State-Variable Filter (KHN) Calculator

William J. Kerwin, Lawrence P. Huelsman, and Robert W. Newcomb published the state-variable filter in a 1967 IEEE paper while at UC Berkeley. Their topology implements an analog state-space realization of a 2nd-order transfer function — three state variables (output of each integrator plus the summer output), each providing a different filter response.

The KHN filter became the defining topology of analog synthesizer VCFs (Voltage-Controlled Filters). Bob Moog\'s 1971 ladder-filter patent used a related but different topology (transistor ladder), but many competing synths from ARP, Oberheim, and Korg used state-variable designs. The Moog vs state-variable filter debate has animated synthesizer culture ever since.

Modern DSP VCF emulators (Korg Volca, Arturia, Roland digital) implement state-variable topology in code rather than op-amps, but the 1967 equations and signal-flow graph remain unchanged. The three-op-amp circuit is still the classic teaching example for state-space filter design.

Enter center frequency f₀, Q, and integrator capacitor value C. The tool computes two matched integrator resistors R = 1/(2π·f₀·C) and the Q-setting feedback resistor R_q = R · (3Q-1). Use matched ±1% resistors and COG/NP0 caps for tight Q.

The three outputs appear simultaneously without reconfiguring: summing-amp output = HP, after 1st integrator = BP, after 2nd integrator = LP. Sum LP + HP to get a notch at f₀. For voltage-controlled tuning (VCF), replace integrator resistors with OTAs or LDR-opto-isolators. Everything runs client-side.