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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…

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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.

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Floating Inductor Properties Current in equals current out Impedance equals sL

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Floating Inductor Replacement GIC 1GIC 2 R sK

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Floating Inductor Replacement

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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)

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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)

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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

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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…

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Realising a Grounded FDNR

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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.

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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.

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

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

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Passive Filters Component Scaling 01 Normalised 0 Practical 2fC2fC i.e. divide all inductances and capacitances by the desired cut-off frequency (in rad/s).

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Passive Filters Component Scaling II 01 Normalised 0 Practical – High Pass 2fC2fC i.e. capacitors become inductors and vice versa.

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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.

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