The swing that pushes itself: what resonance really is
Every child on a playground discovers resonance before they know its name. Push the swing at random and nothing builds; push it once per swing, at the top of each arc, and your small efforts add up until the chains are nearly horizontal. The swing has one natural frequency — set by the length of the chains and gravity — and energy sloshes between two forms: height (potential) at the ends, speed (kinetic) at the bottom. A series RLC circuit is the electrical twin of that swing. The capacitor stores energy in its electric field, the inductor stores it in its magnetic field, and at one special frequency they pass the same packet of energy back and forth, twice per cycle, with only the resistor leaking a little away as heat.
We can be precise. From the previous rung you know the impedances: an inductor is Z_L = jωL, a capacitor is Z_C = 1/(jωC). The inductor's reactance climbs with frequency; the capacitor's falls. Somewhere they must cross — and at that crossing they are equal in size but opposite in sign, so they cancel. That cancellation is resonance. Setting ωL = 1/(ωC) and solving gives the resonant frequency, the single most important formula in this guide.
Reactance vs frequency (series RLC):
|X| X_L = ωL (rises)
| /
| /
| X_C=1/ωC /
| \ /
| \ /
| \ / <-- they cross here: ω = ω0
+-------*------------------------> ω
ω0
Set ωL = 1/(ωC)
=> ω0^2 = 1/(LC)
=> ω0 = 1 / sqrt(L*C) [rad/s]
f0 = ω0 / (2*pi) [Hz]
Worked: L = 100 uH, C = 100 pF
ω0 = 1/sqrt(1e-4 * 1e-10) = 1/sqrt(1e-14) = 1e7 rad/s
f0 = 1e7 / 6.283 = 1.59 MHz (an AM-band station)Series vs parallel: a short circuit and an open circuit in disguise
At resonance the inductor and capacitor cancel, but *how* they cancel depends on whether they sit in series or in parallel — and the two cases behave like opposites. In a series RLC circuit driven by a voltage source, the two reactances subtract along the same current path, so the total impedance collapses to just R. The circuit looks like its smallest possible impedance — almost a short circuit — so current peaks. Series resonance is an acceptor: at ω₀ it eagerly accepts current at one frequency and rejects the rest.
A parallel RLC circuit (often called a *tank*) does the mirror image. The inductor and capacitor swap their stored energy around the loop between themselves, and at ω₀ they draw equal and opposite currents from the source — which nearly cancel. The tank looks like its *largest* possible impedance, almost an open circuit, so the voltage across it peaks while the line current it demands goes to a minimum. Parallel resonance is a rejector: it presents a brick wall to one frequency. This is exactly why a tank circuit sits in the collector of an oscillator or a radio's tuned amplifier — it offers huge gain at f₀ and almost none elsewhere.
SERIES RLC PARALLEL RLC (tank)
R L C +----L----+
o-/\/--UUU--||--o | |
(driven by V) o----+----+----C----+----+----o
| (driven by I)
At ω0: Z = R (minimum) |
-> current is MAXIMUM At ω0: Z = high (maximum)
-> 'acceptor' circuit -> voltage is MAXIMUM
-> line current MINIMUM
-> 'rejector' circuit
Mnemonic: series = LOW Z at f0 (lets f0 through)
parallel = HIGH Z at f0 (blocks f0 from the line)Quality factor Q: how sharp is your tuning?
Two radios can both resonate at 100.3 MHz, yet one nails that one station while the other smears in the neighbours. The difference is the quality factor Q — a single dimensionless number that measures how lightly damped, how *sharp*, a resonance is. Physically, Q is the ratio of energy stored in the L–C pair to energy lost per cycle, scaled by 2π. A high-Q circuit is a near-frictionless swing: push once and it rings for many cycles. A low-Q circuit is a swing in molasses: one good push and it's done.
Q ties directly to the width of the resonance peak. Define bandwidth BW as the span of frequencies where the response stays within −3 dB (a factor of 1/√2 ≈ 0.707 in amplitude, or half-power) of the peak. Then the relationship is beautifully simple: BW = f₀ / Q. Double the Q and you halve the bandwidth — your filter gets twice as picky. This is the whole game of tuning: choose f₀ with L and C, then choose Q (mostly with R) to set how wide a window you keep.
Quality factor & bandwidth:
Series RLC: Q = (1/R) * sqrt(L/C) = ω0*L/R = 1/(ω0*R*C)
Parallel RLC: Q = R * sqrt(C/L) = R/(ω0*L) = ω0*R*C
Bandwidth (the -3 dB / half-power width):
BW = f0 / Q [Hz]
Response shape:
|H|
1.0 | .-^-. <- peak at f0
0.707|.......|...|....... <- -3 dB line
| / \
| __/ \__
+--|----+----|--------> f
f_lo f0 f_hi
|<-- BW = f_hi - f_lo -->|
Worked (FM tuner): f0 = 100.3 MHz, want BW = 200 kHz
Q = f0 / BW = 100.3e6 / 200e3 = 501 (a sharp, high-Q tank)From resonance to filters: carving the spectrum
A filter is just a circuit that treats different frequencies differently — and you already have all the parts. Stop thinking of L and C as components and start thinking of them as frequency-dependent valves. Because a capacitor's impedance falls with frequency, an RC pair where the output is taken across the capacitor passes low frequencies and blocks high ones: a low-pass filter. Swap them and you get a high-pass. Put a series LC in line and you get a band-pass that lets only a window through; put a parallel LC tank in line and you get a band-stop (notch) that kills one frequency — perfect for zapping a 50/60 Hz mains hum.
