Floating Inductors A single Generalised Impedance Convertor (GIC) can simulate a grounded inductor. This is fine for high-pass filters. The inductors in a low-pass filter are floating…

Three Pole Example Three pole high-pass filter. One grounded inductor and two capacitors. Three pole low-pass filter. One floating inductor and two capacitors.

Floating Inductor Properties Current in equals current out Impedance equals sL

Floating Inductor Replacement GIC 1GIC 2 R sK

Floating Inductor Replacement

Floating Inductor Drawback A floating inductor requires two GIC circuits, i.e. four op-amps. An N-th order low pass filter requires N/2 floating inductors = 2N op-amps. An N-th order high pass filter requires only N op-amps. Solution : Frequency Dependent Negative Resistors (FDNRs)

Component Scaling The frequency response of a passive network depends on the ratios between the impedances. If all impedances are multiplied by the same factor, the frequency response is unchanged. NB. Impedance of a capacitor = 1/sC (Cut-off frequency = 10 kHz)

Component Scaling II What if all impedances are scaled by a factor of 1/s ? BeforeAfter Resistor, Z = RCapacitor, Z = R/s Inductor, Z = sLResistor, Z = L Capacitor, Z = 1/sC“Super Capacitor”, Z = 1/s 2 C

Frequency Dependant Negative Resistance Impedance is real – i.e. a resistance It is also negative… …and inversely proportional to the square of frequency Hence – Frequency Dependent Negative Resistance (FDNR) Unfortunately, it doesn’t exist…

Realising a Grounded FDNR

Low Pass Filter Design using FDNRs Original passive filter 3 pole Butterworth LPF, f c = 10 kHz All impedances scaled by 1/s. All impedances scaled by 10 7.

Very Low Frequency Performance Low pass filters should work all the way down to 0 Hz (d.c.) At 0 Hz… Theoretically, gain is still unity. In practice, gain is undefined – dominated by the leakage resistances of the capacitors. Solution – add a known loss resistance.

Low Frequency Stability At cut-off frequency (10 kHz) Loss resistors, r, should be much bigger (at least 50 times bigger in practice)

Practical Design The input and output impedances are now (predominantly) capacitive. For practical use, buffer amplifiers are required on the input and output.

Passive Filters Component Scaling 01  Normalised 0  Practical 2fC2fC i.e. divide all inductances and capacitances by the desired cut-off frequency (in rad/s).

Passive Filters Component Scaling II 01  Normalised 0  Practical – High Pass 2fC2fC i.e. capacitors become inductors and vice versa.

Summary Two GICs can be used to simulate a floating inductor. A more efficient approach scales all impedances by 1/s. Then, the only components requiring synthesis are FDNRs. Op-amp requirements are one-per-pole (rather than two-per-pole for floating inductor synthesis) NB. Component simulation techniques can be used on more complex passive networks – see tutorial for example.