A Generalized Impedance Converter (GIC) configured as an Antoniou simulated inductor using four 10 kΩ precision resistors and a 10 nF capacitor converts that capacitance into an equivalent inductance of 1.0 Henry (H). The governing formula for the Antoniou topology is Leq = (R1 × R3 × R5 × C4) / R2. Substituting our benchmark values: Leq = (10,000 × 10,000 × 10,000 × 10 × 10-9) / 10,000 = 1.0 H. This exact conversion assumes ideal op-amp behavior, zero parasitic trace inductance, and a stable C0G/NP0 ceramic dielectric. If you are designing active audio crossovers or synthetic LC filters, this 1.0 H simulated inductor saves you from sourcing a bulky, expensive, and microphonic physical choke.
The Antoniou GIC Conversion Matrix
To understand how component tolerances affect your synthetic inductance, here is the conversion output for a ±20% variance in the timing capacitor (C4), assuming all four resistors remain locked at exactly 10 kΩ. This is critical when prototyping with 10% or 20% tolerance ceramics before dialing in 1% film caps.
| C4 Value (±20% Variance) | R1, R2, R3, R5 | Simulated Leq | Reactance at 1 kHz (XL) |
|---|---|---|---|
| 8 nF | 10 kΩ | 0.8 H | 5,026 Ω |
| 9 nF | 10 kΩ | 0.9 H | 5,654 Ω |
| 10 nF (Nominal) | 10 kΩ | 1.0 H | 6,283 Ω |
| 11 nF | 10 kΩ | 1.1 H | 6,911 Ω |
| 12 nF | 10 kΩ | 1.2 H | 7,539 Ω |
The assumption that fixes this answer is the ideal op-amp model. In reality, the op-amp's finite Gain-Bandwidth Product (GBW) and slew rate introduce a parallel parasitic resistance and a series parasitic capacitance. Selecting the right active component dictates whether your 1.0 H inductor behaves like an inductor, or devolves into a lossy resistor at higher frequencies.
| Op-Amp Model | GBW Product | Max Simulated Freq (1% error) | Slew Rate | Input Bias Current |
|---|---|---|---|---|
| TL072 (JFET) | 3 MHz | ~15 kHz | 13 V/µs | 50 pA |
| OPA2134 (FET) | 8 MHz | ~40 kHz | 10 V/µs | 2 pA |
| NE5532 (Bipolar) | 10 MHz | ~50 kHz | 9 V/µs | 200 nA |
| ADA4898-1 (Bipolar) | 65 MHz | ~300 kHz | 100 V/µs | 3.5 µA |
| LT1028 (Ultra-low noise) | 50 MHz | ~250 kHz | 2.2 V/µs | 90 nA |
For audio applications (20 Hz to 20 kHz), the Analog Devices MT-086 Tutorial on Generalized Impedance Converters highly recommends JFET or CMOS input op-amps like the TL072 or OPA2134 to minimize DC bias currents flowing through the high-value 10 kΩ resistors, which would otherwise create severe offset voltages.
Mains Voltage vs. Signal Level: Why 120V/230V/3-Phase is Meaningless
A frequent point of confusion for makers bridging power electronics and signal theory is asking how this 1.0 H simulated inductance shifts when applied to 120V, 230V, or 3-phase 480V mains. The direct answer: the conversion is physically meaningless and catastrophically dangerous at mains voltages.
A GIC is strictly a low-voltage signal processing topology. Standard op-amps operate on ±15V rails (30V total maximum). Applying 120V RMS (which peaks at 170V) or 230V RMS (peaking at 325V) will instantly punch through the silicon junctions, short the power rails, and potentially cause a localized fire. Furthermore, a simulated 1.0 H inductor cannot store magnetic energy; it only mimics the V/I phase relationship of an inductor using active feedback. If you need 1.0 H of inductance for a 120V/230V AC line filter, power factor correction, or motor starting, you must use a physical laminated iron-core or ferrite choke capable of handling the magnetic flux saturation and thermal dissipation of mains current. Never attempt to place a GIC in series with a mains branch circuit.
When the Conversion Becomes Meaningless (High-Frequency & Tolerance Limits)
While the math yields a perfect 1.0 H on paper, the physical conversion becomes meaningless under three specific bench conditions:
- Beyond the GBW Limit: As the operating frequency approaches roughly 1/50th of the op-amp's Gain-Bandwidth Product, the open-loop gain drops. The simulated inductor loses its inductive reactance (jωL) and begins to exhibit a negative resistance component, leading to severe Q-enhancement and eventual oscillation. A TL072-based 1.0 H GIC will fail above 15 kHz.
- High-K Dielectric Capacitors: If you substitute the 10 nF C0G capacitor with a standard X7R or Z5U ceramic, the conversion is ruined. X7R dielectrics exhibit a severe voltage coefficient; a 10 nF X7R cap might drop to 4 nF under a mere 5V signal swing, collapsing your 1.0 H inductor down to 0.4 H dynamically. Always use C0G/NP0 ceramics or polypropylene film caps for GIC timing elements.
- Resistor Mismatch: The Antoniou topology relies on the ratio of R1/R2 and R3/R5. If these pairs are not matched to at least 0.1% tolerance, the simulated inductor will exhibit a large parasitic series resistance (ESR), destroying the high-Q response needed for tight active filter designs.
Frequently Asked Questions
Can a GIC simulate a negative resistor or capacitor?
Yes. By swapping the positions of the resistors and the capacitor in the GIC network, you can create a Negative Impedance Converter (NIC) or a Frequency Dependent Negative Resistor (FDNR), which is heavily used in Texas Instruments active filter design topologies to create steep brick-wall low-pass filters without using physical inductors.
Does power factor (PF) affect the GIC calculation?
No. Power factor is a macroscopic AC power metric relating real power (Watts) to apparent power (VA) in loads drawing real current. Because a GIC simulates a purely reactive component (an ideal inductor has a PF of 0), the concept of load power factor is irrelevant to the small-signal impedance conversion math.
What is the maximum current a simulated 1.0 H inductor can handle?
It is limited entirely by the op-amp's output current capability, typically 20 mA to 40 mA for standard DIP-8 packages. If your circuit demands 100 mA through the 'inductor', the op-amp will current-limit, clipping the waveform and instantly destroying the simulated impedance profile.