The natural language for a filter is the transfer function H(ω) = V_out/V_in — a complex number for every frequency that tells you both the gain (how big) and the phase (how shifted) of the output. Plot its magnitude in decibels against a logarithmic frequency axis and you get a Bode plot, the universal one-page portrait of what a block does to a signal. The −3 dB point marks the cutoff; the slope past it (−20 dB/decade per pole) tells you how brutally it rolls off. This is the same frequency-response thinking that runs through signals, control and RF — learn to read a Bode plot and you can read almost any analog block at a glance.
First-order RC low-pass (output across C):
Vin o---/\/\/---+---o Vout
R |
=== C
|
GND
H(ω) = Vout/Vin = (1/jωC) / (R + 1/jωC)
= 1 / (1 + jωRC)
cutoff: ω_c = 1/(RC), f_c = 1/(2πRC)
at f_c: |H| = 1/√2 = -3 dB, phase = -45°
Bode magnitude:
0 dB |________
| \__ <- -20 dB/decade slope
-3 dB |.........\.\..... <- corner at f_c
| \__
+----|-------|------> log f
f_c 10*f_c
Worked: R = 1.6 kΩ, C = 100 nF
f_c = 1/(2π · 1600 · 100e-9) ≈ 1.0 kHz audio low-passTwo-port networks: modeling any black box by its terminals
Sooner or later a circuit gets too big to care about every internal node — an amplifier might hide a hundred transistors, but to the rest of the system it's just a box with two wires in and two wires out. The engineer's escape from complexity is the two-port network: forget the insides, characterise the box entirely by what happens at its terminals. You have an input port (voltage V₁, current I₁) and an output port (V₂, I₂), and four numbers relate them. Which four depends on which variables you choose as causes and which as effects — and that choice gives you the family of two-port parameter sets.
The Z-parameters (impedance, open-circuit) treat the two currents as causes and the two voltages as effects: V = Z·I. The Y-parameters (admittance, short-circuit) do the reverse. But the most famous in transistor work are the [[ee-h-parameters|hybrid h-parameters]] — a clever mix that takes input *current* and output *voltage* as the inputs, because those are exactly the quantities you can hold fixed in a transistor with a short or an open. The result, h₁₁ in ohms, h₁₂ and h₂₁ dimensionless, h₂₂ in siemens, is why old datasheets list a BJT's current gain as h_fe.
Two-port: only the terminals matter
I1 -> <- I2
+-----+---------+-----+
V1 | | black | | V2
| | box | |
+-----+---------+-----+
input port output port
h-parameter equations (the transistor favourite):
V1 = h11·I1 + h12·V2 h11 = input impedance (Ω)
I2 = h21·I1 + h22·V2 h21 = forward current gain (β, dimensionless)
h12 = reverse voltage ratio (≈ 0, ideal)
h22 = output admittance (S)
How you'd measure them:
h11 = V1/I1 with V2 = 0 (output SHORTED)
h21 = I2/I1 with V2 = 0 (output SHORTED) <- this is hfe!
h12 = V1/V2 with I1 = 0 (input OPEN)
h22 = I2/V2 with I1 = 0 (input OPEN)
Same box, other dialects:
Z (V = Z·I, open-circuit) Y (I = Y·V, short-circuit)
ABCD (cascade chains) S-parameters (RF, at high freq)Putting it together: from impedance to design
Step back and look at the arc of this whole track. You started with voltage and current and a single resistor. You learned Kirchhoff's laws, then systematic nodal and mesh analysis, then the Thévenin and superposition shortcuts. You met capacitors and inductors as energy stores, and the phasor that tamed AC by turning sinusoids into complex numbers. This rung is where all of it converges into *design*: impedance was the key that unlocked resonance, resonance gave you Q and bandwidth, those gave you filters, filters gave you the transfer-function and Bode view, and the two-port abstraction lets you hide any of it inside a reusable block.
- Fix the frequency. Choose L and C so ω₀ = 1/√(LC) lands where you want — the station, the cutoff, the hum to kill.
- Fix the sharpness. Pick R (or the topology's loss) to set Q, and check BW = f₀/Q is wide enough to pass your real signal but narrow enough to reject neighbours.
- Shape the response. Decide low/high/band-pass/notch, write H(ω), and sketch its Bode plot to confirm the gain and roll-off before you build anything.
- Box it up. Characterise the finished block by its two-port parameters (h, Z, Y or S) so the next stage sees a clean, predictable interface instead of your internals.
Where does this point? Everywhere. The transfer-function-and-Bode habit is the gateway to signals and systems and to control, where stability lives in the same poles and zeros. Tuned circuits, Q and S-parameters are the foundation of RF and microwave design — radios, antennas, the matching networks that squeeze power into an aerial. Active filters open the door to analog electronics and the op-amp. You have, in seven rungs, gone from reading your first schematic to holding the conceptual toolkit of a working circuit engineer. The components never changed — your way of seeing them did.