalent that would deliver the same power. For a pure sine, V_RMS = V_peak/√2 ≈ 0.707·V_peak, so the "230 V" on a mains socket is an RMS figure, not the peak.

Here is where AC differs fundamentally from DC. A resistor opposes current by the same R at every frequency, but a capacitor and an inductor oppose it by an amount that depends on frequency — and we call that opposition reactance, measured in ohms just like resistance. Capacitive reactance is X_C = 1/(2πfC): it is large at low frequency (so a capacitor blocks DC) and shrinks toward zero as frequency rises (so at high frequency it behaves like a short). Inductive reactance is the mirror image, X_L = 2πfL: small at low frequency (an inductor passes DC) and growing toward an open as frequency rises. A capacitor therefore passes AC but not DC; an inductor passes DC but resists change. They are opposites, and that opposition is the raw material of every filter.

When R, L and C are combined, the total opposition is called impedance, Z, still in ohms but "complex" because it carries a phase as well as a size. For a simple series combination the magnitude is |Z| = √(R² + (X_L − X_C)²). The phase relationship is the part that surprises beginners: in a pure resistor voltage and current are in step; in a pure capacitor the current leads the voltage by 90°; in a pure inductor the voltage leads the current by 90°. A mnemonic that sticks is “ELI the ICE man” — going around the loop, in an L (inductor) E(mf) leads I, and in a C (capacitor) I leads E. This phase shift is what lets a capacitor and inductor cancel each other (resonance) and is the seed of filter behaviour, so hold onto it.

2. Resonance

Because X_L grows with frequency and X_C shrinks with frequency, there is exactly one frequency where they are equal in magnitude: X_L = X_C. That frequency is the resonant frequency, and setting 2πfL = 1/(2πfC) and solving for f gives f₀ = 1/(2π√(LC)). Notice what this means: at f₀ the inductor and capacitor "fight" with equal and opposite reactance, so their effects cancel and only the resistance is left. This is resonance — the point where a circuit is most sensitive to one frequency and relatively indifferent to its neighbours.

In a series RLC circuit the consequence is dramatic. At f₀ the net reactance is zero, so the impedance is at its minimum (|Z| = R) and the current is at its maximum. Below f₀ the circuit is net‑capacitive; above f₀ it is net‑inductive. How tall and narrow that peak is depends on the Q factor (quality factor), which for a series circuit is Q = (1/R)·√(L/C). A high Q means low loss relative to the energy sloshing between L and C, giving a sharp, tall resonance; a low Q gives a broad, flat one. Q also tells you how "selective" the circuit is — a radio with a high‑Q tuned circuit separates two close stations better.

In a parallel (tank) circuit the picture inverts: at resonance the impedance is at its maximum and the current drawn from the source is at its minimum, even though a large current can circulate internally between L and C. The practical span a resonant circuit responds over is its bandwidth, BW — the frequency range between the two points where the power falls to half (the −3 dB points) — and it is tied to Q by BW = f₀/Q: higher Q, narrower bandwidth. Resonance is everywhere in real electronics: selecting a radio frequency, setting a filter's corner, the "ringing" you see on an oscilloscope after a step, and the heart of nearly every oscillator. Designing one means choosing L and C to land on your f₀ and choosing R to set the Q (and therefore the bandwidth) you want.

3. Filter design

A filter is any circuit whose gain depends on frequency — it passes some frequencies and rejects others. The simplest is the first‑order RC low‑pass from the figure at the top: a resistor in series, a capacitor to ground. At low frequency the capacitor's reactance is high, so the output (taken across the capacitor) closely follows the input. At high frequency X_C is low, shunting the signal to ground, and the output falls. The cutoff frequency f_c is defined as the frequency where the output has fallen to 1/√2 ≈ 0.707 of its low‑frequency value — the −3 dB point — and for a single RC stage, f_c = 1/(2πRC). Below f_c the response is essentially flat; above it, it rolls off.

