From Ohm's Law to LC Resonance: The Circuit Math That Tunes Your Radio
This tutorial gives a developer a single, code-reproducible path from Ohm's law to a working radio design, with every formula verified numerically. For anyone touching RF, antennas, filters, or impedance matching, the Q and resonance math here is the foundation, and the included Python lets you rerun all the calculations as you build.
The chapter opens with the tuning knob on an old radio: it turns a variable capacitor inside an LC tank that selects one frequency from the thousands arriving at the antenna. From there it builds up the full circuit toolkit. Ohm's and Kirchhoff's laws, maximum power transfer (matching at 50% efficiency, conjugate matching for reactive sources), capacitor and inductor behavior with time constants, and the Wheeler formula for air-core coils all get worked out with numerical tables. Complex impedance makes AC circuits as straightforward as DC, and an RC low-pass filter shows where the -3 dB cutoff and -20 dB/decade rule come from.
Series and parallel resonance are treated in depth: resonant frequency, quality factor Q, bandwidth B = f0/Q, and the voltage magnification across reactive elements. The energy definition of Q ties to free oscillation decay, and a numerical RK4 simulation measures Q from the logarithmic decrement. The chapter ends by designing a real tuning circuit for a medium-wave crystal radio: a 240 μH inductor and a 365 pF variable capacitor cover 530-1600 kHz, with coil turns computed via Wheeler's formula. Python code in the appendix reproduces every number.
The tutorial deliberately grounds every theoretical result in a concrete build (the crystal radio), which makes the math stick because each formula maps to a physical choice like a capacitor value or a turn count.
The 50% efficiency at impedance matching is a sharp reminder that matching maximizes power transfer, not efficiency; power systems choose efficiency instead, which is why the same rule applies differently in RF versus mains distribution.
The energy definition of Q is more intuitive than the component-reactance formula; it explains why quartz crystals with Q in the millions make such stable oscillators and narrow filters.