There are four basic filter shapes, all built from the same parts. A low‑pass passes below f_c and blocks above (the RC above). A high‑pass is its mirror — capacitor in series, resistor to ground — passing above f_c and blocking below (it uses the same f_c = 1/(2πRC)). A band‑pass passes a band between two cutoffs, which you can make from a resonant RLC or by cascading a low‑pass and a high‑pass. A band‑stop (or notch) does the opposite, rejecting a band while passing below and above — a parallel resonant circuit in line with the signal is a classic example. The steepness of the transition is the roll‑off: a single pole gives 20 dB per decade (6 dB per octave), and stacking poles steepens it (40 dB/dec for two, and so on).

Finally, passive versus active. Passive filters use only R, L and C — simple and needing no supply — but they load the source, can't provide gain, and inductors are bulky and lossy. Active filters use op‑amps with just resistors and capacitors: they offer gain, a high input impedance (so they don't load the previous stage), and precise, repeatable cutoffs without any inductor, at the cost of a power supply and a finite bandwidth. When you design one in practice: pick f_c from 1/(2πRC) and choose a standard R and C that get close; remember the source and load impedances will shift the real corner; and expect the tolerance of R and C to set how exact the cutoff actually is. The Sallen‑Key topology (a two‑pole active filter) is a common next step and appears in the op‑amp discussion.

4. Practical op‑amp topologies

An operational amplifier is a very high‑gain differential amplifier that, in real designs, is almost always wired with negative feedback to turn that huge raw gain into a small, precise, predictable one. The "ideal" op‑amp model that makes the maths easy has four traits: infinite open‑loop gain, infinite input impedance (no current enters the input pins), zero output impedance, and — the payoff of negative feedback — the two input voltages are driven to be equal. That last point is the virtual short: you may assume V₊ ≈ V₋ and that no current flows into either input. Nearly every standard topology falls out of just those two assumptions plus Ohm's Law, so master them and the circuits stop being memorisation.

The two workhorse amplifiers differ only in where the signal enters. The inverting amplifier feeds the signal through a resistor R_in to the − input, with a feedback resistor R_f from the output back to −, and the + input tied to ground. The virtual short holds − at ground (a "virtual ground"), so the input and feedback currents are equal and opposite, giving a gain of A_v = −R_f/R_in — the minus sign meaning the output is 180° out of phase. The non‑inverting amplifier applies the signal to the + input and uses a resistor divider (R_f and R_in) from the output to the − input; here the gain is A_v = 1 + R_f/R_in, with no inversion. Both let you dial in an exact gain with just two resistors and an inexpensive precision part.

Beyond basic amplification, three more topologies cover most needs. The voltage follower (buffer) ties the output straight back to the − input, giving A_v = 1: no voltage gain, but a very high input impedance and very low output impedance, which is exactly what you need to isolate one stage from the next without loading it. The summing amplifier puts several inputs, each with its own resistor, into the − node; the output is the weighted sum, V_o = −(R_f/R1·V1 + R_f/R2·V2 + …), which is the principle behind audio mixing and digital‑to‑analog weighting. The differential (subtractor) amplifier uses four resistors to amplify only the difference between two inputs — with matched ratios, V_o = (R_f/R_in)·(V2 − V1) — rejecting whatever the two inputs share. That common‑mode rejection is quantified by the CMRR and is the reason differential inputs are used in instrument and sensor circuits.

Real op‑amps deviate from ideal in ways that matter. The gain‑bandwidth product (GBW) is roughly constant, so closed‑loop gain times bandwidth ≈ GBW: ask for more gain and you get less bandwidth, which is the hidden limit on how fast your filter or amplifier really is. The slew rate caps how quickly the output can move, so a large, fast signal gets "slewed" (a flattened, distorted top). Outputs can't reach the supply rails, leaving headroom; exceed it and the output saturates (clips) against a rail. And small input offset voltage and bias currents cause errors that bite in precision or high‑source‑impedance circuits. So before you build: check GBW against your highest frequency, leave output headroom, and pick a part whose offset and slew rate suit the job — the topology tells you the gain, the part's specs tell you whether it can actually deliver it at that speed.